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[ "<title>Introduction</title>", "<p>Thrombotic microangiopathy (TMA) is a serious pathological state where there is microvascular thrombosis leading to the mechanical destruction of red blood cells and consumption of platelets resulting in microangiopathic hemolytic anemia (MAHA) and thrombocytopenia, respectively [##UREF##0##1##]. The thrombotic occlusion of the small blood vessels preferentially affects the kidneys, brain, and heart, and this leads to the organ's dysfunction with significant morbidity and mortality [##UREF##0##1##,##REF##28242109##2##]. The systemic process of TMA is coupled with a number of biochemical findings such as thrombocytopenia due to platelet aggregation and thrombi formation, anemia and presence of schistocytes due to fragmentation of red blood cells, raised lactate dehydrogenase (LDH) due to tissue ischemia and cell lysis, and low plasma haptoglobin due to intravascular hemolysis as it binds to free hemoglobin [##REF##36074708##3##].</p>", "<p>Several causes for TMA have been reported, both hereditary and acquired [##UREF##0##1##]. Pregnancy and the postpartum period are well-recognized triggers for TMA and it is thought that this is due to the increase in the production of Von Willebrand factor (VWF), which in turn increases consumption of ADAMTS13 with subsequent thrombosis [##REF##36074708##3##]. The spectrum of pregnancy-associated TMAs includes disorders that are relatively common and cause secondary forms of TMA, such as pre-eclampsia, eclampsia, and HELLP (hemolysis, elevated liver enzymes, and low platelets) syndrome. These disorders are part of the same syndrome with different presentations and severity [##REF##26342729##4##]. In addition, autoimmune conditions, such as systematic lupus erythematosus (SLE) and catastrophic antiphospholipid syndrome (CAPS) can also present with TMA. The activation of both classical and alternative complement pathways appears to play key roles in the SLE-associated TMA [##REF##36074708##3##]. During pregnancy, other serious but less common causes of TMAs are hemolytic uremic syndrome (HUS), thrombotic thrombocytopenic purpura (TTP), and atypical hemolytic uremic syndrome (aHUS).</p>", "<p>The clinical presentation of TMA in pregnancy is challenging and the presence of active autoimmune disorders such as SLE at the same time can make it a diagnostic dilemma. We present a case of severe acute kidney injury (AKI) due to TMA in a young female with active SLE in pregnancy and the postpartum period.</p>" ]

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[ "<title>Discussion</title>", "<p>TMA is a serious systemic illness with progressively life-threatening thrombocytopenia, MAHA, and renal dysfunction [##REF##34269998##5##]. TMA can be classified as primary, characterized by a complement mutation or complement autoantibodies, such as TTP and aHUS, or secondary due to infections, pregnancy, and autoimmune disorders such as SLE [##UREF##1##6##]. During pregnancy, the different causes of TMA have common clinical and laboratory findings which make it challenging to distinguish them apart. The frequency of the disorder and the timing of the presentation can serve as practical clues, as pre-eclampsia or HELLP syndrome have a relatively high incidence of one per 20 and one per 1000 pregnancies, respectively, while syndromes of primary TMA such as HUS or TTP are much less common at one per 25000 and one per 200000 pregnancies, respectively [##REF##27648610##7##]. In addition, pregnancy-associated HUS is the only form of TMA to occur most frequently in the postpartum period and up to three months post-delivery in almost three-fourths of cases, and TMA starting in the postpartum of an uneventful pregnancy is very suggestive of complement-mediated aHUS [##REF##28911789##8##,##REF##20203157##9##]. Moreover, checking the activity testing of ADAMTS13 can be diagnostic in pregnancy-associated TTP [##REF##32808006##10##]. However, this should be done prior to commencing the plasma exchange as the activity level of the test will change significantly and correspond to the clinical improvement once the therapy is started [##REF##14982878##11##]. Unfortunately, testing might not always be available or requires sending to expert reference centers, which can extend the timing for diagnosis to days or even weeks. The PLASMIC (Platelet count; combined hemoLysis variable; absence of Active cancer; absence of Stem-cell or solid-organ transplant; Mean corpuscular volume; International normalised ratio; Creatinine) score has been developed to assist clinicians in deciding the likelihood of severe ADAMTS13 deficiency when the result of ADAMTS13 is not available [##UREF##2##12##]. In our case, the targeted testing was not done prior to the plasma exchange as there were no clear systemic signs of TMA. This highlights the need for clinicians to be aware of this diagnosis in this cohort of patients. The calculated PLASMIC score was 4 points, which is considered low and gives a 0% risk of severe ADAMTS13 deficiency. However, this didn't delay the management with plasma exchange which eventually resulted in a good recovery.</p>", "<p>The kidney biopsy showed clear evidence of TMA with glomerular capillaries filled with thrombi; however, different forms of TMA are often indistinguishable based on the kidney biopsy findings. Moreover, immunofluorescence is usually negative apart from positive staining for fibrinogen with glomerular capillaries, arterioles, and small arteries [##REF##24799306##13##]. In our case, although the first biopsy had granular C3 staining, this was felt to be non-specific as only a few reports have signaled that immunostaining might indicate complement activation in TMA. In addition, no specific or sensitive markers of complement activation are yet known for this entity [##REF##35791743##14##].</p>", "<p>For some forms of pregnancy-associated TMA such as pre-eclampsia, eclampsia, and HELLP syndrome, the rapid delivery can be sufficient to control the disorder and for other forms such as TTP and HUS, it can help achieve more rapid remission [##REF##32808006##10##]. Other lines of management such as plasma exchange should be considered especially when there is an atypical presentation of pregnancy-associated TMAs, life-threatening neurological or cardiac findings, or profound thrombocytopenia (&lt;30g/L) [##REF##32808006##10##]. Expectant management with close monitoring would be reasonable if improvement in hemolysis markers, platelet levels, and no deterioration of renal function. If aHUS diagnosis is made by exclusion of other possibilities, then anti-C5 monoclonal antibodies should be initiated instead of plasma exchange [##REF##32808006##10##]. Although the safety of the use of anti-c5 treatment in pregnancy has not been assessed in controlled clinical trials, the limited initial data suggest its safety, especially when considering the potentially catastrophic effects of uncontrolled TMA in pregnancy. Renal TMA can happen in the context of active LN and plays an important role in its natural history [##UREF##1##6##]. In SLE, the histopathological presence of TMA in the kidneys is a hallmark of severe and active renal disease with worse outcomes [##UREF##3##15##,##REF##20535626##16##].</p>", "<p>In this case, the presence of SLE and makers of activity both biochemically and clinically brought about an even bigger challenge as it can also provide a potential trigger for TMA even without clear systemic markers of hemolysis. The finding of TMA on the renal biopsy without conclusive LN was helpful in changing the route of management, especially as the renal clinical abnormalities were persisting prior to the plasma exchange.</p>" ]

[ "<title>Conclusions</title>", "<p>TMA in pregnancy and the postpartum period is a complex and serious disorder that requires a high index of suspicion and a prompt course of action. Other coexisting elements such as autoimmune disorders or infections can make the diagnosis a real challenge. The natural history of the illness especially in relation to delivery along with targeted testing can aid the diagnosis and management. Histopathological investigations can provide very valuable information and should be pursued, especially when renal involvement is suspected.</p>" ]

[ "<p>Thrombotic microangiopathy (TMA) is a severe systemic disorder with multiorgan manifestations due to thrombosis of the microvasculature. Pregnancy and post-partum are particularly high-risk periods for many forms of TMA. The disease progression is rapid and can lead to organ failure and even death; therefore, urgent recognition and treatment are paramount. The presence of other triggers such as infections or autoimmune diseases like systematic lupus erythematosus (SLE) can add further complexity, which emphasizes the need for definitive diagnostic investigations such as kidney biopsy to promptly direct further diagnosis and management. We describe a case of a 27-year-old female with post-partum severe acute kidney injury and nephrotic range proteinuria. She had a new diagnosis of active SLE and was found to have TMA on kidney biopsy without conclusive features of lupus nephritis. She was managed successfully with plasma exchange with rapid improvement of her kidney markers.</p>" ]

[ "<title>Case presentation</title>", "<p>A 27-year-old female, gravida 5, para 4, with no history of illness prior to her last pregnancy in 2021, presented in the third trimester with arthritis, headache, and generalized fatigue. The patient was found to have hypertension, renal impairment, and proteinuria. She was managed as pre-eclampsia with antihypertensives and had an early induction of vaginal delivery at 32 weeks of gestation. Her labs revealed antinuclear antibodies (ANA) +4, positive anti-double strand DNA, serum creatinine of 140 mmol/L, low complements' levels, 24-hour urine protein of 4377 mg, erythrocyte sedimentation rate (ESR) of 135 mm per hour, and C-reactive protein (CRP) of 9.5 mg/dl. At that point, she was diagnosed as SLE with likely lupus nephritis (LN) and started on methylprednisolone 1 gram IV for three doses, followed by oral prednisolone 1 mg/kg/day along with hydroxychloroquine 200 mg once daily. The kidney biopsy was deferred at that point due to the postpartum status.</p>", "<p>After discharge, she presented again at 40 days postpartum on February 23, 2022, with pleuritic chest pain, dyspnea, generalized fatigue, and myalgia. Her labs showed hemoglobin of 5.7 g/dL, serum platelets of 105,000 per microliter, and serum creatinine of 283 mmol/L with an estimated glomerular filtration rate (eGFR) of 20 ml/minute/1.73 m² by Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) equation, 24-hour urine protein 4972 mg, and low levels of complements 3 and 4 at 54.5 mg/dl and 10 mg/dl, respectively. During her admission, she was managed initially as a relapse of SLE with LN and was given methylprednisolone 1 g per day for four days, hydroxychloroquine 200 mg once daily, and mycophenolate mofetil 1500 mg twice a day. Interestingly, the makers of hemolysis were not elevated, the LDH was 213 unit/L, no schistocytes on blood film, and the total bilirubin was 2.5 µmol/L. Unfortunately, ADAMTS13 activity level, complement antibody, and gene mutation testing were not available at that time.</p>", "<p>Her kidney tests remained markedly deranged despite the initial therapy; therefore, we proceeded with kidney biopsy (Figure ##FIG##0##1##) which revealed several glomeruli showing capillary lumina filled with thrombi, double contour of glomerular capillary basement membrane, and mesangiolysis. Arteriolar and arterial fibrinoid necrosis associated with apoptotic bodies was seen. These features were consistent with thrombotic microangiopathy. Also, there were few glomeruli with endocapillary hypercellularity that were suggestive of LN; however, the direct immunofluorescence study demonstrated only mesangial granular deposit for complement C3 and no deposit for all the immunoglobulins. Therefore, 10 days after commencing the treatment for presumed LN with maximum immunosuppression as described above, we elected to manage the patient as postpartum TMA and to commence her on plasma exchange; her serum creatinine remained markedly elevated at 252 mmol/L at that point. The patient received plasmapheresis with 1.5 plasma volume of fresh frozen plasma on alternate days with significant improvement of her serum creatinine from 252 to 80 mmol/L. After completing four sessions, the 24-hour urine protein improved from 4972.8 mg to 2978 mg and reached 1288 mg within a month. </p>", "<p>In the following year, the level of proteinuria increased to 11036.4 mg/24 hours and serum creatinine was 104 mmol/L. A new kidney biopsy was performed (Figure ##FIG##1##2##), and 17 out of 33 viable glomeruli showed endocapillary hypercellularity associated with karyorrhexis or leukocyte infiltration. Twelve glomeruli revealed global sclerosis and three glomeruli revealed cellular crescent. The direct immunofluorescence study demonstrates intense granular staining for all the immunoglobulin; IgG, IgA, IgM, and for the complements C3 and C1q mainly along the peripheral capillary wall. Some mesangial immune deposits were also noted in the C3 and the IgA. No features of TMA were seen. The biopsy was diagnosed as class III/IV LN.</p>", "<p>The patient was treated with a pulse of IV methylprednisolone 1000 mg for three days followed by oral prednisolone at 1 mg /kg/day for two months and mycophenolate mofetil increased again to 1500 mg twice a day. After that, the levels of proteinuria decreased to 3100 mg/24 hours and serum creatinine improved to 93 mmol/L three months later (Figures ##FIG##2##3##, ##FIG##3##4##). Her blood work during the two admissions post delivery is summarized in Table ##TAB##0##1##.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>First kidney biopsy</title><p>(A) The glomerular vascular pole is showing fibrinoid necrosis. A thrombus is also seen in one capillary lumen. The glomerulus has fibrillary appearance with duplication of the glomerular capillary basement membrane (H&amp;E stain, 400x). (B) The glomerulus is showing mesangiolysis and the capillary lumina are occluded by homogenous eosinophilic thrombi (H&amp;E stain, 400x). (C) There is fibrinoid necrosis with apoptotic bodies in the wall of this interlobular artery (H&amp;E stain, 400x). (D) Few glomeruli are showing endocapillary hypercellularity that is associated with karyorrhexis (H&amp;E stain, 400x)</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>Second kidney biopsy</title><p>Features of lupus nephritis with endocapillary hypercellularity that is associated with karyorrhexis and wire loop lesion. No features of TMA are seen (H&amp;E stain, 400x)</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG3\"><label>Figure 3</label><caption><title>Change in proteinuria</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG4\"><label>Figure 4</label><caption><title>Change in serum creatinine</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Summary of blood tests</title><p>C3: Complement 3; C4: Complement 4</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Test</td><td colspan=\"2\" rowspan=\"1\">First admission post delivery  </td><td colspan=\"2\" rowspan=\"1\">Second admission post delivery  </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">March 1, 2022</td><td rowspan=\"1\" colspan=\"1\">March 10, 2022</td><td rowspan=\"1\" colspan=\"1\">August, 2023</td><td rowspan=\"1\" colspan=\"1\">November,2023</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Hemoglobin (g/dL)</td><td rowspan=\"1\" colspan=\"1\">6.17\n</td><td rowspan=\"1\" colspan=\"1\">7.37</td><td rowspan=\"1\" colspan=\"1\">13.1</td><td rowspan=\"1\" colspan=\"1\">14.5</td></tr><tr><td rowspan=\"1\" colspan=\"1\">White blood count (/uL)</td><td rowspan=\"1\" colspan=\"1\">6.97 ´10³\n</td><td rowspan=\"1\" colspan=\"1\">4.53 ´10³\n</td><td rowspan=\"1\" colspan=\"1\">9.01 ´10³\n</td><td rowspan=\"1\" colspan=\"1\">7.32 ´10³\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Platelets (/uL)</td><td rowspan=\"1\" colspan=\"1\">105 ´10³\n</td><td rowspan=\"1\" colspan=\"1\">129 ´10³\n</td><td rowspan=\"1\" colspan=\"1\">248 ´10³\n</td><td rowspan=\"1\" colspan=\"1\">217 ´10³\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Red blood count (/uL)</td><td rowspan=\"1\" colspan=\"1\">2.29 ´10⁶\n</td><td rowspan=\"1\" colspan=\"1\">2.61 ´10⁶\n</td><td rowspan=\"1\" colspan=\"1\">4.93 ´10⁶\n</td><td rowspan=\"1\" colspan=\"1\">5.4 ´10⁶\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Lactate dehydrogenase (U/L)</td><td rowspan=\"1\" colspan=\"1\">226</td><td rowspan=\"1\" colspan=\"1\">266</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Serum creatinine (umol/L)</td><td rowspan=\"1\" colspan=\"1\">252</td><td rowspan=\"1\" colspan=\"1\">91</td><td rowspan=\"1\" colspan=\"1\">104</td><td rowspan=\"1\" colspan=\"1\">72</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Blood urea nitrogen (umol/L)</td><td rowspan=\"1\" colspan=\"1\">17.2</td><td rowspan=\"1\" colspan=\"1\">15.3</td><td rowspan=\"1\" colspan=\"1\">8</td><td rowspan=\"1\" colspan=\"1\">5.4</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Sodium (mmol/L)</td><td rowspan=\"1\" colspan=\"1\">136</td><td rowspan=\"1\" colspan=\"1\">136</td><td rowspan=\"1\" colspan=\"1\">141</td><td rowspan=\"1\" colspan=\"1\">138</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Potassium (mmol/L)</td><td rowspan=\"1\" colspan=\"1\">5.23</td><td rowspan=\"1\" colspan=\"1\">3.5</td><td rowspan=\"1\" colspan=\"1\">4.17</td><td rowspan=\"1\" colspan=\"1\">3.75</td></tr><tr><td rowspan=\"1\" colspan=\"1\">C3 (mg/dL)</td><td rowspan=\"1\" colspan=\"1\">60.5</td><td rowspan=\"1\" colspan=\"1\">56.7</td><td rowspan=\"1\" colspan=\"1\">81.9</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">C4 (mg/dL)</td><td rowspan=\"1\" colspan=\"1\">13.4</td><td rowspan=\"1\" colspan=\"1\">10.7</td><td rowspan=\"1\" colspan=\"1\">12.4</td><td rowspan=\"1\" colspan=\"1\"> </td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Basil Alnasrallah, Eman Alabbad, Mohammed M. Aljishi, Zainab A. Alkhuraidah, Sumayah Alsabaa</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Basil Alnasrallah, Eman Alabbad, Mohammed M. Aljishi, Zainab A. Alkhuraidah, Sumayah Alsabaa</p><p><bold>Drafting of the manuscript:</bold>  Basil Alnasrallah, Eman Alabbad, Mohammed M. Aljishi, Zainab A. Alkhuraidah, Sumayah Alsabaa</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Basil Alnasrallah, Eman Alabbad, Mohammed M. Aljishi, Zainab A. Alkhuraidah, Sumayah Alsabaa</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Postural stability is defined as the ability to maintain the centre of mass of a body within the base of support with minimal postural sway through somatosensory information (##REF##31629327##Pino-Ortega et al., 2020##), and is commonly assessed through static and dynamic balance (##UREF##4##Shumway-Cook &amp; Woollacott, 2001##). Static balance is defined as the ability to maintain the line of gravity (vertical line from the centre of mass) of a body within the base of support (BoS) with minimal postural sway. While, dynamic balance consists of the ability to move the centre of pressure (CoP) within the BoS and to move CoP from one BoS to another BoS (##REF##32082023##Kusumoto et al., 2020##; ##REF##33599066##Reina et al., 2022##). These assessments are routinely used in sports and clinical settings to identify balance disorders. For instance, a poor balance in sports is associated with lower limb injuries (such as muscle injuries or ligament sprains) (##REF##11086748##McGuine et al., 2000##; ##REF##20547668##Emery &amp; Meeuwisse, 2010##; ##REF##28828077##Brachman et al., 2017##), while in the elderly population it is the most important factor associated with the risk of falls (##REF##20056721##Muir et al., 2010##). Given its importance, the use of effective lower limb-injury detection tools is needed in order to reduce the injury rate, downtime, and health care costs associated with short- and long-term treatment of lower limb injuries (##REF##29190658##Marcoux et al., 2017##).</p>", "<p>Monopodal postural stability is a widely used test to assess static and dynamic balance; several tools with varying levels of difficulty have been proposed in order to adapt to the target population (##REF##3685116##Horak, 1987##; ##REF##15921472##Emery et al., 2005##; ##REF##31598406##Powden, Dodds &amp; Gabriel, 2019##). On the one hand, laboratory balance measures (<italic toggle=\"yes\">e.g</italic>., stabilometry or motion analysis) provide multiple objective values related to stability, but require the use of equipment that is costly, highly technical, and often not portable (##REF##3685116##Horak, 1987##; ##REF##2929827##Fridén et al., 1989##; ##REF##15921472##Emery et al., 2005##; ##REF##31598406##Powden, Dodds &amp; Gabriel, 2019##). On the other hand, other measurement tools have been developed for use in the clinical and sports setting, such as the three-directions modified Star Excursion Balance Test (mSEBT) or the Emery balance test (EBT), which are faster to perform and require less time (##REF##15921472##Emery et al., 2005##; ##REF##31598406##Powden, Dodds &amp; Gabriel, 2019##).</p>", "<p>The mSEBT is the simplification in three directions of the initial eight-direction Star Excursion Balance Test described by ##UREF##2##Gray (1995)##. It evaluates single-leg balance, dynamic neuromuscular control, proprioception, flexibility, core stability, ROM and strength while an individual reaches three directions (anterior, posteromedial, and posterolateral) with the non-stance leg (##REF##22892416##Gribble, Hertel &amp; Plisky, 2012##). The EBT was specifically designed to assess dynamic balance on an unstable surface with eyes closed in young adults and adolescents (##REF##15921472##Emery et al., 2005##). The reliability and validity of these tests have been described in healthy adolescents and asymptomatic adults (##REF##15921472##Emery et al., 2005##; ##REF##24183777##Shaffer et al., 2013##; ##REF##31598406##Powden, Dodds &amp; Gabriel, 2019##).</p>", "<p>These tests are reported in the literature to reflect changes after an intervention, but dissimilar results have been observed when these tests have been used simultaneously (##REF##30849606##Blasco et al., 2019##). The clinimetric analysis of measurement instruments is of great importance in the clinical and sports settings since the change in a specific measurement can reflect a change in the patient’s clinical situation, which is essential for evaluating the effectiveness of interventions (##REF##21794541##de Yébenes Prous, Rodríguez Salvanés &amp; Carmona Ortells, 2008##). The metric property that analyses this effect is responsiveness, which is defined as the ability of a tool to detect meaningful clinical changes over time (##REF##20298572##Mokkink et al., 2010##). Even so, the responsiveness of monopodal postural stability measurements through stabilometry, mSEBT, and EBT has not been evaluated after an instability training programme or analysed using multiple statistical indicators of responsiveness. Furthermore, while studies use the dominant/non-dominant (<italic toggle=\"yes\">i.e</italic>., trained/untrained) lower limb comparison to detect within-subject changes in stability after an intervention (##REF##36863090##Temporiti et al., 2023##), the external responsiveness (<italic toggle=\"yes\">i.e</italic>., discriminative ability) of the tests has not been previously examined. Therefore, the main aim of this study was to analyse the responsiveness of the three monopodal postural stability tests.</p>" ]

[ "<title>Materials and Methods</title>", "<title>Study design</title>", "<p>A single-group pretest-posttest design was used, which involved repeated monopodal postural stability assessment of the dominant and non-dominant lower limb before and after a 4-week intervention (three weekly sessions) consisting of dominant lower limb instability training. This study was conducted from April 2020 to June 2021, starting the recruitment phase in November 2020. All measurements were performed in the clinical research laboratory of the Department of Physiotherapy (University of Valencia). A physiotherapist with experience in applying the test (M.S-B) evaluated the participants. This examiner was blinded during the measurement process, not being aware of which limb had received the intervention. Before participation, participants were informed of the study procedures and their possible associated risks. All of them provided written informed consent. This study was completed following the principles outlined in the Declaration of Helsinki, and it was approved by the Human Research Ethics Committee of the Ethics Committee on Experimental Research of the University of Valencia (Comité Ético de Investigación en Humanos de la Comisión de Ética en Investigación Experimental de la Universitat de Valencia), in Spain (1271077).</p>", "<title>Subjects</title>", "<p>Thirty healthy recreational athletes (21 males/nine females; mean age: 22.7 ± 2.7 years; weight: 70.13 ± 12.39 kg; height: 172.5 ± 8.1 cm; weekly physical activity: 438.0 ± 170.4 min) volunteered in this study, of which 27 completed the entire intervention and evaluations and were included in the analysis.##SUPPL##2## Appendix S1## contains the flow chart of the study participants. Participants were physiotherapy students recruited by email using the University of Valencia Intranet. For inclusion, they had to be between 18 and 30 years old, have no history of lower limb injury or pain during the year preceding the study, and perform at least 90 min of physical activity per week. The established exclusion criteria were to have previously participated in any balance improvement or lower limb proprioception programme or presenting any known balance disorder, such as vertigo, or vestibular or central nervous system alterations.</p>", "<title>Instruments</title>", "<title>Stabilometry</title>", "<p>For the stabilometric assessment of monopodal stability, the Dinascan/IBV P600 force platform (digital signal with a sampling frequency of 1,000 Hz) was used with its software application NedSVE/IBV (Valencia, Spain). The participants were asked to place the foot of the leg to be measured on the mark on the platform, with the knee of the other leg flexed 90° and their arms alongside the body (##FIG##0##Fig. 1A##). The participants, with their eyes closed, were asked to maintain that position for 15 s, during which the platform recorded the variations in balance (##REF##24567674##Romero-Franco et al., 2014##), and rested 30 s before the next measurement. Three measurements were taken. Subsequently, the process was repeated with the contralateral leg (##REF##30589387##Powden et al., 2019##). The values analysed were the CoP displacement (lateral displacement and anteroposterior displacement), the swept area (mm²), and the average speed (m/s). In subsequent analyses, as there is no consensus in the literature on how to process the data (##REF##24567674##Romero-Franco et al., 2014##; ##REF##30589387##Powden et al., 2019##), stabilometry values were analysed based on four variants: the mean of the three measurements, the first measurement, the lowest, and the highest.</p>", "<title>mSEBT</title>", "<p>mSEBT consists of standing on one leg while, with the contralateral leg, reaching as far as possible in three different directions (anterior, posteromedial and posterolateral) (##REF##17193868##Plisky et al., 2006##; ##REF##22892416##Gribble, Hertel &amp; Plisky, 2012##). Adhesive tape was placed on the floor to delimit two posterior diagonals with a 90° angle between them, with a 135° angle with respect to the anterior line (##FIG##0##Fig. 1B##). The distance covered in each attempt was normalised with the length of the leg, for which both lower limbs of each participant were measured in the supine position, taking as reference the anterior superior iliac spine and the internal malleolus of the same leg (##UREF##3##Gribble &amp; Hertel, 2003##). Next, each participant was allowed to make four attempts with each leg and in each direction to practice, followed by three more attempts that were registered (##UREF##3##Gribble &amp; Hertel, 2003##; ##UREF##1##Granacher et al., 2014##). They first performed the anterior direction with their dominant leg, then the posteromedial, and finally the posterolateral. Afterwards, the same procedure was repeated with the non-dominant leg. A 15-s rest was allowed between attempts in the same position (##UREF##1##Granacher et al., 2014##), resting 5 min between different directions (##UREF##3##Gribble &amp; Hertel, 2003##; ##UREF##1##Granacher et al., 2014##). The values of the last three attempts were recorded to calculate the average value later.</p>", "<p>All measurements were made barefoot and with hands placed on hips. In turn, for the anterior measurements, the stance foot was aligned at the most distal aspect of the toes, while for the backward directions, it was aligned at the most posterior aspect of the heel (##REF##22892416##Gribble, Hertel &amp; Plisky, 2012##). Attempts were not considered valid, and the movement was repeated, if the participant failed to touch the line with the mobile foot, moved the supporting foot, dropped hands from hips, lost balance at some point supporting the mobile foot, failed to maintain the start or end position for at least one second, or placed weight on the moving foot at the end of the run (##UREF##1##Granacher et al., 2014##).</p>", "<title>EBT</title>", "<p>Another test used to assess the dynamic balance of a participant was the EBT, which is widely used in athletes and adolescents due to its greater complexity. Participants had to close their eyes and then stand on one leg on an Airex® Balance Pad, barefoot and with their hands placed on their hips (##REF##15921472##Emery et al., 2005##; ##REF##30849606##Blasco et al., 2019##). The participants were asked to remain as stable as possible for a maximum time of 180 s (##REF##9974192##Hahn et al., 1999##). They made three attempts with each leg and rested 15 s between them. A handheld stopwatch was used to measure the time the participant held the position. A test time of 15 s was given to the participants before starting the measurements so that they became familiar with the pad (##REF##15921472##Emery et al., 2005##). The supporting leg should be slightly flexed at the knee (about 30°), and the contralateral leg should be at 45° knee flexion (##FIG##0##Fig. 1C##) (##UREF##1##Granacher et al., 2014##; ##REF##30849606##Blasco et al., 2019##). The recorded value was the best time obtained in the three attempts for each leg (##REF##30849606##Blasco et al., 2019##). The timer was stopped when a participant dropped hands from hips, touched the ground with the contralateral leg, moved the supporting foot, moved the pad from its original position, or opened his eyes (##REF##15921472##Emery et al., 2005##; ##UREF##1##Granacher et al., 2014##).</p>", "<title>Blackboard</title>", "<p>The instability device selected for the instability programme was the Blackboard (Blackboard Training, Innenstadt, Germany), which is a device designed to work on monopodal stability, consisting of two wooden boards joined together by tape. At its base, it has a Velcro surface on which half-cylindrical wooden bars can be freely placed. Depending on the position in which they are placed, one or other type of instability will be obtained (<italic toggle=\"yes\">e.g</italic>., lateromedial or anteroposterior instability or forefoot and rearfoot only or both). The Blackboard was used in its complete instability configuration, with two bars placed in the centre of each board to create instability in both the forefoot and rearfoot (##FIG##1##Fig. 2B##).</p>", "<title>Procedures</title>", "<p>Before starting the instability training programme, height was measured using a 1-millimeter sensitivity flexible tape measure, while weight and body mass index (BMI) were assessed using a standardised body composition analyser (Tanita BC 418 MA; Tanita Corp, Tokyo, Japan). In that same session, monopodal postural stability was evaluated using stabilometry, mSEBT, and EBT tests performed randomly.</p>", "<p>A familiarisation session was then carried out in which the participants performed two to three repetitions of static single-leg support for 20 s, as needed, to become familiar with Blackboard (##FIG##1##Fig. 2A##). Next, following the same setup for the training sessions, participants performed five 40-s repetitions of training only with their dominant leg followed by 60 s of rest (##REF##31855076##Wright, Nauman &amp; Bosh, 2020##). The edges of the Blackboard were allowed to contact the ground and participant could slightly shift their position, but always reaching the proposed 40 s of training. Finally, a 4-week programme including three weekly sessions of instability training in order to improve the stability of the participants was performed. The duration, frequency, and dosage of the programme sessions were based on previous literature on balance training programmes (##UREF##0##Cain, Garceau &amp; Linens, 2017##; ##REF##30192681##Anguish &amp; Sandrey, 2018##; ##REF##30589387##Powden et al., 2019##), and it was carried out in a research laboratory of the Faculty of Physiotherapy of the University of Valencia.</p>", "<title>Statistical analysis</title>", "<p>Baseline data were summarised as means and standard deviations (SD) for continuous variables and as absolute and relative frequencies for categorical variables. Variables were checked for normality with the Kolmogorov-Smirnov test and homogeneity of variances with Levene’s test.</p>", "<p>Responsiveness was quantified based on internal and external responsiveness. On the one hand, internal responsiveness was determined by the paired t-test and supplemented with an effect size statistic, as recommended by ##REF##10812317##Husted et al. (2000)## and similar to what was carried out by other studies (##REF##2366602##Liang, Fossel &amp; Larson, 1990##; ##REF##26750541##Choi et al., 2016##; ##REF##29327598##Navarro-Pujalte et al., 2019##; ##REF##35123394##Pajari et al., 2022##). For this analysis, we used the standardised response mean (SRM) as an effect size statistic, which estimates the magnitude of change that is not influenced by sample size (##REF##10812317##Husted et al., 2000##; ##REF##29327598##Navarro-Pujalte et al., 2019##). Values of 0.20, 0.50, and 0.80 or higher have been proposed in the literature to represent small, medium, and large responsiveness, respectively (##REF##10812317##Husted et al., 2000##).</p>", "<p>On the other hand, external responsiveness was determined by receiver operating characteristic (ROC) curves (##REF##10812317##Husted et al., 2000##; ##REF##28595612##Rysstad et al., 2017##; ##REF##29948606##Wan et al., 2018##; ##REF##33787088##Yee et al., 2022##). We dichotomised the values for ROC curves between the dominant and non-dominant lower limb (<italic toggle=\"yes\">i.e</italic>., experimental and control lower limb), assuming that the values for the dominant lower limb tests had changed after the intervention. This was done from the perspective of the responsiveness to observed change, which is quantified when scores are compared in situations where variation in the attribute is expected but not verified explicitly as having occurred (##REF##11246687##Beaton et al., 2001##). In particular, for the circumstance of change observed before and after a treatment/intervention (usually of “known efficacy”) (##REF##11246687##Beaton et al., 2001##). We calculated the area under the ROC curve (AUC), which represents the probability of the measure correctly classifying participants. An AUC &gt; 0.70 was used as a generic benchmark to consider its discriminant ability acceptable (##REF##8863764##Stratford, Binkley &amp; Riddle, 1996##). The person responsible for the statistical analysis for external responsiveness (R.M-SA) was blinded with respect to the limb in which the intervention was carried out.</p>", "<p>An <italic toggle=\"yes\">a priori</italic> sample size calculation was developed based on a medium effect size (d = 0.50), using an α value of 0.05 and a power of 0.8. The sample size was estimated at 27 subjects. Assuming losses of 10% of the sample in the follow-up measurement, an initial sample of 30 subjects was calculated as necessary.</p>" ]

[ "<title>Results</title>", "<title>Changes associated with instability interventions</title>", "<p>##TAB##0##Table 1## shows the changes associated with an instability training programme measured with three monopodal postural stability tests. The dynamic balance for the dominant lower limb, as measured with the mSEBT and EBT, showed significant time improvements and distance reached, respectively, after the interventions. For the non-dominant lower limb, a significant change was observed in the total score of the mSEBT test and in the postero-medial and postero-lateral directions. Conversely, platform measures suggested that neither limb presented significant changes in the CoP excursions after the interventions, except for the X-axis for the dominant lower limb of the first measurement recorded. Furthermore, relative changes showed the greatest improvements for EBT of the dominant leg, with a 46.2% improvement over baseline time. ##SUPPL##2##Appendix S2## shows individual values for all participants and tests (of the dominant lower limb).</p>", "<title>Internal and external responsiveness</title>", "<p>Internal responsiveness to instability training of the three monopodal stability tests is shown in ##TAB##1##Table 2##. Internal responsiveness statistics suggest that EBT and all parameters in mSEBT for the dominant lower limb showed large internal responsiveness (SRM &gt; 0.8) among participants after instability training. Furthermore, mSEBT values for the non-dominant lower limb (except anterior displacement) also experienced significant changes with an associated large internal responsiveness. Finally, none of the stabilometry platform parameters showed a significant change in response after the intervention.</p>", "<p>The ability of the EBT to discriminate between the dominant and non-dominant lower limb (<italic toggle=\"yes\">i.e</italic>., trained <italic toggle=\"yes\">vs</italic> untrained, respectively) was generally acceptable (AUCs = 0.708) (##TAB##2##Table 3##). However, none of the parameters of the mSEBT test showed an acceptable AUC to distinguish between trained and untrained lower limbs after the intervention (AUC &lt; 0.6). Ultimately, none of the stabilometry parameters showed acceptable AUC either.</p>" ]

[ "<title>Discussion</title>", "<p>To our knowledge, this is the first study that analyses the responsiveness of different monopodal stability tests in healthy participants after an instability training programme. We found that only EBT showed both internal and external responsiveness, while the mSEBT showed acceptable internal responsiveness. In contrast, none of the stabilometry platform measures exhibited responsiveness.</p>", "<p>This study presents novel findings, as it is the first study that has used multiple statistical methods to assess the internal responsiveness (paired t-test and SRM) and external responsiveness (ROC) of three measures of monopodal stability in healthy recreational athletes. This study shows that the EBT is the only monopodal stability measure that detects changes after an instability training programme, with an acceptable internal and external responsiveness. Until now, no study had analysed this psychometric ability of the EBT. However, previous studies have identified changes in stability measured using this test after an instability training programme, as reported by ##REF##30849606##Blasco et al. (2019)##. These authors found improvements in the time of the EBT (ranging between 3.3 and 6.1 s) similar to those found in our study (5.52 s) (##REF##30849606##Blasco et al., 2019##).</p>", "<p>Regarding the dynamic stability measured with the mSEBT, our study shows a high internal but not external responsiveness. Both the intervention and control lower limb improved for all directions, except for the anterior direction of the control side. For the intervention lower limb, all mSEBT parameters showed significant improvements. Similar results have been reported in the total score of mSEBT by ##REF##30849606##Blasco et al. (2019)##, with slightly smaller improvements (ranging between 3.2% and 4.5%) than those observed in our study (5.3% intervention lower limb). Even so, the control lower limb also exhibited similar improvements (3.8%), which, together with the lack of external responsiveness, would suggest that mSEBT is not a suitable test to monitor changes in dynamic balance using the non-dominant lower limb as control. A possible explanation is that the balance intervention on the dominant lower limb favours it going further during the mSEBT when it is not the support lower limb. Another possible mechanism is the effect of cross-education, which is defined as adaptation of an untrained limb after unilateral training of the contralateral limb (##UREF##5##Son &amp; Kang, 2020##) and whose improvements appear to reflect use-dependent plasticity within the central nervous system (<italic toggle=\"yes\">i.e</italic>., interhemispheric communication in the brain, primarily through the <italic toggle=\"yes\">corpus</italic> callosum) (##REF##36395371##Lawry-Popelka, Chung &amp; McCann, 2022##).</p>", "<p>Another important finding of our study is that none of the stabilometry platform measures were able to detect a change in monopodal stability after the instability training programme. This is consistent with other authors who, after instability training, have found no changes in either healthy individuals (##REF##30849606##Blasco et al., 2019##) or participants with chronic ankle instability (CAI) (##REF##18799992##McKeon et al., 2008##). In this latter case, they concluded that CoP-based measures most likely lacked the sensitivity to detect improvements in postural control associated with a balance training programme in patients with CAI (##REF##18799992##McKeon et al., 2008##). The fact that only the dynamic measurements showed responsiveness compared to the measurements obtained with the stabilometric platform could be due to the fact that a healthy participant’s capacity for improvement in static balance is minimal, and there is a ceiling effect for the measurements of the stabilometric platform. On the other hand, the improvement capacity for dynamic balance is possibly greater in those participants and therefore, dynamic balance-related tests detect changes.</p>", "<p>Among the strengths, this research primarily evaluated the responsiveness of several monopodal stability tests in healthy participants. The clinical importance of this study lies in the fact that a simple and rapid dynamic test, such as the EBT, can detect changes in healthy participants after an instability training programme. This could offer a practical application in sports, where most participants are healthy. Therefore, it could be a tool used to identify whether injury prevention programmes aimed at improving monopodal stability are efficient. This study had limitations that should be considered. First, there is a limitation associated with the lack of generalisability. Thus, the sample included only healthy and young recreational athletes, so these findings cannot be extended to identify changes concerning recovery from injuries, such as knee or ankle sprains, or extrapolated to unhealthy or older populations. Even so, in view of the studies that use such tests in healthy subjects, we consider this analysis necessary, and future studies should replicate this metric platform analysis in specific populations. Secondly, the protocol used to measure stabilometry is not standardised as there is no consensus in the literature, making it difficult to compare our findings with other studies. However, we rely on the protocol proposed by ##REF##24567674##Romero-Franco et al. (2014)## to assess stabilometry measurements (##REF##24567674##Romero-Franco et al., 2014##) while analysing stabilometry values for different variants.</p>" ]

[ "<title>Conclusions</title>", "<p>According to the results, a positive responsiveness of the EBT to changes in monopodal stability after instability training in healthy participants can be concluded. In contrast, mSEBT only showed internal responsiveness, and none of the stabilometry platform measures were able to identify these changes, so the stabilometry platform would not be recommended in healthy participants, as well as the mSEBT for those cases where they carry out comparisons between lower limb intra-subject.</p>" ]

[ "<title>Background</title>", "<p>Stabilometry, the modified Star Excursion Balance Test (mSEBT) or the Emery balance test (EBT) are reported in the literature to reflect changes after an intervention in monopodal postural stability. Even so, the responsiveness of those tests has not been evaluated after an instability training programme or analysed using multiple statistical indicators of responsiveness. The main aim of this study was to analyse the responsiveness of the stabilometry, mSEBT or EBT.</p>", "<title>Methods</title>", "<p>Thirty healthy recreational athletes performed a 4-week programme with three weekly sessions of instability training of the dominant lower limb and were evaluated using stabilometry, mSEBT, and EBT tests. Responsiveness was quantified based on internal and external responsiveness.</p>", "<title>Results</title>", "<p>EBT and all parameters in mSEBT for the dominant lower limb showed large internal responsiveness (SRM &gt; 0.8). Furthermore, mSEBT values for the non-dominant lower limb (except anterior displacement) also experienced significant changes with an associated large internal responsiveness. None of the stabilometry platform parameters showed a significant change after the intervention. The ability of the EBT to discriminate between the dominant and non-dominant lower limb (<italic toggle=\"yes\">i.e</italic>., trained <italic toggle=\"yes\">vs</italic> untrained, respectively) was generally acceptable (AUCs = 0.708). However, none of the parameters of the mSEBT test showed an acceptable AUC.</p>", "<title>Conclusions</title>", "<p>EBT showed a positive responsiveness after instability training compared to mSEBT, which only showed internal responsiveness, or stabilometry platform measures, whose none of the parameters could identify these changes.</p>" ]

[ "<title>Supplemental Information</title>" ]

[ "<title>Additional Information and Declarations</title>" ]

[ "<fig position=\"float\" id=\"fig-1\"><object-id pub-id-type=\"doi\">10.7717/peerj.16765/fig-1</object-id><label>Figure 1</label><caption><title>Monopodal postural stability measured by (A) stabilometry, (B) modified Star Excursion Balance test, and (C) Emery balance test.</title></caption></fig>", "<fig position=\"float\" id=\"fig-2\"><object-id pub-id-type=\"doi\">10.7717/peerj.16765/fig-2</object-id><label>Figure 2</label><caption><title>(A) Stability training position using Blackboard and (B) Blackboard setup.</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"table-1\"><object-id pub-id-type=\"doi\">10.7717/peerj.16765/table-1</object-id><label>Table 1</label><caption><title>Differences in the dominant and non-dominant lower limb for the three monopodal stability tests after instability training.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\" content-type=\"text\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\"/><th colspan=\"3\" rowspan=\"1\">Dominant lower limb</th><th colspan=\"3\" rowspan=\"1\">Non-dominant lower limb</th></tr><tr><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">Pre</th><th rowspan=\"1\" colspan=\"1\">Post</th><th rowspan=\"1\" colspan=\"1\">Differences</th><th rowspan=\"1\" colspan=\"1\">Pre</th><th rowspan=\"1\" colspan=\"1\">Post</th><th rowspan=\"1\" colspan=\"1\">Differences</th></tr><tr><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">Mean (SD)</th><th rowspan=\"1\" colspan=\"1\">Mean (SD)</th><th rowspan=\"1\" colspan=\"1\">Mean (95% CI)</th><th rowspan=\"1\" colspan=\"1\">Mean (SD)</th><th rowspan=\"1\" colspan=\"1\">Mean (SD)</th><th rowspan=\"1\" colspan=\"1\">Mean (95% CI)</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>EBT (s)</bold>\n</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">11.95 (7.55)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">17.48 (9.83)</td><td rowspan=\"1\" colspan=\"1\">−5.52 [−8.93 to −2.12]<xref rid=\"table-1fn2\" ref-type=\"table-fn\">*</xref></td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">11.60 (8.22)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">11.39 (8.69)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.21 [−2.34 to 2.77]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>mSEBT</bold>\n</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">ANT (%)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">65.0 (5.24)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">66.3 (4.83)</td><td rowspan=\"1\" colspan=\"1\">−1.19 [−2.23 to −0.15]<xref rid=\"table-1fn2\" ref-type=\"table-fn\">*</xref></td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">65.6 (5.0)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">66.4 (4.8)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.76 [−1.99 to 0.47]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">PM (%)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">84.0 (12.5)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">93.6 (12.1)</td><td rowspan=\"1\" colspan=\"1\">−9.54 [−12.48 to −0.61]<xref rid=\"table-1fn2\" ref-type=\"table-fn\">*</xref></td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">85.0 (11.6)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">92.1 (11.7)</td><td rowspan=\"1\" colspan=\"1\">−6.60 [−9.24 to −3.96]<xref rid=\"table-1fn2\" ref-type=\"table-fn\">*</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\">PL (%)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">91.4 (11.0)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">97.4 (11.23)</td><td rowspan=\"1\" colspan=\"1\">−5.17 [−7.5 to −2.85]<xref rid=\"table-1fn2\" ref-type=\"table-fn\">*</xref></td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">92.1 (13.0)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">97.0 (11.1)</td><td rowspan=\"1\" colspan=\"1\">−4.14 [−6.45 to −1.83]<xref rid=\"table-1fn2\" ref-type=\"table-fn\">*</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\">Total (%)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">80.1 (8.2)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">85.8 (8.1)</td><td rowspan=\"1\" colspan=\"1\">−5.30 [−6.88 to −3.72]<xref rid=\"table-1fn2\" ref-type=\"table-fn\">*</xref></td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">81.0 (8.5)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">85.1 (7.7)</td><td rowspan=\"1\" colspan=\"1\">−3.83 [−5.38 to −2.3]<xref rid=\"table-1fn2\" ref-type=\"table-fn\">*</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Stabilometry</bold>\n</td><td colspan=\"2\" rowspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Mean of 3 measurements</bold>\n</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">Area (mm²)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">420.82 (125.92)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">418.51 (143.43)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">2.31 [−46.54 to 51.17]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">412.59 (110.26)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">437.89 (117.30)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−25.29 [−68.53 to 17.93]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Velocity (m/s)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.076 (0.01)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.075 (0.019)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.001 [−0.00 to 0.00]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.07 (0.01)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.06 (0.01)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.003 [0.00–0.00]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Xmean (mm)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">41.38 (5.47)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">40.34 (5.63)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">1.04 [−1.40 to 3.48]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">40.09 (4.54)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">39.20 (5.18)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.88 [−0.81 to 2.58]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ymean (mm)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">55.96 (11.89)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">54.60 (9.17)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">1.35 [−4.67 to 7.38]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">54.56 (9.55)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">55.21 (11.14)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.64 [−4.19 to 2.89]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>1st measure</bold>\n</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">Area (mm²)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">465.90 (205.77)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">394.74 (105.89)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">71.16 [−20.02 to 162.34]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">405.96 (125.96)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">425.59 (152.19)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−19.63 [86.38–47.12]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Velocity (m/s)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.07 (0.01)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.07 (0.01)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0 [−0.00 to 0.00]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.07 (0.01)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.07 (0.01)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.00 [0.00–0.00]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Xmean (mm)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">41.81 (5.93)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">40.35 (5.88)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">1.46 [−1.75 to 4.66]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">40.58 (6.64)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">39.86 (6.29)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.71 [−1.74 to 3.17]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ymean (mm)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">63.0 (22.41)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">52.75 (8.72)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">10.25 [0.96–19.53]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">55.24 (12.13)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">57.59 (14.68)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−2.35 [−9.66 to 4.96]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Highest measure</bold>\n</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">Area (mm²)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">544.34 (212.62)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">452.44 (210.97)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">91.9 [−16.35 to 200.16]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">514.22 (149.12)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">450.88 (204.9)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">63.34 [−16.51 to 143.20]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Velocity (m/s)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.08 (0.01)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.07 (0.03)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.01 [−0.00 to 0.01]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.08 (0.01)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.07 (0.02)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.008 [−0.00 to 0.02]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Xmean (mm)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">45.98 (6.44)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">39.73 (14.67)<xref rid=\"table-1fn2\" ref-type=\"table-fn\">*</xref></td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">6.25 [0.87–11.63]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">43.77 (5.33)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">39.13 (14.45)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">4.64 [−0.49 to 9.77]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ymean (mm)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">67.7 (20.77)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">58.44 (23.47)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">9.26 [−0.289 to 21.41]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">67.62 (13.64)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">59.47 (24.55)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">8.15 [−1.64 to 17.94]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Lowest measure</bold>\n</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">Area (mm²)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">314.86 (87.52)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">282.28 (128.83)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">32.58 [−20.38 to 85.54]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">326.09 (101.82)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">271.0 (130.27)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">55.09 [3.89–106.29]<xref rid=\"table-1fn2\" ref-type=\"table-fn\">*</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\">Velocity (m/s)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.068 (0.01)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.061 (0.02)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.007 [−0.00 to 0.01]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.06 (0.01)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.05 (0.02)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.007 [−0.00 to 0.01]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Xmean (mm)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">36.84 (5.33)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">33.28 (12.44)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">3.56 [−1.44 to 8.57]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">36.24 (4.79)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">31.41 (11.97)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">4.82 [0.43–9.22]<xref rid=\"table-1fn2\" ref-type=\"table-fn\">*</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ymean (mm)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">45.95 (9.20)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">40.77 (15.88)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">5.18 [−1.75 to 12.11]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">46.25 (8.56)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">40.91 (16.9)</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">5.33 [−1.66 to 12.33]</td></tr></tbody></table></alternatives></table-wrap>", "<table-wrap position=\"float\" id=\"table-2\"><object-id pub-id-type=\"doi\">10.7717/peerj.16765/table-2</object-id><label>Table 2</label><caption><title>Internal responsiveness statistics for the three monopodal stability tests after instability training.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\" content-type=\"text\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\"/><th colspan=\"2\" rowspan=\"1\">Dominant lower limb</th><th colspan=\"2\" rowspan=\"1\">Non-dominant lower limb</th></tr><tr><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">Paired <italic toggle=\"yes\">t</italic>-test (<italic toggle=\"yes\">p</italic>)</th><th rowspan=\"1\" colspan=\"1\">SRM (95% CI)</th><th rowspan=\"1\" colspan=\"1\">Paired <italic toggle=\"yes\">t</italic>-test (<italic toggle=\"yes\">p</italic>)</th><th rowspan=\"1\" colspan=\"1\">SRM (95% CI)</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>EBT</bold>\n</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.003</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">2.43 [1.69–3.09]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.864</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.45 [−0.98 to 0.10]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>mSEBT</bold>\n</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">ANT</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.026</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">1.00 [0.42–1.55]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.215</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">1.00 [0.42–1.55]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">PM</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.001</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">9.00 [7.11–10.63]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.001</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">7.00 [5.49–8.30]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">PL</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.001</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">4.17 [3.17–5.05]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.001</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">2.00 [1.32–2.62]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Total score</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.001</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">5.00 [3.86–6.00]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.001</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">4.00 [3.03–4.86]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Stabilometry</bold>\n</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Mean of 3 measurements</bold>\n</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">Area</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.923</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.13 [−0.64 to 0.37]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.241</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">3.59 [2.74–4.36]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Velocity</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.720</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.23 [−0.73 to 0.28]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.045</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−2.14 [−2.75 to −1.48]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Xmean</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.390</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−6.41 [−7.62 to −5.01]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.294</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−1.39 [−1.96 to −0.78]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ymean</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.648</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.5 [−0.05 to 1.03]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.709</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.41 [−0.14 to 0.94]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>1st measure</bold>\n</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">Area</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.121</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.71 [0.15–1.25]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.551</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.75 [0.19–1.29]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Velocity</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.966</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.00 [−0.52 to 0.52]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.295</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.34 [−0.87 to 0.20]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Xmean</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.359</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">29.96 [23.93–35.12]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.556</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">2.06 [1.37–2.68]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ymean</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.032</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.75 [0.19–1.29]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.515</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.92 [0.35–1.47]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Highest measure</bold>\n</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">Area</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.093</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">55.7 [45.13–64.83]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.116</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−1.14 [−1.67 to −0.58]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Velocity</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.233</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.51 [−1.02 to 0.01]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.162</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.62 [−1.12 to −0.09]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Xmean</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.024</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.76 [−1.27 to −0.23]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.075</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.51 [−1.02 to 0.01]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ymean</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.130</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−3.43 [−4.18 to −2.6]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.099</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.75 [−1.26 to −0.21]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Lowest measure</bold>\n</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">Area</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.218</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.79 [−1.30 to −0.25]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.036</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−1.94 [−2.52 to −1.30]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Velocity</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.154</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.55 [−1.06 to −0.03]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.096</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.64 [−1.15 to −0.11]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Xmean</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.156</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.50 [−1.01 to 0.02]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.032</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.67 [−1.18 to −0.14]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ymean</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.137</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.78 [−1.29 to −0.24]</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.130</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">−0.64 [−1.15 to −0.11]</td></tr></tbody></table></alternatives></table-wrap>", "<table-wrap position=\"float\" id=\"table-3\"><object-id pub-id-type=\"doi\">10.7717/peerj.16765/table-3</object-id><label>Table 3</label><caption><title>External responsiveness by areas under curve (AUC) for Emery balance test and modified Star Excursion Balance Test.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\" content-type=\"text\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\">Test</th><th rowspan=\"1\" colspan=\"1\">Area under curve</th><th rowspan=\"1\" colspan=\"1\">95% confidence interval</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">EBT</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.708</td><td rowspan=\"1\" colspan=\"1\">[0.57–0.84]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">mSEBT</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">Anterior</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.561</td><td rowspan=\"1\" colspan=\"1\">[0.40–0.71]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Posteromedial</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.617</td><td rowspan=\"1\" colspan=\"1\">[0.46–0.76]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Posterolateral</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.557</td><td rowspan=\"1\" colspan=\"1\">[0.40–0.71]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Total score</td><td align=\"char\" char=\".\" rowspan=\"1\" colspan=\"1\">0.460</td><td rowspan=\"1\" colspan=\"1\">[0.31–0.62]</td></tr></tbody></table></alternatives></table-wrap>" ]

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[ "<supplementary-material id=\"supp-1\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16765/supp-1</object-id><label>Supplemental Information 1</label><caption><title>Raw data of all variables for each subject.</title></caption></supplementary-material>", "<supplementary-material id=\"supp-2\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16765/supp-2</object-id><label>Supplemental Information 2</label><caption><title>Codebook to convert numbers to their respective factors.</title></caption></supplementary-material>", "<supplementary-material id=\"supp-3\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16765/supp-3</object-id><label>Supplemental Information 3</label><caption><title>Supplementary Appendix.</title></caption></supplementary-material>" ]

[ "<table-wrap-foot><fn id=\"table-1fn\"><p>\n<bold>Notes:</bold>\n</p></fn><fn id=\"table-1fn1\" fn-type=\"other\"><p>M, mean; SD, standard deviation; mSEBT, modified Star Excursion Balance Test; EBT, Emery balance test; ANT, anterior; PM, posteromedial; PL, posterolateral.</p></fn><fn id=\"table-1fn2\" fn-type=\"other\"><label>*</label><p>Statistically significant differences between pre and post measurements.</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"table-2fn\"><p>\n<bold>Note:</bold>\n</p></fn><fn id=\"table-2fn1\" fn-type=\"other\"><p>SRM, standardised response mean; CI, confidence interval; mSEBT, modified Star Excursion Balance Test; EBT, Emery balance test; ANT, anterior; PM, posteromedial; PL, posterolateral.</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"table-3fn\"><p>\n<bold>Note:</bold>\n</p></fn><fn id=\"table-3fn1\" fn-type=\"other\"><p>mSEBT, modified Star Excursion Balance Test; EBT, Emery balance test.</p></fn></table-wrap-foot>", "<fn-group content-type=\"competing-interests\"><title>Competing Interests</title><fn fn-type=\"COI-statement\" id=\"conflict-1\"><p>The authors declare that they have no competing interests.</p></fn></fn-group>", "<fn-group content-type=\"author-contributions\"><title>Author Contributions</title><fn fn-type=\"con\" id=\"contribution-1\"><p><xref rid=\"author-1\" ref-type=\"contrib\">Mª Piedad Sánchez Martínez</xref> performed the experiments, analyzed the data, authored or reviewed drafts of the article, and approved the final draft.</p></fn><fn fn-type=\"con\" id=\"contribution-2\"><p><xref rid=\"author-2\" ref-type=\"contrib\">Mariana Sánchez-Barbadora</xref> conceived and designed the experiments, performed the experiments, analyzed the data, authored or reviewed drafts of the article, and approved the final draft.</p></fn><fn fn-type=\"con\" id=\"contribution-3\"><p><xref rid=\"author-3\" ref-type=\"contrib\">Noemi Moreno-Segura</xref> analyzed the data, authored or reviewed drafts of the article, and approved the final draft.</p></fn><fn fn-type=\"con\" id=\"contribution-4\"><p><xref rid=\"author-4\" ref-type=\"contrib\">Patricia Beltrá</xref> performed the experiments, analyzed the data, prepared figures and/or tables, authored or reviewed drafts of the article, and approved the final draft.</p></fn><fn fn-type=\"con\" id=\"contribution-5\"><p><xref rid=\"author-5\" ref-type=\"contrib\">Adrian Escriche-Escuder</xref> analyzed the data, authored or reviewed drafts of the article, and approved the final draft.</p></fn><fn fn-type=\"con\" id=\"contribution-6\"><p><xref rid=\"author-6\" ref-type=\"contrib\">Rodrigo Martín-San Agustín</xref> conceived and designed the experiments, performed the experiments, analyzed the data, authored or reviewed drafts of the article, and approved the final draft.</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn id=\"addinfo-1\"><p>The following information was supplied relating to ethical approvals (<italic toggle=\"yes\">i.e</italic>., approving body and any reference numbers):</p><p>This study was approved by the Ethics Committee of the University of Valencia (Spain) (Ethical Application Ref: 1271077).</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Data Availability</title><fn id=\"addinfo-2\"><p>The following information was supplied regarding data availability:</p><p>The raw measurements are available in the <xref rid=\"supplemental-information\" ref-type=\"sec\">Supplemental Files</xref>.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"peerj-12-16765-s001.sav\"><caption><p>Click here for additional data file.</p></caption></media>", "<media xlink:href=\"peerj-12-16765-s002.docx\"><caption><p>Click here for additional data file.</p></caption></media>", "<media xlink:href=\"peerj-12-16765-s003.pdf\"><caption><p>Click here for additional data file.</p></caption></media>" ]

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[ "<title>Introduction</title>", "<p>Birds, which encompass a remarkable diversity of over 11,000 species, are a captivating and highly valued part of the natural world (##UREF##2##BirdLife International, 2018##). Their intricate variety ranges from the tiniest to the largest, and the slowest to the swiftest flyers. Each bird species possesses a unique presence, habits, and habitat preferences (##UREF##2##BirdLife International, 2018##). This remarkable diversity showcases itself in both the vast numbers of some species, like the 8,421 species classified as least concern, and the scarcity of others, with a mere handful of surviving individuals (##UREF##18##IUCN, 2018##). The International Union for Conservation of Nature (IUCN) red list categories further categorize birds, with 1,470 species classified as threatened, and among them, 223 critically endangered, 461 endangered, and 786 vulnerable (##UREF##18##IUCN, 2018##)</p>", "<p>In this tapestry of avian diversity, Ethiopia emerges as a hotspot, harboring 872 distinct bird species, 18 of which are endemic, and another 67 represented as endemic sub-species (##UREF##24##Mengistu, 2002##). With 851 of its bird species evaluated within the IUCN red list categories, Ethiopia underscores the global importance of preserving avian populations (##UREF##18##IUCN, 2018##). As they traverse the world’s diverse habitats, birds leave their ecological footprints, indicating the health of ecosystems. Birds, being excellent indicators of environmental health, offer a window into the impacts of pollution and climate change (##REF##15601765##Sekercioglu, Daily &amp; Ehrlich, 2004##).</p>", "<p>The interplay between birds and their habitats is fundamental in shaping distribution patterns. Habitats, often shaped by vegetation and complemented by other factors, determine where birds thrive. Recognizing the significance of this dynamic, Important Bird and Biodiversity Areas (IBAs) and Key Biodiversity Areas (KBAs) have emerged as key tools for global conservation efforts. These designated areas, which number over 13,000 across more than 200 countries, act as crucial bastions for the conservation of biodiversity (##UREF##2##BirdLife International, 2018##).</p>", "<p>Beyond their ecological roles, birds provide an array of essential ecosystem services. They diligently contribute to pollination, insect pest control, seed dispersal, and nutrient cycling, all which ripple through ecosystems, benefiting both nature and human society. Bird activity knits together ecosystems and influences the abundance of other species (##REF##15601765##Sekercioglu, Daily &amp; Ehrlich, 2004##; ##UREF##34##Wenny et al., 2011##). For example, frugivorous birds maintain gene flow and enhance restoration efforts through seed dispersal. In this context, birds can be regarded as ecological engineers, shaping landscapes, and fostering ecosystem resilience (##UREF##34##Wenny et al., 2011##).</p>", "<p>However, the intricate web of avian diversity and its contributions to ecosystems faces a looming threat. Birds have become bioindicators of environmental changes, and their declining populations serve as a stark warning (##UREF##3##Bonisoli-Alquati et al., 2022##; ##UREF##23##Mekonen, 2017##). The IUCN red list data reveals a steady deterioration in the status of the world’s bird species (##UREF##18##IUCN, 2018##). Human activities, from agricultural expansion and logging to pollution and invasive species introduction, are driving these declines (##UREF##22##Malhotra, 2022##). Furthermore, the long-term specter of climate change hovers, potentially amplifying these threats (##UREF##10##de Moraes et al., 2020##).</p>", "<p>The decline in avian diversity worldwide due to human activities and climate change poses a threat to the ecosystem services that birds provide. Therefore, there is an urgent need for conservation efforts to preserve avian diversity and safeguard these ecosystem services for the benefit of both nature and humanity. The objective of this study was to identify species diversity and relative abundance as baseline information through a survey or census of bird populations in Dodola forest. Initial surveillance or inventory of bird species has not been specifically conducted in the study area. The area is experiencing habitat disturbance, and the status of bird populations remains largely unknown, making this a critical concern. Therefore, it is essential to assess the composition, abundance, and presence or absence of birds across different habitats. This information is crucial for ongoing monitoring and evaluation of bird statuses in the study area. This baseline information would be used to inform conservation efforts and monitor changes in bird populations over time.</p>" ]

[ "<title>Materials and Methods</title>", "<title>Description of the study area</title>", "<title>Location</title>", "<p>The Dodola natural forest habitat is part of the Adaba Dodola Jalo forest which is one of the 61 National Forest Priority Areas (NFPA) of the country that covers approximately 530 km² (##UREF##16##Gelashe, 2017##). The Ericaceous sub-afro alpine habitat is found at higher elevations to the natural forest, while the plantation forest below the dry evergreen afro-montane forest. Dodola forest is located West Arsi zone of the Oromia regional state, southeastern Ethiopia (##FIG##0##Fig. 1##). The study area is adjacent the Bale mountains massif and occurs at 325 km from Addis Ababa towards the southeast, 70 km from Shashemene. The area is bordered by the Kofale district to the west, the Adaba district to the east, the Nensabo and Kokossa districts to the south, and the Asasa district to the north. The geographical location ranges between 6°39′E38°57′N and 7°0′E39°24′N. The altitude range varies from 2,400–3,712 m.a.s.l. The area is a part of tropical forest and tropical shrub land that consists of natural forest (Dry evergreen Afromontane Forest), Ericaceous vegetation (sub-afro alpine habitat) and community plantation forest of a total of 738.30.24 km².</p>", "<title>Climate and vegetation</title>", "<p>The study area has a four-month as dry season (November–February) and an eight-month as wet season (March–October) (##UREF##17##Hundera, Bekele &amp; Kelbessa, 2007##). The characteristics of the forest are categorized as upland dry evergreen forests of Afromontane forests (##UREF##15##Friis, Rasmussen &amp; Vollesen, 1982##). The Dodola region’s forest landscape changes with altitude. Between 2,565 to 2,800 m, conifer forests become dominant, with Podocarpus and Juniperus as the prevailing species. Moving to the middle altitude zone of 2,804–3,115 m, Juniperus procera takes the lead, alongside other broadleaf hardwood species, while Podocarpus falcatus becomes less common and sporadically found at the lower boundary of this zone. In the upper elevation range of 3,120–3,400 m (##UREF##5##Brooks, 2009##), the forest is similar in ecological characteristics to Bale Mountain National Park, featuring highland forest habitat and sub-afro alpine terrain with Ericaceous vegetation (##UREF##12##Evangelista, Swartzinski &amp; Waltermire, 2007##). The Erica trimera dominates at higher elevations, while Erica arborea prevails at lower elevations. Additionally, the Dodola region’s forest includes native species like Hagenia abyssinica, <italic toggle=\"yes\">Hypericum lanceolatum</italic>, and Erica arborea, as well as introduced exotic species like Eucalyptus and Cupressus lusitanica in peripheral areas. Juniperus procera is noteworthy for its susceptibility to wildfires and preference for well-drained, nearly neutral pH soils, thriving within specific altitude, precipitation, and temperature conditions in the study area (##UREF##16##Gelashe, 2017##).</p>", "<title>Socioeconomic information</title>", "<p>The total population of the district is about 194,000. The urban population of 35,000 (18%) is one of the largest in the zone (##UREF##11##Ethiopian Central Statistical Agency, 2007##). Subsistence agriculture and animal husbandry are the main activities in and outside of the forest delineation area.</p>", "<title>Methods</title>", "<title>Preliminary survey</title>", "<p>Preliminary assessment was carried out for identification of key habitats during September 2018. To observe habitat type, age effect, topography, and climatic factors for survey design preconditions. During this period, waypoints were collected using GPS in each habitat type (##UREF##25##QGIS.org, 2018##). A pilot survey was also conducted for sample size information.</p>", "<title>Sampling design</title>", "<p>A point transect sampling method was used to investigate bird species composition, relative abundance, and habitat association (##UREF##6##Buckland et al., 1993##). Based on the preliminary survey, the study area was stratified into three dominant habitat types: the sub-afro alpine Ericaceous scrubland habitat; dry evergreen Afromontane Forest; and mixed plantation forest using QGIS. In each habitat type, systematic sampling design was employed. There are eleven blocks: five Erica, five forest and one plantation. The total block area was 128.839 km² area, which is 17.5% of the study area. A systematic point grid of a 1.5-kilometer fixed dimension was randomly superimposed (##FIG##1##Fig. 2##), and rotation onto the survey region employed proportionally in each habitat type (##UREF##6##Buckland et al., 1993##). The required number of sample points in the survey region calculated as where k₀ and n₀ are roughly estimated in a pilot survey, and the value of <italic toggle=\"yes\">b</italic> = 3 (##UREF##6##Buckland et al., 1993##). In the pilot survey there were five points and 54 individual bird observations. The required number of points was 111 points; 42 points in Erica, 64 in forest and five in plantation (##FIG##1##Fig. 2##). In cluster: \n</p>", "<title>Data collection</title>", "<p>Field guidebooks were tools for identification of the type of bird species exist in the area (birds of the horn of Africa, birds of the East of Africa, birds of Lake Tana, and important bird areas of Ethiopia) (##UREF##28##Redman, Stevenson &amp; Fanshawe, 2016##). The data collected was carried out for two seasons during the months of July and August for the wet season and December and January for the dry season. Per season, data collection was conducted in two sessions/visits. Data collection was carried out early in the afternoon and late in the afternoon. Detection distances was measured from the point to detected object (##UREF##6##Buckland et al., 1993##). All observation beyond 70 m sighting distance were truncated. Birds’ songs were used for most elusive forest birds (##UREF##7##Buckland et al., 2001##). Identification and counting of most bird species were assisted by binoculars. Points taken 200 m distance inside from edge to avoid edge effect. Duration a point count lasts from 2 min to 20 min (##UREF##1##Bibby, Jones &amp; Marsden, 1998##).</p>", "<title>Data analysis</title>", "<p>Lists of information about habitat type, season, visit, block, point, cluster size and species code were organized in a single data frame. With the help of R software data organizing functions, for similarity and diversity analysis, the data was organized in form of data frame where rows as species list, and columns as the presence and absence data. One column for a single habitat, and one column for a sample point to similarity and species accumulative curve data analysis respectively.</p>", "<p>Data analyzed based on distance sampling method distance 7.3 software (##REF##20383262##Thomas et al., 2010##), and the mark recapture distance sampling (MRDS) analysis engine supplemented by R software (##UREF##27##R Core Team, 2019##). R software was used to analyze ANOVA test using the Car package, and similarity and diversity indicies were analyzed with the Simba and Vegan package (##UREF##27##R Core Team, 2019##). AIC and the chi-square statistical test were applied to obtain the best-fitted models (##UREF##7##Buckland et al., 2001##; ##UREF##6##Buckland et al., 1993##). The result was analyzed based on the data recorded on 111 sample points and 222 total efforts of two replication or visit during both seasons. The analysis of distance was based on the formula described by ##FORMU##0##Eqs. (1)##–##FORMU##5##(4)## (##UREF##7##Buckland et al., 2001##; ##UREF##6##Buckland et al., 1993##).</p>", "<p>For point transects analyses MRDS always uses the P3 estimator for encounter rate variance. where <italic toggle=\"yes\">ti</italic> is the number of times point <italic toggle=\"yes\">i</italic> was visited, andis the number of objects detected at point <italic toggle=\"yes\">i</italic> on visit <italic toggle=\"yes\">j</italic>.</p>", "<p>Relative abundance of avian species determined using encounter rates calculated for each species by dividing the number of birds recorded(n) by the number points (k) multiply time of visit or effort (t) (##UREF##7##Buckland et al., 2001##).</p>", "<p>The encounter rate (ER) was estimated as: Encounter rate data was classified into crude ordinal categories of abundance (<italic toggle=\"yes\">e.g.</italic>, abundant, common, frequent, uncommon, and rare) (##TAB##0##Table 1##).</p>", "<p>The number of individuals per total effort were ≤0.01, 0.01–0.2, 0.2–1,1-4 and &gt;4. For each interval, the following abundance labels is given rare, uncommon, frequent, common, and abundant, respectively. Therefore, the relative abundance of each bird species was determined by Excel if function of rare, uncommon, frequent, common, and abundant. For example, if the encounter rate is ≤0.01, the species is considered as rare. Analysis were prepared for two type of data selection steps in multispecies analysis options. the first is setting individual species analysis using data filter, the second was not based on individual species; thus, birds as one taxonomic categories of class of Aves as compared to species taxa. In both steps, habitats were stratum whereas seasons were analyzed by using data filter separately.</p>", "<p>A two-way ANOVA was used to analyze density and number of individual observation effect of three factors through season, habitat, and species. The ANOVA type III error to investigate interaction effect (Model 1). The ANOVA type II error was used for incasing of non-interaction effect (Model 2) \nwhere, μ = the overall mean of species observed, <italic toggle=\"yes\">α</italic>_i, <italic toggle=\"yes\">β</italic>_j and <italic toggle=\"yes\">γ</italic>_j are the i^th, j^th and k^th habitat, season and species effects, respectively. where <italic toggle=\"yes\">δ</italic>_{<italic toggle=\"yes\">ij</italic>} is interaction term (##UREF##29##Searle, Speed &amp; Milliken, 1980##). <italic toggle=\"yes\">Post-hoc</italic> test used for separate group analysis for interaction effect results. Estimated marginal means (emmeans) was used for non-interaction effect pairwise comparison of groups. Differences were considered statistically significant at 5% (##UREF##8##Chambers &amp; Hastie, 1992##). Unbiased sim was calculated as , Simpson’s index D = Simpson’s Simpson returns <italic toggle=\"yes\">1-D</italic> and inv Simpson returns <italic toggle=\"yes\">1/D</italic> (##REF##28973811##Hurlbert, 1971##) , where <italic toggle=\"yes\">n</italic>_{<italic toggle=\"yes\">i</italic>} denotes number of individuals in the i ^th species (<italic toggle=\"yes\">n</italic>_{<italic toggle=\"yes\">i</italic>} = 1,2,3…., <italic toggle=\"yes\">n</italic> and <italic toggle=\"yes\">n</italic>1 + <italic toggle=\"yes\">n</italic>2…<italic toggle=\"yes\">n</italic> = <italic toggle=\"yes\">N</italic>), S = total number of species (##UREF##30##Shannon, 2001##). In Fisher’s logarithmic series the expected number of species f with n observed individuals is The parameter <italic toggle=\"yes\">α</italic> is used as a diversity index. The parameter <italic toggle=\"yes\">x</italic> is taken as a nuisance parameter which is not estimated separately but taken to be n/(n+ <italic toggle=\"yes\">α</italic>) (##UREF##13##Fisher, Corbet &amp; Williams, 1943##). The species discovery curve was used species richness/number of species discovered across each sample points based on the sample-based rarefaction formula for adequate sample size for a multi-species survey. A collection on <italic toggle=\"yes\">n</italic> samples, the rarefaction curve is the plot of against <italic toggle=\"yes\">i</italic> (<italic toggle=\"yes\">i</italic> = 1, …, <italic toggle=\"yes\">n</italic>), where <italic toggle=\"yes\">S</italic>_{<italic toggle=\"yes\">i</italic>} indicates the arithmetic mean, <italic toggle=\"yes\">S</italic>_{<italic toggle=\"yes\">n</italic>} denotes the total number of observed species, <italic toggle=\"yes\">nk</italic> denotes the number of samples containing at least one individual species <italic toggle=\"yes\">k</italic> ∈ <italic toggle=\"yes\">G</italic> (##UREF##9##Chiarucci et al., 2008##).</p>", "<p>The diversity and relative abundance presented by tables, qq plot and detection function plot. Statistical difference presented through ggplot2 supported by narrative descriptions. Habitat association of number of species were computed for Sorenson’s similarity index (SI) among habitats under two seasons by using the following formula. SI = 2a/2a+b+c; where 2a = number of species common to two habitats, b = number of species in first habitat, c = number of species in the second habitat (##UREF##31##Sorensen, 1948##).</p>" ]

[ "<title>Results</title>", "<title>Species composition</title>", "<p>Over the course of two distinct climatic periods (dry and wet), a total of 78 species of birds were recorded. Within the recorded species, the Abyssinian Catbird (<italic toggle=\"yes\">Parophasma galinieri</italic>), Ethiopian Siskin (<italic toggle=\"yes\">Serinus nigriceps</italic>), and Yellow Fronted Parrot (<italic toggle=\"yes\">Poicephalus flavifrons</italic>) have been identified as endemic. Furthermore, there exists a subset of ten species, inclusive of the Wattled Ibis <italic toggle=\"yes\">(Bostrychia carunculate</italic>), the black-winged lovebird (<italic toggle=\"yes\">Agapornis taranta</italic>) and Rouget’s Rail (<italic toggle=\"yes\">Rouget‘s rougetii</italic>), which are recognized as endemic to both Ethiopia and Eritrea (##SUPPL##3##Appendix S1##). Based on the lowest AIC value of MRDS analysis engine, the fitted model was single observer distance model and half-normal key function with model for scale parameters is a constant (CDS).</p>", "<p>##FIG##2##Figure 3## shows the species discovery curve and ##FIG##3##Fig. 4## shows the extrapolation curve with increasing number of species in the <italic toggle=\"yes\">y</italic> axis with sample points in the <italic toggle=\"yes\">x</italic> axis; the curve turns as asymptote shape indicates that the species discover is adequate. The asymptote predicts 86 species to be discovered, which means that over 90% of the species in the area were discovered with a slope 2.62 (the more the slope close to zero, a few or none of species in the area are left detected) (##FIG##2##Fig. 3##).</p>", "<p>The Quantile-Quantile (QQ) plot, which shows the fitted cumulative distribution function (cdf) against the empirical distribution function (edf), represents the number of observed bird species. The dots on the plot correspond to these observations. The line in the QQ plot represents the expected distribution if the model fit was perfect. The proximity of the dots to the line indicates the fit of the model. In this case, the dots surrounding the line suggest that the model is well-fitted (##FIG##4##Figs. 5## and ##FIG##5##6##). The detection function plots illustrate the expected probability density function of frequencies divided by distance. The curve in these plots represents the expected distribution, while the histograms display the number of observations. The unweighted Cramer-von Mises tests a <italic toggle=\"yes\">p</italic>-value was less than 0.001 in both seasons (##FIG##6##Figs. 7## and ##FIG##7##8##). It is important to note that the detection function depicted in ##FIG##5##Figs. 6## and ##FIG##7##8## represents the overall class Aves. This means it does not account for individual bird species observed in the study.</p>", "<p>The species composition of birds during the wet and dry seasons was not significantly different (F, Season = 0.004, <italic toggle=\"yes\">p</italic> &gt; 0.05) which was 0.95. On the other side, there was a significant difference among habitats (F, Habitat = 12.78, <italic toggle=\"yes\">p</italic> &lt; 0.05) which was 7.466e−06 ***. There was no season and habitat interaction effect (F2, Habitat: Season = 2.28, <italic toggle=\"yes\">p</italic> &gt; 0.05) which was 0.11. The estimated marginal means, also known as least-squares means, revealed a significant difference in the mean number of species across two habitat types: Erica and forest. The mean number of species in the Erica habitat was 24 (±3.16 SE), while in the forest habitat it was 22 (±2.33 SE). However, in the plantation habitat, the estimated marginal mean value was −0.8 (±4.3 SE) (##FIG##8##Fig. 9##).</p>", "<p>According to a Tukey pairwise comparison test with a 95% confidence interval, there was no significant difference in the mean number of species between the Erica and forest habitats. Plantation had the least mean number of species. The <italic toggle=\"yes\">P</italic> value for Erica <italic toggle=\"yes\">vs</italic> forest was &gt;0.05. The <italic toggle=\"yes\">P</italic> value for Erica <italic toggle=\"yes\">vs</italic> plantation and forest <italic toggle=\"yes\">vs</italic> plantation was &lt;0.01.</p>", "<p>The highest species diversity (D) during the wet and dry seasons were observed in Forest habitat (dry evergreen afromontane forest), followed by the Erica (sub-afroalpine) habitat with (0.951 &amp; 0.949) and (0.929 &amp; 0.926) respectively, while the mixed plantation habitat had the least with (0.905 &amp; 0.887). The highest species evenness was observed in the Erica habitat. For the entire season, the forest habitat had the highest species diversity (0.943), while Erica habitat had the highest species evenness (0.85) (##TAB##1##Table 2##).</p>", "<title>Species relative abundance</title>", "<p>In the dry season, a total of 2,639 individual birds were recorded, while in the wet season, 2,410 individual birds of 78 species were observed (##TAB##2##Table 3##). In the 2018 IUCN red list categories, six species faced global threats, three species neared the threat status, and a total of 69 species were classified as least concern.</p>", "<p>The mixed plantation forest habitat recorded the highest relative abundance of Aves, with 15.7 and 16.7 during the dry and wet seasons respectively. This was followed by Erica in the dry season with 13.14, and the forest in the wet season with 10.67. The dry season exhibited a higher overall seasonal relative abundance of 11.89 (##TAB##2##Table 3##).</p>", "<p>The relative abundance of individual species in stratified habitat is shown in ##TAB##3##Tables 4##, ##TAB##4##5## and ##TAB##5##6## in Erica, forest and plantation habitat, respectively. In the Erica (sub-afroalpine habitat), the encounter rate was calculated as number of individual observations in the Erica per Erica point samples times number of a point visit (n/84). In the Erica habitat, the Red-wing Starling had the highest relative abundance during the dry season (1.95), while the Scare Swift had the highest relative abundance in the wet season (1.02). The Chestnut-napped Francolin and Common Buzzard were not recorded in the dry season, and similarly, the Yellow-billed Kite and White-headed Vulture were not recorded in the wet season. During the dry season, four, 10 and 21 species were classified as common, frequent, and uncommon, respectively. During the wet season, one, 17 and 18 species were common, frequent, and uncommon, respectively. Rare and abundance species were not recorded under the two seasons (##TAB##3##Table 4##). No species were recorded as rare or abundant in either of the two seasons (##TAB##3##Table 4##).</p>", "<p>In the dry afromontane forest habitat, the encounter rate was calculated as number of individual observation per the dry afromontane forest habitat effort (n/128). Montane White-eye was the highest relative abundance during both seasons (1.88 and 1.57). Mouse-colored Penduline Tit, Variable Sunbird, Abyssinian owl, African Stonechat and Common Buzzard were not recorded in the dry season, while in the wet season, Yellow-billed Kite and White-headed Vulture were not recorded. During the dry season two, 14, 39 and two species were common, frequent, uncommon, and rare respectively. During the wet season, one, 19 and 34 species were common, frequent and uncommon respectively. Rare and abundant species were not recorded under the wet season (##TAB##4##Table 5##). In the plantation forest habitat, the encounter rate was calculated as number of individual observations per plantation forest habitat Effort (n/10). Ground Scarper Thrush was the highest relative abundance during dry seasons (3.1). In the wet season, the yellow crown canary was the highest (2.00). During the dry season two, 10 and 7 species were common, Frequent and Uncommon, respectively. During the wet season, two, four and 14 species were common, Frequent and Uncommon respectively. Rare and abundance species were not recorded under two seasons. six species in the dry season and five species in the wet season were isolated record in the season (##TAB##5##Table 6##).</p>", "<title>Habitat association of bird species</title>", "<p>Not all species were distributed in all habitat type. Of 39 different bird species, 15 species specific to Ericaceous sub-afro alpine scrubland vegetation (##TAB##3##Table 4##), while in Afromontane forest habitat, a number of 59 different bird species were founded, about twenty six species were specific to the habitat (##TAB##4##Table 5##), In the plantation forest habitat 26 bird species were recorded, there three species specific to community plantation forest habitat (##TAB##5##Table 6##) and the rest 34 species were recorded either in three or in only the two habitats (##SUPPL##3##Appendix S1##). A forest habitat accounts for a high number of bird species and high specific species. The Erica and forest habitats share more species that are common in the dry season (##FIG##9##Fig. 10## and ##TAB##6##Table 7##). Plantation and forest share more species that are common in the wet season (##FIG##10##Fig. 11## and ##TAB##6##Table 7##). Plantation and Erica share the lowest common species (##TAB##6##Table 7##).</p>" ]

[ "<title>Discussion</title>", "<title>Low detection frequencies and detection probabilities</title>", "<p>Some species, including the African Black Swift, Pied Crow, and Common Buzzard, exhibited low detection frequencies, falling below the standard recommended threshold of 60–80 observations. This could introduce bias into results, as insufficient data for these species may hinder robust analyses (##UREF##6##Buckland et al., 1993##). Additionally, certain species, particularly those specialized in woodland habitats, displayed lower detectability. This lower detectability, especially for species in closed habitats, may be influenced by various factors, including habitat structure and observer bias (##UREF##19##Johnston et al., 2014##). For multispecies surveys, it is crucial to account for local habitat effects on all species, not just those with abundant data (##UREF##36##Zipkin et al., 2010##).</p>", "<title>Seasonal variability</title>", "<p>The research unveiled substantial seasonal fluctuations in bird abundance, with marked differences between the dry and wet seasons. While data collection was successful in both seasons, some species exhibited stronger presence during the wet season. This observation aligns with existing research emphasizing the influence of seasonal changes in resource availability and weather conditions on avian populations (##UREF##14##French &amp; Rockwell, 2011##; ##UREF##21##Li et al., 2022##). However, the effect of seasonality on avian species composition may be less pronounced in tropical regions. Seasonal changes in bird populations, feeding habits, and migration patterns are more prominent in temperate regions (##UREF##33##Ward, 1969##; ##UREF##35##White, Warren &amp; Baines, 2015##). Many birds in the study area are resident breeders, with limited migration during seasonal shifts, possibly contributing to the insignificant effect of seasons on bird species composition (##SUPPL##3##Appendix S1##). Migratory birds in Ethiopia are primarily associated with aquatic, wetland, and riverine habitats (##UREF##5##Brooks, 2009##).</p>", "<title>Habitat influence on bird composition and structure</title>", "<p>Habitat strongly influenced bird species composition, with distinct preferences observed among different species. Some species displayed specific habitat associations, such as the White-backed Black Tit in forest habitat, Thekla Lark in Erica, and Semi-colored Flycatcher in plantation habitat (##TAB##3##Tables 4##, ##TAB##4##5## and ##TAB##5##6##). This suggests that avian community composition in the Ethiopian Highlands is intricately linked to habitat types. These findings align with previous studies highlighting the importance of habitat characteristics in shaping avian communities (##UREF##0##Aynalem &amp; Bekele, 2008##). This suggests that various bird species have adapted to distinct ecological niches within the Ethiopian Highlands. The high species abundance in natural forest habitats further underscores their significance in avian biodiversity conservation. These findings resonate with studies by ##REF##32188954##Hendershot et al. (2020)##, which underscored the importance of preserving diverse habitat types for effective avian biodiversity conservation. The heightened encounter rate observed within plantation forests (##TAB##5##Table 6##) suggests the presence of edge effects, particularly as influenced by adjacent agricultural areas. These edge effects are known to attract generalist bird species (##UREF##20##Khamcha et al., 2018##), which typically exploit transitional zones. However, it is noteworthy that the elevated encounter rate is primarily attributed to a select few bird species that have specifically adapted to the plantation forest habitat.</p>", "<p>Microclimate and habitat structure emerged as major drivers influencing avian community composition within specific habitats (##UREF##26##Rajpar &amp; Zakaria, 2015##). The relationship between habitat and species composition was further evident in the similarity index results, which indicated higher similarity between neighboring habitats (##TAB##4##Table 5##). Microclimate, habitat structure, and environmental gradients likely contribute to species distribution patterns and preferences within the Ethiopian Highlands.</p>", "<p>Altitudinal gradients played a role in avian diversity, with the highest species composition recorded in middle elevation zones, primarily within forest habitats (##REF##29466335##Quintero &amp; Jetz, 2018##). Decreases in diversity at higher altitudes may be attributed to factors such as lower speciation or higher extinction rates, potentially influenced by smaller areas or lower temperatures (##REF##29466335##Quintero &amp; Jetz, 2018##).</p>", "<p>The size of the habitat patch and edge effects may also influence avian species composition. Edge-sensitive, neutral, and preferring species respond differently to habitat edges (##UREF##4##Brand &amp; George, 2001##). Forest species exhibit sensitivity to the contrast between natural and anthropogenic habitats (##UREF##37##Zurita et al., 2012##). Plantation habitats, characterized by smaller areas, displayed lower species composition estimates (##REF##29466335##Quintero &amp; Jetz, 2018##). As habitat destruction is a significant concern, particularly in forested areas, preserving diverse habitats and their associated bird species should be a conservation priority (##REF##28515875##Girma et al., 2017##; ##UREF##32##Wang et al., 2017##).</p>", "<p>Future research in this region should address the limitations of our study. Long-term monitoring with extended survey periods, including intermediate seasons, can provide a more comprehensive understanding of avian population dynamics. In addition, expanding taxonomic coverage and accounting for external factors such as climate change and invasive species will enhance our understanding of avian biodiversity in the Ethiopian Highlands.</p>" ]

[ "<title>Conclusion</title>", "<p>The study revealed the presence of three unique endemic bird species, constituting a notable 20% of Ethiopia’s endemic avian population. Moreover, within the study area, an impressive 71% of Ethiopia and Eritrea’s endemic bird species call this region home, highlighting the exceptional levels of endemism present. These findings create an opportunity for the development of community-based ecotourism initiatives. Significantly, bird observation within various blocks of the study area is a crucial aspect. Rather than being influenced by seasonal fluctuations, these blocks are distinguished by differences in elevation, vegetation types, and the presence or absence of bird species. These variations in elevation generate microclimates, each nurturing distinct bird communities. However, this localized endemism also presents challenges, including the concentration of endemic species and potential resource constraints that could pose risks to specific bird populations. The study underscores the critical need for sustained surveillance and conservation strategies, particularly targeting forest-dependent and Erica-specific avian species. These proactive measures are imperative to address the potential risks associated with resource limitations and to safeguard the continued existence of these distinct bird communities. Our findings serve as a call to action for conservationists and policy makers, emphasizing the importance of preserving these unique ecosystems for future generations.</p>", "<p>Thus, it is vital to prioritize dedicated conservation efforts, incorporating a multifaceted approach involving community-based ecotourism development and landscape restoration projects. This study represents the inaugural avian survey within this ecologically significant region. It establishes the groundwork for future research endeavors that can explore various aspects of this ecosystem. These future inquiries may investigate relationships between forest fragmentation and bird density, the impact of human disturbance on bird populations, and the intricate interplay between vegetation and bird communities. As this initial exploration concludes, it opens doors to a wealth of forthcoming insights and discoveries aimed at preserving the Dodola dry afromontane forest and ericaceous scrubland ecosystems.</p>" ]

[ "<title>Background</title>", "<p>Birds’ functional groups are useful for maintaining fundamental ecological processes, ecosystem services, and economic benefits. Negative consequences of loss of functional groups are substantial. Birds are usually found at a high trophic level in food webs and are relatively sensitive to environmental change.</p>", "<title>Methods</title>", "<p>The first surveillance bird study was carried out southeast of Ethiopia adjacent to Bale Mountain National Park aimed at investigating the composition, relative abundance, and distribution of Aves. Using regular systematic point transact sampling, the density and species composition were analyzed through the mark recapture distance sampling engine assisted by R statistical software.</p>", "<title>Results</title>", "<p>This study recorded a total of seventy-eight bird species over two distinct seasons. Among these, fifteen species were exclusive to Erica habitats, twenty-six were found in natural forest habitats, and three were specific to plantation forest habitats. The study also discovered three endemic species. Based on the 2018 IUCN Red List categories, six of the species are globally threatened, three are near threatened, and the remaining sixty-nine are classified as least concern. The relative abundance of birds did not significantly differ across habitats and seasons, but variations were observed among blocks. Bird density was found to fluctuate across the three habitats and two seasons; however, these habitat differences were not influenced by seasonal changes.</p>", "<title>Conclusion</title>", "<p>The findings of this study reveal that the differences in composition and relative abundance are not merely seasonal changes in the forest and Erica habitats. Instead, these habitats create microclimates that cater to specific bird species. However, this localized endemism also presents challenges. The concentration of endemic species and potential resource constraints could pose a threat to these habitat-specialist birds.</p>" ]

[ "<title>Supplemental Information</title>" ]

[ "<title>Additional Information and Declarations</title>" ]

[ "<fig position=\"float\" id=\"fig-1\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/fig-1</object-id><label>Figure 1</label><caption><title>Location map of the study area.</title><p>Map credit: Zenebe Ageru Yilma.</p></caption></fig>", "<fig position=\"float\" id=\"fig-2\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/fig-2</object-id><label>Figure 2</label><caption><title>Sampling design.</title><p>Map credit: Zenebe Ageru Yilma.</p></caption></fig>", "<fig position=\"float\" id=\"fig-3\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/fig-3</object-id><label>Figure 3</label><caption><title>Species accumulation curve.</title></caption></fig>", "<fig position=\"float\" id=\"fig-4\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/fig-4</object-id><label>Figure 4</label><caption><title>Species extrapolation curve.</title></caption></fig>", "<fig position=\"float\" id=\"fig-5\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/fig-5</object-id><label>Figure 5</label><caption><title>Detection function QQ plot during wet season.</title></caption></fig>", "<fig position=\"float\" id=\"fig-6\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/fig-6</object-id><label>Figure 6</label><caption><title>Detection function QQ plot during dry season.</title></caption></fig>", "<fig position=\"float\" id=\"fig-7\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/fig-7</object-id><label>Figure 7</label><caption><title>Detection function/plot: detection probability class of birds during wet season.</title><p>Points indicate probability of detection for a given observation and lines indicate the detection function.</p></caption></fig>", "<fig position=\"float\" id=\"fig-8\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/fig-8</object-id><label>Figure 8</label><caption><title>Detection function/plot: detection probability class of birds during dry season.</title><p>Points indicate probability of detection for a given observation and lines indicate the detection function.</p></caption></fig>", "<fig position=\"float\" id=\"fig-9\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/fig-9</object-id><label>Figure 9</label><caption><title>Number of species observed in different habitats during the dry and wet seasons.</title></caption></fig>", "<fig position=\"float\" id=\"fig-10\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/fig-10</object-id><label>Figure 10</label><caption><title>Similarity index among habitat types in the wet season.</title><p>The numbers indicate 1 for the Erica habitat, 2 for the natural forest habitat and three for the plantation forest habitat, where NBX and NBY represents two comparable habitats .</p></caption></fig>", "<fig position=\"float\" id=\"fig-11\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/fig-11</object-id><label>Figure 11</label><caption><title>Similarity index among habitat types in the dry season.</title><p>The numbers indicate 1 for the Erica habitat, 2 for the natural forest habitat and 3 for the plantation forest habitat, where NBX and NBY represents two comparable habitats.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"table-1\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/table-1</object-id><label>Table 1</label><caption><title>Encounter rates to provide a crude ordinal scale of abundance (##UREF##1##Bibby, Jones &amp; Marsden, 1998##).</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\">\n<bold>Abundance category (Number of individuals per 100 field hours)</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Abundance score</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Ordinal scale</bold>\n</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">&lt;0.1</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">Rare</td></tr><tr><td rowspan=\"1\" colspan=\"1\">0.1–2.0</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td></tr><tr><td rowspan=\"1\" colspan=\"1\">2.1–10.0</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">Frequent</td></tr><tr><td rowspan=\"1\" colspan=\"1\">10.1–40.0</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">Common</td></tr><tr><td rowspan=\"1\" colspan=\"1\">40.0+</td><td rowspan=\"1\" colspan=\"1\">5</td><td rowspan=\"1\" colspan=\"1\">Abundant</td></tr></tbody></table></alternatives></table-wrap>", "<table-wrap position=\"float\" id=\"table-2\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/table-2</object-id><label>Table 2</label><caption><title>Birds species diversity during wet and dry seasons.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\">Habitat</th><th rowspan=\"1\" colspan=\"1\">Season</th><th rowspan=\"1\" colspan=\"1\">No. of species</th><th rowspan=\"1\" colspan=\"1\">No. of individuals</th><th rowspan=\"1\" colspan=\"1\">D</th><th rowspan=\"1\" colspan=\"1\">H</th><th rowspan=\"1\" colspan=\"1\">Inv</th><th rowspan=\"1\" colspan=\"1\">unbiased sim</th><th rowspan=\"1\" colspan=\"1\">alpha</th><th rowspan=\"1\" colspan=\"1\">H/log(S)</th></tr></thead><tbody><tr><td rowspan=\"3\" colspan=\"1\">Erica</td><td rowspan=\"1\" colspan=\"1\">Dry</td><td rowspan=\"1\" colspan=\"1\">36</td><td rowspan=\"1\" colspan=\"1\">1,104</td><td rowspan=\"1\" colspan=\"1\">0.926</td><td rowspan=\"1\" colspan=\"1\">2.94</td><td rowspan=\"1\" colspan=\"1\">13.57</td><td rowspan=\"1\" colspan=\"1\">0.93</td><td rowspan=\"1\" colspan=\"1\">7.13</td><td rowspan=\"1\" colspan=\"1\">0.82</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Wet</td><td rowspan=\"1\" colspan=\"1\">36</td><td rowspan=\"1\" colspan=\"1\">877</td><td rowspan=\"1\" colspan=\"1\">0.949</td><td rowspan=\"1\" colspan=\"1\">3.20</td><td rowspan=\"1\" colspan=\"1\">19.51</td><td rowspan=\"1\" colspan=\"1\">0.95</td><td rowspan=\"1\" colspan=\"1\">7.57</td><td rowspan=\"1\" colspan=\"1\">0.89</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Total</td><td rowspan=\"1\" colspan=\"1\">39</td><td rowspan=\"1\" colspan=\"1\">1,981</td><td rowspan=\"1\" colspan=\"1\">0.941</td><td rowspan=\"1\" colspan=\"1\">3.11</td><td rowspan=\"1\" colspan=\"1\">17.06</td><td rowspan=\"1\" colspan=\"1\">0.94</td><td rowspan=\"1\" colspan=\"1\">6.89</td><td rowspan=\"1\" colspan=\"1\">0.85</td></tr><tr><td rowspan=\"3\" colspan=\"1\">Forest</td><td rowspan=\"1\" colspan=\"1\">Dry</td><td rowspan=\"1\" colspan=\"1\">58</td><td rowspan=\"1\" colspan=\"1\">1,378</td><td rowspan=\"1\" colspan=\"1\">0.929</td><td rowspan=\"1\" colspan=\"1\">3.28</td><td rowspan=\"1\" colspan=\"1\">14.17</td><td rowspan=\"1\" colspan=\"1\">0.93</td><td rowspan=\"1\" colspan=\"1\">12.26</td><td rowspan=\"1\" colspan=\"1\">0.81</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Wet</td><td rowspan=\"1\" colspan=\"1\">54</td><td rowspan=\"1\" colspan=\"1\">1,366</td><td rowspan=\"1\" colspan=\"1\">0.951</td><td rowspan=\"1\" colspan=\"1\">3.46</td><td rowspan=\"1\" colspan=\"1\">20.44</td><td rowspan=\"1\" colspan=\"1\">0.95</td><td rowspan=\"1\" colspan=\"1\">11.23</td><td rowspan=\"1\" colspan=\"1\">0.87</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Total</td><td rowspan=\"1\" colspan=\"1\">59</td><td rowspan=\"1\" colspan=\"1\">2,744</td><td rowspan=\"1\" colspan=\"1\">0.943</td><td rowspan=\"1\" colspan=\"1\">3.41</td><td rowspan=\"1\" colspan=\"1\">17.52</td><td rowspan=\"1\" colspan=\"1\">0.94</td><td rowspan=\"1\" colspan=\"1\">10.61</td><td rowspan=\"1\" colspan=\"1\">0.84</td></tr><tr><td rowspan=\"3\" colspan=\"1\">Plantation</td><td rowspan=\"1\" colspan=\"1\">Dry</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">157</td><td rowspan=\"1\" colspan=\"1\">0.887</td><td rowspan=\"1\" colspan=\"1\">2.47</td><td rowspan=\"1\" colspan=\"1\">8.88</td><td rowspan=\"1\" colspan=\"1\">0.89</td><td rowspan=\"1\" colspan=\"1\">5.25</td><td rowspan=\"1\" colspan=\"1\">0.85</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Wet</td><td rowspan=\"1\" colspan=\"1\">22</td><td rowspan=\"1\" colspan=\"1\">167</td><td rowspan=\"1\" colspan=\"1\">0.905</td><td rowspan=\"1\" colspan=\"1\">2.62</td><td rowspan=\"1\" colspan=\"1\">10.57</td><td rowspan=\"1\" colspan=\"1\">0.91</td><td rowspan=\"1\" colspan=\"1\">6.78</td><td rowspan=\"1\" colspan=\"1\">0.85</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Total</td><td rowspan=\"1\" colspan=\"1\">25</td><td rowspan=\"1\" colspan=\"1\">324</td><td rowspan=\"1\" colspan=\"1\">0.901</td><td rowspan=\"1\" colspan=\"1\">2.63</td><td rowspan=\"1\" colspan=\"1\">10.14</td><td rowspan=\"1\" colspan=\"1\">0.90</td><td rowspan=\"1\" colspan=\"1\">6.32</td><td rowspan=\"1\" colspan=\"1\">0.82</td></tr></tbody></table></alternatives></table-wrap>", "<table-wrap position=\"float\" id=\"table-3\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/table-3</object-id><label>Table 3</label><caption><title>The encounter rate of the three habitats during both seasons number of birds/total effort.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\"/><th align=\"center\" colspan=\"5\" rowspan=\"1\">Dry</th><th align=\"center\" colspan=\"5\" rowspan=\"1\">Wet</th></tr><tr><th rowspan=\"1\" colspan=\"1\">Region</th><th rowspan=\"1\" colspan=\"1\">Effort</th><th rowspan=\"1\" colspan=\"1\">N</th><th rowspan=\"1\" colspan=\"1\">ER</th><th rowspan=\"1\" colspan=\"1\">se.ER</th><th rowspan=\"1\" colspan=\"1\">cv.ER</th><th rowspan=\"1\" colspan=\"1\">Effort</th><th rowspan=\"1\" colspan=\"1\">N</th><th rowspan=\"1\" colspan=\"1\">ER</th><th rowspan=\"1\" colspan=\"1\">se.ER</th><th rowspan=\"1\" colspan=\"1\">cv.ER</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">Erica</td><td rowspan=\"1\" colspan=\"1\">84</td><td rowspan=\"1\" colspan=\"1\">1,104</td><td rowspan=\"1\" colspan=\"1\">13.14</td><td rowspan=\"1\" colspan=\"1\">1.88</td><td rowspan=\"1\" colspan=\"1\">0.14</td><td rowspan=\"1\" colspan=\"1\">84</td><td rowspan=\"1\" colspan=\"1\">877</td><td rowspan=\"1\" colspan=\"1\">10.44</td><td rowspan=\"1\" colspan=\"1\">1.15</td><td rowspan=\"1\" colspan=\"1\">0.11</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Forest</td><td rowspan=\"1\" colspan=\"1\">128</td><td rowspan=\"1\" colspan=\"1\">1,378</td><td rowspan=\"1\" colspan=\"1\">10.77</td><td rowspan=\"1\" colspan=\"1\">0.91</td><td rowspan=\"1\" colspan=\"1\">0.08</td><td rowspan=\"1\" colspan=\"1\">128</td><td rowspan=\"1\" colspan=\"1\">1,366</td><td rowspan=\"1\" colspan=\"1\">10.67</td><td rowspan=\"1\" colspan=\"1\">0.93</td><td rowspan=\"1\" colspan=\"1\">0.09</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Plantation</td><td rowspan=\"1\" colspan=\"1\">10</td><td rowspan=\"1\" colspan=\"1\">157</td><td rowspan=\"1\" colspan=\"1\">15.70</td><td rowspan=\"1\" colspan=\"1\">4.38</td><td rowspan=\"1\" colspan=\"1\">0.28</td><td rowspan=\"1\" colspan=\"1\">10</td><td rowspan=\"1\" colspan=\"1\">167</td><td rowspan=\"1\" colspan=\"1\">16.70</td><td rowspan=\"1\" colspan=\"1\">3.92</td><td rowspan=\"1\" colspan=\"1\">0.23</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Total</td><td rowspan=\"1\" colspan=\"1\">222</td><td rowspan=\"1\" colspan=\"1\">2,639</td><td rowspan=\"1\" colspan=\"1\">11.89</td><td rowspan=\"1\" colspan=\"1\">0.91</td><td rowspan=\"1\" colspan=\"1\">0.08</td><td rowspan=\"1\" colspan=\"1\">222</td><td rowspan=\"1\" colspan=\"1\">2,410</td><td rowspan=\"1\" colspan=\"1\">10.86</td><td rowspan=\"1\" colspan=\"1\">0.72</td><td rowspan=\"1\" colspan=\"1\">0.07</td></tr></tbody></table></alternatives></table-wrap>", "<table-wrap position=\"float\" id=\"table-4\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/table-4</object-id><label>Table 4</label><caption><title>Encounter rate(n/point) of individual species in different abundance categories during both season/ number per Erica habitat effort.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\">Species</th><th align=\"center\" colspan=\"3\" rowspan=\"1\">DRY</th><th align=\"center\" colspan=\"3\" rowspan=\"1\">WET</th></tr><tr><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">ER</th><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">cv.ER</th><th rowspan=\"1\" colspan=\"1\">ER</th><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">cv.ER</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">Red-wing Starling</td><td rowspan=\"1\" colspan=\"1\">1.95</td><td rowspan=\"1\" colspan=\"1\">Common</td><td rowspan=\"1\" colspan=\"1\">0.56</td><td rowspan=\"1\" colspan=\"1\">0.79</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.62</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ethiopian Siskin</td><td rowspan=\"1\" colspan=\"1\">1.64</td><td rowspan=\"1\" colspan=\"1\">Common</td><td rowspan=\"1\" colspan=\"1\">0.40</td><td rowspan=\"1\" colspan=\"1\">0.44</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.26</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Moorland Chat</td><td rowspan=\"1\" colspan=\"1\">1.38</td><td rowspan=\"1\" colspan=\"1\">Common</td><td rowspan=\"1\" colspan=\"1\">0.21</td><td rowspan=\"1\" colspan=\"1\">0.85</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.30</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Thekla lark</td><td rowspan=\"1\" colspan=\"1\">1.17</td><td rowspan=\"1\" colspan=\"1\">Common</td><td rowspan=\"1\" colspan=\"1\">0.24</td><td rowspan=\"1\" colspan=\"1\">0.65</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.28</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ground-Scarper Thrush</td><td rowspan=\"1\" colspan=\"1\">0.73</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.36</td><td rowspan=\"1\" colspan=\"1\">0.69</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.37</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Wattled Ibis</td><td rowspan=\"1\" colspan=\"1\">0.70</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.37</td><td rowspan=\"1\" colspan=\"1\">0.81</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.27</td></tr><tr><td rowspan=\"1\" colspan=\"1\">African Stonechat</td><td rowspan=\"1\" colspan=\"1\">0.57</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.25</td><td rowspan=\"1\" colspan=\"1\">0.25</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.28</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Scaly Francolin</td><td rowspan=\"1\" colspan=\"1\">0.45</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.25</td><td rowspan=\"1\" colspan=\"1\">0.36</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.36</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Streaky Seedeater</td><td rowspan=\"1\" colspan=\"1\">0.43</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.21</td><td rowspan=\"1\" colspan=\"1\">0.37</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.33</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Mountain Thrush</td><td rowspan=\"1\" colspan=\"1\">0.43</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.32</td><td rowspan=\"1\" colspan=\"1\">0.27</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.29</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ethiopian Cistocola</td><td rowspan=\"1\" colspan=\"1\">0.40</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.34</td><td rowspan=\"1\" colspan=\"1\">0.14</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.43</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Thick-billed Raven</td><td rowspan=\"1\" colspan=\"1\">0.35</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.44</td><td rowspan=\"1\" colspan=\"1\">0.45</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.35</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Rouget’s Rail</td><td rowspan=\"1\" colspan=\"1\">0.27</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.48</td><td rowspan=\"1\" colspan=\"1\">0.27</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.45</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Brown Rumped-seedeater</td><td rowspan=\"1\" colspan=\"1\">0.27</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.35</td><td rowspan=\"1\" colspan=\"1\">0.21</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.35</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Brown Parisoma</td><td rowspan=\"1\" colspan=\"1\">0.19</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.48</td><td rowspan=\"1\" colspan=\"1\">0.17</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.35</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Common House Martin</td><td rowspan=\"1\" colspan=\"1\">0.18</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.29</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Moorland Francolin</td><td rowspan=\"1\" colspan=\"1\">0.17</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.87</td><td rowspan=\"1\" colspan=\"1\">0.10</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.48</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Cinnamon Bracken Warbler</td><td rowspan=\"1\" colspan=\"1\">0.14</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.55</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.70</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Abyssinian Slaty Flycatcher</td><td rowspan=\"1\" colspan=\"1\">0.14</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.74</td><td rowspan=\"1\" colspan=\"1\">0.12</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.42</td></tr><tr><td rowspan=\"1\" colspan=\"1\">White-Collard Pigeon</td><td rowspan=\"1\" colspan=\"1\">0.12</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.71</td><td rowspan=\"1\" colspan=\"1\">0.19</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.72</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Pallid Harrier</td><td rowspan=\"1\" colspan=\"1\">0.12</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.49</td><td rowspan=\"1\" colspan=\"1\">0.08</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.45</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Cap Crow</td><td rowspan=\"1\" colspan=\"1\">0.10</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.70</td><td rowspan=\"1\" colspan=\"1\">0.24</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.57</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Tacazze Sunbird</td><td rowspan=\"1\" colspan=\"1\">0.07</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.74</td><td rowspan=\"1\" colspan=\"1\">0.35</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.23</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Lammergier</td><td rowspan=\"1\" colspan=\"1\">0.07</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.74</td><td rowspan=\"1\" colspan=\"1\">0.14</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.45</td></tr><tr><td rowspan=\"1\" colspan=\"1\">African Snipe</td><td rowspan=\"1\" colspan=\"1\">0.07</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.07</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Hooded Vulture</td><td rowspan=\"1\" colspan=\"1\">0.06</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.71</td><td rowspan=\"1\" colspan=\"1\">0.04</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Mottled Swift</td><td rowspan=\"1\" colspan=\"1\">0.05</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.21</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Montane Nightjar</td><td rowspan=\"1\" colspan=\"1\">0.04</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.74</td><td rowspan=\"1\" colspan=\"1\">0.18</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.49</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Masachet Sunbird</td><td rowspan=\"1\" colspan=\"1\">0.04</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.10</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.70</td></tr><tr><td rowspan=\"1\" colspan=\"1\">African Dusky flycatcher</td><td rowspan=\"1\" colspan=\"1\">0.04</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.74</td><td rowspan=\"1\" colspan=\"1\">0.07</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.56</td></tr><tr><td rowspan=\"1\" colspan=\"1\">White Headed vulture</td><td rowspan=\"1\" colspan=\"1\">0.04</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.74</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Scare Swift</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.70</td><td rowspan=\"1\" colspan=\"1\">1.02</td><td rowspan=\"1\" colspan=\"1\">Common</td><td rowspan=\"1\" colspan=\"1\">0.64</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Rupplis Vulture</td><td rowspan=\"1\" colspan=\"1\">0.01</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Yellow Billed Kite</td><td rowspan=\"1\" colspan=\"1\">0.01</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Augur Buzzard</td><td rowspan=\"1\" colspan=\"1\">0.01</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.08</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.45</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Chestnut-napped Francolin</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0.07</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Common Buzzard</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0.01</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">African Citril</td><td rowspan=\"1\" colspan=\"1\">0.01</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.01</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Sacred Ibis</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.02</td></tr></tbody></table></alternatives></table-wrap>", "<table-wrap position=\"float\" id=\"table-5\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/table-5</object-id><label>Table 5</label><caption><title>Encounter rate(n/point) of individual species in different abundance categories during both season/number per natural forest habitat effort.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\">Species</th><th align=\"center\" colspan=\"3\" rowspan=\"1\">Dry</th><th align=\"center\" colspan=\"3\" rowspan=\"1\">Wet</th></tr><tr><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">ER</th><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">cv.ER</th><th rowspan=\"1\" colspan=\"1\">ER</th><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">cv.ER</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">Montane White-eye</td><td rowspan=\"1\" colspan=\"1\">1.88</td><td rowspan=\"1\" colspan=\"1\">Common</td><td rowspan=\"1\" colspan=\"1\">0.19</td><td rowspan=\"1\" colspan=\"1\">1.57</td><td rowspan=\"1\" colspan=\"1\">Common</td><td rowspan=\"1\" colspan=\"1\">0.21</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Red-wing Starling</td><td rowspan=\"1\" colspan=\"1\">1.70</td><td rowspan=\"1\" colspan=\"1\">Common</td><td rowspan=\"1\" colspan=\"1\">0.37</td><td rowspan=\"1\" colspan=\"1\">0.88</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.58</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Black-winged Lovebird</td><td rowspan=\"1\" colspan=\"1\">0.49</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.38</td><td rowspan=\"1\" colspan=\"1\">0.59</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.35</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Abyssinian Catbird</td><td rowspan=\"1\" colspan=\"1\">0.42</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.27</td><td rowspan=\"1\" colspan=\"1\">0.34</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.28</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Streaky Seedeater</td><td rowspan=\"1\" colspan=\"1\">0.42</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.29</td><td rowspan=\"1\" colspan=\"1\">0.30</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.24</td></tr><tr><td rowspan=\"1\" colspan=\"1\">White-backed Black Tit</td><td rowspan=\"1\" colspan=\"1\">0.35</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.28</td><td rowspan=\"1\" colspan=\"1\">0.27</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.34</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Brown Rumped-seedeater</td><td rowspan=\"1\" colspan=\"1\">0.35</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.29</td><td rowspan=\"1\" colspan=\"1\">0.23</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.32</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Mountain Thrush</td><td rowspan=\"1\" colspan=\"1\">0.31</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.28</td><td rowspan=\"1\" colspan=\"1\">0.26</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.32</td></tr><tr><td rowspan=\"1\" colspan=\"1\">White-checked Turaco</td><td rowspan=\"1\" colspan=\"1\">0.30</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.31</td><td rowspan=\"1\" colspan=\"1\">0.32</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.28</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Wattled Ibis</td><td rowspan=\"1\" colspan=\"1\">0.30</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.40</td><td rowspan=\"1\" colspan=\"1\">0.53</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.48</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Thick-billed Raven</td><td rowspan=\"1\" colspan=\"1\">0.26</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.35</td><td rowspan=\"1\" colspan=\"1\">0.40</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.28</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Mouse-colored Penduline-tit</td><td rowspan=\"1\" colspan=\"1\">0.25</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.77</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Baglafecht Weaver</td><td rowspan=\"1\" colspan=\"1\">0.24</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.38</td><td rowspan=\"1\" colspan=\"1\">0.62</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.30</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Tacazze Sunbird</td><td rowspan=\"1\" colspan=\"1\">0.23</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.30</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.29</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Yellow-bellied Waxbill</td><td rowspan=\"1\" colspan=\"1\">0.23</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.48</td><td rowspan=\"1\" colspan=\"1\">0.17</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.50</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Common Bulbul</td><td rowspan=\"1\" colspan=\"1\">0.22</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.45</td><td rowspan=\"1\" colspan=\"1\">0.21</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.33</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Cap Crow</td><td rowspan=\"1\" colspan=\"1\">0.19</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.51</td><td rowspan=\"1\" colspan=\"1\">0.22</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.46</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ethiopian Siskin</td><td rowspan=\"1\" colspan=\"1\">0.19</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.53</td><td rowspan=\"1\" colspan=\"1\">0.19</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.53</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Red-collard Widow Bird</td><td rowspan=\"1\" colspan=\"1\">0.16</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.16</td><td rowspan=\"1\" colspan=\"1\">0.22</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.77</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Dusky turtle Dove</td><td rowspan=\"1\" colspan=\"1\">0.14</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.42</td><td rowspan=\"1\" colspan=\"1\">0.21</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.34</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Red-eyed Dove</td><td rowspan=\"1\" colspan=\"1\">0.13</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.36</td><td rowspan=\"1\" colspan=\"1\">0.09</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.39</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Eastern grey woodpecker</td><td rowspan=\"1\" colspan=\"1\">0.13</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.42</td><td rowspan=\"1\" colspan=\"1\">0.16</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.40</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Moorland Chat</td><td rowspan=\"1\" colspan=\"1\">0.11</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.77</td><td rowspan=\"1\" colspan=\"1\">0.06</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.60</td></tr><tr><td rowspan=\"1\" colspan=\"1\">African Olive Pigeon</td><td rowspan=\"1\" colspan=\"1\">0.10</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.50</td><td rowspan=\"1\" colspan=\"1\">0.16</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.37</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Variable Sunbird</td><td rowspan=\"1\" colspan=\"1\">0.10</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.50</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Chestnut-napped Francolin</td><td rowspan=\"1\" colspan=\"1\">0.09</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.23</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.58</td></tr><tr><td rowspan=\"1\" colspan=\"1\">African Citril</td><td rowspan=\"1\" colspan=\"1\">0.09</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.46</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.31</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Yellow-fronted Parrot</td><td rowspan=\"1\" colspan=\"1\">0.09</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.74</td><td rowspan=\"1\" colspan=\"1\">0.14</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.59</td></tr><tr><td rowspan=\"1\" colspan=\"1\">African Dusky flycatcher</td><td rowspan=\"1\" colspan=\"1\">0.09</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.52</td><td rowspan=\"1\" colspan=\"1\">0.13</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.37</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Tawny flanked Prina</td><td rowspan=\"1\" colspan=\"1\">0.09</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.38</td><td rowspan=\"1\" colspan=\"1\">0.07</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.42</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Abyssinian Slaty Flycatcher</td><td rowspan=\"1\" colspan=\"1\">0.09</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.38</td><td rowspan=\"1\" colspan=\"1\">0.03</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.70</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Speckled Mousebird</td><td rowspan=\"1\" colspan=\"1\">0.08</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.72</td><td rowspan=\"1\" colspan=\"1\">0.16</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.61</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Abyssinian Woodpecker</td><td rowspan=\"1\" colspan=\"1\">0.08</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.59</td><td rowspan=\"1\" colspan=\"1\">0.11</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.46</td></tr><tr><td rowspan=\"1\" colspan=\"1\">White-Collard Pigeon</td><td rowspan=\"1\" colspan=\"1\">0.08</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.72</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.74</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ethiopian Boubou</td><td rowspan=\"1\" colspan=\"1\">0.06</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.58</td><td rowspan=\"1\" colspan=\"1\">0.10</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.43</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Scaly Francolin</td><td rowspan=\"1\" colspan=\"1\">0.05</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.74</td><td rowspan=\"1\" colspan=\"1\">0.15</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.50</td></tr><tr><td rowspan=\"1\" colspan=\"1\">African Paradise Flycatcher</td><td rowspan=\"1\" colspan=\"1\">0.05</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.50</td><td rowspan=\"1\" colspan=\"1\">0.06</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.45</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Cinnamon Bracken Warbler</td><td rowspan=\"1\" colspan=\"1\">0.05</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.46</td><td rowspan=\"1\" colspan=\"1\">0.06</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.38</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Yellow Crown Canary</td><td rowspan=\"1\" colspan=\"1\">0.05</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.74</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.47</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Abyssinian Forest Oriole</td><td rowspan=\"1\" colspan=\"1\">0.05</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.74</td><td rowspan=\"1\" colspan=\"1\">0.19</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.50</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Northern Puff back</td><td rowspan=\"1\" colspan=\"1\">0.05</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.74</td><td rowspan=\"1\" colspan=\"1\">0.06</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.60</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Augur Buzzard</td><td rowspan=\"1\" colspan=\"1\">0.04</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.66</td><td rowspan=\"1\" colspan=\"1\">0.05</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.57</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Yellow Wagtail</td><td rowspan=\"1\" colspan=\"1\">0.03</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.07</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.71</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Eurasian Hoopoe</td><td rowspan=\"1\" colspan=\"1\">0.03</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.70</td><td rowspan=\"1\" colspan=\"1\">0.06</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.49</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Abyssinian Ground Hornbill</td><td rowspan=\"1\" colspan=\"1\">0.03</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.05</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.74</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Narnia Trogon</td><td rowspan=\"1\" colspan=\"1\">0.03</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.60</td><td rowspan=\"1\" colspan=\"1\">0.05</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.46</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Africa Wooded Owl</td><td rowspan=\"1\" colspan=\"1\">0.03</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.60</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.70</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Mountain Wagtail</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.57</td><td rowspan=\"1\" colspan=\"1\">0.05</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.57</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Grey-backed Camaroptera</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.57</td><td rowspan=\"1\" colspan=\"1\">0.03</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.49</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Swanson’s Sparrow</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ruppell’s Robin-Chat</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.04</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.59</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Tawny Eagle</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.70</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.57</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Hooded Vulture</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Abyssinian Owl</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">African Stonechat</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Common Buzzard</td><td rowspan=\"1\" colspan=\"1\">0.01</td><td rowspan=\"1\" colspan=\"1\">Rare</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Abyssinian Ground Thrush</td><td rowspan=\"1\" colspan=\"1\">0.01</td><td rowspan=\"1\" colspan=\"1\">Rare</td><td rowspan=\"1\" colspan=\"1\">0.53</td><td rowspan=\"1\" colspan=\"1\">0.04</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.52</td></tr><tr><td rowspan=\"1\" colspan=\"1\">White Headed vulture</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0.05</td><td rowspan=\"1\" colspan=\"1\">uncommon</td><td rowspan=\"1\" colspan=\"1\">0.52</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Yellow Billed Kite</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr></tbody></table></alternatives></table-wrap>", "<table-wrap position=\"float\" id=\"table-6\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/table-6</object-id><label>Table 6</label><caption><title>Encounter rate(n/point) of individual species in different abundance categories during both season/number per plantation forest habitat effort.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\">Species</th><th align=\"center\" colspan=\"3\" rowspan=\"1\">Dry</th><th align=\"center\" colspan=\"3\" rowspan=\"1\">Wet</th></tr><tr><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">ER</th><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">cv.ER</th><th rowspan=\"1\" colspan=\"1\">ER</th><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">cv.ER</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">Ground-Scarper Thrush</td><td rowspan=\"1\" colspan=\"1\">3.10</td><td rowspan=\"1\" colspan=\"1\">Common</td><td rowspan=\"1\" colspan=\"1\">0.67</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.47</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Red-eyed Dove</td><td rowspan=\"1\" colspan=\"1\">2.80</td><td rowspan=\"1\" colspan=\"1\">Common</td><td rowspan=\"1\" colspan=\"1\">0.29</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.31</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Montane White-eye</td><td rowspan=\"1\" colspan=\"1\">2.20</td><td rowspan=\"1\" colspan=\"1\">Common</td><td rowspan=\"1\" colspan=\"1\">0.62</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.61</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Yellow Crown Canary</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.63</td><td rowspan=\"1\" colspan=\"1\">2.00</td><td rowspan=\"1\" colspan=\"1\">Common</td><td rowspan=\"1\" colspan=\"1\">0.51</td></tr><tr><td rowspan=\"1\" colspan=\"1\">African Black Swift</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Baglafecht Weaver</td><td rowspan=\"1\" colspan=\"1\">0.80</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">1.80</td><td rowspan=\"1\" colspan=\"1\">Common</td><td rowspan=\"1\" colspan=\"1\">0.67</td></tr><tr><td rowspan=\"1\" colspan=\"1\">African Citril</td><td rowspan=\"1\" colspan=\"1\">0.60</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.60</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Tacazze Sunbird</td><td rowspan=\"1\" colspan=\"1\">0.50</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.63</td><td rowspan=\"1\" colspan=\"1\">0.60</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.67</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Eurasian Hoopoe</td><td rowspan=\"1\" colspan=\"1\">0.40</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Speckled Mousebird</td><td rowspan=\"1\" colspan=\"1\">0.40</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.73</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Brown Rumped-seedeater</td><td rowspan=\"1\" colspan=\"1\">0.40</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Swanson’s Sparrow</td><td rowspan=\"1\" colspan=\"1\">0.40</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Semi-colored flycatcher</td><td rowspan=\"1\" colspan=\"1\">0.30</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">1.50</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ethiopian Boubou</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.40</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">0.73</td></tr><tr><td rowspan=\"1\" colspan=\"1\">African Dusky flycatcher</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.30</td><td rowspan=\"1\" colspan=\"1\">Frequent</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Abyssinian Slaty Flycatcher</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Dusky turtle Dove</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ethiopian Siskin</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Streaky Seedeater</td><td rowspan=\"1\" colspan=\"1\">0.18</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.67</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.43</td></tr><tr><td rowspan=\"1\" colspan=\"1\">White Headed vulture</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">0.70</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Common Bulbul</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Mountain Thrush</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Pied Crow</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Thick-billed Raven</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0.20</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Augur Buzzard</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0.10</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Yellow Billed Kite</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">0.10</td><td rowspan=\"1\" colspan=\"1\">Uncommon</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr></tbody></table></alternatives></table-wrap>", "<table-wrap position=\"float\" id=\"table-7\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/table-7</object-id><label>Table 7</label><caption><title>Species similarity index (SI) (##UREF##31##Sorensen, 1948##) among the three habitat types.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">Dry season</th><th rowspan=\"1\" colspan=\"1\">Wet season</th></tr><tr><th rowspan=\"1\" colspan=\"1\">Association</th><th align=\"center\" colspan=\"2\" rowspan=\"1\">Sim</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">Erica <italic toggle=\"yes\">vs</italic> forest</td><td rowspan=\"1\" colspan=\"1\">0.425</td><td rowspan=\"1\" colspan=\"1\">0.400</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Erica <italic toggle=\"yes\">vs</italic> plantation</td><td rowspan=\"1\" colspan=\"1\">0.296</td><td rowspan=\"1\" colspan=\"1\">0.276</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Planation <italic toggle=\"yes\">vs</italic> forest</td><td rowspan=\"1\" colspan=\"1\">0.395</td><td rowspan=\"1\" colspan=\"1\">0.500</td></tr></tbody></table></alternatives></table-wrap>" ]

[ "<disp-formula id=\"eqn-1\"><label>(1)</label><alternatives><tex-math id=\"tex-eqn-1\">\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{upgreek}\n\\usepackage{mathrsfs}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}\n\\begin{eqnarray*} \\frac{b}{(cv(D))^{2}} \\ast \\frac{k0}{n0} \\end{eqnarray*}\\end{document}</tex-math><mml:math id=\"mml-eqn-1\" overflow=\"scroll\"><mml:mstyle displaystyle=\"true\"><mml:mfrac><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mfenced separators=\"\" open=\"(\" close=\")\"><mml:mi>c</mml:mi><mml:mi>v</mml:mi><mml:mrow><mml:mfenced separators=\"\" open=\"(\" close=\")\"><mml:mi>D</mml:mi></mml:mfenced></mml:mrow></mml:mfenced></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>∗</mml:mo><mml:mfrac><mml:mrow><mml:mi>k</mml:mi><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:mfrac></mml:mstyle></mml:math></alternatives>\n</disp-formula>", "<disp-formula id=\"eqn-2\"><label>(2)</label><alternatives><tex-math id=\"tex-eqn-2\">\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{upgreek}\n\\usepackage{mathrsfs}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}\n\\begin{eqnarray*}k= \\frac{k0\\{ b+[sd(s)/s]^{2}\\} }{nocv{t}^{2}} .\\end{eqnarray*}\\end{document}</tex-math><mml:math id=\"mml-eqn-2\" overflow=\"scroll\"><mml:mstyle displaystyle=\"true\"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>k</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mfenced separators=\"\" open=\"{\" close=\"}\"><mml:mi>b</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mfenced separators=\"\" open=\"[\" close=\"]\"><mml:mi>s</mml:mi><mml:mi>d</mml:mi><mml:mrow><mml:mfenced separators=\"\" open=\"(\" close=\")\"><mml:mi>s</mml:mi></mml:mfenced></mml:mrow><mml:mo>/</mml:mo><mml:mi>s</mml:mi></mml:mfenced></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mfenced></mml:mrow></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>o</mml:mi><mml:mi>c</mml:mi><mml:mi>v</mml:mi><mml:msup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mstyle></mml:math></alternatives>\n</disp-formula>", "<disp-formula id=\"eqn-3\"><label>(3)</label><alternatives><tex-math id=\"tex-eqn-3\">\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{upgreek}\n\\usepackage{mathrsfs}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}\n\\begin{eqnarray*}v{a}^{\\wedge }{r}_{p3} \\frac{n}{T} = \\frac{1}{T(k-1)} =\\sum _{i=1}^{k}ti( \\frac{ni}{ti} - \\frac{n}{T} )\\end{eqnarray*}\\end{document}</tex-math><mml:math id=\"mml-eqn-3\" overflow=\"scroll\"><mml:mstyle displaystyle=\"true\"><mml:mi>v</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>∧</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>T</mml:mi><mml:mrow><mml:mfenced separators=\"\" open=\"(\" close=\")\"><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:munderover><mml:mrow><mml:mo mathsize=\"big\" movablelimits=\"false\"> ∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:munderover><mml:mi>t</mml:mi><mml:mi>i</mml:mi><mml:mrow><mml:mfenced separators=\"\" open=\"(\" close=\")\"><mml:mfrac><mml:mrow><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mstyle></mml:math></alternatives>\n</disp-formula>", "<inline-formula><alternatives><tex-math id=\"tex-ieqn-11\">\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{upgreek}\n\\usepackage{mathrsfs}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}\n$T={\\mathop{\\sum }\\nolimits }_{I=1}^{K}ti,ni={\\mathop{\\sum }\\nolimits }_{j=1}^{t}nij$\\end{document}</tex-math><mml:math id=\"mml-ieqn-11\" overflow=\"scroll\"><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>I</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>K</mml:mi></mml:mrow></mml:msubsup><mml:mi>t</mml:mi><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula><alternatives><tex-math id=\"tex-ieqn-12\">\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{upgreek}\n\\usepackage{mathrsfs}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}\n$n={\\mathop{\\sum }\\nolimits }_{i=1}^{k}ninj$\\end{document}</tex-math><mml:math id=\"mml-ieqn-12\" overflow=\"scroll\"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>j</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"eqn-4\"><label>(4)</label><alternatives><tex-math id=\"tex-eqn-4\">\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{upgreek}\n\\usepackage{mathrsfs}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}\n\\begin{eqnarray*}ER= \\frac{n}{kt} OR \\frac{n}{K} .\\end{eqnarray*}\\end{document}</tex-math><mml:math id=\"mml-eqn-4\" overflow=\"scroll\"><mml:mstyle displaystyle=\"true\"><mml:mi>E</mml:mi><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mi>O</mml:mi><mml:mi>R</mml:mi><mml:mfrac><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>K</mml:mi></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mstyle></mml:math></alternatives>\n</disp-formula>", "<disp-formula id=\"eqn-5\"><label>(Model 1)</label><alternatives><tex-math id=\"tex-eqn-5\">\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{upgreek}\n\\usepackage{mathrsfs}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}\n\\begin{eqnarray*}{\\mathrm{\\mu }}_{ijk}=\\mathrm{\\mu }+{\\alpha }_{i}+{\\beta }_{j}+{\\gamma }_{k}+{\\delta }_{ij}\\ldots \\ldots \\ldots \\ldots .\\end{eqnarray*}\\end{document}</tex-math><mml:math id=\"mml-eqn-5\" overflow=\"scroll\"><mml:mstyle displaystyle=\"true\"><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">μ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">μ</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>δ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>…</mml:mo><mml:mo>…</mml:mo><mml:mo>…</mml:mo><mml:mo>…</mml:mo><mml:mo>.</mml:mo></mml:mstyle></mml:math></alternatives>\n</disp-formula>", "<disp-formula id=\"eqn-6\"><label>(Model 2)</label><alternatives><tex-math id=\"tex-eqn-6\">\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{upgreek}\n\\usepackage{mathrsfs}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}\n\\begin{eqnarray*}{\\mathrm{\\mu }}_{ijk}=\\mathrm{\\mu }+{\\alpha }_{i}+{\\beta }_{j}+{\\gamma }_{k}\\ldots \\ldots \\ldots \\ldots .\\end{eqnarray*}\\end{document}</tex-math><mml:math id=\"mml-eqn-6\" overflow=\"scroll\"><mml:mstyle displaystyle=\"true\"><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">μ</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">μ</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>…</mml:mo><mml:mo>…</mml:mo><mml:mo>…</mml:mo><mml:mo>…</mml:mo><mml:mo>.</mml:mo></mml:mstyle></mml:math></alternatives>\n</disp-formula>", "<inline-formula><alternatives><tex-math id=\"tex-ieqn-31\">\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{upgreek}\n\\usepackage{mathrsfs}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}\n$\\tau ={\\mathop{\\sum }\\nolimits }_{i=1}^{S}( \\frac{{n}_{i}({n}_{i}-1)}{{N}_{i}({N}_{i}-1)} )$\\end{document}</tex-math><mml:math id=\"mml-ieqn-31\" overflow=\"scroll\"><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>S</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mfenced separators=\"\" open=\"(\" close=\")\"><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mfenced separators=\"\" open=\"(\" close=\")\"><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mfenced separators=\"\" open=\"(\" close=\")\"><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mrow></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula><alternatives><tex-math id=\"tex-ieqn-33\">\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{upgreek}\n\\usepackage{mathrsfs}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}\n$\\sum \\left( { \\frac{ni}{N} }^{2} \\right) $\\end{document}</tex-math><mml:math id=\"mml-ieqn-33\" overflow=\"scroll\"><mml:mo>∑</mml:mo><mml:mfenced separators=\"\" open=\"(\" close=\")\"><mml:mrow><mml:msup><mml:mrow><mml:mfrac><mml:mrow><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:math></alternatives></inline-formula>", "<inline-formula><alternatives><tex-math id=\"tex-ieqn-34\">\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{upgreek}\n\\usepackage{mathrsfs}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}\n$H=-\\sum ( \\frac{ni}{N} )\\log \\frac{ni}{N} $\\end{document}</tex-math><mml:math id=\"mml-ieqn-34\" overflow=\"scroll\"><mml:mi>H</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mo>∑</mml:mo><mml:mrow><mml:mfenced separators=\"\" open=\"(\" 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"<inline-formula><alternatives><tex-math id=\"tex-ieqn-43\">\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{upgreek}\n\\usepackage{mathrsfs}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}\n${f}_{n}=\\alpha \\frac{{x}^{n}}{n} $\\end{document}</tex-math><mml:math id=\"mml-ieqn-43\" overflow=\"scroll\"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>α</mml:mi><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:mfrac></mml:math></alternatives></inline-formula>", "<disp-formula id=\"NONUM-d2e1414\">\n<alternatives><tex-math 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id=\"tex-ieqn-49\">\\documentclass[12pt]{minimal}\n\\usepackage{amsmath}\n\\usepackage{wasysym}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsbsy}\n\\usepackage{upgreek}\n\\usepackage{mathrsfs}\n\\setlength{\\oddsidemargin}{-69pt}\n\\begin{document}\n$\\bar {Si}$\\end{document}</tex-math><mml:math id=\"mml-ieqn-49\" overflow=\"scroll\"><mml:mover accent=\"true\"><mml:mrow><mml:mi>S</mml:mi><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo> ¯</mml:mo></mml:mrow></mml:mover></mml:math></alternatives></inline-formula>" ]

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[ "<supplementary-material id=\"supp-1\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/supp-1</object-id><label>Supplemental Information 1</label><caption><title>The study data and analysis in distance software</title></caption></supplementary-material>", "<supplementary-material id=\"supp-2\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/supp-2</object-id><label>Supplemental Information 2</label><caption><title>Species accumulation r source code</title></caption></supplementary-material>", "<supplementary-material id=\"supp-3\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/supp-3</object-id><label>Supplemental Information 3</label><caption><title>Codes for each species discovered during sampling and number of these species’ observation</title></caption></supplementary-material>", "<supplementary-material id=\"supp-4\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16775/supp-4</object-id><label>Supplemental Information 4</label><caption><title>Checklist of The Birds of Dodola Dry Evergreen Afromontane Forest and Sub-Afro Alpine Scrubland Vegetation</title></caption></supplementary-material>" ]

[ "<table-wrap-foot><fn id=\"table-3fn\"><p>\n<bold>Notes.</bold>\n</p></fn><fn id=\"table-3fn1\" fn-type=\"other\"><p>\n</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"table-4fn\"><p>\n<bold>Notes.</bold>\n</p></fn><fn id=\"table-4fn1\" fn-type=\"other\"><p>\n</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"table-5fn\"><p>\n<bold>Notes.</bold>\n</p></fn><fn id=\"table-5fn1\" fn-type=\"other\"><p>\n</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"table-6fn\"><p>\n<bold>Notes.</bold>\n</p></fn><fn id=\"table-6fn1\" fn-type=\"other\"><p>\n</p></fn></table-wrap-foot>", "<fn-group content-type=\"competing-interests\"><title>Competing Interests</title><fn id=\"conflict-1\" fn-type=\"COI-statement\"><p>The authors declare there are no competing interests.</p></fn></fn-group>", "<fn-group content-type=\"author-contributions\"><title>Author Contributions</title><fn id=\"contribution-1\" fn-type=\"con\"><p><xref rid=\"author-1\" ref-type=\"contrib\">Zenebe Ageru Yilma</xref> conceived and designed the experiments, performed the experiments, analyzed the data, prepared figures and/or tables, authored or reviewed drafts of the article, and approved the final draft.</p></fn><fn id=\"contribution-2\" fn-type=\"con\"><p><xref rid=\"author-2\" ref-type=\"contrib\">Girma Mengesha</xref> conceived and designed the experiments, authored or reviewed drafts of the article, and approved the final draft.</p></fn><fn id=\"contribution-3\" fn-type=\"con\"><p><xref rid=\"author-3\" ref-type=\"contrib\">Zerihun Girma</xref> conceived and designed the experiments, performed the experiments, authored or reviewed drafts of the article, and approved the final draft.</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Data Availability</title><fn id=\"addinfo-1\"><p>The following information was supplied regarding data availability:</p><p>The code is available in the <xref rid=\"supplemental-information\" ref-type=\"sec\">Supplementary File</xref>.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"peerj-12-16775-s001.zip\"><caption><p>Click here for additional data file.</p></caption></media>", "<media xlink:href=\"peerj-12-16775-s002.zip\"><caption><p>Click here for additional data file.</p></caption></media>", "<media xlink:href=\"peerj-12-16775-s003.csv\"><caption><p>Click here for additional data file.</p></caption></media>", "<media xlink:href=\"peerj-12-16775-s004.docx\"><caption><p>Click here for additional data file.</p></caption></media>" ]

{ "acronym": [ " n", " ER", " se.ER", " cv.ER" ], "definition": [ "total number of observations", "Encounter rate", "Standard error for ER", "coefficient variation of ER" ] }

[ "<title>Introduction</title>", "<p>The systematization of taxa of endemic wild mountain plant species is an urgent issue in the latest taxonomy of the Rosaceae Juss family; one of the most prominent examples is the systematization of representatives of endemic mountain shrubs of the almond subgenus. The almond is one of the most essential cultivated and wild plant species worldwide. Shrub forms are often used in introductory and subsequent material for landscaping large cities in Kazakhstan. Studying isolated endemic populations will allow for the comparison of genetic variation among the genus’ general distribution area species, as some medicinal properties of almonds are also known. Specimens from pristine mountain populations are potential carriers of valuable biological and chemical compounds (##UREF##18##Gradziel, 2011##). Almond plants are members of the genus <italic toggle=\"yes\">Prunus</italic> L. (tribe <italic toggle=\"yes\">Amygdaleae</italic>), one of 65 genera of the subfamily <italic toggle=\"yes\">Prunoideae</italic> of the complex family Rosaceae Juss. The genus is represented by four subgenera (subgen. <italic toggle=\"yes\">Amygdalus</italic> (L.) Focke., <italic toggle=\"yes\">Cerasus</italic> (Mill.) A.Gray., <italic toggle=\"yes\">Emplectocladus</italic> (Torr.) A.Gray., and <italic toggle=\"yes\">Prunus</italic> L.) and includes about 254 species. The subgenus <italic toggle=\"yes\">Amygdalus</italic> consists of six sections: <italic toggle=\"yes\">Amygdalopsis</italic> (Carr.) Linsz., <italic toggle=\"yes\">Cerasioides</italic> (Carr.) Linsz., <italic toggle=\"yes\">Chamaeamygdalus</italic> Spach, <italic toggle=\"yes\">Euamygdalus</italic> Spach., <italic toggle=\"yes\">Lycioides</italic> Spach., and <italic toggle=\"yes\">Spartioides</italic> Spach. One of the least studied sections is the pygmy almond <italic toggle=\"yes\">Chamaeamygdalus</italic>, which has low yield, a special protection status (endemic and rare plant species), and ornamental properties (##UREF##5##Artemov et al., 2009##; ##UREF##9##Browicz &amp; Zohary, 1996##). According to the list of vascular plants in Kazakhstan (##UREF##0##Abdulina, 1999##; ##UREF##21##Komorov, 1941##), there are three species in the flora of Kazakhstan: <italic toggle=\"yes\">Chamaeamygdalus</italic>—steppe almond (<italic toggle=\"yes\">Prunus tenella</italic> Batsch syn. <italic toggle=\"yes\">Amygdalus nana</italic> L.), Ledebour’s almond (<italic toggle=\"yes\">Prunus ledebouriana</italic> (Schlecht.) YY Yao syn. <italic toggle=\"yes\">Amygdalus ledebouriana</italic> (##UREF##38##Schlechtendal, 1854##)), and Pettunnikov’s almond (<italic toggle=\"yes\">Prunus petunnikowii</italic> (Litv.) Rehder syn. <italic toggle=\"yes\">Amygdalus petunnikowii</italic> Litw.). In addition, the list of flora of Kazakhstan includes one species of the section <italic toggle=\"yes\">Lycioides</italic> (<italic toggle=\"yes\">Amygdalus communis</italic> L. syn. <italic toggle=\"yes\">Prunus dulcis</italic> (Mill.) D.A. Webb.) and one species of the section <italic toggle=\"yes\">Euamygdalus</italic> (<italic toggle=\"yes\">Amygdalus spinosissima</italic> Bunge. syn. <italic toggle=\"yes\">Prunus spinosissima</italic> (Bunge) Franch.). <italic toggle=\"yes\">P. tenella</italic> is widespread in Southern Europe and the European part of Asia, mainly in the steppe zones, and is available for cultivation. According to the flora of the Kazakh Soviet Socialist Republic (Kazakh SSR), <italic toggle=\"yes\">P. ledebouriana</italic> is an endemic species for Kazakhstan, growing in the Altai, Tarbagatai, and Dzungarian Alatau mountains and replacing <italic toggle=\"yes\">P. tenella</italic> in east Kazakhstan (##UREF##31##Pavlov, 1961##). <italic toggle=\"yes\">P. ledebouriana</italic> is listed in the Red Book of the Republic of Kazakhstan (##UREF##7##Baitulin, 2014##) and the Book of Woody Plants of Central Asia (##UREF##14##Eastwood, Lazkov &amp; Newton, 2009##).</p>", "<p>Various international databases have interpreted the status of <italic toggle=\"yes\">P. ledebouriana</italic> (##UREF##47##The Plant List## (TPL) and ##UREF##50##World Flora Online## (WFO)) and attempted to determine the taxonomy of this species (##UREF##16##GBIF, 2020##). <italic toggle=\"yes\">P. ledebouriana</italic> is closely related to the steppe almond <italic toggle=\"yes\">P. tenella</italic>, as they have similar morphological features and are distinguished by a relatively large habitus (plant height), sizes of leaves, and fruits (##UREF##29##Orazov et al., 2020##). <italic toggle=\"yes\">P. tenella</italic> is the southernmost species of section <italic toggle=\"yes\">Chamaeamygdalus</italic>, has a wide distribution from the northern Balkans to Kazakhstan and China (##UREF##23##Ladizinsky, 1999##), and is often used for cultivation as an ornamental species. These two related species have no boundaries and the genetic differences are poorly understood.</p>", "<p>According to a report on the flora of China, <italic toggle=\"yes\">P. tenella</italic> has a synonymous name, <italic toggle=\"yes\">A. nana</italic> (syn.) (##UREF##24##Lu et al., 2003##). Across various literary sources, two synonymous species have been recorded in the territory of East Kazakhstan (an administrative region of Kazakhstan bordering Russia and China), adding complexity to identifying the species. The distribution of <italic toggle=\"yes\">P. tenella</italic> among studied territories is not uniform since the area consists of several isolated populations. According to various sources, <italic toggle=\"yes\">P. tenella</italic> predominates in the steppes and low hills of the low mountains of the Kalba and Ulba ranges in the Altai Mountains (##UREF##33##Planetarium##).</p>", "<p><italic toggle=\"yes\">P. ledebouriana</italic> is found in the cold and xerophytic mountain areas adjacent to Russia (Narym Range of the Altai Mountains) and China (foothills of the Tarbagatai Range) (##UREF##29##Orazov et al., 2020##). The population in the foothills of Tarbagatai is the most extensive and is included in the Red Book of Kazakhstan (##UREF##42##Stepanova, 1962##; ##UREF##7##Baitulin, 2014##). Natural populations of <italic toggle=\"yes\">P. ledebouriana</italic> are declining due to habitat degradation, frequent droughts, changes in the fire regime (succession), overgrazing, and urbanization (##UREF##44##Sumbembaev, 2018##; ##UREF##45##Sumbembayev et al., 2021##; ##UREF##2##Aidarkhanova et al., 2022##; ##UREF##22##Kusmangazinov et al., 2023##). Therefore, by the Decree of the Government of the Republic of Kazakhstan in 2018, the Tarbagatai State National Natural Park (East Kazakhstan region, Urdzhar district) was adopted and the preservation of this rare and endangered species was recommended (##UREF##36##Republic of Kazakhstan, 2018##). The morphological similarity of the two species complicates the protection of this endemic plant species (##UREF##34##Potter et al., 2002##). In this regard, making a clear distinction between the two species of wild almond populations in Eastern Kazakhstan is crucial.</p>", "<p><italic toggle=\"yes\">P. ledebouriana</italic> primarily reproduces vegetatively in addition to sexual reproduction. It usually flowers from April to May and bears fruit from June to July (##UREF##31##Pavlov, 1961##; ##UREF##8##Bin et al., 2008##; ##UREF##30##Orazov et al., 2022##). Isolating factors include the presence of Lake Zaisan and the Irtysh River that feeds it; the location of the Zaisan basin between Altai and Tarbagatai and the non-contiguous Ulba, Kalba, Narym ridges of the Altai mountains; and the remote location of the Tarbagatai ridge of the Saur-Tarbagatai mountains (##UREF##15##Egorina, Zinchenko &amp; Zinchenko, 2003##).</p>", "<p>Applying molecular methods in botany and plant systematics has provided opportunities for identifying and confirming species and their taxonomic position in the genus (##REF##14607101##Andersen &amp; Lübberstedt, 2003##). Various types of DNA markers have been successfully used to assess the genetic diversity of species of <italic toggle=\"yes\">Prunus</italic>. These studies included the use of random amplified polymorphism of DNA (RAPD) (##UREF##10##Casas et al., 1999##), inter simple simples sequence repeats (ISSR) (##REF##12827440##Martins, Tenreiro &amp; Oliveira, 2003##), amplified fragment length polymorphism (AFLP) (##UREF##43##Struss et al., 2003##), and simple sequence repeats (SSR) (##REF##12647055##Aranzana et al., 2003##) markers. One of the most informative types of DNA markers are microsatellite markers (SSR), which are characterized by a high level of polymorphism and codominant inheritance (##UREF##20##Kalendar, 2011##; ##REF##33738387##Genievskaya et al., 2020##). Analysis of population genetics using microsatellite markers provides information on the overall levels of genetic diversity, genetic structure, and effective population size, which are typically critical when developing effective management strategies for the research of genetic resources of endemic plant species (##UREF##48##Turuspekov &amp; Abugalieva, 2015##; ##UREF##1##Abugalieva &amp; Turuspekov, 2017##; ##UREF##4##Almerekova et al., 2018##; ##UREF##3##Almerekova et al., 2020##; ##REF##33738387##Genievskaya et al., 2020##). Various microsatellite markers have been successfully used to study the phylogenetic relationships between various cultivated almonds (common almond <italic toggle=\"yes\">P. dulcis</italic> (Mill.) DA Webb.) and their wild relatives (##REF##15644967##Xu et al., 2004##; ##REF##16307227##Xie et al., 2006##; ##UREF##39##Shiran et al., 2007##; ##UREF##41##Sorkheh et al., 2007##; ##REF##28701907##Zhang et al., 2018##; ##UREF##52##Zargar et al., 2023##). However, there have been limited studies on the genetic analysis of natural populations of wild species in the subgenus <italic toggle=\"yes\">Amygdalus</italic> (##REF##15629858##Varshney, Graner &amp; Sorrells, 2005##; ##UREF##46##Tahan et al., 2009##). This study is one of the first to explore the mountain populations of <italic toggle=\"yes\">P. ledebouriana</italic> and obtain genetic information. The application of SSR markers in population genetic analysis for the narrowly endemic species <italic toggle=\"yes\">P. ledebouriana</italic> can be successfully used to study the genetic diversity and population structure of the natural population in Eastern Kazakhstan.</p>" ]

[ "<title>Materials &amp; Methods</title>", "<title>Study of genetic structure using SSR</title>", "<p>This study investigated three isolated populations of <italic toggle=\"yes\">P. ledebouriana</italic> from two mountain geographic ranges (Altai and Tarbagatai) and one <italic toggle=\"yes\">P. tenella</italic> from Eastern Kazakhstan. Among these four populations, the first one (1-UR) was collected on several isolated gorges in the state national natural park “Tarbagatai” at the height of the shrub belt. The materials of the second and third populations were collected in the cold and xerophytic mountain regions of the Kalba (2-KO) and Narym (3-KA) ridges. The population of <italic toggle=\"yes\">P. tenella</italic> (4-UK) was collected from a small hilly plain (steppe) zone in the outlines of the Kalbinskiy and Ulbinskiy ridges in the border zone of the city of Ust-Kamenogorsk and the village of Novo-Akhmirovo (##SUPPL##0##Table S1##). The plant height of <italic toggle=\"yes\">P. ledebouriana</italic> was measured according to ##UREF##17##Goloskokov (1972)##. In total, 20 leaves from each of the 60 <italic toggle=\"yes\">P. ledebouriana</italic> and 20 <italic toggle=\"yes\">P. tenella</italic> plant populations were collected. The distances between populations were at least 100 kilometers, and plants within a population were selected at a distance of at least 50 m from each other.</p>", "<p>Fresh plant leaves were used for DNA extraction. Total DNA was isolated from crushed leaf powder according to the Cetyl trimethyl ammonium bromide (CTAB) protocol with double purification with chloroform (##UREF##12##Doyle &amp; Doyle, 1987##). The quality and concentration of DNA were assessed using a NanoDrop 2000 spectrophotometer (Thermo Fisher Scientific, Waltham, MA, USA) and electrophoresis in 1% agarose gel. DNA concentration was normalized to the working concentration for further analysis.</p>", "<p>Twenty-two SSR markers of the nuclear genome were used as DNA markers and selected according to ##UREF##25##Mnejja et al. (2005)##. PCR amplification was performed in 10 ul reaction volume containing 20 ng template DNA, one PCR buffer, 1.5 mM of MgCl2, 0.2 mM of dNTPs, 0.4 mM of each primer, and one unit of Taq DNA polymerase (Sileks, Badenweiler, Germany). PCR conditions were set at 95 °C for 1 min, followed by 35 cycles of 94 °C for 30 s, 50–65 °C for 30 s, 72 °C for 30 s, and 5 min at 72 °C for final elongation. PCR products were separated on a 6% polyacrylamide gel using 0.5x Tris-borate-EDTA buffer. DNA fragments were identified using an ethidium bromide staining procedure. Alleles were determined using 100-bp ladders (Thermo Fisher Scientific). Visualization was performed using the GelDoc XR+ gel documentation system (Bio-Rad, Hercules, CA, USA).</p>", "<title>Statistical analysis and polymorphism information content</title>", "<p>Descriptive statistics and t-tests (SPSS Statistics v.27.0; <ext-link xlink:href=\"https://www.ibm.com/products/spss-statistics\" ext-link-type=\"uri\">https://www.ibm.com/products/spss-statistics</ext-link>) were used to describe morphological features and identify population differences (plant height across different populations <italic toggle=\"yes\">of P. ledebouriana</italic> and <italic toggle=\"yes\">P. tenella</italic>).</p>", "<p>Each DNA fragment obtained was treated as a separate character and evaluated as a discrete variable. Accordingly, rectangular binary data matrices were obtained for SSR markers.</p>", "<p>To assess the effectiveness of markers, the following polymorphism indices were used: marker index (MI), resolving power (Rp), observed heterozygosity (HO), expected heterozygosity (HE), polymorphism information content (PIC values), inbreeding coefficient (FIS), and fixation index, (Fst) as estimated by ##REF##28563791##Weir &amp; Cockerham (1984)##.</p>", "<p>The genetic diversity of <italic toggle=\"yes\">P. ledebouriana</italic>, including the Shannon diversity index (I), was assessed using PopGene (##UREF##51##Yeh et al., 2000##). Wright’s F-statistics was calculated for each SSR locus using the GenAlEx 6.5 program (##UREF##32##Peakall &amp; Smouse, 2006##). Analysis of molecular variance (AMOVA) was performed across organizations using GenAlEx 6.5 (##UREF##32##Peakall &amp; Smouse, 2006##). Principal coordinate analysis (PCA) was performed using the Numerical Taxonomy and Multivariate Analysis System (NTSYS-pc) program (##UREF##37##Rohlf, 1998##). The STRUCTURE program also applied Bayesian cluster analysis (##REF##10835412##Pritchard, Stephens &amp; Donnelly, 2000##). The dendrogram was constructed using the PAST program and unweighted pair group method with arithmetic mean (UPGMA) algorithm and Boot N:1,000 (##UREF##19##Joshi et al., 2000##). Cluster analysis of <italic toggle=\"yes\">P. ledebouriana</italic> and <italic toggle=\"yes\">P. tenella</italic> was studied using STRUCTURE. An admixture model was used, which made it possible to analyze the frequencies of admixtures and correlated alleles. Five independent simulations were run, each including 100,000 burn-in steps and a subsequent 100,000 Markov chain Monte Carlo (MCMC) iterations.</p>" ]

[ "<title>Results</title>", "<title>Variability of plant height among populations</title>", "<p>The samples of three <italic toggle=\"yes\">P. ledebouriana</italic> and one population of <italic toggle=\"yes\">P. tenella</italic> were assessed according to plant height (##FIG##0##Fig. 1##). The most significant plant height was recorded for the 3-KA population (2.09 ± 0.06 m), which was the mountain population’s highest elevation above sea level (##SUPPL##0##Table S1##). Similar parameters for plant height were recorded in representatives from two other mountain populations: 2-KO and 1-UR, 1.79 ± 0.03 m and 1.78 ± 0.03 m, respectively. The lowest height values of plants in the steppe populations of <italic toggle=\"yes\">P. tenella</italic> were recorded in 4-UK (1.41 ± 1.04 m). The <italic toggle=\"yes\">T</italic>-test confirmed a statistically significant difference between the plant height of the three populations of <italic toggle=\"yes\">P. ledebouriana</italic>: 3-KA, 2-KO, and 1-UR (<italic toggle=\"yes\">P</italic> value = 2.3e−15). The <italic toggle=\"yes\">t</italic>-test suggested that all three mountainous populations had significantly higher plants than the steppe population <italic toggle=\"yes\">P. tenella</italic> 4-UK (<italic toggle=\"yes\">P</italic> &lt; 0.0001).</p>", "<title>Allelic variation of SSRs</title>", "<p>The assessment of allelic variations in <italic toggle=\"yes\">P. ledebouriana</italic> and <italic toggle=\"yes\">P. tenella</italic> showed that 19 out of the studied 22 markers were polymorphic (##SUPPL##1##Table S2##). ##FIG##1##Figure 2## shows two gel snapshots of different populations with high polymorphism. It appeared that CPDCT038 had three loci. Information on these 19 polymorphic SSR loci, including the sizes of bands, is presented in ##TAB##0##Table 1##.</p>", "<p>Results showed that the three SSR loci had two alleles, 11 SSR loci had three, three SSR markers had four, and two loci had five. The results of genotyping using 19 SSR loci and their statistical details are shown in ##TAB##1##Table 2##. It was determined that 19 loci, according to the polymorphism level (PIC values) could be separated into three subpopulations. The first group was comprised of five SSR loci with a PIC above 0.5, the second group of eight loci with a PIC between 0.3 and 0.5, and the third group of the remaining six loci with a PIC below 0.3. The mean PIC value was 0.38 (##TAB##2##Table 3##).</p>", "<p>The genetic diversity assessment using Nei’s index suggested that the highest genetic diversity was in <italic toggle=\"yes\">P. tenella</italic> 4-UK (0.622), followed by three populations of <italic toggle=\"yes\">P. ledebouriana</italic>: 1-UR, 2-KO, and 3-KA (##TAB##1##Table 2##). The average genetic diversity index for the three populations of <italic toggle=\"yes\">P. ledebouriana</italic> was 0.501. To compare inter-population diversity, two main scenarios were used. The first included the assumption of the unity of four populations as representatives of the species <italic toggle=\"yes\">P. ledebouriana</italic>, and the second scenario assumed the presence of three populations of <italic toggle=\"yes\">P. ledebouriana</italic> and one population of <italic toggle=\"yes\">P. tenella</italic>. The AMOVA suggested that the total genetic diversity in <italic toggle=\"yes\">P. ledebouriana</italic> could be partitioned as 73% within populations and 27% between populations. The evaluation of the partitioning of the genetic variation across four populations (including 4-UK <italic toggle=\"yes\">P. tenella</italic>) resulted in a decrease in the level of variation within populations (63%) and an increase in variation between populations (37%).</p>", "<p>The <italic toggle=\"yes\">t-</italic>test was applied to test associations between SSR markers and plant height in samples of both species. The results showed that nine SSR loci were statistically associated with plant height (##TAB##3##Table 4##).</p>", "<title>Population structure in samples of four studied populations using SSR markers</title>", "<p>All <italic toggle=\"yes\">P. ledebouriana</italic> and <italic toggle=\"yes\">P. tenella</italic> samples were analyzed for population structure using the package STRUCTURE based on genotyping data from 19 polymorphic SSR markers. The structure was assessed using results from <italic toggle=\"yes\">K</italic> = 2 to <italic toggle=\"yes\">K</italic> = 10. The assessment of K plots suggested that starting from <italic toggle=\"yes\">K</italic> = 3 and <italic toggle=\"yes\">K</italic> = 4 (##FIG##2##Fig. 3##), plants from population 4-UK were separated from three populations of <italic toggle=\"yes\">P. ledebouriana</italic>. Interestingly, the population 3-KA, growing at the highest elevation and characterized by the lowest level of genetic diversity within four studied populations (##TAB##1##Table 2##), was drifting apart from other groups of plants in step <italic toggle=\"yes\">K</italic> = 2. In the analysis of principal coordinates (PCoA), it was determined that the first and second principal coordinates described 49.05% and 41.16% of the variability, respectively (##FIG##3##Fig. 4##). PC1 effectively separated 3-KA from the other three populations. At the same time, PC2 allowed for the differentiation of 4-UK from 1-UR and 2-KO. In addition, the UPGMA dendrogram was built based on the genotyping results for samples in four populations. The results suggested that population 4-UK formed a distinct cluster, and only one sample from that population (4-UK_07) was positioned close to the cluster with the domination of samples from population 3-KA. Likewise, in PCoA analysis, the UPGMA dendrogram distinguished 3-KA from 1-UR and 2-KO. At the same time, the latter populations had a mix of samples in several clades (##FIG##4##Fig. 5##).</p>" ]

[ "<title>Discussion</title>", "<title>The phylogenetic relationship between <italic toggle=\"yes\">P. tenella</italic> and <italic toggle=\"yes\">P. ledebouriana</italic></title>", "<p>The phylogeny of the genus <italic toggle=\"yes\">Prunus</italic> is complex. Previous reports provided contradictory results on the relationships of plum species (##REF##24173059##Badenes &amp; Parfitt, 1995##). Nevertheless, it has been well established that <italic toggle=\"yes\">P. tenella</italic> belongs to the section <italic toggle=\"yes\">Amygdalopsis</italic> within the subgenus <italic toggle=\"yes\">Amygdalus</italic> (##UREF##6##Avdeev, 2016##). The complexity of taxonomy in species within this section could be more precise, as there are questions about the relationship among different taxa. These questions include the taxonomic relationship between <italic toggle=\"yes\">P. tenella</italic> and <italic toggle=\"yes\">P. ledebouriana</italic>, where the former is widespread in the Eurasian continent, and the latter is limited to mountainous populations of East Kazakhstan, particularly in the Altai mountains (##UREF##54##Zhukovsky, 1971##; ##UREF##13##Dzhangaliev, Salova &amp; Turekhanova, 2003##; ##UREF##28##Myrzagalieva et al., 2015##; ##UREF##27##Myrzagalieva &amp; Orazov, 2018##). Several reports proposed that these two species have minor differences and could be considered one species (##UREF##40##Sokolov, Svyazeva &amp; Kubli, 1980##; ##UREF##35##Qiu et al., 2012##).</p>", "<p>Conversely, publications suggest that <italic toggle=\"yes\">P. ledebouriana</italic> and <italic toggle=\"yes\">P. tenella</italic> have sufficient morphological differences to separate them into two distinct species (##UREF##53##Zaurov et al., 2015##; ##UREF##29##Orazov et al., 2020##). Plant height is one of the most prominent traits used to differentiate these species (##UREF##26##Mushegyan, 1962##; ##UREF##49##Vintereoller, 1976##). In this work, all plants from the three <italic toggle=\"yes\">P. ledebouriana</italic> populations and one population <italic toggle=\"yes\">of P. tenella</italic> were measured for plant height. The results clearly showed that samples from the two species have a distinct separation based on this trait (<italic toggle=\"yes\">P</italic> &lt; 0.0001). Similar results for different morphological characteristics confirm the difference among other representatives of the genus (##UREF##11##Devi, Singh &amp; Thakur, 2018##). Hence, evaluating plant height can be a reliable way of distinguishing between two closely genetically related species within the section <italic toggle=\"yes\">Amygdalopsis</italic>. The performance of this trait in plants is reliably related to the elevation above sea level (a.s.l.), as the lowest altitude for <italic toggle=\"yes\">P. ledebouriana</italic> samples was higher than 1.7 m a.s.l. and the highest altitude for <italic toggle=\"yes\">P. tenella</italic> samples was lower than 1.5 m a.s.l. (##SUPPL##0##Table S1##). We assessed samples from two species using 19 polymorphic SSR loci to confirm the conclusion based on the plant height study. The application of SSR markers resulted in a constructed UPGMA phylogenetic tree that separated 20 samples of <italic toggle=\"yes\">P. tenella</italic> (4-UK) from 60 samples of <italic toggle=\"yes\">P. ledebouriana</italic> (1-UR, 2-KO, 3-KA) (##FIG##3##Fig. 4##). This result was also supported by the PCoA plot, where PC2 (41.2%) split 4-UK samples from 1-UR and 2-KO.</p>", "<p>In contrast, PC1 (49.1%) separated 3-KA from the remaining three populations (##FIG##2##Fig. 3##). Interestingly, the clusterization in ##FIG##3##Fig. 4## suggests that, generally, there is a low level of admixture between populations, which supports the model of “isolation by distance” (##REF##22574758##Meirmans, 2012##). The genetic heterozygosity index (Nei’s index) assessment showed that the highest genetic diversity was registered in the population of <italic toggle=\"yes\">P. tenella</italic> (0.606). In contrast, the lowest index was recorded in population 3-KA (0.449), representing the area with the highest sampling elevation. High altitude is a sufficiently solid environmental factor that negatively influences genetic variation in <italic toggle=\"yes\">P. ledebouriana</italic>. Nevertheless, the separation of 3-KA from 1-UR and 2-KO supported a more significant genetic variation within the species. The high genetic differentiation between mountain and steppe populations is most likely due to the factor of the steppe zone and the presence of anthropogenic pressure on the <italic toggle=\"yes\">P. tenella</italic> (4-UK) population. In turn, mountain populations are distinguished by mountain isolation of ridges. The analysis of samples by DNA genotyping using SSR markers suggested that five loci were characterized as markers with the highest polymorphism level (##TAB##2##Table 3##). These five SSR loci could also be recommended for the discrimination of <italic toggle=\"yes\">Prunus</italic> species in other studies. In addition, a <italic toggle=\"yes\">t</italic>-test was applied to test the association of 19 polymorphic SSR loci with plant height (##TAB##3##Table 4##). It was concluded that nine out of 19 SSR loci were significantly associated with plant height. This result may not be a direct reflection of associations between SSRs and plant height but an indication of the genetic differences between <italic toggle=\"yes\">P. ledebouriana</italic> and <italic toggle=\"yes\">P. tenella,</italic> as these two species have significantly differed in plant height (##FIG##0##Fig. 1##). Therefore, these nine SSR loci can be efficiently used in further studies of discrimination between <italic toggle=\"yes\">P. ledebouriana</italic> and <italic toggle=\"yes\">P. tenella</italic>. The assessment of the population structure using the STRUCTURE package suggested that populations of two species started separating at steps <italic toggle=\"yes\">K</italic> = 3 and <italic toggle=\"yes\">K</italic> = 4, which is another indication that <italic toggle=\"yes\">P. ledebouriana</italic> and <italic toggle=\"yes\">P. tenella</italic> are two different species. The evaluation of samples in four clusters at <italic toggle=\"yes\">K</italic> = 4 (##FIG##2##Fig. 3##) showed little admixture level, supporting the model of isolation by distance with a limited gene flow among the populations. Mantel tests revealed a positive correlation between geographic and genetic distance among populations (<italic toggle=\"yes\">r</italic> = 0.4387), demonstrating consistency with the isolation-by-distance model.</p>" ]

[ "<title>Conclusion</title>", "<p>Discriminating the endemic species <italic toggle=\"yes\">P. ledebouriana</italic> from wild almond <italic toggle=\"yes\">P. tenella</italic> has been a poorly studied issue for the genus <italic toggle=\"yes\">Prunus</italic>. In this work, two different approaches were conducted to analyze the genetic relationship between these two closely related species. In the first approach, the detailed analysis of plant height from one population of <italic toggle=\"yes\">P. tenella</italic> and three populations of <italic toggle=\"yes\">P. ledebouriana</italic> allowed a significant separation (<italic toggle=\"yes\">P</italic> &lt; 0.0001) between the two species. In the second approach, the samples of these four populations were genotyped using 19 polymorphic SSR loci. The NJ phylogenetic tree and PCoA plot also showed a significant separation of two species on two groups of clusters. Also, the UPGMA dendrogram and PCoA plot have demonstrated that within <italic toggle=\"yes\">P. ledebouriana</italic>, the population 3-KA sharply differed from populations 1-UR and 2-KO, supporting a high diversity level within the species. The assessment of the connections between SSR loci and plant height showed that nine out of 19 loci were associated with the studied morphological trait, suggesting that these loci can be efficiently used in DNA discrimination of two species. The population structure analysis suggested that samples in two species were separated starting from step <italic toggle=\"yes\">K</italic> = 3. The assessment of plants in clusters at steps <italic toggle=\"yes\">K</italic> = 3 and <italic toggle=\"yes\">K</italic> = 4 suggested a limited admixture level between populations, supporting the model of isolation by distance. Thus, the analysis of plant height and application of SSR markers were successfully used to discriminate <italic toggle=\"yes\">P. tenella</italic> and <italic toggle=\"yes\">P. ledebouriana</italic> and study the genetic diversity and population structure of the endemic species <italic toggle=\"yes\">P. ledebouriana</italic>. A clear distinction between similar plants from different populations makes it possible to delineate the boundary of mutual replacement of the <italic toggle=\"yes\">P. tenella</italic> and <italic toggle=\"yes\">P. ledebouriana</italic> species. It will be possible to accurately separate precious endemic populations of <italic toggle=\"yes\">P. ledebouriana</italic> from the simple <italic toggle=\"yes\">P. tenella</italic>, and then clarify the taxonomy of the genus and organize conservation measures in the mountainous zones of Eastern Kazakhstan. The obtained data on SSR will allow further research at a higher level. It is proposed to use markers such as ITS, s6pdh, trnL-trnF, and trnS-trnG to compare several species of shrub almonds from Central Asia and use whole-genome methods.</p>" ]

[ "<title>Background</title>", "<p>Genetic differences between isolated endemic populations of plant species and those with widely known twin species are relevant for conserving the biological diversity of our planet’s flora. <italic toggle=\"yes\">Prunus ledebouriana</italic> (Schlecht.) YY Yao is an endangered and endemic species of shrub almond from central Asia. Few studies have explored this species, which is closely related and morphologically similar to the well-known <italic toggle=\"yes\">Prunus tenella</italic> Batsch. In this article, we present a comparative analysis of studies of three <italic toggle=\"yes\">P. ledebouriana</italic> populations and one close population of <italic toggle=\"yes\">P. tenella</italic> in Eastern Kazakhstan in order to determine the particular geographic mutual replacement of the two species.</p>", "<title>Methods</title>", "<p>The populations were collected from different ecological niches, including one steppe population near Ust-Kamenogorsk (<italic toggle=\"yes\">P. tenella</italic>) and three populations (<italic toggle=\"yes\">P. ledebouriana</italic>) in the mountainous area. Estimation of plant height using a <italic toggle=\"yes\">t</italic>-test suggested a statistically significant difference between the populations and the two species (<italic toggle=\"yes\">P</italic> &lt; 0.0001). DNA simple sequence repeat (SSR) markers were applied to study the two species’ genetic diversity and population structure.</p>", "<title>Results</title>", "<p>A total of 19 polymorphic SSR loci were analyzed, and the results showed that the population collected in mountainous areas had a lower variation level than steppe populations. The highest level of Nei’s genetic diversity index was demonstrated in the 4-UK population (0.622) of <italic toggle=\"yes\">P. tenella</italic>. The lowest was recorded in population 3-KA (0.461) of <italic toggle=\"yes\">P. ledebouriana</italic>, collected at the highest altitude of the four populations (2,086 meters above sea level). The total genetic variation of <italic toggle=\"yes\">P. ledebouriana</italic> was distributed 73% within populations and 27% between populations. STRUCTURE results showed that two morphologically similar species diverged starting at step <italic toggle=\"yes\">K</italic> = 3, with limited population mixing. The results confirmed the morphological and genetic differences between <italic toggle=\"yes\">P. tenella</italic> and <italic toggle=\"yes\">P. ledebouriana</italic> and described the level of genetic variation for <italic toggle=\"yes\">P. ledebouriana</italic>. The study’s results proved that the steppe zone and mountain altitude factor between <italic toggle=\"yes\">P. tenella</italic> and isolated mountain samples of <italic toggle=\"yes\">P. ledebouriana</italic>.</p>" ]

[ "<title> Supplemental Information</title>" ]

[ "<p>The authors express their gratitude to the head of the Department of Science, Information, and Monitoring of the State National Natural Park, Tarbagatai Alemseitova Janylkan Kabikyzy, for organizing field scientific expeditions in the territory of the National Park.</p>", "<title>Additional Information and Declarations</title>" ]

[ "<fig position=\"float\" id=\"fig-1\"><object-id pub-id-type=\"doi\">10.7717/peerj.16735/fig-1</object-id><label>Figure 1</label><caption><title>Column diagram of plant heights in populations of <italic toggle=\"yes\">P. ledebouriana</italic> (A—Urdzhar, B—Kokpekty, C—Katon-Karagai-blue) and <italic toggle=\"yes\">P. tenella</italic> (D—Ulansky-yellow).</title></caption></fig>", "<fig position=\"float\" id=\"fig-2\"><object-id pub-id-type=\"doi\">10.7717/peerj.16735/fig-2</object-id><label>Figure 2</label><caption><title>Gel images of the results of electrophoresis of two polymorphic SSR loci, (A) <italic toggle=\"yes\">P. ledebouriana</italic> (CPDCT045) and (B) <italic toggle=\"yes\">P. tenella</italic> (CPDCT025).</title></caption></fig>", "<fig position=\"float\" id=\"fig-3\"><object-id pub-id-type=\"doi\">10.7717/peerj.16735/fig-3</object-id><label>Figure 3</label><caption><title>Bayesian clustering of 60 <italic toggle=\"yes\">P. ledebouriana</italic> (A—Urdzhar, B—Kokpekty, C—Katon-Karagay) and 20 <italic toggle=\"yes\">P. tenella</italic> (D—Ulansky) plants at K = 2, K = 3, K = 4, and K = 10 step.</title></caption></fig>", "<fig position=\"float\" id=\"fig-4\"><object-id pub-id-type=\"doi\">10.7717/peerj.16735/fig-4</object-id><label>Figure 4</label><caption><title>Principal coordinate analysis (PCoA) for populations of <italic toggle=\"yes\">P. ledebouriana</italic> (A—Urdzhar, B—Kokpekty, C—Katon-Karagay) and <italic toggle=\"yes\">P. tenella</italic> (D—Ulansky) using polymorphic SSR loci.</title></caption></fig>", "<fig position=\"float\" id=\"fig-5\"><object-id pub-id-type=\"doi\">10.7717/peerj.16735/fig-5</object-id><label>Figure 5</label><caption><title>Unweighted pair group method with arithmetic mean (UPGMA) dendrogram of <italic toggle=\"yes\">P. ledebouriana</italic> (Pop1—Urdzhar, Pop2—Kokpekty, Pop3—Katon-Karagay) and <italic toggle=\"yes\">P. tenella</italic> (Pop4—Ulansky) using polymorphic SSR loci.</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"table-1\"><object-id pub-id-type=\"doi\">10.7717/peerj.16735/table-1</object-id><label>Table 1</label><caption><title>The list of polymorphic SSR loci selected from ##UREF##25##Mnejja et al. (2005)## and analyzed in plants of <italic toggle=\"yes\">P. ledebouriana</italic> and <italic toggle=\"yes\">P. tenella</italic>.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\">\n<bold>#</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Loci</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Number of alleles</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Motif</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Annealing temperature optimized (°C)</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Size obtained (bp)</bold>\n</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">1.</td><td rowspan=\"1\" colspan=\"1\">CPDCT005</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">(CT)₁₄</td><td rowspan=\"1\" colspan=\"1\">64⁺</td><td rowspan=\"1\" colspan=\"1\">96, 98, 101</td></tr><tr><td rowspan=\"1\" colspan=\"1\">2.</td><td rowspan=\"1\" colspan=\"1\">CPDCT007</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">(GA)₁₉</td><td rowspan=\"1\" colspan=\"1\">48⁺</td><td rowspan=\"1\" colspan=\"1\">163, 172, 183</td></tr><tr><td rowspan=\"1\" colspan=\"1\">3.</td><td rowspan=\"1\" colspan=\"1\">CPDCT008</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">(GA)₁₈</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">185, 188, 192</td></tr><tr><td rowspan=\"1\" colspan=\"1\">4.</td><td rowspan=\"1\" colspan=\"1\">CPDCT012</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">(GA)₁₂</td><td rowspan=\"1\" colspan=\"1\">56⁺</td><td rowspan=\"1\" colspan=\"1\">157, 166</td></tr><tr><td rowspan=\"1\" colspan=\"1\">5.</td><td rowspan=\"1\" colspan=\"1\">CPDCT015</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">(CT)₂₀</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">214, 215, 216, 217</td></tr><tr><td rowspan=\"1\" colspan=\"1\">6.</td><td rowspan=\"1\" colspan=\"1\">CPDCT016</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">(GA)₁₉</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">173, 178, 193</td></tr><tr><td rowspan=\"1\" colspan=\"1\">7.</td><td rowspan=\"1\" colspan=\"1\">CPDCT022</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">(CT)₁₇</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">152, 158, 159</td></tr><tr><td rowspan=\"1\" colspan=\"1\">8.</td><td rowspan=\"1\" colspan=\"1\">CPDCT023</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">(GA)₉</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">171, 176, 179</td></tr><tr><td rowspan=\"1\" colspan=\"1\">9.</td><td rowspan=\"1\" colspan=\"1\">CPDCT025</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">(CT)₁₀</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">184, 188, 194, 195</td></tr><tr><td rowspan=\"1\" colspan=\"1\">10.</td><td rowspan=\"1\" colspan=\"1\">CPDCT027</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">(CT)₁₉</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">174, 180, 190</td></tr><tr><td rowspan=\"1\" colspan=\"1\">11.</td><td rowspan=\"1\" colspan=\"1\">CPDCT035</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">(GA)₁₇</td><td rowspan=\"1\" colspan=\"1\">52⁺</td><td rowspan=\"1\" colspan=\"1\">162, 163, 164, 165</td></tr><tr><td rowspan=\"1\" colspan=\"1\">12.</td><td rowspan=\"1\" colspan=\"1\">CPDCT038/1</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">(GA)₂₅</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">156, 160</td></tr><tr><td rowspan=\"1\" colspan=\"1\">13.</td><td rowspan=\"1\" colspan=\"1\">CPDCT038/2</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">(GA)₂₅</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">181, 189</td></tr><tr><td rowspan=\"1\" colspan=\"1\">14.</td><td rowspan=\"1\" colspan=\"1\">CPDCT038/3</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">(GA)₂₅</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">274, 311, 329</td></tr><tr><td rowspan=\"1\" colspan=\"1\">15.</td><td rowspan=\"1\" colspan=\"1\">CPDCT039</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">(GA)₁₅</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">154, 155, 156</td></tr><tr><td rowspan=\"1\" colspan=\"1\">16.</td><td rowspan=\"1\" colspan=\"1\">CPDCT040</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">(GA)₂₄</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">144, 145, 146</td></tr><tr><td rowspan=\"1\" colspan=\"1\">17.</td><td rowspan=\"1\" colspan=\"1\">CPDCT043</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">(GA)₂₁</td><td rowspan=\"1\" colspan=\"1\">48⁺</td><td rowspan=\"1\" colspan=\"1\">71, 76, 102</td></tr><tr><td rowspan=\"1\" colspan=\"1\">18.</td><td rowspan=\"1\" colspan=\"1\">CPDCT045</td><td rowspan=\"1\" colspan=\"1\">5</td><td rowspan=\"1\" colspan=\"1\">(GA)₁₆</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">139, 144, 149, 159, 167</td></tr><tr><td rowspan=\"1\" colspan=\"1\">19.</td><td rowspan=\"1\" colspan=\"1\">CPDCT046</td><td rowspan=\"1\" colspan=\"1\">5</td><td rowspan=\"1\" colspan=\"1\">(GA)₂₁</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">149, 150, 151, 152, 153</td></tr></tbody></table></alternatives></table-wrap>", "<table-wrap position=\"float\" id=\"table-2\"><object-id pub-id-type=\"doi\">10.7717/peerj.16735/table-2</object-id><label>Table 2</label><caption><title>Assessment of genetic diversity in studied populations of <italic toggle=\"yes\">P. ledebouriana</italic> and <italic toggle=\"yes\">P. tenella</italic>.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\">\n<bold>ID</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Species</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Mean/SE</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Na</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Ne</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>uHe</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Fst</bold>\n</th></tr></thead><tbody><tr><td rowspan=\"2\" colspan=\"1\">1-UR</td><td rowspan=\"2\" colspan=\"1\"><italic toggle=\"yes\">P. ledebouriana</italic> (<italic toggle=\"yes\">n</italic> = 20)</td><td rowspan=\"1\" colspan=\"1\">Mean</td><td rowspan=\"1\" colspan=\"1\">4.158</td><td rowspan=\"1\" colspan=\"1\">2.427</td><td rowspan=\"1\" colspan=\"1\">0.560</td><td rowspan=\"2\" colspan=\"1\">0,060</td></tr><tr><td rowspan=\"1\" colspan=\"1\">SE</td><td rowspan=\"1\" colspan=\"1\">0.906</td><td rowspan=\"1\" colspan=\"1\">0.177</td><td rowspan=\"1\" colspan=\"1\">0.037</td></tr><tr><td rowspan=\"2\" colspan=\"1\">2-KO</td><td rowspan=\"2\" colspan=\"1\"><italic toggle=\"yes\">P. ledebouriana</italic> (<italic toggle=\"yes\">n</italic> = 20)</td><td rowspan=\"1\" colspan=\"1\">Mean</td><td rowspan=\"1\" colspan=\"1\">3.316</td><td rowspan=\"1\" colspan=\"1\">2.106</td><td rowspan=\"1\" colspan=\"1\">0.483</td><td rowspan=\"2\" colspan=\"1\">0,113</td></tr><tr><td rowspan=\"1\" colspan=\"1\">SE</td><td rowspan=\"1\" colspan=\"1\">0.991</td><td rowspan=\"1\" colspan=\"1\">0.158</td><td rowspan=\"1\" colspan=\"1\">0.045</td></tr><tr><td rowspan=\"2\" colspan=\"1\">3-KA</td><td rowspan=\"2\" colspan=\"1\"><italic toggle=\"yes\">P. ledebouriana</italic> (<italic toggle=\"yes\">n</italic> = 20)</td><td rowspan=\"1\" colspan=\"1\">Mean</td><td rowspan=\"1\" colspan=\"1\">3.316</td><td rowspan=\"1\" colspan=\"1\">2.108</td><td rowspan=\"1\" colspan=\"1\">0.461</td><td rowspan=\"2\" colspan=\"1\">0,057</td></tr><tr><td rowspan=\"1\" colspan=\"1\">SE</td><td rowspan=\"1\" colspan=\"1\">1.000</td><td rowspan=\"1\" colspan=\"1\">0.179</td><td rowspan=\"1\" colspan=\"1\">0.056</td></tr><tr><td rowspan=\"2\" colspan=\"1\">4-UK</td><td rowspan=\"2\" colspan=\"1\"><italic toggle=\"yes\">P. tenella</italic> (<italic toggle=\"yes\">n</italic> = 20)</td><td rowspan=\"1\" colspan=\"1\">Mean</td><td rowspan=\"1\" colspan=\"1\">4.632</td><td rowspan=\"1\" colspan=\"1\">2.868</td><td rowspan=\"1\" colspan=\"1\">0.622</td><td rowspan=\"2\" colspan=\"1\">0,066</td></tr><tr><td rowspan=\"1\" colspan=\"1\">SE</td><td rowspan=\"1\" colspan=\"1\">1.041</td><td rowspan=\"1\" colspan=\"1\">0.338</td><td rowspan=\"1\" colspan=\"1\">0.025</td></tr></tbody></table></alternatives></table-wrap>", "<table-wrap position=\"float\" id=\"table-3\"><object-id pub-id-type=\"doi\">10.7717/peerj.16735/table-3</object-id><label>Table 3</label><caption><title>Assessment of genetic diversity of SSR loci in the analysis of <italic toggle=\"yes\">P. ledebouriana</italic> and <italic toggle=\"yes\">P. tenella</italic> populations.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\">\n<bold>#</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Locus</bold>\n</th><th rowspan=\"1\" colspan=\"1\"><bold>H</bold>_{<bold>O</bold>}/ <bold>H</bold>_{<bold>E</bold>}</th><th rowspan=\"1\" colspan=\"1\">\n<bold>FIS</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>PIC</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>Rp</bold>\n</th><th rowspan=\"1\" colspan=\"1\">\n<bold>MI</bold>\n</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">CPDCT005</td><td rowspan=\"1\" colspan=\"1\">0.719/0.280</td><td rowspan=\"1\" colspan=\"1\">0.252</td><td rowspan=\"1\" colspan=\"1\">0.251</td><td rowspan=\"1\" colspan=\"1\">0.282</td><td rowspan=\"1\" colspan=\"1\">0.279</td></tr><tr><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">CPDCT007</td><td rowspan=\"1\" colspan=\"1\">0.711/0.288</td><td rowspan=\"1\" colspan=\"1\">0.400</td><td rowspan=\"1\" colspan=\"1\">0.283</td><td rowspan=\"1\" colspan=\"1\">0.307</td><td rowspan=\"1\" colspan=\"1\">0.304</td></tr><tr><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">CPDCT008</td><td rowspan=\"1\" colspan=\"1\">0.793/0.206</td><td rowspan=\"1\" colspan=\"1\">0.288</td><td rowspan=\"1\" colspan=\"1\">0.194</td><td rowspan=\"1\" colspan=\"1\">0.207</td><td rowspan=\"1\" colspan=\"1\">0.205</td></tr><tr><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">CPDCT012</td><td rowspan=\"1\" colspan=\"1\">0.584/0.415</td><td rowspan=\"1\" colspan=\"1\">0.342</td><td rowspan=\"1\" colspan=\"1\">0.345</td><td rowspan=\"1\" colspan=\"1\">0.432</td><td rowspan=\"1\" colspan=\"1\">0.427</td></tr><tr><td rowspan=\"1\" colspan=\"1\">5</td><td rowspan=\"1\" colspan=\"1\">CPDCT015</td><td rowspan=\"1\" colspan=\"1\">0.703/0.296</td><td rowspan=\"1\" colspan=\"1\">0.104</td><td rowspan=\"1\" colspan=\"1\">0.314</td><td rowspan=\"1\" colspan=\"1\">0.333</td><td rowspan=\"1\" colspan=\"1\">0.329</td></tr><tr><td rowspan=\"1\" colspan=\"1\">6</td><td rowspan=\"1\" colspan=\"1\">CPDCT016</td><td rowspan=\"1\" colspan=\"1\">0.570/0.429</td><td rowspan=\"1\" colspan=\"1\">0.122</td><td rowspan=\"1\" colspan=\"1\">0.345</td><td rowspan=\"1\" colspan=\"1\">0.432</td><td rowspan=\"1\" colspan=\"1\">0.427</td></tr><tr><td rowspan=\"1\" colspan=\"1\">7</td><td rowspan=\"1\" colspan=\"1\">CPDCT022</td><td rowspan=\"1\" colspan=\"1\">0.856/0.143</td><td rowspan=\"1\" colspan=\"1\">0.124</td><td rowspan=\"1\" colspan=\"1\">0.158</td><td rowspan=\"1\" colspan=\"1\">0.166</td><td rowspan=\"1\" colspan=\"1\">0.164</td></tr><tr><td rowspan=\"1\" colspan=\"1\">8</td><td rowspan=\"1\" colspan=\"1\">CPDCT023</td><td rowspan=\"1\" colspan=\"1\">0.502/0.497</td><td rowspan=\"1\" colspan=\"1\">0.694</td><td rowspan=\"1\" colspan=\"1\">0.403</td><td rowspan=\"1\" colspan=\"1\">0.500</td><td rowspan=\"1\" colspan=\"1\">0.494</td></tr><tr><td rowspan=\"1\" colspan=\"1\">9</td><td rowspan=\"1\" colspan=\"1\">CPDCT025</td><td rowspan=\"1\" colspan=\"1\">0.390/0.609</td><td rowspan=\"1\" colspan=\"1\">0.407</td><td rowspan=\"1\" colspan=\"1\">0.524</td><td rowspan=\"1\" colspan=\"1\">0.612</td><td rowspan=\"1\" colspan=\"1\">0.605</td></tr><tr><td rowspan=\"1\" colspan=\"1\">10</td><td rowspan=\"1\" colspan=\"1\">CPDCT027</td><td rowspan=\"1\" colspan=\"1\">0.387/0.612</td><td rowspan=\"1\" colspan=\"1\">0.563</td><td rowspan=\"1\" colspan=\"1\">0.535</td><td rowspan=\"1\" colspan=\"1\">0.616</td><td rowspan=\"1\" colspan=\"1\">0.609</td></tr><tr><td rowspan=\"1\" colspan=\"1\">11</td><td rowspan=\"1\" colspan=\"1\">CPDCT035</td><td rowspan=\"1\" colspan=\"1\">0.502/0.497</td><td rowspan=\"1\" colspan=\"1\">0.491</td><td rowspan=\"1\" colspan=\"1\">0.461</td><td rowspan=\"1\" colspan=\"1\">0.500</td><td rowspan=\"1\" colspan=\"1\">0.494</td></tr><tr><td rowspan=\"1\" colspan=\"1\">12</td><td rowspan=\"1\" colspan=\"1\">CPDCT038/1</td><td rowspan=\"1\" colspan=\"1\">0.635/0.364</td><td rowspan=\"1\" colspan=\"1\">0.327</td><td rowspan=\"1\" colspan=\"1\">0.296</td><td rowspan=\"1\" colspan=\"1\">0.366</td><td rowspan=\"1\" colspan=\"1\">0.362</td></tr><tr><td rowspan=\"1\" colspan=\"1\">13</td><td rowspan=\"1\" colspan=\"1\">CPDCT038/2</td><td rowspan=\"1\" colspan=\"1\">0.860/0.139</td><td rowspan=\"1\" colspan=\"1\">0.243</td><td rowspan=\"1\" colspan=\"1\">0.129</td><td rowspan=\"1\" colspan=\"1\">0.140</td><td rowspan=\"1\" colspan=\"1\">0.138</td></tr><tr><td rowspan=\"1\" colspan=\"1\">14</td><td rowspan=\"1\" colspan=\"1\">CPDCT038/3</td><td rowspan=\"1\" colspan=\"1\">0.445/0.554</td><td rowspan=\"1\" colspan=\"1\">0.179</td><td rowspan=\"1\" colspan=\"1\">0.511</td><td rowspan=\"1\" colspan=\"1\">0.590</td><td rowspan=\"1\" colspan=\"1\">0.582</td></tr><tr><td rowspan=\"1\" colspan=\"1\">15</td><td rowspan=\"1\" colspan=\"1\">CPDCT039</td><td rowspan=\"1\" colspan=\"1\">0.432/0.567</td><td rowspan=\"1\" colspan=\"1\">0.086</td><td rowspan=\"1\" colspan=\"1\">0.468</td><td rowspan=\"1\" colspan=\"1\">0.570</td><td rowspan=\"1\" colspan=\"1\">0.563</td></tr><tr><td rowspan=\"1\" colspan=\"1\">16</td><td rowspan=\"1\" colspan=\"1\">CPDCT040</td><td rowspan=\"1\" colspan=\"1\">0.434/0.565</td><td rowspan=\"1\" colspan=\"1\">0.317</td><td rowspan=\"1\" colspan=\"1\">0.467</td><td rowspan=\"1\" colspan=\"1\">0.568</td><td rowspan=\"1\" colspan=\"1\">0.561</td></tr><tr><td rowspan=\"1\" colspan=\"1\">17</td><td rowspan=\"1\" colspan=\"1\">CPDCT043</td><td rowspan=\"1\" colspan=\"1\">0.485/0.514</td><td rowspan=\"1\" colspan=\"1\">0.359</td><td rowspan=\"1\" colspan=\"1\">0.453</td><td rowspan=\"1\" colspan=\"1\">0.517</td><td rowspan=\"1\" colspan=\"1\">0.511</td></tr><tr><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">CPDCT045</td><td rowspan=\"1\" colspan=\"1\">0.398/0.601</td><td rowspan=\"1\" colspan=\"1\">0.120</td><td rowspan=\"1\" colspan=\"1\">0.529</td><td rowspan=\"1\" colspan=\"1\">0.615</td><td rowspan=\"1\" colspan=\"1\">0.607</td></tr><tr><td rowspan=\"1\" colspan=\"1\">19</td><td rowspan=\"1\" colspan=\"1\">CPDCT046</td><td rowspan=\"1\" colspan=\"1\">0.309/0.690</td><td rowspan=\"1\" colspan=\"1\">0.135</td><td rowspan=\"1\" colspan=\"1\">0.631</td><td rowspan=\"1\" colspan=\"1\">0.695</td><td rowspan=\"1\" colspan=\"1\">0.686</td></tr><tr><td align=\"center\" colspan=\"2\" rowspan=\"1\">Mean</td><td rowspan=\"1\" colspan=\"1\">0.586/0.413</td><td rowspan=\"1\" colspan=\"1\">0.308</td><td rowspan=\"1\" colspan=\"1\">0.384</td><td rowspan=\"1\" colspan=\"1\">0.445</td><td rowspan=\"1\" colspan=\"1\">0.693</td></tr></tbody></table></alternatives></table-wrap>", "<table-wrap position=\"float\" id=\"table-4\"><object-id pub-id-type=\"doi\">10.7717/peerj.16735/table-4</object-id><label>Table 4</label><caption><title>Results of association assessment between SSR loci and plant height in populations of <italic toggle=\"yes\">P. ledebouriana</italic> and <italic toggle=\"yes\">P. tenella</italic> using <italic toggle=\"yes\">t</italic>-tests.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\">#</th><th rowspan=\"1\" colspan=\"1\">Marker ID</th><th rowspan=\"1\" colspan=\"1\">Allele</th><th rowspan=\"1\" colspan=\"1\">\n<italic toggle=\"yes\">n</italic>\n</th><th rowspan=\"1\" colspan=\"1\">Average PH</th><th rowspan=\"1\" colspan=\"1\"><italic toggle=\"yes\">p</italic>-value</th></tr></thead><tbody><tr><td rowspan=\"2\" colspan=\"1\">1</td><td rowspan=\"2\" colspan=\"1\">CPDCT005</td><td rowspan=\"1\" colspan=\"1\">C</td><td rowspan=\"1\" colspan=\"1\">67</td><td rowspan=\"1\" colspan=\"1\">182.8</td><td rowspan=\"2\" colspan=\"1\">0.000179<xref rid=\"table-4fn4\" ref-type=\"table-fn\">^***</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\">B</td><td rowspan=\"1\" colspan=\"1\">11</td><td rowspan=\"1\" colspan=\"1\">144.6</td></tr><tr><td rowspan=\"3\" colspan=\"1\">2</td><td rowspan=\"3\" colspan=\"1\">CPDCT007</td><td rowspan=\"1\" colspan=\"1\">B</td><td rowspan=\"1\" colspan=\"1\">66</td><td rowspan=\"1\" colspan=\"1\">183.5</td><td rowspan=\"3\" colspan=\"1\">0.000125<xref rid=\"table-4fn4\" ref-type=\"table-fn\">^***</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\">A</td><td rowspan=\"1\" colspan=\"1\">9</td><td rowspan=\"1\" colspan=\"1\">140.2</td><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">C</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">150.7</td></tr><tr><td rowspan=\"3\" colspan=\"1\">3</td><td rowspan=\"3\" colspan=\"1\">CPDCT025</td><td rowspan=\"1\" colspan=\"1\">B</td><td rowspan=\"1\" colspan=\"1\">33</td><td rowspan=\"1\" colspan=\"1\">166.8</td><td rowspan=\"3\" colspan=\"1\">9.77e−05<xref rid=\"table-4fn4\" ref-type=\"table-fn\">^***</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\">C</td><td rowspan=\"1\" colspan=\"1\">21</td><td rowspan=\"1\" colspan=\"1\">180.5</td><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">A</td><td rowspan=\"1\" colspan=\"1\">8</td><td rowspan=\"1\" colspan=\"1\">151.5</td></tr><tr><td rowspan=\"3\" colspan=\"1\">4</td><td rowspan=\"3\" colspan=\"1\">CPDCT027</td><td rowspan=\"1\" colspan=\"1\">C</td><td rowspan=\"1\" colspan=\"1\">27</td><td rowspan=\"1\" colspan=\"1\">179.7</td><td rowspan=\"3\" colspan=\"1\">9.78e−05<xref rid=\"table-4fn4\" ref-type=\"table-fn\">^***</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\">A</td><td rowspan=\"1\" colspan=\"1\">25</td><td rowspan=\"1\" colspan=\"1\">197.6</td><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">B</td><td rowspan=\"1\" colspan=\"1\">14</td><td rowspan=\"1\" colspan=\"1\">172.2</td></tr><tr><td rowspan=\"4\" colspan=\"1\">5</td><td rowspan=\"4\" colspan=\"1\">CPDCT035</td><td rowspan=\"1\" colspan=\"1\">A</td><td rowspan=\"1\" colspan=\"1\">55</td><td rowspan=\"1\" colspan=\"1\">186</td><td rowspan=\"4\" colspan=\"1\">1.81e−08<xref rid=\"table-4fn4\" ref-type=\"table-fn\">^***</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\">D</td><td rowspan=\"1\" colspan=\"1\">10</td><td rowspan=\"1\" colspan=\"1\">136.5</td><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\">C</td><td rowspan=\"1\" colspan=\"1\">8</td><td rowspan=\"1\" colspan=\"1\">147.8</td><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">B</td><td rowspan=\"1\" colspan=\"1\">7</td><td rowspan=\"1\" colspan=\"1\">195.7</td></tr><tr><td rowspan=\"3\" colspan=\"1\">6</td><td rowspan=\"3\" colspan=\"1\">CPDCT038_3</td><td rowspan=\"1\" colspan=\"1\">C</td><td rowspan=\"1\" colspan=\"1\">44</td><td rowspan=\"1\" colspan=\"1\">170.8</td><td rowspan=\"3\" colspan=\"1\">0.00883<xref rid=\"table-4fn3\" ref-type=\"table-fn\">^**</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\">A</td><td rowspan=\"1\" colspan=\"1\">26</td><td rowspan=\"1\" colspan=\"1\">192.3</td><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">B</td><td rowspan=\"1\" colspan=\"1\">7</td><td rowspan=\"1\" colspan=\"1\">156.7</td></tr><tr><td rowspan=\"3\" colspan=\"1\">7</td><td rowspan=\"3\" colspan=\"1\">CPDCT040</td><td rowspan=\"1\" colspan=\"1\">C</td><td rowspan=\"1\" colspan=\"1\">41</td><td rowspan=\"1\" colspan=\"1\">188.8</td><td rowspan=\"3\" colspan=\"1\">0.000729<xref rid=\"table-4fn4\" ref-type=\"table-fn\">^***</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\">B</td><td rowspan=\"1\" colspan=\"1\">33</td><td rowspan=\"1\" colspan=\"1\">166.1</td><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">A</td><td rowspan=\"1\" colspan=\"1\">6</td><td rowspan=\"1\" colspan=\"1\">153.6</td></tr><tr><td rowspan=\"3\" colspan=\"1\">8</td><td rowspan=\"3\" colspan=\"1\">CPDCT043</td><td rowspan=\"1\" colspan=\"1\">A</td><td rowspan=\"1\" colspan=\"1\">52</td><td rowspan=\"1\" colspan=\"1\">166.5</td><td rowspan=\"3\" colspan=\"1\">1.39e−05<xref rid=\"table-4fn4\" ref-type=\"table-fn\">^***</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\">B</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">204.1</td><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">C</td><td rowspan=\"1\" colspan=\"1\">10</td><td rowspan=\"1\" colspan=\"1\">181.1</td></tr><tr><td rowspan=\"5\" colspan=\"1\">9</td><td rowspan=\"5\" colspan=\"1\">CPDCT046</td><td rowspan=\"1\" colspan=\"1\">D</td><td rowspan=\"1\" colspan=\"1\">32</td><td rowspan=\"1\" colspan=\"1\">193.7</td><td rowspan=\"5\" colspan=\"1\">0.000134<xref rid=\"table-4fn4\" ref-type=\"table-fn\">^***</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\">A</td><td rowspan=\"1\" colspan=\"1\">29</td><td rowspan=\"1\" colspan=\"1\">172.3</td><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\">B</td><td rowspan=\"1\" colspan=\"1\">10</td><td rowspan=\"1\" colspan=\"1\">157</td><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\"/><td rowspan=\"1\" colspan=\"1\">E</td><td rowspan=\"1\" colspan=\"1\">5</td><td rowspan=\"1\" colspan=\"1\">143.8</td><td rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">C</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">165.2</td></tr></tbody></table></alternatives></table-wrap>" ]

[]

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[ "<supplementary-material id=\"supp-1\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16735/supp-1</object-id><label>Table S1</label><caption><title>Geographic locations of collected three populations of <italic toggle=\"yes\">P. ledebouriana</italic> and one population of <italic toggle=\"yes\">P. tenella</italic> in Eastern Kazakhstan</title></caption></supplementary-material>", "<supplementary-material id=\"supp-2\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16735/supp-2</object-id><label>Table S2</label><caption><title>Simple sequence repeat</title></caption></supplementary-material>" ]

[ "<table-wrap-foot><fn id=\"table-1fn\"><p>\n<bold>Notes.</bold>\n</p></fn><fn id=\"table-1fn1\" fn-type=\"other\"><p>⁺—Optimized annealing temperature (°C).</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"table-2fn\"><p>\n<bold>Notes.</bold>\n</p></fn><fn id=\"table-2fn1\" fn-type=\"other\"><p>\n</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"table-3fn\"><p>\n<bold>Notes.</bold>\n</p></fn><fn id=\"table-3fn1\" fn-type=\"other\"><p>\n</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"table-4fn\"><p>\n<bold>Notes.</bold>\n</p></fn><fn id=\"table-4fn1\" fn-type=\"other\"><p><italic toggle=\"yes\">P</italic>—values are provided with significance level indicated by the asterisks.</p></fn><fn id=\"table-4fn2\"><label>*</label><p><italic toggle=\"yes\">P</italic> &lt; 0.05.</p></fn><fn id=\"table-4fn3\"><label>**</label><p><italic toggle=\"yes\">P</italic> &lt; 0.01.</p></fn><fn id=\"table-4fn4\"><label>***</label><p><italic toggle=\"yes\">P</italic> &lt; 0.001.</p></fn></table-wrap-foot>", "<fn-group content-type=\"competing-interests\"><title>Competing Interests</title><fn id=\"conflict-1\" fn-type=\"COI-statement\"><p>The authors declare there are no competing interests.</p></fn></fn-group>", "<fn-group content-type=\"author-contributions\"><title>Author Contributions</title><fn id=\"contribution-1\" fn-type=\"con\"><p><xref rid=\"author-1\" ref-type=\"contrib\">Aidyn Orazov</xref> conceived and designed the experiments, performed the experiments, analyzed the data, prepared figures and/or tables, getting the main results, and approved the final draft.</p></fn><fn id=\"contribution-2\" fn-type=\"con\"><p><xref rid=\"author-2\" ref-type=\"contrib\">Moldir Yermagambetova</xref> performed the experiments, analyzed the data, prepared figures and/or tables, getting the main results, and approved the final draft.</p></fn><fn id=\"contribution-3\" fn-type=\"con\"><p><xref rid=\"author-3\" ref-type=\"contrib\">Anar Myrzagaliyeva</xref> analyzed the data, authored or reviewed drafts of the article, scientific advice, and approved the final draft.</p></fn><fn id=\"contribution-4\" fn-type=\"con\"><p><xref rid=\"author-4\" ref-type=\"contrib\">Nashtay Mukhitdinov</xref> conceived and designed the experiments, authored or reviewed drafts of the article, scientific advice, and approved the final draft.</p></fn><fn id=\"contribution-5\" fn-type=\"con\"><p><xref rid=\"author-5\" ref-type=\"contrib\">Shynar Tustubayeva</xref> analyzed the data, prepared figures and/or tables, getting the main results, and approved the final draft.</p></fn><fn id=\"contribution-6\" fn-type=\"con\"><p><xref rid=\"author-6\" ref-type=\"contrib\">Yerlan Turuspekov</xref> conceived and designed the experiments, analyzed the data, authored or reviewed drafts of the article, getting the main results, and approved the final draft.</p></fn><fn id=\"contribution-7\" fn-type=\"con\"><p><xref rid=\"author-7\" ref-type=\"contrib\">Shyryn Almerekova</xref> conceived and designed the experiments, performed the experiments, analyzed the data, prepared figures and/or tables, getting the main results, and approved the final draft.</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Data Availability</title><fn id=\"addinfo-1\"><p>The following information was supplied regarding data availability:</p><p>Additional genetic and general information about populations are available in the <xref rid=\"supplemental-information\" ref-type=\"sec\">Supplemental Files</xref>.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"peerj-12-16735-s001.docx\"><caption><p>Click here for additional data file.</p></caption></media>", "<media xlink:href=\"peerj-12-16735-s002.xlsx\"><caption><p>Click here for additional data file.</p></caption></media>" ]

{ "acronym": [ "n", " Na", " Ne", " uHe", " Fst", "HO", "HE", " FIS", " Rp", " MI" ], "definition": [ "number of plants studied in the population", "number of alleles per locus", "effective number of alleles", "Unbiased Expected Heterozygosity", "fixation index", "Observed heterozygosity", "Expected heterozygosity", "Fixation index", "Resolving power", "Marker index" ] }

[ "<title>Introduction</title>", "<p>Faecal indicator bacteria (FIB) are a group of bacteria used to evaluate water faecal contamination. Ideally, FIB should be of faecal origin only and not grow in the extraintestinal environment (##REF##25941519##Rochelle-Newall et al., 2015##). Furthermore, the abundance of FIB should correlate with the presence of faecal contamination-related pathogen. Compared with direct detection of these pathogens, FIB are more abundant in the water and thus easier to detect (##UREF##23##Tortora, Funke &amp; Case, 2013##).</p>", "<p>Globally, <italic toggle=\"yes\">Escherichia coli</italic> has been used as a FIB since the last century (##UREF##24##USEPA, 1986##). Due to its wide application, extensive studies have been done on its survival in water. The survival of <italic toggle=\"yes\">E. coli</italic> in aquatic habitats is affected by both biotic and abiotic factors (##REF##28383815##Jang et al., 2017##). For example, biotic factors include biofilm formation and the presence of other microorganisms (##REF##24747902##Korajkic et al., 2014##; ##UREF##21##Stocker et al., 2019##), whereas abiotic factors include temperature, pH, salinity, sunlight and nutrient availability (##UREF##18##Petersen &amp; Hubbart, 2020##; ##REF##37010795##Moon et al., 2023##). Therefore, seasonal variations with changes in temperature, precipitation and anthropogenic activity could also affect <italic toggle=\"yes\">E. coli</italic> abundance and their survival. ##REF##12442800##An, Kampbell &amp; Peter Breidenbach (2002)## reported lowest <italic toggle=\"yes\">E. coli</italic> density in summer and attributed this to the lower loading of faecal material, more vigorous grazing, and poor survival of <italic toggle=\"yes\">E. coli</italic> in warm water. However, ##REF##27959871##Durham et al. (2016)## reported highest <italic toggle=\"yes\">E. coli</italic> abundance in summer, suggesting that site-specific factors are also relevant.</p>", "<p>Nevertheless, there remain doubts about <italic toggle=\"yes\">E. coli</italic>’s reliability as a FIB, as studies have revealed that sediments are an environmental reservoir of <italic toggle=\"yes\">E. coli</italic> in freshwater habitats (##REF##16391098##Ishii et al., 2006##; ##REF##21558695##Ishii &amp; Sadowsky, 2008##; ##UREF##2##Cho et al., 2010##; ##REF##20219232##Garzio-Hadzick et al., 2010##; ##REF##25816691##Tymensen et al., 2015##; ##UREF##7##Fluke, González-Pinzón &amp; Thomson, 2019##). Relative to the water column, sediments generally have higher nutrient levels, lower dissolved oxygen, and lower UV intensity, which helps <italic toggle=\"yes\">E. coli</italic> survive in sediments (##REF##15758111##Jamieson et al., 2005##; ##REF##18574188##Koirala et al., 2008##; ##UREF##13##Lorke &amp; MacIntyre, 2009##; ##REF##25941519##Rochelle-Newall et al., 2015##). Studies have also reported that habitat transition of sediment <italic toggle=\"yes\">E. coli</italic> to the water column, increases <italic toggle=\"yes\">E. coli</italic> abundance in the water; for example, during resuspension of sediment by mechanical effects like precipitation or water flow (##REF##16980417##Whitman, Nevers &amp; Byappanahalli, 2006##; ##UREF##2##Cho et al., 2010##; ##REF##28295001##Abia et al., 2017##). Apart from resuspension, habitat transition should theoretically also occur as <italic toggle=\"yes\">E. coli</italic> grows in the sediments (##REF##16391098##Ishii et al., 2006##). For instance, <italic toggle=\"yes\">E. coli</italic> that thrives on sediment biofilms can be released into the water due to biofilm sloughing (##REF##29358142##Mackowiak et al., 2018##).</p>", "<p>Previous habitat transition studies focused more on sediment resuspension induced by mechanical effects. These mechanical effects included anthropogenic vessel activity and precipitation caused by seasonal variation. ##REF##12442800##An, Kampbell &amp; Peter Breidenbach (2002)## revealed the resuspension of sediment caused by motorboat leads to water quality deterioration. Precipitation can also cause the resuspension of sediment, causing <italic toggle=\"yes\">E. coli</italic> habitat transition from sediment to the upper water column (##REF##37149158##Li, Filippelli &amp; Wang, 2023##). However, increase in <italic toggle=\"yes\">E. coli</italic> due to sediment resuspension will quickly return to pre-resuspension concentrations (##REF##16980417##Whitman, Nevers &amp; Byappanahalli, 2006##; ##REF##28295001##Abia et al., 2017##). Besides that, our literature review revealed no report that measured sediment <italic toggle=\"yes\">E. coli</italic> habitat transition rates to the overlying waters.</p>", "<p>As the habitat transition rate could be an important process that contributes to <italic toggle=\"yes\">E. coli</italic> prevalence in the waters, we designed experiments to measure the habitat transition rate of <italic toggle=\"yes\">E. coli</italic> in sediment samples from lakes. In this study, five tropical urban lake waters were selected, as lake waters are generally more static and have less sediment resuspension (##REF##29571392##Lim et al., 2018##; ##UREF##1##Bong et al., 2020##). The absence of mechanical effects in the lake waters will help clarify the role of <italic toggle=\"yes\">E. coli</italic> habitat transition. Since the abundance of <italic toggle=\"yes\">E. coli</italic> in the upper water column is also affected by its growth or decay, we concurrently carried out habitat transition experiments with size-fractionation decay experiments according to ##REF##21146847##Lee et al. (2011)##. Our results helped shed light on the possible reasons for the persistence of <italic toggle=\"yes\">E. coli</italic> in urban tropical lakes as shown earlier by ##REF##36327936##Wong et al. (2022)##, and could help improve the current water surveillance strategies.</p>" ]

[ "<title>Materials &amp; Methods</title>", "<title>Study sites and environmental variables</title>", "<p>A total of 35 water and 21 sediment samples were collected regularly at five independent urban lakes (Tasik Varsiti, Tasik Taman Jaya, Tasik Aman, Tasik Kelana and Tasik Central Park Bandar Utama), located 2–7 km between each other in the Klang Valley, Peninsular Malaysia, from May 2022 until November 2022 (##FIG##0##Fig. 1##). The sampling dates with respective coordinates and experiments conducted for each sampling are listed in ##SUPPL##0##Table S1##. To avoid effects of precipitation, sampling was carried out when there was no rain. Surface water samples (≈ 0.1 m) were collected using autoclaved bottles (121 °C at 15 psi for 15 min) whereas surface sediment samples (≈ three cm depth) were taken with a shovel and collected using UV sterilized (at 245 nm wavelength for 20 min, intensity 550 µW cm⁻²) plastic zip lock bags. All samples were transferred on ice to the laboratory within 3 h for further analysis.</p>", "<p>A conductivity probe (YSI Pro 30, Yellow Springs, OH, USA) and a pH meter (Hach HQ11d, Loveland, CO, USA) were used to measure <italic toggle=\"yes\">in-situ</italic> water temperature and pH, respectively. For dissolved oxygen (DO), water samples were collected with DO bottles in triplicates, and fixed with manganese chloride and alkaline iodide solution, before titration with sodium thiosulphate solution according to Winkler’s method (##UREF##9##Grasshoff, Kremling &amp; Ehrhardt, 1999##). Total suspended solids (TSS) was determined by filtering a known volume of water sample through a pre-combusted glass fibre filter (GF/F) (Sartorius, Goettingen, Germany) and measuring the weight increase after drying at 70 °C for a week. Particulate organic matter (POM) was determined by the weight loss after combustion at 500 °C for 2 h (HYSC MF-05, Seoul, Korea). Chlorophyll <italic toggle=\"yes\">a</italic> (Chl <italic toggle=\"yes\">a</italic>) was also concentrated on the GF/F filter and extracted with 90% (v/v) ice-cold acetone at −20 °C overnight. Chl <italic toggle=\"yes\">a</italic> concentration was then measured <italic toggle=\"yes\">via</italic> a spectrofluorometer (PerkinElmer LS55, Waltham, MA, USA) (##UREF##17##Parsons, Maita &amp; Lalli, 1984##). The filtrate from the filtration was kept frozen until the determination of ammonium (NH₄) and phosphate (PO₄). These dissolved inorganic nutrients were determined on a spectrophotometer (Hitachi U-1900, Tokyo, Japan) <italic toggle=\"yes\">via</italic> methods described by ##UREF##17##Parsons, Maita &amp; Lalli (1984)##.</p>", "<p>The sediment sample collected was dried in a freeze dryer (Labconco FreeZone 6 Liter, Kansas City, MO, USA). For sediment particle sizing, about 10 cm³ of dried sediments were mixed with distilled water until a final volume of 40 mL. Then 10 mL of sodium hexametaphosphate (20% final concentration) was added to disperse the sediment particles (##UREF##16##Mil-Homens et al., 2006##). The prepared sample was then homogenized and left overnight before analysis with the Beckman Coulter LS230 Particle Size Analyzer (Brea, CA, USA). For sediment organic matter content, the freeze-dried sediment was combusted at 500 °C for 3 h, and the organic matter content was measured <italic toggle=\"yes\">via</italic> the loss on ignition method (##UREF##11##Heiri, Lotter &amp; Lemcke, 2001##).</p>", "<title>Enumeration of coliform and <italic toggle=\"yes\">E. coli</italic> in water and sediment samples</title>", "<p>For water samples, both coliform and <italic toggle=\"yes\">E. coli</italic> were measured whereas for sediment samples, only <italic toggle=\"yes\">E. coli</italic> was measured. The additional coliform measurement in the water samples helped in the classification of the lake waters according to the National Water Quality Standards for Malaysia (##UREF##4##Department of Environment, 2008##). Membrane filter technique (MFT) was used to enumerate both coliform and <italic toggle=\"yes\">E. coli</italic> in water where a known volume of water sample (0.01 mL to 10 mL) was filtered through a sterile 47 mm diameter, 0.45 µm pore-size nitrocellulose membrane filter (Millipore, Burlington, MA, USA). For volumes &lt;1.0 mL, the filtration vessel was filled with 5 mL sterile saline (0.85% sodium chloride (NaCl) final concentration) before addition of sample. After filtration, the membrane filter was placed on the CHROMagar™ ECC agar (CHROMagar, Paris, France) and incubated at 37 °C for 24 h. All blue and mauve-coloured colonies were counted as total coliform, whereas only blue colonies were counted as <italic toggle=\"yes\">E. coli</italic> (##UREF##3##Chromagar, 2019##).</p>", "<p>For sediment samples, 2 g of fresh sediment sample was mixed with 18 mL of sterile saline and then sonicated for 50 s with an ultrasonicator (220 W, 2 mm probe; SASTEC ST-JY98-IIIN, Subang Jaya, Malaysia) (##UREF##5##Epstein &amp; Rossel, 1995##). After allowing the mixture to settle for 10 min, the suspension was pipetted and used as inoculum in the MFT described above. The membrane filter was then placed on m-TEC agar (Sigma-Aldrich, Burlington, MA, USA) and incubated at 44.5 °C for 24 h. Purple- or magenta-coloured colonies were counted as <italic toggle=\"yes\">E. coli</italic> (##UREF##15##Merck KGaA, 2018##).</p>", "<title>Measuring <italic toggle=\"yes\">E. coli</italic> decay or growth rates</title>", "<p>Using the size fractionation method, the water sample was divided into three size fractions: total or unfiltered, &lt;20 µm and &lt;0.2 µm fractions (##REF##21146847##Lee et al., 2011##). The &lt;20 µm fraction was collected after filtration through a nylon net with a 20 µm mesh opening size, whereas the &lt;0.2 µm fraction was collected after filtration with a 0.2 µm pore-size membrane filter (Millipore GTTP filter, Burlington, MA, USA).</p>", "<p>As <italic toggle=\"yes\">E. coli</italic> counts in the water was sometimes too low, we used a laboratory strain of <italic toggle=\"yes\">E. coli</italic> (isolated from Tasik Varsiti) for the decay or growth experiment. A fresh <italic toggle=\"yes\">E. coli</italic> culture was adjusted to 0.5 McFarland standard (about 1.5 ×10⁸ cfu mL⁻¹) before further serial dilution to 10⁵ cfu mL⁻¹. About 198 mL of each size fraction was then inoculated with 2 mL of 10⁵ cfu mL⁻¹\n<italic toggle=\"yes\">E. coli</italic> culture for a final concentration of about 10³ cfu mL⁻¹. Inoculated size fractions were then incubated at 30 ° C for 24 h in the dark. The abundance of <italic toggle=\"yes\">E. coli</italic> was determined as cfu mL⁻¹ every 6 h through MFT on m-TEC agar. The cfu data was then transformed <italic toggle=\"yes\">via</italic> natural logarithm and plotted against incubation time. A positive gradient of the best-fit regression line indicates <italic toggle=\"yes\">E. coli</italic> growth whereas a negative gradient shows decay rate (##REF##21146847##Lee et al., 2011##).</p>", "<p>As protists are the major bacterial predators (##REF##776086##Enzinger &amp; Cooper, 1976##), we also enumerated protists (##REF##16346372##Caron, 1983##). A 50 mL water sample was preserved with glutaraldehyde (1% final concentration) during each sampling. At the laboratory, 1 to 2 mL preserved sample was filtered onto a black 0.8 µm polycarbonate filter (Millipore ATTP filter, Burlington, MA, USA) with a GF/A filter (Whatman, Little Chalfont, UK) as a backing filter. Filters were then rinsed twice with 0.1 M pH 4.0 Trizma-hydrochloride before being flooded with two mL of primulin solution (250 mg L⁻¹) for 15 mins. After staining, the solution was removed gently by vacuum filtration. The black filter was then placed on one drop of immersion oil on a clean glass slide, and the prepared slide was observed under an epifluorescence microscope (Olympus BX60F-3, Tokyo, Japan) with U-MWU filter cassette (excitor 330–385 nm, dichroic mirror 400 nm, barrier 420 nm).</p>", "<title>Habitat transition experiment for <italic toggle=\"yes\">E. coli</italic></title>", "<p>We added 2 g of fresh sediment sample to the bottom of an autoclaved universal bottle (121 °C at 15 psi for 15 min), carefully avoiding any contact with the inner wall of the bottle. Then 0.4% (w/v) sterile soft agar (Difco, East Rutherford, NJ, USA) (kept at 45 °C) was added to the bottle until it covers approximately one cm level above the sediment. After the agar solidified, 18 mL of sterile saline was added slowly (##FIG##1##Fig. 2##). To check for contamination, a blank without addition of sediment sample was also carried out. The habitat transition experiment was then incubated at 30 °C for 24 h in dark. The abundance of <italic toggle=\"yes\">E. coli</italic> in the overlying saline was enumerated every 6 h <italic toggle=\"yes\">via</italic> MFT on m-TEC agar. The cfu data was then natural logarithm transformed and plotted against incubation time where the gradient of the best-fit regression line was determined as <italic toggle=\"yes\">E. coli</italic> increase rate (µ_increase).</p>", "<p>As <italic toggle=\"yes\">E. coli</italic> also undergoes intrinsic growth in the saline environment (##UREF##12##Hrenović &amp; Ivanković, 2009##), we setup a microcosm experiment by replacing raw sediment sample with autoclaved sediment sample from Tasik Kelana (<italic toggle=\"yes\">n</italic> = 2) and Tasik Central Park Bandar Utama (<italic toggle=\"yes\">n</italic> = 2). The sediment was autoclaved to replicate possible nutrient contribution from the sediment but prevent adding bacteria to the microcosm. We then inoculated 10³ cfu mL⁻¹ of <italic toggle=\"yes\">E. coli</italic> to the sterile saline. The microcosm was then incubated at 30 °C for 24 h in the dark and the abundance of <italic toggle=\"yes\">E. coli</italic> in the saline was enumerated every 6 h <italic toggle=\"yes\">via</italic> MFT on m-TEC. The best-fit linear slope was determined as <italic toggle=\"yes\">E. coli</italic> intrinsic growth (µ_intrinsic), and the habitat transition rate was finally estimated by the following equation: µ_increase–µ_intrinsic.</p>", "<title>Data analysis</title>", "<p>All data were reported in this study as mean ± standard deviation (SD) unless stated otherwise. Values beyond mean ± 2 × SD were determined as outliers, and the coefficient of variation (<italic toggle=\"yes\">CV</italic>) was used to measure the dispersion of data. Before statistical analysis, bacterial cfus were transformed by log (cfu + 1), whereas for growth or decay rate estimations, bacterial cfus were transformed <italic toggle=\"yes\">via</italic> natural logarithm. Correlation analysis was carried out to identify relationships among variables, whereas linear regression was used for rate analysis. Student’s <italic toggle=\"yes\">t</italic>-test was used to compare between groups, whereas one-way ANOVA (analysis of variance) with Tukey’s <italic toggle=\"yes\">post-hoc</italic> analysis was used to determine the differences among lakes, and <italic toggle=\"yes\">p</italic> ≤ 0.05 was considered significant. PAST (PAleontological STatistics) software (version 4.09) for Windows (##UREF##10##Hammer, Harpper &amp; Ryan, 2001##) was used to perform the statistical analyses, whereas plots were made in GraphPad Prism (version 9.5.1.733) for Windows (##UREF##22##Swift, 1997##).</p>" ]

[ "<title>Results</title>", "<title>Environmental variables</title>", "<p>##TAB##0##Table 1## lists the size and land use of five lakes, and physico-chemical variables measured in the water samples collected at the five lakes in this study. Surface water temperature and pH varied little among the five lakes and ranged from 28.7 ± 0.8 °C to 29.9 ± 1.2 °C (<italic toggle=\"yes\">CV</italic> = 3%) and 6.8 ± 0.4 to 7.2 ± 0.3 (<italic toggle=\"yes\">CV</italic> = 5%), respectively. In contrast, DO levels varied among the five lakes, from 2.65 ± 1.78 mg L⁻¹ at Tasik Taman Jaya to 9.65 ± 2.00 mg L⁻¹ at Tasik Central Park Bandar Utama (ANOVA: <italic toggle=\"yes\">n</italic> = 18, <italic toggle=\"yes\">F</italic> (4,13) = 3.08, <italic toggle=\"yes\">p</italic> = 0.05).</p>", "<p>In contrast, TSS and POM concentrations were different among the lakes, and ranged from 21 ± 7 mg L⁻¹ to 65 ± 13 mg L⁻¹ (ANOVA: <italic toggle=\"yes\">n</italic> = 18, <italic toggle=\"yes\">F</italic> (4,13) = 14.06, <italic toggle=\"yes\">p</italic> &lt; 0.001), and from 13 ± 2 mg L⁻¹ to 43 ± 5 mg L⁻¹ (ANOVA: <italic toggle=\"yes\">n</italic> = 18, <italic toggle=\"yes\">F</italic> (4,13) = 34.8, <italic toggle=\"yes\">p</italic> &lt; 0.001), respectively. TSS and POM concentrations were highest at Tasik Central Park Bandar Utama (Tukey’s HSD: TSS: <italic toggle=\"yes\">q</italic> &gt; 6.65, <italic toggle=\"yes\">p</italic> &lt; 0.01; POM: <italic toggle=\"yes\">q</italic> &gt; 6.41, <italic toggle=\"yes\">p</italic> &lt; 0.01). Chl <italic toggle=\"yes\">a</italic> concentration also varied among lakes (ANOVA: <italic toggle=\"yes\">n</italic> = 18, <italic toggle=\"yes\">F</italic> (4,13) = 14.14, <italic toggle=\"yes\">p</italic> &lt; 0.001), and was highest at Tasik Aman (90.63 ± 15.14 µg L⁻¹) (Tukey’s HSD: <italic toggle=\"yes\">q</italic> &gt; 5.06, <italic toggle=\"yes\">p</italic> &lt;0.03). For dissolved inorganic nutrients, NH₄ varied among five lakes (ANOVA: <italic toggle=\"yes\">n</italic> = 18, <italic toggle=\"yes\">F</italic> (4,13) = 16.85, <italic toggle=\"yes\">p</italic> &lt; 0.001), ranged from 0.30 to 98.83 µM and was highest at Tasik Taman Jaya (Tukey’s HSD: <italic toggle=\"yes\">q</italic> &gt; 4.70, <italic toggle=\"yes\">p</italic> &lt; 0.04), whereas PO₄was similar among the lakes (ANOVA: <italic toggle=\"yes\">n</italic> = 18, <italic toggle=\"yes\">F</italic> (4,13) = 1.80, <italic toggle=\"yes\">p</italic> = 0.19), and varied from 0.15 to 0.60 µM.</p>", "<p>For the physico-chemical properties of sediments (##TAB##1##Table 2##), average particle size ranged from 55.2 ± 26.2 to 613.4 ± 124.2 µm, and were different among the five lakes (ANOVA: <italic toggle=\"yes\">n</italic> = 18, <italic toggle=\"yes\">F</italic> (4,13) = 21.62, <italic toggle=\"yes\">p</italic> &lt; 0.001) with the largest average particle size at Tasik Kelana (Tukey’s HSD: <italic toggle=\"yes\">q</italic> &gt; 5.52, <italic toggle=\"yes\">p</italic> &lt; 0.02). The sediment texture at Tasik Varsiti was mainly loam, whereas in other lakes were mainly sand. Average sediment organic matter measured ranged from 3 to 72 mg g⁻¹ and was not different among the five lakes (ANOVA: <italic toggle=\"yes\">n</italic> = 18, <italic toggle=\"yes\">F</italic> (4,13) = 2.96, <italic toggle=\"yes\">p</italic> = 0.06).</p>", "<title>Biotic variables</title>", "<p>Total coliform and <italic toggle=\"yes\">E. coli</italic> were detected in all five urban lakes (##FIG##2##Fig. 3##, ##SUPPL##1##Table S2##). Total coliform ranged from 21 to 4,600 cfu mL⁻¹, and <italic toggle=\"yes\">E. coli</italic> ranged from 1 to 2,300 cfu mL⁻¹. Total coliform in the water was different among the five lakes (ANOVA: <italic toggle=\"yes\">n</italic> = 20, <italic toggle=\"yes\">F</italic> (4,15) = 3.58, <italic toggle=\"yes\">p</italic> = 0.03) but <italic toggle=\"yes\">E. coli</italic> count was not different (ANOVA: <italic toggle=\"yes\">n</italic> = 20, <italic toggle=\"yes\">F</italic> (4,15) = 2.52, <italic toggle=\"yes\">p</italic> = 0.09). The highest total coliform was detected at Tasik Kelana (Tukey’s HSD: <italic toggle=\"yes\">q</italic> = 4.97, <italic toggle=\"yes\">p</italic> = 0.02). For the urban lake sediments, <italic toggle=\"yes\">E. coli</italic> was present in all five lake sediments (##FIG##3##Fig. 4##). Its abundance ranged from below detection to 12,000 cfu g⁻¹, and there was no difference among the five lakes (ANOVA: <italic toggle=\"yes\">n</italic> = 20, <italic toggle=\"yes\">F</italic> (4,15) = 2.69, <italic toggle=\"yes\">p</italic> = 0.07).</p>", "<title><italic toggle=\"yes\">E. coli</italic> decay or growth rates</title>", "<p>Generally, the abundance of <italic toggle=\"yes\">E. coli</italic> in the larger fractions (total and &lt;20 µm fraction) decreased with incubation time, while the &lt;0.2 µm fraction increased (##FIG##4##Figs. 5## and ##FIG##5##6##, ##SUPPL##2##Table S3##). Decay rates among the five lakes in the total fraction (ANOVA: <italic toggle=\"yes\">n</italic> = 19, <italic toggle=\"yes\">F</italic> (4,14) = 7.85, <italic toggle=\"yes\">p</italic> &lt; 0.01) and in the &lt;20 µm fraction (ANOVA: <italic toggle=\"yes\">n</italic> = 18, <italic toggle=\"yes\">F</italic> (4,13) = 4.89, <italic toggle=\"yes\">p</italic> = 0.01) were different (##FIG##6##Fig. 7##, ##SUPPL##3##Table S4##). The highest decay rates in both the total fraction (Tukey’s HSD: <italic toggle=\"yes\">q</italic> &gt; 4.65, <italic toggle=\"yes\">p</italic> &lt; 0.04) and &lt;20 µm fraction (Tukey’s HSD: <italic toggle=\"yes\">q</italic> = 5.77, <italic toggle=\"yes\">p</italic> = 0.01) were observed at Tasik Taman Jaya. The decay rates measured in the total fraction also did not differ from those in the &lt;20 µm fraction (Student’s <italic toggle=\"yes\">t</italic>-test: <italic toggle=\"yes\">n</italic> = 37, <italic toggle=\"yes\">t</italic> (35) = 0.43, <italic toggle=\"yes\">p</italic> = 0.67). As decay was most likely attributed to protistan grazers (##REF##21146847##Lee et al., 2011##), we measured protists abundance in the water samples, and observed that protists counts ranged from 3.04 ×10⁴ cells mL⁻¹ to 6.93 ×10⁴ cells mL⁻¹ but showed no differences among the five lakes (ANOVA: <italic toggle=\"yes\">n</italic> = 15, <italic toggle=\"yes\">F</italic> (4,10) = 2.18, <italic toggle=\"yes\">p</italic> = 0.14). In contrast, <italic toggle=\"yes\">E. coli</italic> grew in the &lt;0.2 µm fraction, and the <italic toggle=\"yes\">E. coli</italic> growth rates varied among five lakes (ANOVA: <italic toggle=\"yes\">n</italic> = 18, <italic toggle=\"yes\">F</italic> (4,13) = 3.65, <italic toggle=\"yes\">p</italic> = 0.03).</p>", "<title>Habitat transition experiment for <italic toggle=\"yes\">E. coli</italic></title>", "<p>In the habitat transition experiment, <italic toggle=\"yes\">E. coli</italic> abundance generally increased with time (##FIG##7##Fig. 8##, ##SUPPL##4##Table S5##). <italic toggle=\"yes\">E. coli</italic> increase rates (µ_increase) in the water column (<italic toggle=\"yes\">p</italic> &lt; 0.05) ranged from 0.40 to 0.59 h⁻¹ at Tasik Varsiti, 0.46 to 0.62 h⁻¹ at Tasik Taman Jaya, 0.41 to 0.74 h⁻¹ at Tasik Aman, 0.61 to 0.71 h⁻¹ at Tasik Kelana and 0.69 to 0.78 h⁻¹ at Tasik Central Park Bandar Utama (##SUPPL##5##Table S6##).</p>", "<p>As the <italic toggle=\"yes\">E. coli</italic> increase rate is a sum of both transition and intrinsic growth, we also measured <italic toggle=\"yes\">E. coli</italic> intrinsic growth rates (##SUPPL##6##Table S7##). The intrinsic growth rates using sterile sediments from Tasik Kelana were 0.39 h⁻¹ and 0.32 h⁻¹, and were similar to Tasik Central Park Bandar Utama (0.41 h ⁻¹ and 0.36 h⁻¹). Although sediments at Tasik Kelana had the lowest organic matter content (3 to 8 mg g ⁻¹), whereas Tasik Central Park Bandar Utama had the highest (19 to 72 mg g⁻¹) among the five lakes, their <italic toggle=\"yes\">E. coli</italic> intrinsic growth rates were not different (ANOVA: <italic toggle=\"yes\">n</italic> = 4, <italic toggle=\"yes\">F</italic> (1,2) = 0.48, <italic toggle=\"yes\">p</italic> = 0.56). Therefore, for the calculation of habitat transition rates, we assumed the average intrinsic growth rate (0.37 ± 0.04 h⁻¹) for all five lakes (##FIG##8##Fig. 9##, ##SUPPL##7##Table S8##). The habitat transition rates were different among five lakes (ANOVA: <italic toggle=\"yes\">n</italic> = 18, <italic toggle=\"yes\">F</italic> (4,13) = 4.01, <italic toggle=\"yes\">p</italic> = 0.02), with the highest at Tasik Central Park Bandar Utama (Tukey’s HSD: <italic toggle=\"yes\">q</italic> = 4.67, <italic toggle=\"yes\">p</italic> = 0.04).</p>" ]

[ "<title>Discussion</title>", "<title>Environmental condition of the urban lakes</title>", "<p>The surface water temperatures recorded at the five lakes were relatively high with low variability, and is typical of tropical waters (##REF##29571392##Lim et al., 2018##). The DO concentrations measured at Tasik Varsiti, Tasik Aman, Tasik Kelana and Tasik Central Park Bandar Utama were at healthy levels, and within the range previously reported for tropical freshwater (##REF##36327936##Wong et al., 2022##). However, for Tasik Taman Jaya, we observed hypoxic levels (2.65 ± 1.78 mg L⁻¹) (##UREF##6##Farrell &amp; Richards, 2009##), which was not surprising as ##REF##36327936##Wong et al. (2022)## had previously classified Tasik Taman Jaya at Class III for total coliform and Class V for faecal coliform, indicative of extensive treatment required for the suitability of water supply (##UREF##4##Department of Environment , 2008##). All lakes were also observed with high Chl <italic toggle=\"yes\">a</italic>, indicating varying levels of eutrophication (##REF##29571392##Lim et al., 2018##).</p>", "<title>Total coliform and <italic toggle=\"yes\">E. coli</italic> in water and sediment of urban lakes</title>", "<p>The total coliform abundance enumerated in five lake waters were within the range previously reported in Malaysia (##REF##36327936##Wong et al., 2022##). According to National Water Quality Standards for Malaysia, Tasik Varsiti, Tasik Taman Jaya and Tasik Kelana were categorized as Class V for total coliform, whereas Tasik Aman and Tasik Central Park Bandar Utama as Class III. Relative to ##REF##36327936##Wong et al. (2022)##, the water quality in these lakes had deteriorated over the last two years.</p>", "<p>In our study, the abundance of <italic toggle=\"yes\">E. coli</italic> in sediment and water were correlated (Pearson correlation: <italic toggle=\"yes\">n</italic> = 40, <italic toggle=\"yes\">r</italic> (38) = 0.53, <italic toggle=\"yes\">p</italic> = 0.02) (##FIG##9##Fig. 10##), similar to ##REF##16980417##Whitman, Nevers &amp; Byappanahalli (2006)## and ##UREF##7##Fluke, González-Pinzón &amp; Thomson (2019)##. Although we did not use the same agar medium to enumerate <italic toggle=\"yes\">E. coli</italic> abundance in water and sediment, we had previously shown that <italic toggle=\"yes\">E. coli</italic> counts on m-TEC and CHROMagar ECC agar were strongly correlated (Regression: <italic toggle=\"yes\">n</italic> = 18, <italic toggle=\"yes\">R</italic>² = 0.99, <italic toggle=\"yes\">F</italic> (1,7) = 561.35, <italic toggle=\"yes\">p</italic> &lt; 0.001) (##FIG##10##Fig. 11##, ##SUPPL##8##Table S9##), and that the abundance of <italic toggle=\"yes\">E. coli</italic> (log cfu mL⁻¹) obtained on m-TEC was on average 149% higher than that on CHROMagar ECC. In order to compare <italic toggle=\"yes\">E. coli</italic> abundance in water and sediment, we corrected the abundance obtained, and found that the abundance of <italic toggle=\"yes\">E. coli</italic> in the sediment was still higher than in the water column (##UREF##20##Stephenson &amp; Rychert, 1982##; ##REF##12442800##An, Kampbell &amp; Peter Breidenbach, 2002##; ##REF##20219232##Garzio-Hadzick et al., 2010##; ##REF##30272806##Pandey et al., 2018##; ##UREF##7##Fluke, González-Pinzón &amp; Thomson, 2019##).</p>", "<p>Relative to the water column, sediment provides <italic toggle=\"yes\">E. coli</italic> with more nutrients (##REF##15758111##Jamieson et al., 2005##); lower UV intensity (##REF##18574188##Koirala et al., 2008##); lesser bacterivore grazing (##REF##16535141##Wright et al., 1995##) and lower oxygen (##UREF##13##Lorke &amp; MacIntyre, 2009##). This host intestinal-like environment helps <italic toggle=\"yes\">E. coli</italic> to survive better in sediments, even at varying climates (##REF##16391098##Ishii et al., 2006##; ##REF##21558695##Ishii &amp; Sadowsky, 2008##; ##REF##20219232##Garzio-Hadzick et al., 2010##; ##REF##25941519##Rochelle-Newall et al., 2015##; ##REF##25816691##Tymensen et al., 2015##; ##UREF##7##Fluke, González-Pinzón &amp; Thomson, 2019##).</p>", "<p>As <italic toggle=\"yes\">E. coli</italic> is dependent upon sediment organic matter for growth (##REF##16391098##Ishii et al., 2006##), higher abundance of <italic toggle=\"yes\">E. coli</italic> in sediments with higher organic matter has been reported (##REF##16793111##Lee et al., 2006##). However in our study, sediment organic matter and sediment <italic toggle=\"yes\">E. coli</italic> abundance were not correlated (Pearson correlation: <italic toggle=\"yes\">n</italic> = 36, <italic toggle=\"yes\">r</italic> (34) = −0.11, <italic toggle=\"yes\">p</italic> = 0.66). One possible reason could be the organic matter replete state in our lakes. The sediment organic matter from this study was relatively higher (Organic matter content: Tasik Varsiti 3%, Tasik Taman Jaya 3.3%, Tasik Aman 2.7%, Tasik Kelana 0.5% and Tasik Central Park Bandar Utama 3.3%), than that reported by ##REF##16793111##Lee et al. (2006)## (<italic toggle=\"yes\">i.e.,</italic> 0.7%–1.1%). In this study, the sediment <italic toggle=\"yes\">E. coli</italic> abundance also did not correlate with particle size (Pearson correlation: <italic toggle=\"yes\">n</italic> = 36, <italic toggle=\"yes\">r</italic> (34) = −0.07, <italic toggle=\"yes\">p</italic> = 0.78), further substantiating that sediment texture may be less important for the survival of sediment <italic toggle=\"yes\">E. coli</italic> (##REF##16793111##Lee et al., 2006##).</p>", "<p>Although sediments could act as reservoirs of <italic toggle=\"yes\">E. coli</italic>, and contribute to the water column <italic toggle=\"yes\">E. coli</italic>, <italic toggle=\"yes\">E. coli</italic> dynamics in the water column is also dependent upon <italic toggle=\"yes\">E. coli</italic> decay or growth rates in the water (##REF##21146847##Lee et al., 2011##). The decay rate in total and &lt;20 µm fractions were higher than the smaller fraction and consistent with previous reports (##REF##21146847##Lee et al., 2011##; ##REF##36327936##Wong et al., 2022##). In the total fraction, the <italic toggle=\"yes\">E. coli</italic> decay rates ranged from 0.02 to 0.16 h⁻¹ (or 0.50 to 3.94 d⁻¹). These decay rates were generally within the range reported by ##REF##3323155##Flint (1987)## measured at 37  °C, and higher than in subtropical water (##UREF##0##Bitton et al., 1983##). Temperature may explain the generally higher rates observed in this study, as microbial activity is at its optimum in tropical aquatic habitats (##REF##24194204##White et al., 1991##).</p>", "<p>In our study, the decay of <italic toggle=\"yes\">E. coli</italic> in the larger fraction is mainly due to protistan bacterivory (##REF##776086##Enzinger &amp; Cooper, 1976##; ##REF##21146847##Lee et al., 2011##; ##REF##36327936##Wong et al., 2022##). However, we found no correlation between decay rate and protist counts (Pearson correlation: <italic toggle=\"yes\">n</italic> = 10, <italic toggle=\"yes\">r</italic> (8) = −0.3, <italic toggle=\"yes\">p</italic> = 0.62). As <italic toggle=\"yes\">E. coli</italic> only accounts for small fraction of the total bacterial community, at about 4.48% of total culturable gram-negative rod in freshwater (##REF##10907419##Goñi Urriza et al., 1999##), this could explain the uncoupling between protists and <italic toggle=\"yes\">E. coli</italic> decay rates. ##REF##21146847##Lee et al. (2011)## have also reported that <italic toggle=\"yes\">E. coli</italic> decay rates have a relatively small impact on the overall bacterivory rate. Although viral lysis could also cause <italic toggle=\"yes\">E. coli</italic> mortality, previous studies have shown that its role is generally minimal (##REF##21146847##Lee et al., 2011##). Moreover in the &lt;0.2 µm fraction, where protists were removed, <italic toggle=\"yes\">E. coli</italic> did not decrease but increased against incubation time, suggesting that viral lysis was not significant (##REF##21146847##Lee et al., 2011##).</p>", "<title>The role of habitat transition rates for <italic toggle=\"yes\">E. coli</italic> persistence in the water column</title>", "<p>Habitat transition experiments in this study showed that <italic toggle=\"yes\">E. coli</italic> in sediments could transition from the sediment to the overlying water column without mechanical effects <italic toggle=\"yes\">i.e.,</italic> turbulence and resuspension. Although seasonal change in precipitation and turbulence can cause sediment resuspension and an increase in <italic toggle=\"yes\">E. coli</italic> abundance in the upper water column (##REF##37149158##Li, Filippelli &amp; Wang, 2023##), the effect should be minimal in lakes as <italic toggle=\"yes\">E. coli</italic> abundance quickly return to pre-resuspension level (##REF##16980417##Whitman, Nevers &amp; Byappanahalli, 2006##; ##REF##28295001##Abia et al., 2017##). Moreover, we have shown net transition rates in laboratory experiments without mechanical effects. Therefore, any precipitation will only increase the impact of habitat transition, and not affect the conclusion from this study.</p>", "<p>The habitat transition rates fluctuated among lakes (<italic toggle=\"yes\">CV</italic> = 53%) and was not correlated with sediment particle size (Pearson correlation: <italic toggle=\"yes\">n</italic> = 32, <italic toggle=\"yes\">r</italic> (30) = 0.38, <italic toggle=\"yes\">p</italic> = 0.14) and organic matter (Pearson correlation: <italic toggle=\"yes\">n</italic> = 32, <italic toggle=\"yes\">r</italic> (30) = −0.08, <italic toggle=\"yes\">p</italic> = 0.78). Although the habitat transition of <italic toggle=\"yes\">E. coli</italic> from sediment to water may be associated with biofilm sloughing (##REF##16793111##Lee et al., 2006##), the mechanisms that drive dispersal in <italic toggle=\"yes\">E. coli</italic> biofilms are complicated, and some are still unknown (##UREF##14##McDougald et al., 2012##).</p>", "<p>In this study, we observed the presence of <italic toggle=\"yes\">E. coli</italic> in all five tropical urban lake waters. Although previous studies have also reported on the survival of <italic toggle=\"yes\">E. coli</italic> in the water column, they did not include the effects of sediment (##REF##21146847##Lee et al., 2011##; ##REF##36327936##Wong et al., 2022##). Given the higher abundance of <italic toggle=\"yes\">E. coli</italic> in the sediments (##REF##20219232##Garzio-Hadzick et al., 2010##; ##UREF##7##Fluke, González-Pinzón &amp; Thomson, 2019##), and that these <italic toggle=\"yes\">E. coli</italic> can transition to the water column, the effects of sediment on the abundance of <italic toggle=\"yes\">E. coli</italic> in the water column could be important.</p>", "<p>In order to evaluate the effect of habitat transition on the abundance of <italic toggle=\"yes\">E. coli</italic> in the water column, we compared <italic toggle=\"yes\">E. coli</italic> habitat transition rates with total fraction decay rates. We found that in most cases (&gt;80%), the habitat transition rates were higher than the total fraction decay rates (##FIG##11##Fig. 12##). Thus, there was a net increase rate of <italic toggle=\"yes\">E. coli</italic> in the water column, calculated by the following equation: µ_habitat_transition–µ_{totalfractiondecay}. The <italic toggle=\"yes\">E. coli</italic> net increase rates ranged up to 0.36 h⁻¹ (0.16 ± 0.13 h⁻¹) in our study. When the habitat transition rate exceeds the total fraction decay rate, using <italic toggle=\"yes\">E. coli</italic> as a faecal indicator could overestimate the faecal contamination level.</p>", "<p>However, these rates were from microcosm-based experiments that did not include other biotic (<italic toggle=\"yes\">e.g.</italic>, biofilm and competition) and abiotic (sunlight) factors that can also affect <italic toggle=\"yes\">E. coli</italic> survivability <italic toggle=\"yes\">in-situ</italic> (##REF##24747902##Korajkic et al., 2014##; ##UREF##21##Stocker et al., 2019##; ##UREF##18##Petersen &amp; Hubbart, 2020##; ##REF##37010795##Moon et al., 2023##). Further studies are therefore needed to understand the role of these factors on <italic toggle=\"yes\">E. coli</italic> habitat transitions. In this study, we showed the role of sediment as reservoirs and habitat transition as a possible explanation for the persistence of <italic toggle=\"yes\">E. coli</italic> in tropical aquatic habitats (##REF##36327936##Wong et al., 2022##).</p>" ]

[ "<title>Conclusions</title>", "<p>Sediments acted as a reservoir of <italic toggle=\"yes\">E. coli</italic> in tropical urban lakes with a higher abundance of <italic toggle=\"yes\">E. coli</italic> than in the water column. The habitat transition of <italic toggle=\"yes\">E. coli</italic> from sediment to the water column affects its abundance in the water column, and could be one of the reasons for the persistence of <italic toggle=\"yes\">E. coli</italic> in tropical urban lakes.</p>" ]

[ "<title>Background</title>", "<p><italic toggle=\"yes\">Escherichia coli</italic> is a commonly used faecal indicator bacterium to assess the level of faecal contamination in aquatic habitats. However, extensive studies have reported that sediment acts as a natural reservoir of <italic toggle=\"yes\">E. coli</italic> in the extraintestinal environment. <italic toggle=\"yes\">E. coli</italic> can be released from the sediment, and this may lead to overestimating the level of faecal contamination during water quality surveillance. Thus, we aimed to investigate the effects of <italic toggle=\"yes\">E. coli</italic> habitat transition from sediment to water on its abundance in the water column.</p>", "<title>Methods</title>", "<p>This study enumerated the abundance of <italic toggle=\"yes\">E. coli</italic> in the water and sediment at five urban lakes in the Kuala Lumpur-Petaling Jaya area, state of Selangor, Malaysia. We developed a novel method for measuring habitat transition rate of sediment <italic toggle=\"yes\">E. coli</italic> to the water column, and evaluated the effects of habitat transition on <italic toggle=\"yes\">E. coli</italic> abundance in the water column after accounting for its decay in the water column.</p>", "<title>Results</title>", "<p>The abundance of <italic toggle=\"yes\">E. coli</italic> in the sediment ranged from below detection to 12,000 cfu g^–1, and was about one order higher than in the water column (1 to 2,300 cfu mL^–1). The habitat transition rates ranged from 0.03 to 0.41 h^–1. In contrast, the <italic toggle=\"yes\">E. coli</italic> decay rates ranged from 0.02 to 0.16 h⁻¹. In most cases (&gt;80%), the habitat transition rates were higher than the decay rates in our study.</p>", "<title>Discussion</title>", "<p>Our study provided a possible explanation for the persistence of <italic toggle=\"yes\">E. coli</italic> in tropical lakes. To the best of our knowledge, this is the first quantitative study on habitat transition of <italic toggle=\"yes\">E. coli</italic> from sediments to water column.</p>" ]

[ "<title> Supplemental Information</title>" ]

[ "<p>We thank Yi You Wong, Kyle Young Low, Walter Aaron and Ee Lean Thiang for their assistance with sampling.</p>", "<title>Additional Information and Declarations</title>" ]

[ "<fig position=\"float\" id=\"fig-1\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/fig-1</object-id><label>Figure 1</label><caption><title>Map of the location of sampling stations.</title><p>Map showing the location of the tropical urban lakes (Tasik Varsiti, Tasik Taman Jaya, Tasik Aman, Tasik Kelana and Tasik Central Park Bandar Utama) sampled at the Kuala Lumpur–Petaling Jaya area, Malaysia. Map data ©2023 Google (##UREF##8##Google, 2023##).</p></caption></fig>", "<fig position=\"float\" id=\"fig-2\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/fig-2</object-id><label>Figure 2</label><caption><title>A flow diagram of the habitat transition experiment.</title><p>(A) Addition of 2 g sediment sample. (B) Addition of 0.4% sterile agar on top of the sediment sample. (C) Addition of 18 mL 0.85% sterile saline after agar solidified. (D) Habitat transition of sediment <italic toggle=\"yes\">E. coli</italic> (not to scale) to the water column during incubation.</p></caption></fig>", "<fig position=\"float\" id=\"fig-3\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/fig-3</object-id><label>Figure 3</label><caption><title>Box-and-whisker plot showing the range and median of <italic toggle=\"yes\">E. coli</italic> and total coliform (log cfu mL⁻¹) in the waters of the five stations in this study.</title><p>Whisker shows the minimal to maximal bacteria abundance and the box shows the interquartile range, while the horizontal line represents the median. The same letters of the alphabet are used to indicate significant differences after Tukey’s pairwise analysis.</p></caption></fig>", "<fig position=\"float\" id=\"fig-4\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/fig-4</object-id><label>Figure 4</label><caption><title>Box-and-whisker plot showing the range and median of <italic toggle=\"yes\">E. coli</italic> (log cfu g⁻¹) in the sediments of the five stations in this study.</title><p>Whisker shows the minimal to maximal bacteria abundance and the box shows the interquartile range, while the horizontal line represents the median.</p></caption></fig>", "<fig position=\"float\" id=\"fig-5\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/fig-5</object-id><label>Figure 5</label><caption><title><italic toggle=\"yes\">E. coli.</italic> decay or growth over time (ln cfu mL⁻¹) in total, &lt;20 µm and &lt;0.2 µm fractions.</title><p>Measured at Tasik Varsiti (<italic toggle=\"yes\">n</italic> = 5), Tasik Taman Jaya (<italic toggle=\"yes\">n</italic> = 3).</p></caption></fig>", "<fig position=\"float\" id=\"fig-6\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/fig-6</object-id><label>Figure 6</label><caption><title><italic toggle=\"yes\">E. coli</italic> decay or growth over time (ln cfu mL⁻¹) in total, &lt;20 µm and &lt;0.2 µm fractions.</title><p>Measured at Tasik Aman (<italic toggle=\"yes\">n</italic> = 4), Tasik Kelana (<italic toggle=\"yes\">n</italic> = 4) and Tasik Central Park Bandar Utama (<italic toggle=\"yes\">n</italic> = 3).</p></caption></fig>", "<fig position=\"float\" id=\"fig-7\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/fig-7</object-id><label>Figure 7</label><caption><title>Scatter dot-plots of <italic toggle=\"yes\">E. coli</italic> decay and growth rates (<italic toggle=\"yes\">p</italic>≤ 0.05) for each station.</title><p>Mean ± SD is represented by a plus symbol with error bars. One asterisk (*) and two asterisks (**) showed significant differences in ANOVA at <italic toggle=\"yes\">p</italic> &lt; 0.05 and <italic toggle=\"yes\">p</italic> &lt; 0.01, respectively, among stations in the fractions. Same letter of the alphabet indicates significant differences after Tukey’s pairwise analysis.</p></caption></fig>", "<fig position=\"float\" id=\"fig-8\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/fig-8</object-id><label>Figure 8</label><caption><title>Increase in <italic toggle=\"yes\">E. coli</italic> abundance into the overlying water (ln cfu mL⁻¹) against incubation time.</title><p>Measured with sediments from Tasik Varsiti (<italic toggle=\"yes\">n</italic> = 6), Tasik Taman Jaya (<italic toggle=\"yes\">n</italic> = 3), Tasik Aman (<italic toggle=\"yes\">n</italic> = 3), Tasik Kelana (<italic toggle=\"yes\">n</italic> = 3) and Tasik Central Park Bandar Utama (<italic toggle=\"yes\">n</italic> = 3).</p></caption></fig>", "<fig position=\"float\" id=\"fig-9\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/fig-9</object-id><label>Figure 9</label><caption><title>Scatter dot-plots of <italic toggle=\"yes\">E. coli</italic> habitat transition rates (<italic toggle=\"yes\">p</italic>&lt; 0.05) for each station.</title><p>Mean ± SD is represented by a plus symbol with error bars. Same letters indicate significant difference after Tukey’s pairwise analysis.</p></caption></fig>", "<fig position=\"float\" id=\"fig-10\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/fig-10</object-id><label>Figure 10</label><caption><title>Correlation between the abundance of <italic toggle=\"yes\">E. coli</italic> in water and sediment.</title></caption></fig>", "<fig position=\"float\" id=\"fig-11\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/fig-11</object-id><label>Figure 11</label><caption><title>Correlation between log m-TEC cfu mL⁻¹ and log ECC cfu mL⁻¹ in the enumeration of <italic toggle=\"yes\">E. coli</italic> in water.</title><p>m-TEC stands for m-TEC agar, ECC stands for CHROMagar ECC agar.</p></caption></fig>", "<fig position=\"float\" id=\"fig-12\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/fig-12</object-id><label>Figure 12</label><caption><title>Scatter dots plot of habitat transition and total fraction decay rates of <italic toggle=\"yes\">E. coli</italic> in water column of five stations measured in this study.</title><p>Mean ± SD is represented by a plus symbol with error bars.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"table-1\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/table-1</object-id><label>Table 1</label><caption><title>The size and land use of five lakes, and water physico-chemical variables (Mean ± SD) measured at five lakes in this study.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">Tasik Varsiti <break/>(<italic toggle=\"yes\">n</italic> = 4)</th><th rowspan=\"1\" colspan=\"1\">Tasik Taman Jaya <break/>(<italic toggle=\"yes\">n</italic> = 3)</th><th rowspan=\"1\" colspan=\"1\">Tasik Aman <break/>(<italic toggle=\"yes\">n</italic> = 4)</th><th rowspan=\"1\" colspan=\"1\">Tasik Kelana <break/>(<italic toggle=\"yes\">n</italic> = 4)</th><th rowspan=\"1\" colspan=\"1\">Tasik Central Park Bandar Utama <break/>(<italic toggle=\"yes\">n</italic> = 3)</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">Area (m²)/Land use</td><td rowspan=\"1\" colspan=\"1\">13,102.74/educational</td><td rowspan=\"1\" colspan=\"1\">28,804.52/residential and commercial</td><td rowspan=\"1\" colspan=\"1\">17,778.91/residential</td><td rowspan=\"1\" colspan=\"1\">67,074.41/residential and commercial</td><td rowspan=\"1\" colspan=\"1\">1,786.64/commercial</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Temperature (°C)</td><td rowspan=\"1\" colspan=\"1\">29.9 ± 0.6</td><td rowspan=\"1\" colspan=\"1\">28.7 ± 0.8</td><td rowspan=\"1\" colspan=\"1\">29.6 ± 1.1</td><td rowspan=\"1\" colspan=\"1\">29.9 ± 1.2</td><td rowspan=\"1\" colspan=\"1\">29.9 ± 0.6</td></tr><tr><td rowspan=\"1\" colspan=\"1\">pH</td><td rowspan=\"1\" colspan=\"1\">7.2 ± 0.3</td><td rowspan=\"1\" colspan=\"1\">6.8 ± 0.4</td><td rowspan=\"1\" colspan=\"1\">7.2 ± 0.3</td><td rowspan=\"1\" colspan=\"1\">6.9 ± 0.3</td><td rowspan=\"1\" colspan=\"1\">7.2 ± 0.6</td></tr><tr><td rowspan=\"1\" colspan=\"1\">DO (mg L⁻¹)</td><td rowspan=\"1\" colspan=\"1\">9.01 ± 0.68</td><td rowspan=\"1\" colspan=\"1\">2.65 ± 1.78</td><td rowspan=\"1\" colspan=\"1\">9.47 ± 3.93</td><td rowspan=\"1\" colspan=\"1\">9.43 ± 4.39</td><td rowspan=\"1\" colspan=\"1\">9.65 ± 2.00</td></tr><tr><td rowspan=\"1\" colspan=\"1\">TSS (mg L⁻¹)<xref rid=\"table-1fn1\" ref-type=\"table-fn\">^***</xref></td><td rowspan=\"1\" colspan=\"1\">26 ± 6<xref rid=\"table-1fn\" ref-type=\"table-fn\">^a</xref></td><td rowspan=\"1\" colspan=\"1\">33 ± 6<xref rid=\"table-1fn\" ref-type=\"table-fn\">^b</xref></td><td rowspan=\"1\" colspan=\"1\">31 ± 8<xref rid=\"table-1fn\" ref-type=\"table-fn\">^c</xref></td><td rowspan=\"1\" colspan=\"1\">20 ± 7<xref rid=\"table-1fn\" ref-type=\"table-fn\">^d</xref></td><td rowspan=\"1\" colspan=\"1\">65 ± 13^abcd</td></tr><tr><td rowspan=\"1\" colspan=\"1\">POM (mg L⁻¹)<xref rid=\"table-1fn1\" ref-type=\"table-fn\">^***</xref></td><td rowspan=\"1\" colspan=\"1\">20 ± 1<xref rid=\"table-1fn\" ref-type=\"table-fn\">^a</xref></td><td rowspan=\"1\" colspan=\"1\">13 ± 2^bc</td><td rowspan=\"1\" colspan=\"1\">25 ± 4^bde</td><td rowspan=\"1\" colspan=\"1\">14 ± 5^df</td><td rowspan=\"1\" colspan=\"1\">43 ± 5^acef</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Chl <italic toggle=\"yes\">a</italic> (µg L⁻¹)<xref rid=\"table-1fn1\" ref-type=\"table-fn\">^***</xref></td><td rowspan=\"1\" colspan=\"1\">50.89 ± 14.54^ab</td><td rowspan=\"1\" colspan=\"1\">31.03 ± 3.62^cd</td><td rowspan=\"1\" colspan=\"1\">90.63 ± 15.14^ace</td><td rowspan=\"1\" colspan=\"1\">55.04 ± 13.44<xref rid=\"table-1fn\" ref-type=\"table-fn\">^e</xref></td><td rowspan=\"1\" colspan=\"1\">84.00 ± 2.60^bd</td></tr><tr><td rowspan=\"1\" colspan=\"1\">NH₄ (µM)<xref rid=\"table-1fn1\" ref-type=\"table-fn\">^***</xref></td><td rowspan=\"1\" colspan=\"1\">0.43 ± 0.18^ab</td><td rowspan=\"1\" colspan=\"1\">79.36 ± 26.27^acde</td><td rowspan=\"1\" colspan=\"1\">1.67 ± 1.18^cf</td><td rowspan=\"1\" colspan=\"1\">37.54 ± 23.38^bdf</td><td rowspan=\"1\" colspan=\"1\">0.97 ± 0.25<xref rid=\"table-1fn\" ref-type=\"table-fn\">^e</xref></td></tr><tr><td rowspan=\"1\" colspan=\"1\">PO₄ (µM)</td><td rowspan=\"1\" colspan=\"1\">0.18 ± 0.03</td><td rowspan=\"1\" colspan=\"1\">0.37 ± 0.04</td><td rowspan=\"1\" colspan=\"1\">0.34 ± 0.18</td><td rowspan=\"1\" colspan=\"1\">0.29 ± 0.16</td><td rowspan=\"1\" colspan=\"1\">0.21 ± 0.02</td></tr></tbody></table></alternatives></table-wrap>", "<table-wrap position=\"float\" id=\"table-2\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/table-2</object-id><label>Table 2</label><caption><title>Sediment composition, particle size and organic matter measured at five stations in this study.</title></caption><alternatives><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/><col span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\">Station</th><th rowspan=\"1\" colspan=\"1\">USDA Textural class<xref rid=\"table-2fn3\" ref-type=\"table-fn\">^a</xref></th><th rowspan=\"1\" colspan=\"1\">Particle size (µm)<xref rid=\"table-2fn1\" ref-type=\"table-fn\">^***</xref></th><th align=\"center\" colspan=\"3\" rowspan=\"1\">Sediment composition (%)</th><th rowspan=\"1\" colspan=\"1\">Organic matter (mg g⁻¹)</th></tr><tr><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\"/><th rowspan=\"1\" colspan=\"1\">Sand</th><th rowspan=\"1\" colspan=\"1\">Clay</th><th rowspan=\"1\" colspan=\"1\">Silt</th><th rowspan=\"1\" colspan=\"1\"/></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">Tasik Varsiti <break/>(<italic toggle=\"yes\">n</italic> = 4)</td><td rowspan=\"1\" colspan=\"1\">Loam</td><td rowspan=\"1\" colspan=\"1\">117.35 ± 80.86^ab</td><td rowspan=\"1\" colspan=\"1\">47.35% ± 18.41%</td><td rowspan=\"1\" colspan=\"1\">14.43% ± 8.96%</td><td rowspan=\"1\" colspan=\"1\">38.23% ± 9.75%</td><td rowspan=\"1\" colspan=\"1\">30 ± 11</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Tasik Taman Jaya <break/>(<italic toggle=\"yes\">n</italic> = 3)</td><td rowspan=\"1\" colspan=\"1\">Coarse sand</td><td rowspan=\"1\" colspan=\"1\">340.47 ± 99.57^ac</td><td rowspan=\"1\" colspan=\"1\">87.08% ± 6.36%</td><td rowspan=\"1\" colspan=\"1\">1.55% ± 2.68%</td><td rowspan=\"1\" colspan=\"1\">11.38% ± 3.79%</td><td rowspan=\"1\" colspan=\"1\">33 ± 10</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Tasik Aman <break/>(<italic toggle=\"yes\">n</italic> = 4)</td><td rowspan=\"1\" colspan=\"1\">Sandy loam</td><td rowspan=\"1\" colspan=\"1\">138.56 ± 102.74<xref rid=\"table-2fn\" ref-type=\"table-fn\">^d</xref></td><td rowspan=\"1\" colspan=\"1\">64.15% ± 20.82%</td><td rowspan=\"1\" colspan=\"1\">16.01% ± 15.80%</td><td rowspan=\"1\" colspan=\"1\">19.84% ± 16.68%</td><td rowspan=\"1\" colspan=\"1\">27 ± 14</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Tasik Kelana <break/>(<italic toggle=\"yes\">n</italic> = 4)</td><td rowspan=\"1\" colspan=\"1\">Coarse sand</td><td rowspan=\"1\" colspan=\"1\">627.14 ± 107.96^bcde</td><td rowspan=\"1\" colspan=\"1\">97.93% ± 1.98%</td><td rowspan=\"1\" colspan=\"1\">0%</td><td rowspan=\"1\" colspan=\"1\">2.07% ± 1.98%</td><td rowspan=\"1\" colspan=\"1\">5 ± 2</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Tasik Central Park Bandar Utama <break/>(<italic toggle=\"yes\">n</italic> = 3)</td><td rowspan=\"1\" colspan=\"1\">Loamy sand</td><td rowspan=\"1\" colspan=\"1\">253.97 ± 38.65<xref rid=\"table-2fn\" ref-type=\"table-fn\">^e</xref></td><td rowspan=\"1\" colspan=\"1\">82.2% ± 2.55%</td><td rowspan=\"1\" colspan=\"1\">5.33% ± 1.32%</td><td rowspan=\"1\" colspan=\"1\">12.47% ± 1.84%</td><td rowspan=\"1\" colspan=\"1\">33 ± 26</td></tr></tbody></table></alternatives></table-wrap>" ]

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[ "<supplementary-material id=\"supp-1\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/supp-1</object-id><label>Supplemental Information 1</label><caption><title>List of lakes coordinates, sampling dates and experiments conducted in this study</title><p>The names of five lakes and their respective coordinates in this study and the type of sample collected in each sampling and the experiments conducted. Conducted experiments were labeled with “ √ ”, whereas “ ×” indicates was not performed. The failed experiments are highlighted in yellow and the reason stated in the “Note” column.</p></caption></supplementary-material>", "<supplementary-material id=\"supp-2\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/supp-2</object-id><label>Supplemental Information 2</label><caption><title>Raw data of the abundance of total coliform and <italic toggle=\"yes\">E. coli</italic> in water and <italic toggle=\"yes\">E. coli</italic> abundance in sediment were measured at five stations in this study</title></caption></supplementary-material>", "<supplementary-material id=\"supp-3\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/supp-3</object-id><label>Supplemental Information 3</label><caption><title>Raw data of the abundance of <italic toggle=\"yes\">E. coli</italic> measured every 6 h until 24 h in different fractions for the <italic toggle=\"yes\">E. coli</italic> decay and growth rates determination</title><p>These data were used to determine the decay and growth rate of <italic toggle=\"yes\">E. coli</italic> in different water fractions.</p></caption></supplementary-material>", "<supplementary-material id=\"supp-4\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/supp-4</object-id><label>Supplemental Information 4</label><caption><title>Raw data of <italic toggle=\"yes\">E. coli d</italic> ecay and growth rates (h⁻¹) determined in different fractions at five stations</title></caption></supplementary-material>", "<supplementary-material id=\"supp-5\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/supp-5</object-id><label>Supplemental Information 5</label><caption><title>Raw data of the abundance of <italic toggle=\"yes\">E. coli</italic> in the overlaying water measured every 6 h in the habitat transition experiment</title><p>These data were used to determine the <italic toggle=\"yes\">E. coli</italic> increase rate.</p></caption></supplementary-material>", "<supplementary-material id=\"supp-6\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/supp-6</object-id><label>Supplemental Information 6</label><caption><title>Raw data of <italic toggle=\"yes\">E. coli</italic> increase rate (h⁻¹) in the overlying water determined at five stations in this study</title></caption></supplementary-material>", "<supplementary-material id=\"supp-7\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/supp-7</object-id><label>Supplemental Information 7</label><caption><title>Raw data of the abundance of <italic toggle=\"yes\">E. coli</italic> measured every 6 h until 24 h for the <italic toggle=\"yes\">E. coli</italic> intrinsic growth rate determination, at Tasik Kelana and Tasik Central Park Bandar Utama</title></caption></supplementary-material>", "<supplementary-material id=\"supp-8\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/supp-8</object-id><label>Supplemental Information 8</label><caption><title>Raw data of the <italic toggle=\"yes\">E. coli</italic> habitat transition rate calculated by using the <italic toggle=\"yes\">E. coli</italic> increase rate to deduct the <italic toggle=\"yes\">E. coli</italic> intrinsic growth rate obtained in this study</title></caption></supplementary-material>", "<supplementary-material id=\"supp-9\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/supp-9</object-id><label>Supplemental Information 9</label><caption><title>Raw data of <italic toggle=\"yes\">E. coli</italic> count obtained from m-TEC and CHROMagar ECC agar in our previous study</title></caption></supplementary-material>", "<supplementary-material id=\"supp-10\" position=\"float\" content-type=\"local-data\"><object-id pub-id-type=\"doi\">10.7717/peerj.16556/supp-10</object-id><label>Supplemental Information 10</label><caption><title>Raw data of environmental variables measured in this study</title><p>Water: temperature, pH, dissolved oxygen, total suspended solid, particular organic matter, Chlorophyll <italic toggle=\"yes\">a</italic>, NH₄, PO₄ and protists count; Sediment: particle size, sediment composition, sediment textural class and organic matter.</p></caption></supplementary-material>" ]

[ "<table-wrap-foot><fn id=\"table-1fn\"><p>\n<bold>Notes.</bold>\n</p></fn><fn id=\"table-1fn1\"><label>***</label><p>Showed significant differences in ANOVA at <italic toggle=\"yes\">p</italic> &lt;0.001.</p></fn><fn id=\"table-1fn2\" fn-type=\"other\"><p>The same superscript letter of the alphabet indicates significant difference after Tukey’s pairwise analysis.</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"table-2fn\"><p>\n<bold>Notes.</bold>\n</p></fn><fn id=\"table-2fn1\"><label>***</label><p>Showed significant difference in ANOVA at <italic toggle=\"yes\">p</italic> &lt;0.001.</p></fn><fn id=\"table-2fn2\" fn-type=\"other\"><p>The same superscript letter of the alphabet indicates significant differences after Tukey’s pairwise analysis.</p></fn><fn id=\"table-2fn3\"><label>a</label><p>Sediment textural class was determined according to the United States Department of Agriculture (##UREF##19##Soil Science Division Staff, 2017##).</p></fn></table-wrap-foot>", "<fn-group content-type=\"competing-interests\"><title>Competing Interests</title><fn id=\"conflict-1\" fn-type=\"COI-statement\"><p>The authors declare there are no competing interests.</p></fn></fn-group>", "<fn-group content-type=\"author-contributions\"><title>Author Contributions</title><fn id=\"contribution-1\" fn-type=\"con\"><p><xref rid=\"author-1\" ref-type=\"contrib\">Boyu Liu</xref> conceived and designed the experiments, performed the experiments, analyzed the data, prepared figures and/or tables, authored or reviewed drafts of the article, and approved the final draft.</p></fn><fn id=\"contribution-2\" fn-type=\"con\"><p><xref rid=\"author-2\" ref-type=\"contrib\">Choon Weng Lee</xref> conceived and designed the experiments, performed the experiments, authored or reviewed drafts of the article, and approved the final draft.</p></fn><fn id=\"contribution-3\" fn-type=\"con\"><p><xref rid=\"author-3\" ref-type=\"contrib\">Chui Wei Bong</xref> conceived and designed the experiments, prepared figures and/or tables, and approved the final draft.</p></fn><fn id=\"contribution-4\" fn-type=\"con\"><p><xref rid=\"author-4\" ref-type=\"contrib\">Ai-Jun Wang</xref> analyzed the data, authored or reviewed drafts of the article, and approved the final draft.</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Data Availability</title><fn id=\"addinfo-1\"><p>The following information was supplied regarding data availability:</p><p>The raw data are available in the <xref rid=\"supplemental-information\" ref-type=\"sec\">Supplemental Files</xref>.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>1. Background</title>", "<p>For thousands of years, narcotics have been used for medicinal and palliative purposes and still have an important role in relieving pain, diarrhea, cough, and other symptoms. Narcotics abuse has risen dramatically in recent years. For instance, Golestan Cohort Study conducted in Golestan province, Iran, reported that 17% (n = 8,487) of the participants' misused opium, with a mean duration of 12.7 years (##REF##22511302##1##). Another study conducted in Fars province, Iran, reported that 8% (n = 339) of the participants misused opium (##REF##17613976##2##). In the United States, 3% to 4% of adults receive long-term opioid treatment (##REF##29976376##3##).</p>", "<p>Along with opioid use and misuse, there are unfavorable side effects, including endocrinopathies due to long-time opioid usage (##REF##29976376##3##). Endocrinopathy of narcotics should be considered for any patient using the equivalent of 100 mg of morphine per day or more. Measuring the response of plasma cortisol levels to intravenous or muscular injections of ACTH is a common screening test for detecting adrenal insufficiency. Various diagnostic criteria have been set according to base cortisol levels, stimulated cortisol levels, or their difference (##REF##14719714##4##).</p>", "<p>Methadone is a synthetic opioid. A complete Mu (µ) receptor agonist may mimic endogenous opioids, enkephalins, and endorphins (##REF##10695444##5##). Methadone is frequently used to relieve pain, especially in the intensive care unit (ICU), and ease quitting opium addiction. This drug is qualitatively equivalent to morphine but has a longer half-life. The plasma half-life of methadone is very long and variable (13 - 100 hours). Despite this feature, many patients need methadone every 4 - 8 hours to maintain the analgesic effects (##REF##16759341##6##).</p>", "<p>Methadone maintenance therapy (MMT) has shown excellent results in managing heroin-dependent patients. However, the researchers questioned whether MMT could improve the function of the hypothalamic-pituitary-adrenal (HPA) axis, which is damaged by heroin dependence, and improve baseline cortisol levels. In this regard, studies are limited and contradictory. For instance, in a 2006 study by Aouizerate et al., methadone reduced serum cortisol levels. However, a 2016 study by Young et al. found increased body cortisol levels following methadone administration (##REF##16759341##6##, ##REF##27010803##7##).</p>", "<p>Detecting adrenal insufficiency is critical, especially in ICU patients and patients undergoing major surgeries. To the best of our knowledge in the available research, adrenal insufficiency in opium-addicted patients on MMT has not been evaluated with a cosyntropin test (ACTH stimulation test) (##REF##20807731##8##).</p>" ]

[ "<title>3. Methods</title>", "<p>This study was conducted in November 2019 at Imam Reza Hospital Rehab Center, Birjand, Iran. The patients were on methadone to ease quitting opium addiction. Our inclusion criteria were an addiction to opium for at least six months, not using corticosteroids in the past year, age of 20 to 45, not having significant co-morbidities such as diabetes or cancer, and no history of quitting opium addiction. According to a study by Annane et al. (##REF##10697064##9##) which reported a mean cortisol level of 13.9 ± 10.3 in its population, a sample size of 42 was calculated with α = 0.01 and β = 0.1.</p>", "<p>Convenience sampling was used to select patients. Eighty patients were assessed for eligibility, 42 of whom were enrolled in the study based on our inclusion criteria. The study procedure was explained to the patients; those who filled out informed consent and met the inclusion criteria were enrolled. A questionnaire was filled out to gather demographic characteristics. Cosyntropin tests were performed at 8-9 AM to minimize the effect of circadian rhythm on cortisol levels. Initially, a 5 mL blood sample was obtained to measure baseline cortisol. Afterward, 250 micrograms of intra-muscular cosyntropin were injected. In 30- and 60-minute intervals, blood samples were taken. The samples were analyzed at the central laboratory of Imam Reza Hospital. Chemiluminescence detection was used to measure cortisol levels with kits from Saluggia company, Italy.</p>", "<p>According to Henry's Clinical Diagnosis and Management by Laboratory Methods (##UREF##0##10##), following the cosyntropin test, cortisol levels should be higher than 18 µg/dL, and lower levels determine adrenal insufficiency. Also, according to a study by Annane et al. (##REF##10697064##9##), a cortisol level change of less than 9 µg/dL is considered adrenal insufficiency. We used these definitions of adrenal insufficiency in our study. Also, based on the chemiluminescence device's reference, which measured cortisol, the mean cortisol level in the standard population is 14 µg/dL (##UREF##1##11##). We compared our population with this value.</p>", "<p>All statistical analyses were performed using SPSS version 16 software (SPSS Inc., Chicago, Illinois, USA). The normal distribution of variables was evaluated using the Kolmogorov–Smirnov test. Descriptive data are shown as the mean ± standard deviation or number (%). One-way analysis of variance (ANOVA) was applied to compare the demographic and clinical features between the groups. Repeated-measures ANOVA was recruited to assess the effect of cosyntropin on cortisol levels. Degrees of freedom were adjusted via Mauchly's W test, followed by a Greenhouse-Geisser correction of P-values.</p>" ]

[ "<title>4. Results</title>", "<p>As shown in ##TAB##0##Table 1##, the mean age of the participants was 34.4 ± 5.2, and most were men (90.5%). Eight of them were cigarette smokers.</p>", "<p>Cortisol levels and response to the cosyntropin test had a normal distribution (P-value = 0.44 and 0.28, respectively).</p>", "<p>The mean serum cortisol level at baseline was 9.46 ± 5.42 µg/dL, significantly different from its normal value of 14 µg/dL (P &lt; 0.001). The mean response to the cosyntropin test (difference from baseline) was 9.34 ± 8.11 µg/dL.</p>", "<p>According to Henry's Clinical Diagnosis and Management by Laboratory Methods (##UREF##0##10##), 21 (50.0%) participants had adrenal insufficiency, and according to the study of Annane et al. (##REF##10697064##9##), 24 (57.1%) participants had adrenal insufficiency.</p>", "<p>There was a significant difference between baseline cortisol levels and cortisol levels at 30- and 60-minute intervals (P-values &lt; 0.001) (##FIG##0##Figure 1##).</p>", "<p>Mean baseline cortisol levels and response to cosyntropin were not associated with age, dose, and duration of methadone usage (##TAB##1##Table 2##).</p>" ]

[ "<title>5. Discussion</title>", "<p>This study investigated changes in cortisol levels and the response to ACTH hormone in former opium addicts on methadone treatment. Considering adrenal insufficiency can affect the management of these patients. Fifty-five percent of our participants had cortisol levels lower than 18 µg/dL following the cosyntropin test, indicating adrenal insufficiency.</p>", "<p>Many earlier studies show that chronic opioid misuse can lead to HPA axis suppression. In this regard, a review by Donegan and Bancos concluded that 9 to 29% of patients receiving long-term opiate therapy may experience opioid-induced adrenal insufficiency. However, our study's prevalence of adrenal insufficiency was significantly higher (##REF##29976376##3##). Whether MMT restores the disrupted HPA axis is still unknown, and studies in this regard are inconsistent.</p>", "<p>Some studies have shown an overactivated HPA axis in MMT patients compared to controls (##REF##12700714##12##, ####REF##1511228##13##, ##REF##11714593##14####11714593##14##). A study by Yang et al. on 52 MMT patients and 41 age-matched controls showed that MMT patients had significantly higher hair cortisol levels than the controls. Likewise, MMT patients showed significantly higher perceived stress levels. The authors imply that this higher stress level may have masked the suppressed HPA axis (##REF##27010803##7##). In contrast, in a study, heroin users showed normal HPA activation with metyrapone, an 11-beta-hydroxylase inhibitor. The same study showed that patients on MMT addicted to cocaine had a hyperactivated HPA response to metyrapone (##REF##11282257##15##). Dackis et al. studied five methadone misusers and 12 controls and observed a decreased response to ACTH stimulation in methadone misusers (##REF##6128489##16##). Some case reports have also shown that chronic use of opioids can cause adrenal insufficiency (##REF##26161260##17##, ##REF##25221675##18##).</p>", "<p>As mentioned, studies on cortisol levels and HPA axis function in patients on MMT are contradictory, and to date, the reasons for this discrepancy are unclear (##REF##26221126##19##). One possible explanation may be that studies have used plasma, saliva, and urine cortisol levels as biological markers to assess basal cortisol levels. These biological markers are prone to circadian rhythms and events before sampling. Recently, endogenous cortisol levels in human hair have been proposed to overcome limitations and indicate cortisol over up to six months (##REF##27010803##7##). Another explanation may be that participants in previous studies have been at different stages of the detoxification reaction. In addition, the activity of the HPA axis in patients on MMT may be affected by negative emotions. For example, patients with depressive symptoms may have higher basal cortisol levels. In addition, psychological and MMT factors may have synergistic effects on HPA axis function (##REF##935296##20##, ##REF##15991000##21##). Differences in opioid receptor affinity due to polymorphisms in different individuals may be another explanation (##REF##11751037##22##). The duration of MMT can affect the result of studies. Kreek et al. showed that metyrapone and ACTH stimulation tests were abnormal in the first two months of MMT but normal after two months (##REF##6099512##23##). In line with this, response to the cosyntropin test was increased in longer MMT durations in our study (##TAB##1##Table 2##); however, this finding was not statistically significant (P-value = 0.40). The mechanism of HPA axis normalization is not clear. One explanation is given by Kling et al., who used positron emission tomography (PET) to study opiate receptors in MMT patients. They observed that only 19 – 32% of opiate receptors were occupied, and the remaining receptors could function normally in the HPA axis (##REF##11082442##24##).</p>", "<p>Adrenal insufficiency can cause hemodynamic disturbances, changes in consciousness, hypoxemia, and ileus. It can be life-threatening if not managed properly (##REF##11779267##25##). However, most of the patients have non-specific symptoms that may mislead clinicians. Therefore, knowing that many opioid abusers and MMT patients may suffer from adrenal insufficiency can help prevent serious complications in case of major medical stress. Cortisol helps maintain the balance of the cardiovascular system during surgical trauma by facilitating the activity of catecholamines. In this regard, Baghaei Wadji et al. examined the effects of opium addiction on the response to the stress of major surgeries. The serum cortisol level of the addict group showed a significant increase compared to the non-addict group 24 hours after surgery, indicating a stronger response of opium addicts to surgical stress (##UREF##2##26##). Cortisol levels during and after surgery are proportional to the severity of the operation, and any disturbances, whether an inappropriate increase like in the mentioned study or an inappropriate decrease like in our study, can be life-threatening (##REF##19385286##27##).</p>", "<p>Most studies confirm that opioid misuse suppresses the HPA axis. However, whether long-term MMT can normalize the HPA axis is still unknown. Larger studies with control groups are needed to answer this question.</p>" ]

[]

[ "<title>Background</title>", "<p>Opium has been used for thousands of years for medical and analgesic purposes, and its misuse has also increased in recent years. Methadone, a synthetic opioid, has been used as an analgesic and to help patients quit opium addiction. However, some evidence suggests that long-term use of opioids can affect the hypothalamic-pituitary-adrenal axis.</p>", "<title>Objectives</title>", "<p>We aimed to evaluate the serum cortisol level and response to the cosyntropin stimulation test in opium addicts on methadone treatment.</p>", "<title>Methods</title>", "<p>The study was conducted in November 2019 at Imam Reza Hospital Rehab Center, Birjand, Iran. Thirty-eight methadone-treated opium addicts participated in the study. A blood sample was initially obtained, then 250 µg intramuscular cosyntropin was injected. After 30 and 60 minutes, two other blood samples were obtained. The data were analyzed using SPSS.</p>", "<title>Results</title>", "<p>There was a significant difference between serum cortisol levels and the normal value in methadone users (9.46 ± 5.42 vs. 14 µg/dL) (P &lt; 0.001). The mean response to the cosyntropin stimulation test in methadone users was 9.34 ± 8.11 µg/dL. Also, 55% of the participants had adrenal insufficiency.</p>", "<title>Conclusions</title>", "<p>Serum cortisol levels significantly differed from normal values in methadone-treated patients. Therefore, we recommend measuring serum cortisol levels in methadone-treated patients before major medical procedures to consider the stress doses of corticosteroids.</p>" ]

[ "<title>2. Objectives</title>", "<p>Regarding the high prevalence of methadone use, we aimed to measure the changes in cortisol levels and the response to the cosyntropin test to determine the extent and prevalence of adrenal insufficiency in opium-addicted patients on methadone treatment.</p>" ]

[ "<p>We would like to thank Mr. Keivan Kalali, who contributed to writing the manuscript and analyzing the data.</p>", "<title>Authors' Contribution: </title>", "<p>Study concept and design: F. Z., M. G, and A. B; Acquisition of data: F. Z., A. K., and A. B; Analysis and interpretation of data: A. B, SA. E., and A. K.; Drafting of the manuscript: F. Z. and SA. E.; Critical revision of the manuscript for important intellectual content: F. Z. and M. G; Statistical analysis: A. B and SA. E.; Administrative, technical, and material support: M. G and A. K.; Study supervision: M. G, A. B, and A. K.</p>", "<title>Conflict of Interests Statement: </title>", "<p>Birjand University of Medical Sciences completely funded the study. The authors are academics, are not involved in any business related to the content of this research, and have no financial interest in favor of any possible result of the study.</p>", "<title>Ethical Approval: </title>", "<p>This study was approved under the ethical approval code of <ext-link xlink:href=\"https://ethics.research.ac.ir/ProposalCertificateEn.php?id=75347\" ext-link-type=\"uri\">IR.BUMS.REC.1398.129</ext-link>.</p>", "<title>Funding/Support: </title>", "<p>Birjand University of Medical Sciences supported the study; no specific fund was granted.</p>", "<title>Informed Consent: </title>", "<p>The study procedure was explained to the patients; those who filled out informed consent and met the inclusion criteria were enrolled.</p>" ]

[ "<fig position=\"float\" id=\"A135206FIG1\" fig-type=\"inline\"><label>Figure 1.</label><caption><title>Changes in serum cortisol level in response to cosyntropin test</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"A135206TBL1\"><label>Table 1.</label><caption><title>Demographic and Clinical Characteristics of Study Participants ^{<xref rid=\"A135206TBL1FN1\" ref-type=\"table-fn\">a</xref>}</title></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th style=\"text-align: left;\" rowspan=\"1\" colspan=\"1\">Variables </th><th rowspan=\"1\" colspan=\"1\">Values</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Age </bold>\n</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">34.4 ± 5.2</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>BMI </bold>\n</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">22.5 ± 3.6</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Gender</bold>\n</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td style=\"padding-left: 20pt;\" rowspan=\"1\" colspan=\"1\">Male</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">38 (90.5)</td></tr><tr><td style=\"padding-left: 20pt;\" rowspan=\"1\" colspan=\"1\">Female</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">4 (9.5)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Smoking</bold>\n</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">8 (19)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Corticosteroid use</bold>\n</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">1 (2.4)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Comorbidity</bold>\n</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td style=\"padding-left: 20pt;\" rowspan=\"1\" colspan=\"1\">Thyroid disease</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">1 (2.4)</td></tr><tr><td style=\"padding-left: 20pt;\" rowspan=\"1\" colspan=\"1\">COPD</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">1 (2.4)</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"A135206TBL2\"><label>Table 2.</label><caption><title>The Relationship Between Baseline Cortisol Levels and Response to Cosyntropin Test with Other Variables</title></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th style=\"text-align: left;\" rowspan=\"1\" colspan=\"1\">Variables</th><th rowspan=\"1\" colspan=\"1\">No. (%)</th><th rowspan=\"1\" colspan=\"1\">Mean Baseline Cortisol (µg/dL)</th><th rowspan=\"1\" colspan=\"1\">P-Value ^{<xref rid=\"A135206TBL2FN1\" ref-type=\"table-fn\">a</xref>}</th><th rowspan=\"1\" colspan=\"1\">Response to Cosyntropin Test (µg/dL)</th><th rowspan=\"1\" colspan=\"1\">P-Value ^{<xref rid=\"A135206TBL2FN1\" ref-type=\"table-fn\">a</xref>}</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Duration of methadone usage, mo</bold>\n</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">0.08</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">0.40</td></tr><tr><td style=\"padding-left: 20pt;\" rowspan=\"1\" colspan=\"1\">&lt; 1 </td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">6 (14.3)</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">9.12 ± 2.75</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">5.75 ± 3.77</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td style=\"padding-left: 20pt;\" rowspan=\"1\" colspan=\"1\">1 – 6 </td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">9 (21.4)</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">12.96 ± 7.72</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">8.30 ± 6.56</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td style=\"padding-left: 20pt;\" rowspan=\"1\" colspan=\"1\">&lt; 6 </td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">27 (64.3)</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">8.36 ± 4.60</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">10.48 ± 9.11</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Dosage of methadone used</bold>\n</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">0.72</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">0.89</td></tr><tr><td style=\"padding-left: 20pt;\" rowspan=\"1\" colspan=\"1\">&lt; 100</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">16 (38.1)</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">9.07 ± 5.57</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">9.13 ± 9.85</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td style=\"padding-left: 20pt;\" rowspan=\"1\" colspan=\"1\">&lt; 100</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">26 (61.9)</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">9.70 ± 5.43</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">9.47 ± 7.03</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<bold>Age, y</bold>\n</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">0.13</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">0.39</td></tr><tr><td style=\"padding-left: 20pt;\" rowspan=\"1\" colspan=\"1\">25 - 30 </td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">5 (11.9)</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">9.64 ± 3.42</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">6.00 ± 5.41</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td style=\"padding-left: 20pt;\" rowspan=\"1\" colspan=\"1\">31 - 35 </td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">7 (16.7)</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">5.39 ± 2.06</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">13.01 ± 10.59</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td style=\"padding-left: 20pt;\" rowspan=\"1\" colspan=\"1\">36 - 40 </td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">18 (42.9)</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">11.06 ± 6.43</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">8.01 ± 5.54</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td style=\"padding-left: 20pt;\" rowspan=\"1\" colspan=\"1\">41 - 45 </td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">12 (28.6)</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">9.35 ± 4.95</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\">10.60 ± 10.36</td><td style=\"text-align: center;\" rowspan=\"1\" colspan=\"1\"/></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><fn id=\"A135206TBL1FN1\"><p>^a Values are expressed as Mean ± SD or No. (%).</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"A135206TBL2FN1\"><p>^a By using one-way ANOVA</p></fn></table-wrap-foot>" ]

[ "<graphic xlink:href=\"aapm-13-3-135206-i001\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p><italic>Streptococcus pneumoniae</italic> (<italic>S. pneumoniae</italic>) is an aerobic gram-positive coccus that causes a broad variety of infections. Non-invasive pneumococcal infections include bronchitis, otitis media, and sinusitis. An infection in which <italic>S. pneumoniae</italic> is isolated from a typically sterile body site is known as an invasive pneumococcal illness. The most frequent presentation is pneumonia, followed by bacteremia (in which no source is identified), meningitis, septic arthritis, spontaneous peritonitis, endocarditis, osteomyelitis, and soft tissue infection. It has long been one of the most relevant bacterial causes of disease in humans, but since 2000, its impact has been blunted by the widespread use of vaccines that largely prevent infection and colonisation in young children [##REF##36153636##1##]. Since there have been over 90 distinct <italic>S. pneumoniae</italic> serovars, the research aims at developing vaccines that deliver broad immunity. Pneumococcal conjugate vaccine (PCV) and pneumococcal polysaccharide vaccine (PPSV) are the two forms of pneumococcal vaccines that are available for clinical use. Both the active components are capsular polysaccharides from pneumococcal serotypes that commonly cause invasive disease [##UREF##0##2##]. The pneumococcal vaccination is indicated for all adults 65 years of age or older, as well as those under 65 who are at risk for pneumococcal infection or severe complications, namely immunocompromised, those with long-term predisposing illnesses (such as lung disease), functional or anatomic asplenia, or a history of invasive pneumococcal disease. In Portugal, the recommended vaccines are the PPSV23 (Pneumovax 23^®), which includes 23 partially purified capsular polysaccharide serotypes (1, 2, 3, 4, 5, 6B, 7F, 8, 9N, 9V, 10A, 11A, 12F, 14, 15B, 17F, 18C, 19A, 19F, 20, 22F, 23F, and 33F), and the 13-valent PCV13 (Prevnar 13^®), which contains capsular polysaccharide antigens covalently linked to a nontoxic protein (covers serotypes 1, 3, 4, 5, 6A, 6B, 7F, 9V, 14, 18C, 19A, 19F, 23F) [##UREF##1##3##]. As serotypes that cause pneumococcal disease continue to change, in the summer of 2021, two new PCVs for use in adults emerged: PCV15 and PCV20 [##REF##37242402##4##]. These are approved and commercialised in Portugal, although they are not yet cited in the national guidelines.</p>" ]

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[ "<title>Discussion</title>", "<p>The risk of infection, sepsis, and sepsis-related mortality appears to be approximately two to three times higher in asplenic patients when compared with the general population [##UREF##2##5##, ####UREF##3##6####3##6##]. Patients with impaired splenic function are at risk for severe and overwhelming infections with encapsulated bacteria, bloodborne parasites, and other infections where the spleen plays an important role. This organ has an abundance of lymphoid tissue, including splenic macrophages that are responsible for the opsonisation and phagocytosis of encapsulated organisms such as <italic>S. pneumoniae</italic>, <italic>Haemophilus influenzae </italic>(<italic>H. influenzae</italic>), and <italic>Neisseria meningitidis (N. meningitidis)</italic>. The spleen is also a major site of early immunoglobulin M production, which is important in the acute clearance of pathogens from the bloodstream. The overall case-fatality rate for <italic>S. pneumoniae</italic> bacteraemia is about 20%, but it may be as high as 60% among patients with asplenia [##UREF##3##6##, ####REF##34088948##7####34088948##7##].</p>", "<p>Reviewing the case, the symptoms reported by the patient, fever, myalgia, and diarrhoea, in a splenectomised patient were the clue to an underlying serious process. Management of fever in asplenic patients includes immediate empiric intravenous antibiotic administration. Ceftriaxone and cefotaxime are both active against most <italic>S. pneumoniae, H. influenzae</italic> type b, and <italic>N. meningitidis</italic> isolates; vancomycin should be added if there is a risk of beta-lactam-resistant <italic>S. pneumoniae</italic>. Because progression to septic shock and respiratory distress can occur rapidly, preparations for fluid resuscitation, vasopressor support, and airway management should be made [##UREF##3##6##]. IVIG is controversial in sepsis and not recommended for the general population. However, as IVIG has the potential to offset the immune deficits of splenectomised patients, it is reasonable to give IVIG to selected patients with sepsis who have impaired splenic function [##REF##34088948##7##, ####UREF##4##8####4##8##].</p>", "<p><italic>S. pneumoniae</italic> has long been one of the most prominent bacterial causes of disease in humans and was one of the first to be identified as a cause of human infection. In spite of the widespread use of vaccines for more than 20 years, this disease is still responsible for approximately 1.6 million deaths worldwide each year [##REF##36153636##1##].</p>", "<p>In Portugal, epidemiological studies have documented a dynamic evolution in the serotypes, with an increase in cases of invasive pneumococci disease by serotypes not included in the PCV13. Most of them, however, are included in the PPSV23. The most common serotypes in Portugal from 2015 to 2018 were eight (19%), three (15%), 22F (7%), 14 (6%), and 19A (5%). As such, during this period, PCV13 covered 44% of the circulating serotypes, and PPSV23 covered 80%. In the population of Northern Portugal, 22F is the most common serotype not covered by PPSV23 or PCV13 [##REF##34066862##9##].</p>", "<p>During the summer of 2021, the United States Food and Drug Administration licensed two new PCVs for use in adults: PCV15 and PCV20. These vaccines target common serotypes causing invasive pneumococcal disease and pneumococcal pneumonia in the United States. PCV15 contains all PCV13 serotypes (1, 3, 4, 5, 6A, 6B, 7F, 9V, 14, 18C, 19A, 19F, 23F) plus 22F and 33F. PCV20 contains all PCV15 serotypes plus 8, 10A, 11A, 12F, and 15B [##REF##35085226##10##].</p>", "<p>In this case, the patient already had two doses of PPSV23 (15 and 10 years ago) and one of PCV13 approximately 11 months before the fatal event. According to the new Centers for Disease Control and Prevention guidelines, the recommendation for adults 19 through 64 years old with immunocompromising conditions who have received PCV13 and two doses of PPSV23 is to give no additional pneumococcal vaccine or to give one dose of PCV20 at least five years after the last pneumococcal vaccine [##UREF##5##11##]. This patient had PCV13 about a year before, so no vaccine is recommended. Due to recent and dubious guidelines with multiple recommendation choices, the authors advise that, when available, local epidemiology should be addressed in order to cover the most common serotypes, especially in high-risk individuals.</p>", "<p>Reviewing European epidemiology, the last report by the European Centre for Disease Prevention and Control dates to 2018. At that time, the proportion of the five most frequent serotypes of <italic>S. pneumoniae</italic> that caused invasive pneumococcal disease in adults aged 65 or older was three (14.7%), eight (14.0%), 19A (7.6%), 22F (7.4%), and 9N (5.4%). As such, the proportion of serotypes covered by the PCV13 vaccine was 29%, and the proportion covered by PPSV23 was 73%. As such, an effort must be made to upgrade epidemiological data, as the frequency of serotypes not covered by PCV13 or PPSV23 should trigger the implementation of the new vaccines [##UREF##6##12##].</p>" ]

[ "<title>Conclusions</title>", "<p>Fever in a patient with impaired splenic function is an emergency, and it should be promptly and adequately identified and managed as it may have a fulminant course.</p>", "<p>Despite the success of PCVs, with the reduction in PCV-serotype nasopharyngeal colonisation rates in children, leading to herd immunity and reduced incidence of invasive disease, serotypes continue to change as vaccine serotypes disappear from the community and other non-vaccine serotypes take their place. These two population-level phenomena, indirect effects (or herd immunity) and the emergence of replacement strains, contribute to a circle difficult to interrupt. The much-awaited broadly serotype-independent vaccine may be the “holy grail” of pneumococcal vaccine development. Until then, local epidemiology should help tailor the vaccination scheme where the recommendations are dubious.</p>", "<p>The report of cases like this demonstrates the need for continuous serotype surveillance and vaccine development, as even with vaccination and all the therapeutic measures, there are still lives that cannot be saved.</p>" ]

[ "<p>Invasive pneumococcal disease is a serious infection with an elevated case-fatality rate that can be even higher among patients with asplenia. Its impact has been blunted by the widespread use of vaccines; even recently, in 2021, two new pneumococcal conjugate vaccines emerged. The authors present a case of a 58-year-old male, splenectomised with the immunisation schedule complete, who died of invasive pneumococcal disease with a fulminant course. It is highlighted that fever in a patient with impaired splenic function is an emergency, and despite the success of immunisation in reducing pneumococcal carriage and invasive disease, serotypes continue to change. Also, the local epidemiology may help guide situations where the immunisation recommendations are dubious regarding the implementation of the new vaccines.</p>" ]

[ "<title>Case presentation</title>", "<p>We report a case of a 58-year-old Caucasian male with a history of surgical splenectomy due to abdominal trauma at the age of 18. He had been previously hospitalised in July 2021 with pneumococcal shock with an unknown portal of entry; he was discharged after 18 days, making a full recovery. He had only done two doses of the 23-valent PPSV at that point (in 2007 and 2012), so he finished the immunisation schedule with the 13-valent PCV one month after being released from the hospital. After 11 months, he returned to the emergency department, reporting fever, shivering, myalgia, abdominal discomfort, and diarrhoea. Objectively, he was normotensive, slightly tachycardic, febrile, eupnoeic, did not need oxygen supply, and was without any other specific sign in physical examination. Laboratory tests showed a white blood cell (WBC) count of 12.120 x 10⁹/L, with 93% neutrophilia, C-reactive protein (CRP) 7.2 mg/L (reference range &lt;0.3 mg/L), platelet count 239 x 10⁹/L, and serum creatinine (sCr) 1.36 mg/dL (Table ##TAB##0##1##). The chest X-ray was normal (Figure ##FIG##0##1##), and the arterial blood gas showed no signs of hyperlactatemia or respiratory failure. The pneumococcal urinary antigen test was positive, and he was discharged home and medicated with amoxicillin/clavulanic acid. After six hours, he returned to the emergency room in shock, with altered mental status, a very agitated, Glasgow Coma Scale score of 12 (E2 V4 M6), hypotensive, tachycardic, with evident signs of hypoperfusion, tachypnoeic with acute respiratory failure, and an exuberant erythematous rash (Figure ##FIG##1##2##). Laboratory tests revealed haemoconcentration with haemoglobin 21 g/dL, WBC 12 x 10⁹/L, and CRP 157 mg/L, aggravating acute kidney injury with sCr 3.60 mg/dL, severe thrombocytopenia with 28 x 10⁹/L, and hypoglycaemia (Table ##TAB##0##1##). Thoracic-abdominopelvic CT showed extensive lung consolidation bilaterally consistent with congestion, without any other alterations (Figure ##FIG##2##3##). A transthoracic echocardiogram confirmed ventricular hyperkinesia with small hypercontractile ventricles and a small inferior vena cava with marked respiratory variation without any other major alteration. He was intubated and ventilated, and he initiated fluid therapy with crystalloid and vasopressor support with norepinephrine (maximum 3.12 µg/kg/min). Due to the severity of the shock, hydrocortisone 200 mg and albumin 40 g were administered, and he started epinephrine. Regarding antibiotherapy, he has begun taking piperacillin, tazobactam, and vancomycin. Due to the exuberant presentation and the suspicion of streptococcal toxic shock, clindamycin 900 mg and intravenous immunoglobulin (IVIG) 1 g/kg (80 g) were also administered. Three fresh frozen plasmas, one platelet pool, and 10 mg of vitamin K were given to him, and he was put on continuous veno-venous haemofiltration. Despite all of these therapeutic interventions, he remained with refractory hypotension, respiratory failure with a PaO2/FiO2 ratio of 60, disseminated intravascular coagulation with uncontrollable bleeding from intravenous lines, catheters, and mucosal surfaces, and metabolic acidaemia. A real-time polymerase chain reaction assay detected the DNA of <italic>S. pneumoniae</italic> in blood and endotracheal aspirate. The detection for first line serotypes (3, 5, 7, 9, 14, 15, 16, 19, 20, 23, 33, 38) was negative. The sample volume was not enough to test for the second line serotypes (7C, 8, 10, 11, 12, 15A, 17F, 18, 19F, 22F, 31, 34, 35B, 35F).</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Chest X-ray on the first admission showing no significant alteration</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>Diffuse, exuberant erythematous rash that blanches on pressure, a rare finding in streptococcal shock</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG3\"><label>Figure 3</label><caption><title>Thoracic-abdominopelvic CT on the second admission showing extensive lung consolidation bilaterally consistent with congestion</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Comparison of laboratory values</title><p>aPTT: activated partial thromboplastin time; PT: prothrombin time</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">Parameters</td><td colspan=\"2\" rowspan=\"1\">Patient’s results</td><td rowspan=\"2\" colspan=\"1\">Reference range</td></tr><tr><td rowspan=\"1\" colspan=\"1\">First admission</td><td rowspan=\"1\" colspan=\"1\">Second admission</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Haemoglobin</td><td rowspan=\"1\" colspan=\"1\">15.5 g/dL</td><td rowspan=\"1\" colspan=\"1\">21.0 g/dL</td><td rowspan=\"1\" colspan=\"1\">13.0-18.0 g/dL</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Platelet count</td><td rowspan=\"1\" colspan=\"1\">239 x 10⁹/L</td><td rowspan=\"1\" colspan=\"1\">28 x 10⁹/L</td><td rowspan=\"1\" colspan=\"1\">150-400 x 10⁹/L</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">White blood count</td><td rowspan=\"1\" colspan=\"1\">1.12 x 10⁹/L</td><td rowspan=\"1\" colspan=\"1\">12.01 x 10⁹/L</td><td rowspan=\"1\" colspan=\"1\">4.00-11.00 x 10⁹/L</td></tr><tr><td rowspan=\"1\" colspan=\"1\">C-reactive protein</td><td rowspan=\"1\" colspan=\"1\">7.2 mg/L</td><td rowspan=\"1\" colspan=\"1\">157.0 mg/L</td><td rowspan=\"1\" colspan=\"1\">&lt;3.0 mg/L</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Urea</td><td rowspan=\"1\" colspan=\"1\">49 mg/dL</td><td rowspan=\"1\" colspan=\"1\">73 mg/dL</td><td rowspan=\"1\" colspan=\"1\">10-50 mg/dL</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Serum creatinine</td><td rowspan=\"1\" colspan=\"1\">1.36 mg/dL</td><td rowspan=\"1\" colspan=\"1\">3.60 mg/dL</td><td rowspan=\"1\" colspan=\"1\">0.67-1.17 mg/dL</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Lactate</td><td rowspan=\"1\" colspan=\"1\">1.87 mmol/L</td><td rowspan=\"1\" colspan=\"1\">17.49 mmol/L</td><td rowspan=\"1\" colspan=\"1\">&lt;2 mmol/L</td></tr><tr><td rowspan=\"1\" colspan=\"1\">aPTT</td><td rowspan=\"1\" colspan=\"1\">23.6 seconds</td><td rowspan=\"1\" colspan=\"1\">&gt;180 seconds</td><td rowspan=\"1\" colspan=\"1\">24.2-36.4 seconds</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">PT</td><td rowspan=\"1\" colspan=\"1\">13.3 seconds</td><td rowspan=\"1\" colspan=\"1\">37.9 seconds</td><td rowspan=\"1\" colspan=\"1\">10.1-14.2 seconds</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Fibrinogen</td><td rowspan=\"1\" colspan=\"1\">331 mg/dL</td><td rowspan=\"1\" colspan=\"1\">48 mg/dL</td><td rowspan=\"1\" colspan=\"1\">180-350 mg/dL</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Liliana Costa, Núria Jorge, Sofia Silva, André Silva-Pinto, José-Artur Paiva</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Liliana Costa, Sofia Silva</p><p><bold>Drafting of the manuscript:</bold>  Liliana Costa, André Silva-Pinto, José-Artur Paiva</p><p><bold>Supervision:</bold>  Liliana Costa, Núria Jorge, Sofia Silva, José-Artur Paiva</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Núria Jorge, Sofia Silva, André Silva-Pinto, José-Artur Paiva</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0016-00000052255-i01\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0016-00000052255-i02\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0016-00000052255-i03\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Tuberculosis (TB) is a major health problem. The World Health Organization has defined latent tuberculosis infection (LTBI) as a state of persistent immune response to stimulation by Mycobacterium tuberculosis antigens without evidence of clinically manifested active TB [##UREF##0##1##]. TB bacteria can remain dormant for years, and in 10% of individuals with LTBI, the infection may progress to active TB. In half of these individuals, the progression occurs within the first two years of acquiring the infection, and in the other half, progression occurs after two years. The overall prevalence rates of LTBI in the Middle East and North African regions are 41.78% and 43.81% of the adult population, respectively [##REF##33564476##2##].</p>", "<p>Many immunosuppressive drugs are potent T-cell inhibitors that can impair the interferon response. Studies have shown that patients undergoing immunosuppression treatment, particularly with regard to tumor necrosis factor alpha (TNF-α), have an increased risk of TB; for example, the relative risk is 29.3 for patients taking adalimumab and 18.6 for those taking infliximab [##REF##19386692##3##,##UREF##1##4##]. Furthermore, a study by the British Society for Rheumatology Biologics Register reported a 3- to 4-fold higher rate of TB with infliximab (144 events/100,000 person-years) and adalimumab (136/100,000 person-years) in comparison with etanercept (39/100,000 person-years) group [##REF##19854715##5##].</p>", "<p>Screening for LTBI and treating individuals who test positive are the cornerstones of TB prevention and are particularly important in high-risk patients, especially those with autoimmune disorders [##REF##19386692##3##].</p>", "<p>This study aimed to evaluate the frequency of positive and indeterminate interferon-gamma release assay (IGRA) tests, the management approach, and the risk of TB reactivation in rheumatologic patients at a tertiary hospital in the United Arab Emirates (UAE).</p>" ]

[ "<title>Materials and methods</title>", "<p>A single-center retrospective observational study was performed at Tawam Hospital, Abu Dhabi, UAE. Ethical approval for this study was obtained from the Tawam Human Research Ethics Committee. The department record system was searched to identify all patients on immunosuppression therapy for recruitment in the study. All adult patients (aged ≥16 years) attending the rheumatology clinic during a 12-year period (October 2010-April 2022) were enrolled. Those with positive and indeterminate IGRA testing were included in the analysis. Patients with negative IGRA results, those lost to follow-up, and those with active TB at the time of diagnosis of an autoimmune disease were excluded. A chart review was performed to gather demographic, radiological, and clinical data and management outcomes of patients with positive and indeterminate IGRA tests. The need for infectious disease (ID) referral and the use of anti-TB medications were evaluated. Moreover, long-term follow-up data were collected to determine the risk of TB reactivation in the cohort.</p>", "<p>Statistical analysis</p>", "<p>Descriptive data are expressed as mean ± standard deviation (SD), median (range), or number and frequency, as applicable. Quantitative variables are expressed as mean and SD or median and quartile. For the comparison of the groups, the Wilcoxon or Mann-Whitney test for the means was used depending on the normality test. Univariate and multivariate logistic regression analyses were performed to identify the factors correlated with positive and indeterminate tests. The significance level was set at P &lt; 0.05. Statistical analysis was performed using the Jamovi 2.3.21.0 program (Jamovi Project, <ext-link xlink:href=\"https://www.jamovi.org\" ext-link-type=\"uri\">https://www.jamovi.org</ext-link>). RStudio (Version 2023.03.0+386) and R (Version 4.2.3) were employed for data cleaning and logistic regression modeling.</p>" ]

[ "<title>Results</title>", "<p>A total of 1,012 positive and 223 indeterminate LTBI tests were identified in the 12-year period, of which 39 indeterminate and 123 positive results met the inclusion criteria.</p>", "<p>Indeterminate IGRA results</p>", "<p>Thirty-nine rheumatologic patients had indeterminate IGRA results. Twenty-four (61.5%) were women and 22 (56.4%) were UAE nationals, and their mean age was 38.6 years (SD=17.1). The predominant rheumatologic conditions in the cohort were systemic lupus erythematosus (SLE) (21, 53.8%), rheumatoid arthritis (RA) (four, 10.3%), psoriatic arthritis (PSA) (four, 10.3%), and small vessel vasculitis (five, 12.8%). The median duration of rheumatologic disease since diagnosis was 7.75 years (6 months-15 years).</p>", "<p>Conventional synthetic disease-modifying antirheumatic drugs (csDMARDs) were used in 13 (33%) of the patients. Corticosteroids were used in 26 (66.7%), and the mean prednisolone dose at the time of the IGRA test was 91 ± 241 ­­mg. Moreover, four (10.3%) of the patients were covered with biologics at the time of the IGRA test for various reasons, including switching immunosuppression, in-patient work-up owing to symptoms, contact with patients having active TB, and periodic medical check-ups, or for no reason at all. Table ##TAB##0##1## provides a comparison between the characteristics of patients with positive and indeterminate IGRA results.</p>", "<p>In almost one-third of the patients (n = 14, 35.9%), the IGRA tests were repeated, which revealed indeterminate results in seven patients, negative results in four patients, and positive results in three patients. Chest radiographs were acquired for two-third of patients (n = 26, 66.7%), and only one-third of patients (n = 13, 33%) required chest computed tomography (CT). Additionally, ID consultations were sought for 43.6% (n = 17) of the cases. A total of eight (20.5%) patients received anti-TB medications owing to a diagnosis of LTBI (isoniazid monotherapy (INH) and vitamin B6 for nine months). The majority of patients were on maintenance immunosuppression medications and biologics during the follow-up period without evidence of reactivation of TB infection.</p>", "<p>Positive IGRA results</p>", "<p>Positive IGRA results were obtained in 123 rheumatologic patients. Their mean age was 55.7 years (SD=16.5), 78 patients (63.4%) were UAE nationals, and the female-to-male ratio was 3:1. The most common rheumatologic conditions were RA (n = 69, 56%), SLE (n = 17, 13.8%), PSA (n = 10, 8.1%), and Bechet disease (n = 6, 4.8%). csDMARDs were used in 65 (52.8%) of the patients. Corticosteroids were used in 43 (34.9%), and the mean dose at the time of the IGRA test was 40 ± 166 mg.</p>", "<p>The IGRA test was repeated in 28 (22.8%) patients, chest radiographs were acquired for half of the patients (n = 67, 54.5%), and chest CT was performed in one-fifth of the cases (n = 25, 20.3%). ID consultation was required for sixty patients (48.8%), and five patients had active TB infection. Seventy-four (60%) of the patients were treated with anti-TB medications, including those with active TB infection (n = 5) and those with LTBI (n = 69). Of note, three patients had a previous history of treatment for TB infection. These patients did not receive repeat courses despite positive IGRA testing, and there was no evidence of active infection during follow-up. The other two patients with active TB were on TNFα inhibitor (Figure ##FIG##0##1##).</p>", "<p>In univariate analysis for RA, current prednisolone use and prednisolone dose ≥ 15 mg were independently associated with positive IGRA results (P &lt; 0.05). In multivariate analysis, male sex (odds ratio [OR] = 7.27; 95% confidence interval [CI]: 2.24-27.83, P = 0.002) and current prednisolone use (OR = 4.31; 95% CI: 1.2-16.14, P = 0.026) were associated with positive IGRA results (Table ##TAB##1##2##).</p>" ]

[ "<title>Discussion</title>", "<p>In this study, 123 positive and 39 indeterminate IGRA testing results were obtained in rheumatologic patients over a period of 12 years, with prevalence rates of 12.2% and 17.5%, respectively, across all subspecialties. These rates are much lower than the rate for the general population in the Middle East (41.78%) [##REF##33564476##2##]. The prevalence rate of LTBI in patients with rheumatic diseases differs from country to country, with 20.4% in India [##UREF##2##6##], 21.6% in Morocco [##REF##34173843##7##], and 29.5% in Brazil [##UREF##2##6##]. These discrepancies in the outcomes could be attributed to variations in the study design and the impact of immunosuppression and corticosteroid usage on interpretation [##UREF##2##6##,##REF##34173843##7##].</p>", "<p>This study found an inverse association between old age and positive IGRA results. With every one-year increase in age, the odds of being in the positive IGRA group decreased by approximately 6% in the univariable analysis (OR = 0.94, P &lt; 0.001) and by approximately 4% in the multivariable analysis (OR = 0.96, P = 0.007). This finding is in line with some studies suggesting a strong association between old age (≥65 years of age) and indeterminate results [##REF##24829238##8##,##UREF##3##9##].</p>", "<p>In our cohort, patients with RA were observed to exhibit higher positive IGRA results, which agrees with previous research studies. RA itself increased the risk of TB infection among patients compared with the general population, and this association was independent of the use of biologics [##REF##24608401##10##,##REF##12858438##11##]. In multivariate analysis, no significant association was found between RA and positive and indeterminate IGRA results (OR = 0.44 (0.09-2.17, P = 0.307). In contrast, SLE was more common in the indeterminate group, which is similar to reports in the literature. Maharani et al. [##REF##32884324##12##] demonstrated that active disease status resulted in indeterminate IGRA results in 12.66% of the patients in the SLE group, which was validated by another study [##REF##21562022##13##]. In the present study, the disease activity status was not included as it was out of the study scope.</p>", "<p>Regarding the subsequent management approach, only half of the positive IGRA group had a chest x-ray (CXR), and approximately one-third proceeded to chest CT. Although there is limited evidence for the usefulness of CXR in screening, it is still recommended to improve the specificity. However, chest CT is considered more effective in identifying active TB in patients with a positive IGRA test, particularly if deemed necessary [##REF##36328476##14##,##REF##34183300##15##].</p>", "<p>Owing to uncertainties in the subsequent steps following an indeterminate IGRA result, repeating the test is recommended. Nearly one-third of the patients in our cohort (n = 14, 35.9%) had a repeat test. Throughout the study duration, two-thirds of the patients had a CXR and one-third required chest CT. It is important to note that clear guidelines regarding the use of either approach are lacking unless the repeat test is positive and when necessary [##UREF##4##16##].</p>", "<p>Immunosuppressants and corticosteroids can worsen an abnormal immune response. In this study, two factors contributed to the likelihood of obtaining an indeterminate result. Alongside a diagnosis of SLE, the utilization of corticosteroids negatively influenced the outcome of the IGRA test. Two third (n=14) of the SLE patients in the indeterminate group were treated with corticosteroids (doses of 15 mg, or &gt;15 mg), which could explain the indeterminate result. Previous studies have documented that corticosteroid use increases the likelihood of having an indeterminate result, but some studies have reported conflicting outcomes [##REF##25186123##17##, ####REF##26659461##18##, ##REF##17644549##19####17644549##19##]. Nevertheless, in this study, no differences were seen in the use of corticosteroid doses ≥15 mg between the two studied groups, which could be a consequence of the small sample size.</p>", "<p>With long-term follow-up, two cases (1.6%) of active TB infections were identified in patients with positive IGRA tests. All patients were treated with TNFα inhibitor (Adalimumab, 40 mg every two weeks) in conjunction with methotrexate (10-20 mg, weekly) for a duration of 208 and 312 weeks. In contrast, there were no cases of active TB in the indeterminate group.</p>", "<p>In any case, TNF plays an important role in the host’s response to infection, by maintaining the integrity of the granuloma that forms as a result of the infection. Thus, the use of TNF antagonists causes disruption of the granuloma integrity resulting in mycobacterial growth and activation. This supports our findings, while data from clinical trials reported negligible TB reactivation in non-anti-TNFa biologic [##UREF##2##6##,##REF##24789001##20##].</p>", "<p>Infectious disease standards of care</p>", "<p>Screening for LTBI using the IGRA or tuberculin skin test (TST) is recommended in high-risk patients, including HIV-positive, solid organ transplant, stem cell transplantation, and immunocompromised patients receiving biological therapy, especially TNFa antagonists [##UREF##5##21##]. The IGRA test is better than the TST in previously BCG-vaccinated immunosuppressed patients, with an estimated sensitivity of 67%-75% and specificity of 93%-99% [##REF##22496318##22##,##UREF##6##23##]. A detailed medical history of signs and symptoms suggestive of active TB infection, history of exposure to patients with active TB, travel or migration from endemic areas, type of immunosuppression medications, and medical comorbid conditions should be obtained. LTBI is diagnosed based on positive IGRA/TST testing, negative CXR or chest CT, and no evidence of active TB infection [##UREF##5##21##,##UREF##7##24##]. Consultation with the ID team is required for LTBI treatment in immunosuppressed patients.</p>", "<p>For several years, isoniazid (5 mg/kg, 300 mg/d) (INH) supplemented with pyridoxine (vitamin B6) for nine months was the standard treatment for latent TB. However, this treatment is associated with a risk of hepatic injury and noncompliance with the therapy duration. Regular monitoring for hepatotoxicity is critical in patients receiving INH therapy, and the medication should be withheld whenever indicated. INH-related hepatotoxicity is defined as an increase in liver aminotransferases &gt;5 times the upper normal limit or &gt;3 times the normal limit with symptoms. Alternatively, a 4-month course of rifampin carries a lower risk for liver injury than INH [##REF##30067931##25##]. Nevertheless, rifampin is a potent cytochrome P450 inducer and can accelerate the metabolism of some immunosuppressive agents (calcineurin inhibitors, including cyclosporine and tacrolimus). Hence, drug-drug interactions should be monitored. Three months of daily INH plus rifampin has also been approved for LTBI therapy [##REF##17504682##26##].</p>", "<p>Sterling et al. reported comparable effectiveness for directly observed, once-weekly therapy with rifapentine plus INH for three months and a self-administered nine-month course of daily INH [##REF##22150035##27##]. The three-month therapy was well tolerated, with lower rates of adverse events and higher compliance. The National Tuberculosis Controllers Association and Centers for Disease Control and Prevention 2020 LTBI treatment guidelines recommend the use of rifamycin-based regimens, including three months of once-weekly INH plus rifapentine, four months of daily rifampin, and three months of daily INH plus rifampin. These are the preferred recommended regimens because of their effectiveness, safety, and high treatment completion rates. Alternative LTBI therapeutic regimens are 6 or 9 months of daily INH [##UREF##7##24##].</p>", "<p>Consensus is lacking on the safe period for starting biological or immunosuppressive therapy in patients with LTBI. Some advocate ruling out active TB infection and commencing LTBI treatment 3 weeks to 2 months prior to the initiation of immunosuppressive medications [##UREF##5##21##]. Potential drug-drug interactions should be monitored. How frequently patients undergoing long-term biological or immunosuppressive therapy should be screened for LTBI is not well established. Immunosuppressed patients who were treated for LTBI previously will require further evaluation and ID referral for optimal risk assessment and for determining whether repeat LTBI therapy is needed [##UREF##5##21##].</p>", "<p>Rheumatology standards of care</p>", "<p>According to the 2018 guidelines of the British Society of Rheumatology [##REF##30137623##28##], all patients must be screened for TB before commencing treatment with biologics. This screening involves a clinical examination, a CXR, and either a TST or an IGRA test. If a positive result is obtained for LTBI, the patient must begin treatment at least 1 month before starting biologic therapy, with monitoring at 3-month intervals. Etanercept should be the first line of treatment for patients who require anti-TNF therapy and are at a high risk of TB reactivation because anti-TNF monoclonal antibody medications (particularly adalimumab and infliximab) have a higher risk of TB reactivation than etanercept [##REF##30137623##28##].</p>", "<p>The European Alliance of Associations for Rheumatology [##REF##36328476##14##] recommends that all patients be screened for LTBI before starting treatment with biologic DMARDs or targeted synthetic DAMRDs. Additionally, if a patient is deemed to be at a high risk owing to factors such as alcohol abuse, smoking, living with people who have TB, or living in endemic countries, screening should also be performed if the patient is considering csDMARDs and/or glucocorticoids. No consensus exists on the recommended dose or duration for glucocorticoid usage, but based on previous studies, screening should preferably be done if the glucocorticoid dose is ≥15 mg/d and if the treatment period exceeds four weeks. This screening can be accomplished via CXR and IGRA. However, guidance on how frequently the test should be performed or when it should be repeated has not been updated. The American College of Rheumatology recommends annual testing for high-risk patients who live or travel to endemic countries [##REF##36328476##14##,##REF##22473917##29##]. However, such recommendations must be regularly updated, especially as new medications are developed. Management is based on international guidelines and the most often utilized regimens mentioned previously.</p>", "<p>This is the first study in the UAE to address the IGRA test results in rheumatologic conditions over the course of more than one decade. Some of the limitations were a small sample size of patients, the single-center experience, and a lack of a comparative group.</p>" ]

[ "<title>Conclusions</title>", "<p>Long-term data on the risk of TB activation in positive and indeterminate IGRA results for rheumatological conditions are low. It is recommended to reassess the choice of using anti-TNF-α, with a positive IGRA result if no other feasible alternatives can be offered. Our findings stress the importance of age, underlying diseases, and immunosuppressive treatments in interpreting IGRA results and guiding patient management. A large multicenter study is needed to understand the differences and outcomes of such patients in TB endemic and nonendemic geographical areas.</p>" ]

[ "<p>Introduction</p>", "<p>Prior to immunosuppression, rheumatology patients are routinely screened for latent tuberculosis (TB) infection using interferon-gamma release assays (IGRAs). Variability in the management of latent and indeterminate IGRA results across institutions limited long-term outcome data. A retrospective study was conducted at Tawam Hospital, United Arab Emirates, to investigate the incidence and management protocols associated with positive and indeterminate IGRA results, as well as TB infection, among patients with rheumatic conditions.</p>", "<p>Methods</p>", "<p>A single-center retrospective observational study was performed at Tawam Hospital, Abu Dhabi, UAE. Ethical approval for this study was obtained from the Tawam Human Research Ethics Committee. Laboratory records and the hospital's electronic medical system were used to obtain information about IGRA results over a 12-year period (April 2010-April 2022). The hospital's electronic medical system was used to obtain patient information and subsequent management approaches of positive and indeterminate IGRAs. Moreover, long-term follow-up data were collected to determine the risk of TB reactivation in the cohort.</p>", "<p>Results</p>", "<p>We found a total of 1,012 positive and 223 indeterminate IGRA test results within the 12-year period. Within the rheumatology department, 123 positive and 39 indeterminate IGRA results were identified. In the indeterminate IGRA group, the majority were women (n = 24, 61.5%) and UAE nationals (n = 22, 56.4%), and their mean age was 38.6 years. Systemic lupus erythematosus was the most prevalent rheumatologic condition (n = 21, 53.8%). Thirteen (33.3%) were on disease-modifying anti-rheumatic drugs (DMARDs) and 26 (66.7%) were on corticosteroids during IGRA testing. A total of eight patients (20.5%) received anti-TB medications. In the positive IGRA group, the mean age was 55.7 years and the female-to-male ratio was 3:1. The most common rheumatologic condition was rheumatoid arthritis (n = 69, 56%). Sixty-five (52.8%) patients were on conventional DMARDs, 43 (34.9%) were on corticosteroids during IGRA testing, and 74 (60%) received anti-TB medications. Two cases (1.6%) of active TB infections were detected among patients with positive IGRA tests, both of whom were receiving anti-tumor necrosis factor alpha inhibitor treatment in combination with methotrexate. No cases of active TB infection were observed in the indeterminate IGRA group.</p>", "<p>Conclusion</p>", "<p>Long-term data on the risk of TB activation in positive and indeterminate IGRA results for rheumatological conditions are low. It is recommended to reassess the choice of using anti-TNF-α, with a positive IGRA result if no other feasible alternatives can be offered. Our findings stress the importance of age, underlying diseases, and immunosuppressive treatments in interpreting IGRA results and guiding patient management. A large multicenter study is needed to understand the differences and outcomes of such patients in TB endemic and nonendemic geographical areas.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Post-IGRA screening of rheumatology patients in Tawam Hospital, 2010-2022</title><p>CT, computed tomography; CXR, chest x-ray; ID, infectious disease; IGRA, interferon-gamma release assay; TB, tuberculosis</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Characteristics of patients with positive and indeterminate interferon-gamma release assay (IGRA) testing</title><p>csDMARDs, conventional disease-modifying antirheumatic drugs; EGPA, eosinophilic granulomatosis with polyangiitis; FMF, familial Mediterranean fever; GPA, granulomatosis with polyangiitis; MCTD, mixed connective tissue disease; PSA, psoriatic arthritis; RA, rheumatoid arthritis; SLE, systemic lupus erythematosus; SPA, spondylarthritis; TB, tuberculosis.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Characteristics</td><td rowspan=\"1\" colspan=\"1\">Positive IGRA</td><td rowspan=\"1\" colspan=\"1\">Indeterminate IGRA</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Patients, n</td><td rowspan=\"1\" colspan=\"1\">123</td><td rowspan=\"1\" colspan=\"1\">39</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Female, n (%)</td><td rowspan=\"1\" colspan=\"1\">90 (73.1)</td><td rowspan=\"1\" colspan=\"1\">24 (61.5)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Male, n (%)</td><td rowspan=\"1\" colspan=\"1\">33 (26.8)</td><td rowspan=\"1\" colspan=\"1\">15 (38.5)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Mean age, years</td><td rowspan=\"1\" colspan=\"1\">55.7</td><td rowspan=\"1\" colspan=\"1\">38.6</td></tr><tr><td rowspan=\"1\" colspan=\"1\">National, n (%)</td><td rowspan=\"1\" colspan=\"1\">78 (63.4)</td><td rowspan=\"1\" colspan=\"1\">22 (56.4)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Types of rheumatology diseases, n (%)</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">SLE</td><td rowspan=\"1\" colspan=\"1\">17 (13.8)</td><td rowspan=\"1\" colspan=\"1\">21 (53.8)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">RA</td><td rowspan=\"1\" colspan=\"1\">69 (56)</td><td rowspan=\"1\" colspan=\"1\">4 (10.3)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">PSA</td><td rowspan=\"1\" colspan=\"1\">10 (8.1)</td><td rowspan=\"1\" colspan=\"1\">4 (10.3)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Bechet</td><td rowspan=\"1\" colspan=\"1\">6 (4.8)</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Vasculitis</td><td rowspan=\"1\" colspan=\"1\">11 (8.9)</td><td rowspan=\"1\" colspan=\"1\">5 (12.8)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">SPA</td><td rowspan=\"1\" colspan=\"1\">4 (3.3)</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Others</td><td rowspan=\"1\" colspan=\"1\">6 (4.9)</td><td rowspan=\"1\" colspan=\"1\">5 (12.8)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">FMF</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Dermatomyositis or polymyositis</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Scleroderma</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Sjögren syndrome</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">1</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Sarcoidosis</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">1</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Still disease</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">1</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">MCTD</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">1</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Previous TB infection and treatment, n</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Immunosuppression medications and biologics, n (%)</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">csDMARDs with and without hydroxychloroquine</td><td rowspan=\"1\" colspan=\"1\">65 (52.8)</td><td rowspan=\"1\" colspan=\"1\">13 (33.3)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Methotrexate</td><td rowspan=\"1\" colspan=\"1\">41 (33.3)</td><td rowspan=\"1\" colspan=\"1\">4 (10.3)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Sulfasalazine</td><td rowspan=\"1\" colspan=\"1\">8 (6.5)</td><td rowspan=\"1\" colspan=\"1\">1(2.6)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Leflunomide</td><td rowspan=\"1\" colspan=\"1\">1 (0.8)</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Azathioprine</td><td rowspan=\"1\" colspan=\"1\">10 (8.1)</td><td rowspan=\"1\" colspan=\"1\">2 (5.1)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Mycophenolate mofetil</td><td rowspan=\"1\" colspan=\"1\">5 (4.1)</td><td rowspan=\"1\" colspan=\"1\">6 (15.4)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Anti-TNF therapy</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">5 (4.1)</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">2 (1.6)</td><td rowspan=\"1\" colspan=\"1\">2 (5.1)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Infliximab</td><td rowspan=\"1\" colspan=\"1\">1 (0.8)</td><td rowspan=\"1\" colspan=\"1\">2 (5.1)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Interleukin 17 inhibitors</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ixekizumab</td><td rowspan=\"1\" colspan=\"1\">1 (0.8)</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">JAK-inhibitors</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Tofacitinib</td><td rowspan=\"1\" colspan=\"1\">1 (0.8)</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Others</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Abatacepts</td><td rowspan=\"1\" colspan=\"1\">1 (0.8)</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Rituximab</td><td rowspan=\"1\" colspan=\"1\">2 (1.6)</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Corticosteroid ≥ 15 mg</td><td rowspan=\"1\" colspan=\"1\">23 (18.7)</td><td rowspan=\"1\" colspan=\"1\">15 (38.5)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Concomitant therapy</td><td rowspan=\"1\" colspan=\"1\">40 (32.5)</td><td rowspan=\"1\" colspan=\"1\">14 (35.9)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Median year of rheumatologic condition diagnosis, years (range)</td><td rowspan=\"1\" colspan=\"1\">18.75 (0.5-37)</td><td rowspan=\"1\" colspan=\"1\">7.75 (6 months to 15 years)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">No smoking, n (%)</td><td rowspan=\"1\" colspan=\"1\">84 (68.2)</td><td rowspan=\"1\" colspan=\"1\">27 (69.2)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">IGRA test median, years (range)</td><td rowspan=\"1\" colspan=\"1\">6.75 (0.5-13)</td><td rowspan=\"1\" colspan=\"1\">6.25 (6 months to 12 years)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Repeat IRGA test results, n (%)</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Repeated</td><td rowspan=\"1\" colspan=\"1\">28 (22.8)</td><td rowspan=\"1\" colspan=\"1\">14 (35.9)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Positive</td><td rowspan=\"1\" colspan=\"1\">21 (17.1)</td><td rowspan=\"1\" colspan=\"1\">3 (7.7)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Indeterminate</td><td rowspan=\"1\" colspan=\"1\">1 (0.8)</td><td rowspan=\"1\" colspan=\"1\">7 (17.9)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Negative</td><td rowspan=\"1\" colspan=\"1\">6 (4.9)</td><td rowspan=\"1\" colspan=\"1\">4 (10.3)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Not repeated</td><td rowspan=\"1\" colspan=\"1\">95 (77.2)</td><td rowspan=\"1\" colspan=\"1\">25 (64.1)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">TB infection on follow-up</td><td rowspan=\"1\" colspan=\"1\">2 (1.6)</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">TB lymphadenitis</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">TB pulmonary/pleural</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Anti-TB medications, n (%)</td><td rowspan=\"1\" colspan=\"1\">74 (60.2)</td><td rowspan=\"1\" colspan=\"1\">8 (20.5)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Active TB infection</td><td rowspan=\"1\" colspan=\"1\">5 (3 with a history of active TB prior to their autoimmune diagnosis and 2 after autoimmune diagnosis)</td><td rowspan=\"1\" colspan=\"1\">0</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Latent TB</td><td rowspan=\"1\" colspan=\"1\">69</td><td rowspan=\"1\" colspan=\"1\">8</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>The comparison between positive and indeterminate conditions. The independent variables or predictors include age, sex, whether the individual has rheumatoid arthritis, systemic lupus erythematosus, psoriatic arthritis, vasculitis, current use of prednisolone, use of corticosteroid (≥15 mg), and use of biologics</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Dependent: type</td><td rowspan=\"1\" colspan=\"1\">Positive</td><td rowspan=\"1\" colspan=\"1\">Indeterminate</td><td rowspan=\"1\" colspan=\"1\">OR (univariable)</td><td rowspan=\"1\" colspan=\"1\">OR (multivariable)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Age, mean (SD)</td><td rowspan=\"1\" colspan=\"1\">55.7 (16.5)</td><td rowspan=\"1\" colspan=\"1\">38.6 (17.1)</td><td rowspan=\"1\" colspan=\"1\">0.94 (0.91-0.96, P &lt; 0.001)</td><td rowspan=\"1\" colspan=\"1\">0.96 (0.92-0.99, P = 0.007)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Sex</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Female</td><td rowspan=\"1\" colspan=\"1\">90 (78.9)</td><td rowspan=\"1\" colspan=\"1\">24 (61.5)</td><td rowspan=\"1\" colspan=\"1\">-</td><td rowspan=\"1\" colspan=\"1\">-</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Male</td><td rowspan=\"1\" colspan=\"1\">33 (68.8)</td><td rowspan=\"1\" colspan=\"1\">15 (38.5)</td><td rowspan=\"1\" colspan=\"1\">1.70 (0.79-3.62, P = 0.168)</td><td rowspan=\"1\" colspan=\"1\">7.27 (2.24-27.83, P = 0.002)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Rheumatoid arthritis</td><td rowspan=\"1\" colspan=\"1\">69 (56)</td><td rowspan=\"1\" colspan=\"1\">4 (10.3)</td><td rowspan=\"1\" colspan=\"1\">0.09 (0.03-0.24, P &lt; 0.001)</td><td rowspan=\"1\" colspan=\"1\">0.44 (0.09-2.17, P = 0.307)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Systemic lupus erythematosus</td><td rowspan=\"1\" colspan=\"1\">17 (13.8)</td><td rowspan=\"1\" colspan=\"1\">21 (53.8)</td><td rowspan=\"1\" colspan=\"1\">7.27 (3.27-16.70, P &lt; 0.001)</td><td rowspan=\"1\" colspan=\"1\">6.51 (1.53-32.67, P = 0.015)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Psoriatic arthritis</td><td rowspan=\"1\" colspan=\"1\">10 (8.1)</td><td rowspan=\"1\" colspan=\"1\">4 (10.3)</td><td rowspan=\"1\" colspan=\"1\">1.29 (0.34-4.13, P = 0.681)</td><td rowspan=\"1\" colspan=\"1\">2.34 (0.37-14.75, P = 0.360)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Vasculitis</td><td rowspan=\"1\" colspan=\"1\">11 (8.9)</td><td rowspan=\"1\" colspan=\"1\">5 (12.8)</td><td rowspan=\"1\" colspan=\"1\">1.50 (0.45-4.43, P = 0.482)</td><td rowspan=\"1\" colspan=\"1\">0.74 (0.13-4.08, P = 0.729)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Current prednisolone</td><td rowspan=\"1\" colspan=\"1\">43 (34.9)</td><td rowspan=\"1\" colspan=\"1\">26 (66.7)</td><td rowspan=\"1\" colspan=\"1\">3.72 (1.77-8.18, P = 0.001)</td><td rowspan=\"1\" colspan=\"1\">4.31 (1.20-16.14, P = 0.026)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Corticosteroid, ≥ 15 mg</td><td rowspan=\"1\" colspan=\"1\">23 (18.7)</td><td rowspan=\"1\" colspan=\"1\">15 (38.5)</td><td rowspan=\"1\" colspan=\"1\">2.72 (1.23-5.98, P = 0.013)</td><td rowspan=\"1\" colspan=\"1\">0.88 (0.24-3.23, P = 0.840)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Biologic /small molecules</td><td rowspan=\"1\" colspan=\"1\">13 (10.6)</td><td rowspan=\"1\" colspan=\"1\">4 (10.3)</td><td rowspan=\"1\" colspan=\"1\">0.76 (0.21-2.25, P = 0.650)</td><td rowspan=\"1\" colspan=\"1\">1.59 (0.31-6.99, P = 0.552)</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Shamma Al Nokhatha, Fatima AlKindi , Merna Abdelsalhen, Fatima AlKhyeli, Maryam Alfalasi</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Shamma Al Nokhatha, Fatima AlKindi , Merna Abdelsalhen, Fatima AlKhyeli, Ahmad R. Alsaber , Maryam Alfalasi</p><p><bold>Drafting of the manuscript:</bold>  Shamma Al Nokhatha, Fatima AlKindi , Merna Abdelsalhen, Fatima AlKhyeli, Ahmad R. Alsaber , Maryam Alfalasi</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Shamma Al Nokhatha, Fatima AlKindi , Ahmad R. Alsaber </p><p><bold>Supervision:</bold>  Shamma Al Nokhatha</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study. Tawam Human Research Ethics Committee issued approval KD/AJ/837</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Animal Ethics</title><fn fn-type=\"other\"><p><bold>Animal subjects:</bold> All authors have confirmed that this study did not involve animal subjects or tissue.</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050581-i01\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Lemierre syndrome is classically characterized by bacterial invasion of the pharyngeal mucosa, often preceded by a bacterial or viral pharyngeal infection [##REF##28698043##1##,##REF##32303484##2##], leading to the development of internal jugular vein (IJV) thrombophlebitis and disseminated septic emboli [##REF##28698043##1##, ####REF##32303484##2##, ##REF##27695351##3####27695351##3##]. The most frequent causative organism is <italic>Fusobacterium necroforum</italic>, an anaerobic gram-negative rod, which has become synonymous with the disease. However, various other bacteria have been isolated in cases of Lemierre syndrome and should be considered when beginning empiric therapy. We describe such a case here.</p>" ]

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[ "<title>Discussion</title>", "<p>Here we report a case of Lemierre syndrome in an otherwise healthy male without obvious signs of oropharyngeal involvement on initial presentation and initial findings consistent with pneumonia rather than septic thrombophlebitis. Lemierre syndrome was first reported by French physicians Courmont and Cade in 1900 [##UREF##0##4##] and was described by French bacteriologist Andre-Alfred Lemierre in 1936 [##UREF##1##5##,##REF##32445216##6##]. The syndrome was far more prevalent in the “pre-antibiotic\" era, where treatments were often limited to IJV excision or ligation [##REF##24152679##7##]. Due to the widespread use of antibiotics, rates of Lemierre syndrome have declined so significantly that some have labeled it a “forgotten disease” [##REF##8427443##8##]. However, in recent decades, the rates of Lemierre syndrome have increased [##REF##9796654##9##,##UREF##2##10##], possibly due to reduced antibiotic use for pharyngitis and improvement in imaging techniques [##REF##26261192##11##]. Incidence ranges from 3 to 14 cases per million persons, depending on the population studied [##REF##32303484##2##,##REF##32445216##6##]. Incidence rates are higher in adolescent and young adult patients [##REF##32303484##2##]. Despite antibiotics, it remains a serious pathology with mortality rates as high as 18% [##REF##24152679##7##].</p>", "<p>Lemierre syndrome typically begins as an infection in the palatine tonsils and peritonsillar tissues [##REF##24152679##7##], although other primary sources, such as sinuses, mastoid, oral, and auricular have been reported [##REF##19944898##12##]. Invasion of the local tissue is not fully understood but may be due to an initial insult from viral or bacterial pharyngitis combined with bacterial-specific factors [##REF##32445216##6##,##REF##24152679##7##]. In many cases, there is no obvious inciting illness or injury, as observed in the case discussed here. Resulting bacteremia results in thrombophlebitis of the IJV [##REF##24152679##7##]. From here, the thrombi embolize multiple tissues, most frequently the lungs, joints, or brain [##REF##12441902##13##,##REF##19554637##14##]. In rare cases, the thrombus may propagate to the subclavian or cranial sinuses [##REF##24152679##7##]. Septic emboli in the lungs may result in abscesses, sterile effusions empyema, and cavitation [##REF##32303484##2##,##REF##31489210##15##]. Indeed, most investigations for Lemierre syndrome begin with chest X-rays, possibly due to associated lung pathology from septic emboli [##REF##24152679##7##].</p>", "<p>Lemierre syndrome occurs most frequently in healthy young adults, often males, in the second and third decades [##REF##32303484##2##]. The reasons for this are unclear but may be due to the frequency of tonsillitis and pharyngitis in this demographic [##REF##34468182##16##]. Diagnosis of Lemierre syndrome relies on diagnostic for identification of IJV thrombophlebitis, with CT the most common non-plain film modality [##REF##19554637##14##]. Several authors note that a high degree of clinical suspicion is often needed to appropriately identify this condition [##REF##32303484##2##], especially as the disease may initially be treated as pharyngitis or pneumonia. As mentioned by Lee et al., the presence of deep neck infections, septicemia, IJV thrombophlebitis, and signs of metastatic infection (such as septic emboli) should raise suspicion for Lemierre syndrome [##REF##32303484##2##], especially if present in an otherwise healthy young adult. Early clinical signs and radiologic findings are crucial as the prolonged growth of anaerobic gram-negative bacteria, such as <italic>F. necrophorum</italic>, may delay diagnosis [##REF##32303484##2##].</p>", "<p>While <italic>F. necrophorum </italic>is the most frequent causative organism (81.7% of cases according to Chirinos et al. [##REF##12441902##13##]), various bacteria have been isolated, though at far lower rates. The <italic>S. anginosus </italic>group (SAG) typically colonizes the reproductive and digestive tracts as well as the respiratory cavity and can cause visceral suppurative infections [##REF##26502716##17##]. They are unique in their tendency to form abscesses and empyema. However, determining whether they are causal in a given infection can be difficult since they are resident oral cavity and respiratory tract flora [##REF##26502716##17##]. A very small number of Lemierre syndrome cases involving the SAG species have been reported in the literature. As such, SAG appears to represent an uncommon group of pathogens in this syndrome [##REF##31315119##18##]. Polymicrobial infections involve up to 30% of cases and, in many cases, are in combination with <italic>F. necrophorum</italic> [##REF##19944898##12##].</p>", "<p>Treatment involves empiric therapy, which is narrowed once bacteria are specified. Given the prevalence of <italic>F. necrophorum</italic> resistance of β-lactams, macrolides, fluoroquinolone, and aminoglycosides, β-lactamase-resistant antibiotics are often the recommended treatment [##REF##27695351##3##]. Treatment length has not been established with randomized controlled trials but treatment for several weeks is often recommended [##REF##15769572##19##]. Anticoagulation has been debated. Some have supported anticoagulation in cases where there are recurrent emboli, thrombus extension, or lack of improvement with antibiotic therapy [##REF##24152679##7##], while others have opposed it due to the risk of bleeding. Multiple retrospective analyses have shown no benefit to anticoagulation. For example, a retrospective study of 394 patients found no difference in mortality [##UREF##3##20##]. Unfortunately, Lemierre syndrome can result in long-term complications; one study noted serious sequelae, such as neurologic deficits, in &gt;10% of patients with Lemierre syndrome, possibly due to complications from septic emboli [##REF##32445216##6##]. </p>" ]

[ "<title>Conclusions</title>", "<p>In sum, we present a unique case of Lemierre syndrome with blood culture positive for <italic>S. constellatus</italic>. Clinicians should be cognizant of Lemierre syndrome as a cause of septic emboli in young, healthy adults and recognize that a variety of pathogens may be causative. In some cases, such as this one, patients may lack obvious clinical signs of oropharyngeal infection on initial presentation. As demonstrated here, infection and emboli of unknown origin may warrant imaging of the neck vasculature for thrombi.</p>" ]

[ "<p>Lemierre syndrome is characterized by thrombophlebitis of the internal jugular vein (IJV) secondary to bacterial pharyngitis or tonsillitis. Though antibiotic use has made this a rarer syndrome, it can nevertheless manifest in patients presenting with pharyngitis. Herein, we describe a 20-year-old male patient with no relevant medical history presenting with signs concerning for pneumonia and was ultimately diagnosed with Lemierre syndrome with <italic>Streptococcus constellatus</italic> bacteremia. Complications included IJV thrombus with presumed septic emboli to the lungs. The patient was discharged on ampicillin/sulbactam with plans to transition to amoxicillin/clavulanate.</p>" ]

[ "<title>Case presentation</title>", "<p>A 20-year-old male with no significant past medical history presented to the emergency department with a four-day history of cough, shortness of breath, non-bloody diarrhea, non-bloody emesis, decreased appetite, body aches, sweats, fevers up to 103º F and significant fatigue. He also reported a recent sore throat which had resolved prior to presentation. No signs of neck or oropharyngeal pathology were noted by the emergency medicine team. During this initial encounter, he was noted to have a leukocytosis (12.5 K/uL), mild anemia (12.7 g/dL), and an elevated D-dimer level (5.8 FEU/mL)(Table ##TAB##0##1##). Although no remarkable findings were seen on the chest X-ray, a CT scan of the chest showed multifocal consolidative changes throughout the middle and bilateral lobes, with no evidence of deep vein thrombosis (DVT). The patient had performed several COVID-19 tests prior to presentation, all of which were negative. He was started on doxycycline for presumed community-acquired pneumonia and was sent home with instructions to return if symptoms worsened.</p>", "<p>Over the next 24 hours, the patient’s shortness of breath progressed and he returned to the emergency department, where labs revealed a worsening leukocytosis (16.2 K/uL) and an elevated pro-BNP (3292 pg/mL). A viral panel including testing for SARS-COV-2, influenza A/B, and RSV was negative. Hazy bilateral infiltrates were now evident on the chest X-ray. A CT scan of the chest continued to demonstrate bilateral multifocal infiltrates consistent with atypical pneumonia and concerning for possible septic emboli (Figure ##FIG##0##1##), ultimately concerning for sepsis. He was started on empiric antibiotic coverage with vancomycin, cefepime, and azithromycin. A transthoracic transesophageal echocardiogram observed a left ventricular ejection fraction of 50% without any vegetation. A CT of the head and brain with and without contrast was unremarkable and without signs of emboli. The following day, worsening bilateral nodular infiltrates were seen on a repeat chest X-ray. Laboratory results showed worsening leukocytosis (17.4 K/uL), anemia (hemoglobin 10.9 g/dL) with a normal haptoglobin and slightly elevated LDH (300 IU/L), thrombocytopenia (platelets 49 K/uL), elevated procalcitonin (27.70), and elevated ferritin (592.9 ng/mL). He was subsequently transferred to the intensive care unit for a higher level of care given the worsening pneumonia and risk of decompensation.</p>", "<p>Upon presentation to the ICU, the patient was febrile (103.1 °F), tachycardic (122 bpm), tachypneic (RR 31), and with oxygen saturation at 94% on 2 liters nasal cannula. Physical exam was notable for bilateral cervical lymphadenopathy and bilateral wheezes throughout the upper lung fields. Oral examination was unremarkable, though sore throat prior to presentation was concerning for possible oral source of infection. Antibiotic coverage was subsequently changed to amoxicillin/clavulanic acid and doxycycline to cover atypical pneumonia, anaerobes, and tick-borne illnesses. The infectious diseases team was consulted and initiated an extensive work-up in addition to previous studies, given the infection of unknown etiology, elevated inflammatory markers, and thrombocytopenia (Table ##TAB##1##2##). Preliminary blood culture results identified <italic>Streptococcus anginosus</italic> group by Verigene. A CT scan of the neck with contrast demonstrated a thrombus in the left IJV (Figure ##FIG##1##2##). Antibiotics were then narrowed to ampicillin/sulbactam to cover Streptococcus species. Given the presence of septic emboli, anticoagulation was discussed with the infectious disease team and was ultimately decided against. A subsequent transesophageal echocardiogram noted an improved ejection fraction (60-65%) without any vegetations or intracardiac shunts. Finalized blood culture results showed revealed <italic>Streptococcus constellatus</italic>. At this time, findings were most consistent with a<italic> S. constellatus</italic> infection with multifocal pneumonia secondary to septic thrombophlebitis (Lemierre’s syndrome). </p>", "<p>The patient subsequently began to defervesce, with decreasing frequency of fevers and improvement of his leukocytosis and other laboratory parameters, including resolution of thrombocytopenia. A repeat chest X-ray continued to show multifocal opacities consistent with septic emboli, but overall interval improvement in aeration. The patient was subsequently discharged with an additional three weeks of intravenous (IV) ampicillin/sulbactam, transitioning to oral amoxicillin/clavulanic acid for an additional three weeks.</p>" ]

[ "<p>We would like to thank Dr. David C. Keyes for his assistance with radiological imaging interpretation.</p>" ]

[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>CT Chest with Pulmonary Septic Emboli</title><p>CT chest without contrast showing pulmonary septic emboli (arrows) and small bilateral pleural effusions. Scattered airspace opacities are most prominent in the lower lobes. Findings are consistent with bilateral pneumonia. </p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>CT Soft Tissue Neck With Thrombus in the Left Internal Jugular Vein</title><p>CT soft tissue neck with contrast showing a filling defect within the left internal jugular vein consistent with a moderate thrombus (arrows). Diffuse fat stranding is also noted throughout, suggesting a degree of anasarca.  No focally compressive phlegmons or abscesses are noted.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Selected Lab during Illness Course</title><p>ED = emergency department, ICU = intensive care unit, WBC = white blood cells, pro-BNP = pro-B-type natriuretic peptide</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Test</td><td colspan=\"4\" rowspan=\"1\">Result</td><td rowspan=\"1\" colspan=\"1\">Reference Range</td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Initial ED Visit</td><td rowspan=\"1\" colspan=\"1\">Subsequent ED Visit and Admission</td><td rowspan=\"1\" colspan=\"1\">Transfer and ICU Upgrade</td><td rowspan=\"1\" colspan=\"1\">Discharge</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">WBC (K/uL)</td><td rowspan=\"1\" colspan=\"1\">12.5</td><td rowspan=\"1\" colspan=\"1\">16.2</td><td rowspan=\"1\" colspan=\"1\">17.4</td><td rowspan=\"1\" colspan=\"1\">7.6</td><td rowspan=\"1\" colspan=\"1\">4.0 – 10.5 </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Hemoglobin (g/dL)</td><td rowspan=\"1\" colspan=\"1\">12.7</td><td rowspan=\"1\" colspan=\"1\">12.5</td><td rowspan=\"1\" colspan=\"1\">11.2</td><td rowspan=\"1\" colspan=\"1\">12.0</td><td rowspan=\"1\" colspan=\"1\">13.0 – 16.0 </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Platelet Count (K/uL)</td><td rowspan=\"1\" colspan=\"1\">81</td><td rowspan=\"1\" colspan=\"1\">56</td><td rowspan=\"1\" colspan=\"1\">49</td><td rowspan=\"1\" colspan=\"1\">596</td><td rowspan=\"1\" colspan=\"1\">130 – 400 </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Lactic Acid (mmol/L)</td><td rowspan=\"1\" colspan=\"1\">1.6</td><td rowspan=\"1\" colspan=\"1\">1.8</td><td rowspan=\"1\" colspan=\"1\">0.9</td><td rowspan=\"1\" colspan=\"1\">-</td><td rowspan=\"1\" colspan=\"1\">0.5 – 2.0 </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Pro-BNP (pg/ml)</td><td rowspan=\"1\" colspan=\"1\">-</td><td rowspan=\"1\" colspan=\"1\">3292.0</td><td rowspan=\"1\" colspan=\"1\">1771.0</td><td rowspan=\"1\" colspan=\"1\">-</td><td rowspan=\"1\" colspan=\"1\">&lt; 125</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>Infectious and Immunological Work-Up</title><p>EIA = Enzyme Immunoassay, PCR = Polymerase chain reaction, Ab = Antibody, Ag = Antigen, CMV = Cytomegalovirus, EBV = Epstein Barr Virus, HIV = Human Immunodeficiency Virus, TB = Tuberculosis</p><p>*Results are indeterminate for response to ESAT-6 and/or CFP-10 test antigens. </p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Test</td><td rowspan=\"1\" colspan=\"1\">Result</td><td rowspan=\"1\" colspan=\"1\">Reference Range</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Blastomyces Ag (EIA)</td><td rowspan=\"1\" colspan=\"1\">None Detected</td><td rowspan=\"1\" colspan=\"1\">None Detected</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">CMV DNA (PCR)</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Coccidioides Ag (EIA)</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Cryptococcal Ag</td><td rowspan=\"1\" colspan=\"1\">Negative</td><td rowspan=\"1\" colspan=\"1\">Negative</td></tr><tr><td rowspan=\"1\" colspan=\"1\">EBV DNA (PCR)</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Histoplasma Ag (EIA)</td><td rowspan=\"1\" colspan=\"1\">None Detected</td><td rowspan=\"1\" colspan=\"1\">None Detected</td></tr><tr><td rowspan=\"1\" colspan=\"1\">HIV-1/2 Ag/Ab (4^th Gen)</td><td rowspan=\"1\" colspan=\"1\">Non-Reactive</td><td rowspan=\"1\" colspan=\"1\">Non-Reactive</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">HIV-1 RNA (PCR)</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Legionella Pneumophila Urine Ag</td><td rowspan=\"1\" colspan=\"1\">Negative</td><td rowspan=\"1\" colspan=\"1\">Negative</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Monotest</td><td rowspan=\"1\" colspan=\"1\">Negative</td><td rowspan=\"1\" colspan=\"1\">Negative</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Quantiferon TB Gold Plus</td><td rowspan=\"1\" colspan=\"1\">Indeterminate*</td><td rowspan=\"1\" colspan=\"1\">Negative</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">          Nil</td><td rowspan=\"1\" colspan=\"1\">0.06 IU/mL</td><td rowspan=\"1\" colspan=\"1\">-</td></tr><tr><td rowspan=\"1\" colspan=\"1\">          Mitogen-Nil</td><td rowspan=\"1\" colspan=\"1\">0.34 IU/mL</td><td rowspan=\"1\" colspan=\"1\">-</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">          TB1-NIL</td><td rowspan=\"1\" colspan=\"1\">&lt;0.00 IU/mL</td><td rowspan=\"1\" colspan=\"1\">-</td></tr><tr><td rowspan=\"1\" colspan=\"1\">          TB2-NIL</td><td rowspan=\"1\" colspan=\"1\">&lt;0.00 IU/mL</td><td rowspan=\"1\" colspan=\"1\">-</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Tickborne panel</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">           <italic>A. Phagocytophilum </italic>DNA (PCR)</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">          <italic> Babesia microti</italic> DNA (PCR)</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td></tr><tr><td rowspan=\"1\" colspan=\"1\">          <italic> B. miyamotoi</italic> DNA (PCR)</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">          <italic> Borrelia</italic> spp DNA (PCR)</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td></tr><tr><td rowspan=\"1\" colspan=\"1\">           <italic>Ehrlichia chaffeensis</italic> DNA (PCR)</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td><td rowspan=\"1\" colspan=\"1\">Not Detected</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Viral Hepatitis</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">          Hepatitis A Ab (Total)</td><td rowspan=\"1\" colspan=\"1\">Reactive</td><td rowspan=\"1\" colspan=\"1\">Non-Reactive</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">          Hepatitis A Ab (IgM)</td><td rowspan=\"1\" colspan=\"1\">Negative</td><td rowspan=\"1\" colspan=\"1\">Negative</td></tr><tr><td rowspan=\"1\" colspan=\"1\">          Hepatitis B Surface Ag</td><td rowspan=\"1\" colspan=\"1\">Non-Reactive</td><td rowspan=\"1\" colspan=\"1\">Non-Reactive</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">          Hepatitis B Surface Ab</td><td rowspan=\"1\" colspan=\"1\">&lt;10.00 mIU/ml</td><td rowspan=\"1\" colspan=\"1\">&lt;10.00 mIU/ml</td></tr><tr><td rowspan=\"1\" colspan=\"1\">          Hepatitis C Ab</td><td rowspan=\"1\" colspan=\"1\">Non-Reactive</td><td rowspan=\"1\" colspan=\"1\">Non-Reactive</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Immunoglobulin Panel</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">          IgM</td><td rowspan=\"1\" colspan=\"1\">74 mg/dL</td><td rowspan=\"1\" colspan=\"1\">50 – 300 mg/dL</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">          IgG</td><td rowspan=\"1\" colspan=\"1\">760 mg/dL</td><td rowspan=\"1\" colspan=\"1\">600 – 1640 mg/dL</td></tr><tr><td rowspan=\"1\" colspan=\"1\">          IgA</td><td rowspan=\"1\" colspan=\"1\">113 mg/dL</td><td rowspan=\"1\" colspan=\"1\">47 – 310 mg/dL</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Christopher J. Peterson, Lauren N. Mazin, Miles Thomas</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Christopher J. Peterson, Lauren N. Mazin, Jonas Rawlins, Miles Thomas</p><p><bold>Drafting of the manuscript:</bold>  Christopher J. Peterson, Lauren N. Mazin, Miles Thomas</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Christopher J. Peterson, Jonas Rawlins, Miles Thomas</p><p><bold>Supervision:</bold>  Jonas Rawlins</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050580-i01\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050580-i02\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>There is an increase in the burden of cancer globally due to population growth, aging, and an increase in risk factors such as obesity, smoking, and diet [##REF##26181261##1##]. Head and neck cancers most commonly occur in the oral cavity. Oral cancer's prognosis and survival rates are poor despite significant advancements in its treatment [##REF##32994033##2##,##REF##17306612##3##]. Cancer in the maxillary arch is an uncommon tumor with higher mortality, and 10% of all oral cancers develop in the oral cavity subsites of the upper gingiva and hard palate [##REF##23733663##4##].</p>", "<p>Based on the tissue from whence they originated, malignant tumors of the maxilla can be categorized as squamous cell carcinoma, salivary gland tumors such as mucoepidermoid carcinomas, mesenchymal tumors such as chondrosarcomas, and other malignancies, including basal cell carcinoma and malignant schwannoma [##REF##16256412##5##].</p>", "<p>Using free flaps and advances in microvascular surgery, many oncology patients with palatal tumors have been able to have their tumors resected and immediately reconstructed after the surgery. A flap with vascularized bone is an ideal option to optimize the future prosthetic bearing area. In the event that the resection site cannot be closed surgically, an obturator must be provided. In addition to improving chewing, swallowing, speech, dental aesthetics, and facial support, the obturator restores the partition between the nasal and oral cavities, thus improving quality of life [##UREF##0##6##].</p>", "<p>Postsurgical maxillary defects can result in several problems, such as hypernasal speech, nasal fluid leakage, the high potential for aspiration, poor aesthetics, and impaired masticatory function [##REF##11580820##7##]. Therefore, treatment of the maxillary defects through surgery or prosthodontics is crucial to these patients' recovery. Some oncology patients may require conventional rehabilitation with an obturator following surgery [##REF##14603449##8##].</p>", "<p>Patients who have had a maxillectomy typically undergo several stages of prosthetic treatment. First, a surgical obturator is made and worn for the first one to four weeks after the procedure. Next, an interim obturator is made and worn for three to six months until the defect is improved, and finally, a long-term obturator is made [##REF##21781182##9##].</p>", "<p>In the initial postoperative phase, a surgical obturator acts as a partition between the oral and nasal cavities, enabling relatively normal speaking and deglutition and minimizing the psychological effects of the operation and the hospital stay. Additionally, it offers a matrix for surgical packing and lowers the chance of surgical wound contamination [##UREF##1##10##].</p>", "<p>After the surgery, the surgical obturator can be adjusted to accommodate changes in the defect and surrounding tissues. In the meantime, an interim obturator can assist with oral functions until the wound has fully healed and the defect has achieved stability in terms of shape and size. Once the maxillary defect has healed and become stable, a permanent obturator can be used for long-term restoration. An effective seal of the defect is crucial to preventing liquid leakage into the nasal canal. Removable prostheses must be constructed with adequate support, retention, and stability to ensure proper functionality. The type and size of the defect, the presence of supporting palatal shelves, and the condition of the remaining dentition are essential factors that influence the movement of the prosthesis during use. In cases of incomplete dentition, the remaining teeth can serve as abutments, improving the prognosis of the prosthesis [##UREF##1##10##].</p>", "<p>Care must be taken to prevent overload of the remaining dentition and to retain these teeth to the best of their ability. Various types of obturators have been used, such as hollow bulbs, full bulbs, and two-piece obturators. Obturators with a hollow design are often preferred for their light weight [##REF##28473930##11##].</p>", "<p>This case report discusses the step-by-step process of creating a cobalt chromium obturator, which is a special type of dental device used to close a gap in the palatal bone of the upper jaw. The report focuses on the clinical stages involved in making a one-part hollow box obturator.</p>" ]

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[ "<title>Discussion</title>", "<p>Individuals who have undergone a maxillectomy often encounter recurring challenges in prosthodontic treatment related explicitly to insufficient support, retention, and stability. The extent of the defect, the number of remaining teeth, the amount of remaining bone structure, the condition of the surrounding mucosa, the impact of radiation therapy, and the patient's ability to adjust to the prosthetic device all play a role in determining the outlook for prosthodontic treatment in these individuals [##REF##33386202##13##]. Saving as many remaining teeth as feasible for patients undergoing unilateral maxillectomy may be vital for optimal prosthesis design and performance [##UREF##3##14##]. The other components are subjected to continual pressure from such a massive, hefty obturator, impairing tissue health, patient function, and comfort.</p>", "<p>After the obturator has been processed into acrylic resin, the bulb component is frequently hollowed out to reduce the overall weight of the prosthesis. The extent of the maxillary defect determines whether a hollow maxillary obturator is appropriate. By incorporating a hollow design, the weight of the prosthesis can be reduced by as much as 33% [##REF##17931364##15##].</p>", "<p>The obturator prosthesis is critical to recovering oral function in postsurgical maxillectomy patients. Framework designs for obturators may differ depending on the defect classification system [##UREF##4##16##]. Removable obturator prostheses should adhere to fundamental prosthodontic principles, which include distributing stress over a wide area, employing a rigid major connector for cross-arch stabilization, and incorporating stabilizing and retaining components at strategic locations within the arch to minimize the risk of displacement due to functional forces. In this case, a tripodal design was chosen. The remaining teeth, palate, and specifically prepared rests offered support for the prosthesis. Rests were created on the left first and second premolars, the first and second molars, and the right canine on the right side. The complete palate was designed to ensure optimal distribution of functional loads across the underlying tissue [##UREF##4##16##]. </p>", "<p>In patients with remaining natural teeth, these teeth play a crucial role in maintaining, supporting, and stabilizing the obturator. Retention can be achieved through various means, such as utilizing the remaining teeth or ridge, the lateral aspect of the defect, the undercut in the soft tissue, and the scar tissue. Components for stabilization and indirect retention need to be carefully positioned to prevent movement of the portion of the prosthesis that covers the defect.</p>", "<p>Occlusion is the key factor in achieving stability for prostheses. It is crucial to ensure that occlusal forces are evenly distributed in both centric and eccentric jaw positions to minimize prosthesis movement and the resulting forces on individual structures. To reduce stress caused by lateral forces, proper selection of an occlusal scheme, elimination of premature occlusal contacts, and the use of stabilizing components that provide broad distribution are essential [##UREF##2##12##,##UREF##5##17##].</p>", "<p>A metal framework obturator prosthesis offers several advantages, including its durability and ability to conduct heat, allowing normal stimulation of the supporting structure [##UREF##6##18##].</p>", "<p>It is essential to wait for the defect site's complete healing and dimensional stability before constructing the definitive obturator. The timeframe for this can vary between 3 and 6 months following the surgery, depending on various factors, including the tumor’s prognosis, the defect's size, the progress of healing, and whether teeth are present [##UREF##4##16##]. The designs of obturators can differ depending on the classification system used to categorize the defects. A tripodal design was chosen in this specific case, considering the support provided by the remaining teeth and palate. The molars, first premolar, and canine were all stabilized to enhance stability. The remaining palate was also covered to ensure proper distribution of functional loads during oral functions.</p>", "<p>Dental implants have revolutionized the field of prosthodontics, playing a crucial role in removable [##REF##28944354##19##, ####REF##28237542##20##, ##REF##29146396##21####29146396##21##], fixed [##REF##32891403##22##, ####REF##31326057##23##, ##REF##26698006##24####26698006##24##], and maxillofacial prosthesis [##REF##20040025##25##,##REF##32639703##26##]. With their ability to provide stability, functionality, and aesthetic appeal, dental implants have transformed the lives of countless individuals, restoring their oral health and overall well-being.</p>", "<p>Enhancing the quality of life for hemimaxillectomy patients is a difficult task compared to patients with conventional prostheses. However, specialists with expertise, knowledge, and experience can achieve this goal. By implementing a team approach, utilizing skills and experience at each stage, and regularly evaluating the patient, the challenges faced by hemimaxillectomy patients can be effectively overcome [##REF##24163568##27##].</p>" ]

[ "<title>Conclusions</title>", "<p>The primary challenge in a maxillectomy patient's recovery is ensuring adequate retention, stability, and support. A thorough understanding of the patient's needs and extensive expertise is critical in effectively rehabilitating these individuals. The patient's masticatory abilities, speech intelligibility, and overall quality of life can be significantly improved by designing a definitive obturator prosthesis with maximum coverage and appropriate design.</p>" ]

[ "<p>Maxillectomy defects can lead to oroantral communication, causing difficulties with chewing, swallowing, speech, and facial appearance. Prosthodontists play a crucial role in rehabilitating such defects using obturators. This case report presents the fabrication of a definitive obturator with a cast metal framework for a patient who had an acquired maxillary defect and previously experienced issues with an ill-fitting obturator. In this clinical report, the patient's canine teeth on both sides and the premolars and molars on the left side were used for rest placement. Retention was achieved by utilizing the remaining teeth, employing two embrasure Aker clasps on the left molars and premolars and a C-wrought wire clasp on the right canine. A complete palate was designed as the major connector to ensure optimal load distribution to the surrounding tissues. Additionally, an indirect retainer was planned for the right canine. This definitive prosthesis rehabilitated the patient, improving masticatory efficiency, enhancing speech clarity, and improving quality of life.</p>" ]

[ "<title>Case presentation</title>", "<p>A 70-year-old male patient was referred to the Department of Prosthodontics, Tabiah University Dental Hospital, Madina, Saudi Arbia, with a chief complaint regarding a previously ill-fitting acrylic maxillary obturator. The Research Ethical Committee of the College of Dentistry, Taibah University, Madinah, Saudi Arabia, approved this study (approval # 14032022). The specific issues reported by the patient were inadequate retention of the old obturator, stability, leakage, and food accumulation underneath the obturator. As a result, the patient desired to replace the obturator with a more suitable alternative. The patient had undergone a right maxillectomy due to the surgical removal of squamous cell carcinoma from the right maxillary sinus. Following the surgery, the patient received postoperative radiotherapy. Approximately six years ago, an obturator was fabricated for the patient to obturate the defect caused by the maxillectomy. </p>", "<p>The extra-oral examination revealed a class III skeletal base, with no abnormalities detected in the examined lymph nodes, temporomandibular joint (TMJ), or face. The intra-oral examination revealed a surgical defect on the right side of the hard palate resulting from a right maxillectomy. According to Aramany's classification of maxillary defects, this defect is classified as Class II [##UREF##2##12##]. The gingiva on the intact side and lower arch appeared healthy, displaying pink, but with generalized recession. The remaining teeth exhibited a 16% bleeding index and a 34% plaque index. The occlusal examination revealed a Class III malocclusion, characterized by a 0.5 mm anterior open bite and a group function occlusion when the obturator was in place during both centric and eccentric occlusions. Additionally, there was a slight midline shift to the right side (Figure ##FIG##0##1##).</p>", "<p>The patient's diagnosis includes an acquired palatal defect resulting from the surgical removal of a tumor, generalized plaque-induced gingivitis, acquired tooth loss, and a sub-optimal maxillary obturator that is causing leaks. The primary goal of the treatment was to close the communication between the oral and nasal cavities using an obturator. This would artificially block the unrestricted transfer of speech sounds, food, and liquids between these cavities. Additionally, the treatment aimed to enhance the aesthetics and function of the patient's oral cavity.</p>", "<p>The proposed course of treatment involved giving the patient oral health instructions (OHI), performing both supra and subgingival scaling and polishing, offering guidance on using floss and interdental brushes, recommending a fluoridated mouthwash with 0.05% sodium fluoride (NaF), and suggesting the use of a toothpaste with a minimum of 1350 parts per million (ppm) of fluoride. Following these interventions, the plan was to provide the patient with a removable cobalt-chrome partial obturator for the maxilla.</p>", "<p>The maxillary and mandibular impressions were taken using irreversible fast-setting hydrocolloids (Tropicalgin, Zhermack) after modifying the upper stock tray to ensure a better fit and to block out undercuts with petrolatum-laden gauze. These impressions were poured with dental stone type IV to produce study casts (Figure ##FIG##1##2##).</p>", "<p>The maxillary cast was duplicated for future reference. The study casts were accurately surveyed to determine the design of the metal framework. Considering his functional and aesthetic requirements, a removable cobalt-chrome partial obturator for the maxillary arch was planned. Following the jaw relation record, the casts were mounted on a semi-adjustable articulator.</p>", "<p>The remaining teeth and palate provided the necessary support. Both sides' canines, left-side premolars, and molars were used for cingulum and occlusal rest placement. Retention was achieved by utilizing the remaining teeth, with two embrasure Aker clasps on the left molars and premolars and a C-wrought wire clasp on the right canine. The placement of cingulum rest as an indirect retainer was planned in the right canine tooth. To ensure the functional load was evenly distributed, it was determined that the remaining palate should be fully covered (Table ##TAB##0##1## and Figure ##FIG##2##3##).</p>", "<p>A special tray was made using cold-cure acrylic resin (Acrostone, Egypt) on the primary cast. Border molding was done using green stick compound (Dental Kerr Impression Compound, USA), and the final impression was taken using polyvinyl siloxane (PVS) material (Addition Silicon, Aquasil, Dentsply). The impression was poured with extra-hard type IV dental stone (Kimberlit, Type IV Dental Stone, Protechno-Spain) to generate the master cast. This master cast was duplicated to generate the refractory cast made of investment material, on which the framework wax-up was performed. The framework was then cast using cobalt-chromium alloy (Metal Brealloy, CO-CR alloy, Breadent-Germany) (Figure ##FIG##3##4##).</p>", "<p>The modified cast technique used PVS impression material to create a precise impression (Figure ##FIG##4##5##). The fit of the framework with the underlying structures was evaluated by placing it in the patient's mouth and using a pressure indicator paste to assist in the assessment. Occlusion rims were fabricated on the framework, and the centric jaw relation was recorded (Figure ##FIG##5##6##). The casts were then mounted on a semi-adjustable articulator (Bio-art semi-adjustable articulator. SM66297. Brazil). Acrylic denture teeth (Trubyte, Dentsply, Gloucestershire, England) were arranged, and the obturator was tested to ensure proper occlusion with the mandibular teeth, aesthetic appearance, and support for the underlying tissues. Subsequently, the obturator was processed, finished, and polished following standard procedures (Figure ##FIG##6##7##). During the insertion, pressure indicator paste (PIP) was used to identify any areas of excessive pressure. The denture was placed in the patient's mouth (Figure ##FIG##7##8##), and instructions were provided on the care and usage of the obturator. The patient underwent monthly evaluations for the first three months, followed by visits every three months for two years.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Intra-oral view of the maxillary defect</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>Primary impression of the maxilla</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG3\"><label>Figure 3</label><caption><title>Design drawn for the removable cobalt-chrome partial obturator </title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG4\"><label>Figure 4</label><caption><title>Metal framework of the obturator with all components in their place in the dental cast.</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG5\"><label>Figure 5</label><caption><title>Metal farmwork used the altered cast impression technique for the defect side using PVS impression material.</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG6\"><label>Figure 6</label><caption><title>Demonstrates the addition of wax to the metal framework, which is then utilized for recording the jaw relation.</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG7\"><label>Figure 7</label><caption><title>Photograph of the final processed definitive obturator on the dental cast.</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG8\"><label>Figure 8</label><caption><title>Intraoral views of the final obturator in place</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Design for the removable cobalt-chrome partial obturator used in the case presented.</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\nRight side\n</td><td rowspan=\"1\" colspan=\"1\">\nLeft side \n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nGuide plane\n</td><td rowspan=\"1\" colspan=\"1\">\nDistal proximal surface of canine\n</td><td rowspan=\"1\" colspan=\"1\">\nNone\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nRests\n</td><td rowspan=\"1\" colspan=\"1\">\nCingulum rest on canine\n</td><td rowspan=\"1\" colspan=\"1\">\nCingulum rests on the canine; mesial and distal occlusal rests on the first premolar, second premolar, and first molar; and mesial occlusal rests on the second molar.\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nClasps\n</td><td rowspan=\"1\" colspan=\"1\">\nwrought wire clasp on canine \n</td><td rowspan=\"1\" colspan=\"1\">\nOcclusally approaching clasps on the first premolar, second premolar, first molar, and second molar.\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nReciprocal components\n</td><td rowspan=\"1\" colspan=\"1\">\nPart of the major connector on palatal surface of canine\n</td><td rowspan=\"1\" colspan=\"1\">\nPart of the major connector on the first premolar, second premolar, first molar, and second molar.\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nMajor connector\n</td><td colspan=\"2\" rowspan=\"1\">\nPalatal plate major connector \n</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Ahmad E. Farghal</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Ahmad E. Farghal</p><p><bold>Drafting of the manuscript:</bold>  Ahmad E. Farghal</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Ahmad E. Farghal</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Schizophrenia symptoms are described as positive, such as hallucinations and delusions, and negative, like decreased emotional expression, lack of motivation, and cognitive decline. Over time, schizophrenia may interfere with an individual's social life, increase the likelihood of unemployment, and decrease life expectancy by 10-20 years. Patients who have poor adherence to their medication regimen are more likely to discontinue and tend to relapse repeatedly. Although investigations into the causes of schizophrenia have been conducted over several decades, the condition remains poorly understood. Genetic involvement is suspected because the incidence of schizophrenia in identical twins is approximately 50% and the incidence of schizophrenia in the offspring is approximately 10 times higher when both parents have schizophrenia. Although various theories regarding the causes of schizophrenia have been reported, they have not yet led to direct clinical application [##REF##26777917##1##].</p>", "<p>The prevalence of schizophrenia is approximately 1%; however, environmental factors are thought to play a role in the onset of the disease. For example, the incidence of schizophrenia is higher in individuals who reside at high latitudes or in densely populated urban areas [##REF##18480098##2##]. Patients with schizophrenia also present with various symptoms, including positive and negative symptoms. Therefore, in terms of treatment, it is difficult to objectively assess the severity of symptoms and determine the appropriate treatment options.</p>", "<p>In cases of severe psychomotor agitation, it is important to determine the appropriate treatment method, such as the intravenous (IV) administration of antipsychotic drugs and modified electroconvulsive therapy. Therefore, in the acute treatment of schizophrenia, prompt and appropriate management strategies should be selected, and biomarkers must be used to objectively determine the severity of symptoms and to predict treatment outcomes.</p>", "<p>One etiological theory of schizophrenia suggests a relationship with neuroinflammation. Neuroinflammation involves the activation of microglial cells and increased peripheral benzodiazepine receptor expression. Postmortem studies have reported that schizophrenia is associated with increased numbers of activated microglial cells. Recent studies measuring peripheral benzodiazepine receptors using positron emission tomography (PET) scans have also reported that neuroinflammation is an important factor affecting the onset of schizophrenia symptoms [##REF##19837763##3##]. Inflammation may increase the permeability of the blood-brain barrier (BBB), permitting localized central nervous system lesions to spread to the periphery and allowing infectious pathogens to invade, further damaging the central nervous system [##UREF##0##4##].</p>", "<p>Brain imaging tests are used to evaluate central nervous system functioning. However, such tests are challenging for acutely agitated patients to complete. Moreover, some psychiatric treatment facilities lack brain imaging equipment. Importantly, most treatment facilities have the necessary equipment to carry out blood tests to determine biomarkers. Among the various blood tests that could be useful, tests that examine immunological and inflammatory mechanisms may help examine patients with schizophrenia. For example, previous studies have examined changes in white blood cells (WBCs), particularly lymphocytes; however, no consensus has been reached.</p>", "<p>The neutrophil-lymphocyte ratio (NLR) has gained recent attention as a new biomarker for many diseases. NLR comprises a simple ratio of neutrophil-to-lymphocyte counts obtained from peripheral blood. NLR is a biomarker that links two aspects of the immune system: the innate immune response by neutrophils and adaptive immunity by lymphocytes. Furthermore, NLR is a prognostic predictor known to correlate independently with mortality in various diseases such as sepsis, coronavirus disease 2019 (COVID-19), and cancer [##UREF##0##4##]. The neutrophil-albumin ratio (NAR), platelet-lymphocyte ratio (PLR), and C-reactive protein (CRP)-albumin ratio (CAR) are other promising biomarkers for use in patients with cancer, sepsis, and heart failure [##REF##18639229##5##, ####REF##30196571##6##, ##UREF##1##7####1##7##].</p>", "<p>Individuals with schizophrenia commonly have higher NLRs than healthy individuals [##REF##25191768##8##]. Furthermore, they present with low lymphocyte counts and high neutrophil counts, indicating an imbalance in leukocyte distribution. Lymphocyte depletion is observed in inflammatory conditions due to the increased apoptosis of lymphocytes [##REF##12519925##9##]. Therefore, these findings may contribute to our understanding of the inflammatory mechanisms associated with schizophrenia. Furthermore, the use of antipsychotic medications will not affect the NLR [##REF##34603107##10##]. Past studies found that the NLR did not decrease in schizophrenic patients who were treatment-resistant but did increase among patients with schizophrenia who were treatment-responsive. These findings suggest that treatment may have an effect on schizophrenia symptoms [##REF##35876785##11##, ####REF##34995832##12##, ##REF##31504969##13####31504969##13##]. NLR has also been shown to be significantly correlated with the Positive and Negative Syndrome Scale (PANSS) and Clinical Global Impression-Severity (CGI-S) scores, as well as aggression, clinical symptoms, and disease severity [##REF##32038330##14##,##REF##35502339##15##].</p>", "<p>We hypothesized that inpatients with higher psychomotor agitation would have a higher NLR and examined the NLR as a biomarker for determining acute severity and selecting acute treatments for patients with schizophrenia. We compared patients admitted for acute treatment of schizophrenia according to whether or not they were treated with IV haloperidol, used for severe symptoms, as a biomarker for assessing severity and determining the appropriate treatment. We retrospectively studied the medical records of patients with acute schizophrenia who required hospitalization and who had severe psychomotor agitation, refused oral medication, and required IV haloperidol treatment.</p>" ]

[ "<title>Materials and methods</title>", "<p>Participants</p>", "<p>The participants were selected from patients admitted to the psychiatric emergency unit of Showa University Northern Yokohama Hospital in Kanagawa, Japan, between January 2014 and December 2019. This hospital provides acute psychiatric inpatient treatment. All patients with schizophrenia were diagnosed according to the criteria of the Diagnostic and Statistical Manual of Mental Disorders, 5th Edition (DSM-5) [##UREF##2##16##]. Patients younger than 16 years were excluded because it is difficult to confirm the diagnosis and adjust the appropriate dosage of antipsychotics in children and adolescents. Patients who were not treated for a physical disease that can significantly affect blood counts were recruited. However, patients who died or were discharged and those with inflammatory diseases, such as infections, metabolic diseases, advanced-stage cancer, trauma, and collagen diseases, and blood hematopoietic diseases, such as leukemia, were excluded to prevent the effect of changes in WBC count and CRP levels.</p>", "<p>We retrospectively analyzed clinical data from the patients' electronic medical records. Following data extraction, we categorized the participants into two groups based on their antipsychotic treatment: those who required IV haloperidol upon admission and those who were administered oral antipsychotic drugs.</p>", "<p>Patients were prescribed IV haloperidol if they refused oral antipsychotic drugs or demonstrated extremely severe symptoms, including (1) an imminent risk of suicide attempts or self-harm; (2) pronounced hyperactivity or restlessness; and/or (3) the patient's condition, without treatment, is life-threatening.</p>", "<p>Clinical data</p>", "<p>We analyzed various demographic and social factors, including age, sex, education, and marital and cohabitation status. We additionally gathered clinical data about the duration of the illness, the number of previous hospitalizations, CGI-S score at admission, acute psychiatric symptoms, chlorpromazine equivalent dose, and blood test results at admission.</p>", "<p>To evaluate the patient's psychiatric symptoms and condition, we used the criteria outlined in the hospitalization form prescribed in the Act on Mental Health and Welfare for Persons with Mental Disorders or Disabilities. The evaluation items included auditory hallucinations, visual hallucinations, delusions, association loosening, disorganized thinking, flat affect, depressed mood, restlessness, increased irritability, agitation, and stupor.</p>", "<p>This was a retrospective study. Hence, it had some limitations. For example, the possibility of psychotropic medication bias, such as the use of other antipsychotics, mood stabilizers, and anxiolytics, and the presence of missing data on symptom rating scale scores and blood test results could not be ruled out.</p>", "<p>Blood cell indexes</p>", "<p>The following blood test results were analyzed: CRP and blood cell count (including WBC count, platelet count, neutrophil count, lymphocyte count, monocyte count, eosinophil count, and basophil count). We also calculated the NLR (neutrophil count/lymphocyte count), NAR (neutrophil count/albumin (g/dL)), CAR (CRP (mg/dL)/albumin (g/dL)), and PLR (platelet count/albumin (g/dL)) from the same blood test results.</p>", "<p>Statistical analysis</p>", "<p>All data analyses were performed using IBM SPSS Statistics for Windows, Version 28.0 (Released 2021; IBM Corp., Armonk, New York, United States). The Shapiro-Wilk test was used to check the normality of the study variables' distributions. Pearson's chi-squared test was used to compare categorical variables. Student's t-test was used to compare normally distributed variables, while the Mann-Whitney U test was used for non-normally distributed variables. Mean±standard deviation (SD) and the number of variables were used to present data. Categorical variables are presented as numbers and percentages. Statistical significance was set at p&lt;0.05.</p>", "<p>Ethical compliance</p>", "<p>This research was carried out using inpatient medical records depicting typical treatment protocols. We used deidentified IDs and computers disconnected from external networks to ensure patient confidentiality. All identifiable data was strictly concealed. Instead of being exempted from the requirement for informed consent, we provided an \"opt-out\" option on our website. This informed users that their medical data would be used for research purposes and allowed them to opt out. Patients were also given the option to consent or opt out with assurances that their decision would not affect the clinical care they received. The study was designed according to the principles of the Declaration of Helsinki and was approved by the Institutional Review Board of Showa University Northern Yokohama Hospital (approval number: 19H042).</p>" ]

[ "<title>Results</title>", "<p>We enrolled 262 inpatients diagnosed with schizophrenia, out of which 10 were excluded due to complications of infection and one was excluded due to death. Therefore, we analyzed 251 patients. Out of these patients, 102 were males and 139 were females. The mean±SD for NLR was 3.27±2.91, and for NAR, it was 1099.17±546.</p>", "<p>The study had 43 patients in the IV haloperidol group and 208 patients in the oral antipsychotic group (Figure ##FIG##0##1##). Following the post-hoc analysis, we rejected the null hypothesis of the population means of these two groups being equal, with a probability (power) of 0.845.</p>", "<p>Demographic characteristics and background information of the participants</p>", "<p>The study cohort consisted of two groups: an IV haloperidol (n=43; 17 males and 26 females) with a mean age of 42.65±13.12 years and an oral antipsychotic group (n=208; 85 males and 123 females) with a mean age of 41.34±13.83 years (Table ##TAB##0##1##). The groups were similar in age and sex and other characteristics such as marital status, smoking habits, family history, and living situation.</p>", "<p>However, the IV haloperidol group showed significantly worse psychiatric symptoms at admission than the oral antipsychotic group, as indicated by the CGI-S score (p&lt;0.05, as shown in Table ##TAB##0##1##). Additionally, the chlorpromazine equivalent dose was significantly lower in the IV haloperidol group than in the oral antipsychotic group (p&lt;0.05, as shown in Table ##TAB##0##1##).</p>", "<p>Acute psychosomatic symptoms</p>", "<p>There were significant between-group differences for the following symptoms: delusions (53.49% IV vs. 72.6% oral), disorganized thinking (32.56% IV vs. 12.98% oral), depressed mood (0% IV vs. 7.69% oral), restlessness (0% IV vs. 11.06% oral), and agitation (25.58% IV vs. 11.06% oral) (Table ##TAB##1##2##). The differences were statistically significant (p&lt;0.05) for delusions, depressed mood, restlessness, and agitation and highly significant (p&lt;0.01) for disorganized thinking (Table ##TAB##1##2##). However, there were no significant between-group differences for auditory hallucinations, visual hallucinations, association loosening, flat affect, increased irritability, and stupor (Table ##TAB##1##2##).</p>", "<p>Comparison of the blood cell indexes</p>", "<p>There were significant differences between the IV haloperidol and oral antipsychotics for WBC count (8175.35±3441.82 IV vs. 6680.58±2566.97 oral, Mann-Whitney U test, p&lt;0.01) and neutrophil count (5992.96±2960.96 IV vs. 4337.54±2221.31 oral, Mann-Whitney U test, p&lt;0.01) (Table ##TAB##2##3##).</p>", "<p>The group that received IV haloperidol had a significantly higher NLR (4.03±3.55 vs. 3.11±2.89, Mann-Whitney U test, p&lt;0.05) and NAR (1383.07±698.64 vs. 1037.67±546, Mann-Whitney U test, p&lt;0.05) than those who received oral antipsychotics. However, the two groups did not significantly differ in terms of lymphocyte, CRP, CAR, or PLR levels (Table ##TAB##2##3##). Moreover, the group that received IV haloperidol had significantly higher WBC and neutrophil counts than the group that received oral antipsychotics. Lymphocyte counts and albumin levels did not differ between the two groups. Therefore, the significant difference between NLR and NAR could be attributed to the neutrophil counts.</p>" ]

[ "<title>Discussion</title>", "<p>This study examined potential biomarkers of imminent psychomotor arousal in patients with schizophrenia. Patients who refused oral antipsychotic drugs were offered IV haloperidol treatment. IV haloperidol was also used with patients who demonstrated extremely severe symptoms, including imminent suicide attempts or self-injurious behavior, hyperactivity, or restlessness, particularly if nontreatment would be life-threatening. In the IV haloperidol group, symptoms such as disorganized thinking and agitation were pronounced, and the CGI-S score was higher than that in the oral antipsychotic group at the time of admission. Similarly, NLR and NAR were higher in the IV haloperidol group than in the oral antipsychotic group. These patients demonstrated psychomotor agitation so imminent that they did not fully understand the need for antipsychotic treatment and, therefore, were administered IV haloperidol.</p>", "<p>We considered blood cell indexes as biomarkers of schizophrenia severity and treatment selection because such tests are easily performed in inpatient facilities. NLR and NAR can be calculated using only biochemistry and blood counts, commonly performed in blood tests and attracting attention as simple, inexpensive markers that can be tested at any facility [##UREF##0##4##].</p>", "<p>Previous studies have suggested that NLR is higher in patients with schizophrenia than in healthy controls [##REF##25191768##8##,##REF##35502339##15##]. In some meta-analyses of NLR in schizophrenia, patients' NLR values were 2.63, and healthy controls' NLR values were 1.78 [##REF##30573407##17##]. Patients' mean NLR was 2.03-3.24, whereas healthy controls had a mean NLR range of 1.6-2 [##REF##30806142##18##]. Our study's mean of patients' NLR was 3.28±2.9. Compared with previous studies, this study's participants were considered typical patients with schizophrenia.</p>", "<p>Previous studies reported on relationships between NLR and schizophrenia symptom severity. In a study of 22 patients with schizophrenia, NLR was significantly correlated with PANSS-total, PANSS-positive, PANSS-general, and CGI scores and was reduced after long-term antipsychotic treatment. Increased NLR is associated with severe schizophrenia symptoms [##REF##32038330##14##]. Labonté et al. found that patients with schizophrenia who were treatment-resistant showed no decrease in NLR; however, NLR did decrease following treatment among treatment-responsive patients with schizophrenia. NLR may, therefore, be useful for determining treatment response and assessing a patient's symptoms [##REF##34995832##12##]. A positive correlation between aggression and NLR has also been reported. Patients with schizophrenia who are highly aggressive have higher pretreatment NLR than their nonaggressive counterparts. Therefore, NLR could be used as a biomarker to assess aggressive behavior [##REF##35502339##15##]. Kulaksizoglu and Kulaksizoglu showed significant correlations between PANSS-total and NLR, indicating that NLR is related not only to the pathophysiology of schizophrenia but also to clinical symptoms [##REF##27574431##19##].</p>", "<p>Patients who required IV haloperidol treatment due to their psychomotor agitation had higher NLR values, CGI-S scores, and rates of symptoms such as perplexing thoughts and agitation. These findings are consistent with previous studies that showed a relationship between NLR and severity. Additionally, the study revealed new insights into the types of symptoms that are severe enough to require IV haloperidol treatment.</p>", "<p>NAR reflects hyperinflammation and is used to predict pathologic complete remission in patients with pancreatic and rectal cancer [##REF##30196571##6##,##REF##27434664##20##,##UREF##3##21##]. A previous study found that patients' NAR was higher compared to that of healthy controls; the authors considered NAR useful in diagnosing schizophrenia [##UREF##3##21##]. However, there are few studies of NAR for psychiatric disorders. The mean of NAR in our participants was higher in the IV haloperidol group than in the oral antipsychotic group. Higher values for NAR and NLR appear to be associated with more severe psychiatric symptoms. Therefore, evaluating NLR and NAR from blood tests and seeing how the values fluctuate relative to psychotic symptom severity may help physicians with treatment planning. NLR values may be useful in determining the need for invasive therapeutic intervention when oral treatment is unavailable.</p>", "<p>Of course, our results should be considered alongside several study limitations. Firstly this was a single-center study. Our cohort was from the psychiatric emergency unit of a general hospital. Hence, it is an incomplete representation of the general population. Moreover, there was a bias toward patients with schizophrenia who had chronic illnesses, and the area had only a few elderly patients with schizophrenia. Secondly, our data were collected retrospectively. Therefore, we could not exclude the possibility of psychotropic medication bias, including the use of other antipsychotics, mood stabilizers, and anxiolytic drugs. Furthermore, some important data could have been missing. Because the data were collected a long time back, there was a possibility of heterogeneity in terms of patient information, such as changes in the tests and treatments during that period of time. Thirdly, some values needed to be added to the symptom assessment. Importantly, only the CGI-S was administered to patients to assess psychiatric symptoms on admission and not after that. Therefore, no pre-to-post comparisons were carried out. The Brief Psychiatric Rating Scale and PANSS were not measured in all patients and were not available to assess schizophrenia symptoms. Lastly, we only assessed NLR and NAR at admission because the timing of blood tests performed after hospitalization was irregular. Moreover, we failed to analyze changes in NLR and NAR after symptom improvement and did not attempt to correlate NLR and NAR levels with changes in symptom severity. This study did not examine patients with schizophrenia who were aged &lt;16 years. Recently, previous studies have reported that patients with schizophrenia who were aged &lt;18 years have a higher NLR than healthy controls and adults [##REF##38141839##22##,##REF##37218478##23##]. Therefore, a study design that excludes age limitations should be considered.</p>", "<p>To tackle the issues at hand, additional research studies that examine the psychiatric symptoms of schizophrenia are needed. Such studies should include the results of blood tests taken at admission, during treatment, and at discharge. Furthermore, the research should analyze the correlation between changes in NLR and NAR and the changes in symptom severity.</p>", "<p>Our study found that patients with higher levels of psychomotor arousal had higher NLR and NAR values. This implies that blood tests may be useful for objectively assessing the level of psychomotor arousal during psychiatric exacerbations. However, we recommend validating these findings within the context of adequately powered, prospective studies.</p>" ]

[ "<title>Conclusions</title>", "<p>Herein, patients with severe psychosomatic schizophrenia who could not receive oral antipsychotics and required hospitalization for IV haloperidol had higher CGI-S and increased NLR and NAR compared with patients who could receive oral treatment. In the IV haloperidol group, patients also had higher rates of psychiatric symptoms such as delusions and agitation compared with patients who could be orally treated.</p>", "<p>In clinical practice, there are many situations in which the lack of objective biomarkers of psychiatric symptoms makes it difficult to decide whether or not to provide IV antipsychotic treatment. In such cases, elevated NLR and NAR were considered useful for IV haloperidol treatment selection as biomarkers that could be easily measured in patients with psychomotor agitation who should be treated.</p>", "<p>This study and previous reports showed that NLR and NAR could be an objective indicator that is useful for disease diagnosis, severity determination, and treatment selection in patients with schizophrenia. Nevertheless, larger, multicenter, prospective studies should be performed to validate our results.</p>" ]

[ "<p>Introduction</p>", "<p>Schizophrenia symptom severity is linked to neuroinflammation. Certain blood cell indexes such as neutrophil-lymphocyte ratio (NLR) and neutrophil-albumin ratio (NAR) have been used as biomarkers in various diseases, including schizophrenia. In acute clinical practice, it is challenging to decide whether to provide intravenous antipsychotic treatment in some cases due to the lack of objective biomarkers of psychiatric symptoms. The NLR of individuals with schizophrenia is thought to be associated with disease severity, and changes in NLR may reflect a patient's response to antipsychotic treatment. We investigated the application of NLR as a biomarker for identifying acute severity and determining acute treatment response in patients with schizophrenia.</p>", "<p>Methods</p>", "<p>We retrospectively examined 251 inpatients diagnosed with schizophrenia and classified them according to treatment (intravenous haloperidol vs. oral antipsychotic medication during the acute phase) and investigated their NLR and NAR while receiving inpatient care.</p>", "<p>Results</p>", "<p>A total of 48 inpatients were given intravenous haloperidol to manage their acute symptoms; 208 were given oral antipsychotics. The intravenous haloperidol group experienced more severe symptoms, such as agitation and disorganized thinking, during the acute phase. Further, those who received intravenous haloperidol had significantly higher Clinical Global Impression-Severity (CGI-S) scores than the oral antipsychotic group. NLR and NAR were also significantly higher in the haloperidol intravenous group.</p>", "<p>Conclusion</p>", "<p>Elevated NLR and NAR could be easily measured in patients with psychomotor agitation who should be treated at any facility. Further, they are useful biomarkers for determining disease severity and the effects of treatment on psychomotor excitement in patients who require intravenous haloperidol.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Classification of participants</title><p>Overall, 262 patients with schizophrenia requiring hospitalization were selected. Overall, one death and 10 patients with infectious complications were excluded. Hence, 251 patients were included in this study. Patients were divided into two groups: those who required IV haloperidol treatment and those who received oral antipsychotic treatment.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Demographic and clinical data for the IV haloperidol (n=43) and oral antipsychotic groups (n=208)</title><p>Statistical analysis was performed using the Mann-Whitney U test (a) and chi-squared test (b).</p><p>NA: not applicable; SD: standard deviation; CGI-S: Clinical Global Impression-Severity</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Characteristic</td><td rowspan=\"1\" colspan=\"1\">IV haloperidol  (n=43)</td><td rowspan=\"1\" colspan=\"1\">Oral antipsychotics  (n=208)</td><td rowspan=\"1\" colspan=\"1\">p-value</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Age^a\n</td><td rowspan=\"1\" colspan=\"1\">42.65±13.12</td><td rowspan=\"1\" colspan=\"1\">41.34±13.83</td><td rowspan=\"1\" colspan=\"1\">0.57</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Sex^b: male; female</td><td rowspan=\"1\" colspan=\"1\">17; 26</td><td rowspan=\"1\" colspan=\"1\">85; 123</td><td rowspan=\"1\" colspan=\"1\">0.87</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Solitary life^b\n</td><td rowspan=\"1\" colspan=\"1\">14</td><td rowspan=\"1\" colspan=\"1\">43</td><td rowspan=\"1\" colspan=\"1\">0.09</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Married^b\n</td><td rowspan=\"1\" colspan=\"1\">13</td><td rowspan=\"1\" colspan=\"1\">63</td><td rowspan=\"1\" colspan=\"1\">0.947</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Divorced or widowed^b\n</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">25</td><td rowspan=\"1\" colspan=\"1\">0.452</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Family history^b\n</td><td rowspan=\"1\" colspan=\"1\">15</td><td rowspan=\"1\" colspan=\"1\">75</td><td rowspan=\"1\" colspan=\"1\">0.700</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Smoking habits^b\n</td><td rowspan=\"1\" colspan=\"1\">7</td><td rowspan=\"1\" colspan=\"1\">31</td><td rowspan=\"1\" colspan=\"1\">0.452</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Duration of education^a\n</td><td rowspan=\"1\" colspan=\"1\">13±2.06</td><td rowspan=\"1\" colspan=\"1\">12.6±2.45</td><td rowspan=\"1\" colspan=\"1\">0.348</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Duration of illness^a\n</td><td rowspan=\"1\" colspan=\"1\">15.95±13.31</td><td rowspan=\"1\" colspan=\"1\">14.72±10.38</td><td rowspan=\"1\" colspan=\"1\">0.839</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Number of hospitalizations^a\n</td><td rowspan=\"1\" colspan=\"1\">2.67±4.15</td><td rowspan=\"1\" colspan=\"1\">2.24±2.39</td><td rowspan=\"1\" colspan=\"1\">0.948</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Chlorpromazine equivalent^a\n</td><td rowspan=\"1\" colspan=\"1\">384.51±462.52</td><td rowspan=\"1\" colspan=\"1\">573.67±657.9</td><td rowspan=\"1\" colspan=\"1\">p&lt;0.05</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">CGI-S</td><td rowspan=\"1\" colspan=\"1\">5.56±1.13</td><td rowspan=\"1\" colspan=\"1\">5.09±1.06</td><td rowspan=\"1\" colspan=\"1\">p&lt;0.05</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Haloperidol total mg (IV)^a\n</td><td rowspan=\"1\" colspan=\"1\">25.33±23.38</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Haloperidol infusion duration (days)^a\n</td><td rowspan=\"1\" colspan=\"1\">2.14±1.75 (median: 1)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>Psychiatric symptoms in the IV haloperidol and oral antipsychotic groups</title><p>Statistical analysis was performed using the chi-squared test.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">IV haloperidol  (n=43)</td><td rowspan=\"1\" colspan=\"1\">Oral antipsychotics  (n=208)</td><td rowspan=\"1\" colspan=\"1\">p-value</td></tr><tr><td rowspan=\"1\" colspan=\"1\">　</td><td rowspan=\"1\" colspan=\"1\">% (n)</td><td rowspan=\"1\" colspan=\"1\">% (n)</td><td rowspan=\"1\" colspan=\"1\">　</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Auditory hallucinations</td><td rowspan=\"1\" colspan=\"1\">39.53 (17)</td><td rowspan=\"1\" colspan=\"1\">53.85 (112)</td><td rowspan=\"1\" colspan=\"1\">0.09</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Visual hallucinations</td><td rowspan=\"1\" colspan=\"1\">2.33 (1)</td><td rowspan=\"1\" colspan=\"1\">4.33 (9)</td><td rowspan=\"1\" colspan=\"1\">0.47</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Delusions</td><td rowspan=\"1\" colspan=\"1\">53.49 (23)</td><td rowspan=\"1\" colspan=\"1\">72.6 (151)</td><td rowspan=\"1\" colspan=\"1\">&lt;0.05</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Association loosening</td><td rowspan=\"1\" colspan=\"1\">6.98 (3)</td><td rowspan=\"1\" colspan=\"1\">14.42 (30)</td><td rowspan=\"1\" colspan=\"1\">0.19</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Disorganized thinking</td><td rowspan=\"1\" colspan=\"1\">32.56 (14)</td><td rowspan=\"1\" colspan=\"1\">12.98 (27)</td><td rowspan=\"1\" colspan=\"1\">&lt;0.01</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Flat affect</td><td rowspan=\"1\" colspan=\"1\">6.98 (3)</td><td rowspan=\"1\" colspan=\"1\">16.83 (35)</td><td rowspan=\"1\" colspan=\"1\">0.10</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Depressed mood</td><td rowspan=\"1\" colspan=\"1\">0 (0)</td><td rowspan=\"1\" colspan=\"1\">7.69 (16)</td><td rowspan=\"1\" colspan=\"1\">&lt;0.05</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Restlessness</td><td rowspan=\"1\" colspan=\"1\">0 (0)</td><td rowspan=\"1\" colspan=\"1\">11.06 (23)</td><td rowspan=\"1\" colspan=\"1\">&lt;0.05</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Increased irritability</td><td rowspan=\"1\" colspan=\"1\">20.93 (9)</td><td rowspan=\"1\" colspan=\"1\">20.67 (43)</td><td rowspan=\"1\" colspan=\"1\">0.97</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Agitation</td><td rowspan=\"1\" colspan=\"1\">25.58 (11)</td><td rowspan=\"1\" colspan=\"1\">11.06 (23)</td><td rowspan=\"1\" colspan=\"1\">&lt;0.05</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Stupor</td><td rowspan=\"1\" colspan=\"1\">9.3 (4)</td><td rowspan=\"1\" colspan=\"1\">3.85 (8)</td><td rowspan=\"1\" colspan=\"1\">0.13</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB3\"><label>Table 3</label><caption><title>Between-group comparison of laboratory variables of interest</title><p>Statistical analysis was performed using the Mann-Whitney U test.</p><p>WBC: white blood cell; CRP: C-reactive protein; NLR: neutrophil-lymphocyte ratio; NAR: neutrophil-albumin ratio; CAR: CRP-albumin ratio; PLR: platelet-lymphocyte ratio; SD: standard deviation</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">IV haloperidol  (n=43)</td><td rowspan=\"1\" colspan=\"1\">Oral antipsychotics  (n=208)</td><td rowspan=\"1\" colspan=\"1\">p-value</td></tr><tr><td rowspan=\"1\" colspan=\"1\">　</td><td rowspan=\"1\" colspan=\"1\">Mean±SD</td><td rowspan=\"1\" colspan=\"1\">Mean±SD</td><td rowspan=\"1\" colspan=\"1\">　</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">WBC count</td><td rowspan=\"1\" colspan=\"1\">8175.35±3441.82</td><td rowspan=\"1\" colspan=\"1\">6680.58±2566.97</td><td rowspan=\"1\" colspan=\"1\">&lt;0.01</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Neutrophil count</td><td rowspan=\"1\" colspan=\"1\">5992.96±2960.96</td><td rowspan=\"1\" colspan=\"1\">4337.54±2221.31</td><td rowspan=\"1\" colspan=\"1\">&lt;0.01</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Lymphocyte count</td><td rowspan=\"1\" colspan=\"1\">1839.53±782.08</td><td rowspan=\"1\" colspan=\"1\">1707.64±886.03</td><td rowspan=\"1\" colspan=\"1\">0.27</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Albumin (g/dL)</td><td rowspan=\"1\" colspan=\"1\">4.31±0.69</td><td rowspan=\"1\" colspan=\"1\">4.23±0.59</td><td rowspan=\"1\" colspan=\"1\">0.61</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">CRP (mg/dL)</td><td rowspan=\"1\" colspan=\"1\">0.77±1.06</td><td rowspan=\"1\" colspan=\"1\">0.64±1.27</td><td rowspan=\"1\" colspan=\"1\">0.08</td></tr><tr><td rowspan=\"1\" colspan=\"1\">NLR</td><td rowspan=\"1\" colspan=\"1\">4.03±3.55</td><td rowspan=\"1\" colspan=\"1\">3.11±2.89</td><td rowspan=\"1\" colspan=\"1\">&lt;0.05</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">NAR</td><td rowspan=\"1\" colspan=\"1\">1383.07±698.64</td><td rowspan=\"1\" colspan=\"1\">1037.67±546</td><td rowspan=\"1\" colspan=\"1\">&lt;0.05</td></tr><tr><td rowspan=\"1\" colspan=\"1\">CAR</td><td rowspan=\"1\" colspan=\"1\">0.2±0.31</td><td rowspan=\"1\" colspan=\"1\">0.17±0.33</td><td rowspan=\"1\" colspan=\"1\">0.11</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">PLR</td><td rowspan=\"1\" colspan=\"1\">5.82±1.68</td><td rowspan=\"1\" colspan=\"1\">5.79±1.97</td><td rowspan=\"1\" colspan=\"1\">0.82</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Hiroi Tomioka, Shutaro Sugita, Akira Iwanami, Atsuko Inamoto, Kenji Sanada, Kensuke Mera</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Hiroi Tomioka, Shutaro Sugita, Hiroki Yamada, Kensuke Mera, Taro Tazaki, Hana Nishiyama</p><p><bold>Drafting of the manuscript:</bold>  Hiroi Tomioka, Shutaro Sugita, Kensuke Mera, Taro Tazaki, Hana Nishiyama</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Hiroi Tomioka, Shutaro Sugita, Hiroki Yamada, Akira Iwanami, Atsuko Inamoto, Kenji Sanada</p><p><bold>Supervision:</bold>  Hiroi Tomioka, Shutaro Sugita, Hiroki Yamada, Akira Iwanami, Atsuko Inamoto, Kenji Sanada</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study. Institutional Review Board of Showa University Northern Yokohama Hospital issued approval 19H042</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Animal Ethics</title><fn fn-type=\"other\"><p><bold>Animal subjects:</bold> All authors have confirmed that this study did not involve animal subjects or tissue.</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0016-00000052181-i01\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Arthritis robustus (rheumatoid robustus) commonly occurs in men over the age of 50, particularly those who are physically active and involved in manual labor. They do not complain of pain, stiffness, disability, or distress, though clinical signs of inflammation, deformity, and radiological erosions are present. Synovial proliferation, subcutaneous nodules, periarticular erosions, and subchondral cysts are common, while periarticular osteopenia is rare compared to classical rheumatoid arthritis (RA) [##REF##30285183##1##].</p>", "<p>We detail a unique case of arthritis robustus with tenosynovitis, review published literature, discuss potential reasons for the unique presentation, and explore the clinical and therapeutic implications.</p>" ]

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[ "<title>Discussion</title>", "<p>Arthritis robustus primarily affects elderly males actively involved in physical labor. Despite the presence of an inflammatory joint disease that aligns with this patient’s clinical profile, the level of pain and stiffness is minimal. Also, a history of smoking was a risk factor for the development of RA [##REF##19174392##2##]. Our patient is a milkman who uses small hand joints during milking. Continuous use of these joints would affect the perception of pain and stiffness, reducing awareness of symptoms.</p>", "<p>De Haas et al. described nine male patients with classical RA who had subcutaneous nodules as well as high titers of seropositivity of both RF/anti-CCP [##REF##4685893##3##]. Despite these characteristics, they were “robust, healthy, and working normally.” The duration of arthritis robustus and joint involvement showed no significant difference compared to controls, who were males experiencing both active disease and remission. However, all of them were involved in strenuous physical work, had a mesomorphic body structure, and demonstrated a tendency toward independence during psychological interviews. These authors interrogated if these clinical features could be explained by the “soft-hearted” treatment of some patients. It was reported, however, that “RA, typus robustus” men needed fewer analgesics and less physiotherapy.</p>", "<p>Chopra and Chib reported a case of arthritis robustus, masquerading as gout [##REF##3733655##4##]. The term “arthritis robustus variant of RA” was applied to categorize four out of 20 young servicemen experiencing chronic inflammatory polyarthritides. It must be noted that these were physically active youthful men, and the prevalence of “robustness” was four out of 15 RA patients [##REF##2636577##5##]. More recently, Prasad et al. reported a 58-year-old telephone wireman with an active lifestyle who had clinically evident arthritis without arthralgia, diagnosed as arthritis robustus only when he presented with myocardial infarction [##REF##25392778##6##]. Thompson and Carr reported 10 patients, out of a randomly selected list of 100 RA patients, who did not complain of pain despite having clinical and biochemical evidence of inflammation [##UREF##0##7##]. Jones takes this discussion forward to propose the role of psychosocial factors in pain perception and management [##REF##9536829##8##]. Earlier, authors have highlighted the correlation between pain threshold and analgesic usage in men with RA, as opposed to those with ankylosing spondylitis, where pain threshold does not influence analgesic requirement [##REF##5082544##9##].</p>", "<p>A large study looked into the discordance of self-reported symptoms with objective disease activity scores/inflammatory markers. This study looked at three cohorts, namely (1) the Early Rheumatoid Arthritis Network (ERAN), (2) the British Society for Rheumatology Biologics Registry (BSRBR), starting therapy with tumor necrosis factor (TNF) inhibitors, as well as (3) those on non-biologic medications. A subset of patients with discordantly better patient-reported outcomes (PRO) compared to inflammation was identified, including 11% in the ERAN cohort, 23% in the BSRBR cohort of TNF inhibitors, and 10% in (BSBR) non-biologic medications. This suggested that non-inflammatory factors may influence the interpretation of inflammation/pain, acknowledging the presence of the typus robustus RA phenotype in this subset. It is worth noting that the authors place a greater emphasis on the needs of another subset that reported discordantly worse (in contrast to arthritis robustus) PRO compared to inflammation (12%, 40%, and 21% in the three cohorts) [##REF##27964757##10##].</p>", "<p>The pathogenesis of RA is multifactorial, with putative inflammatory and non-inflammatory pathways contributing to its clinical phenotype. Individuals with arthritis robustus may have lower interleukin-6-mediated dysfunction, which changes their pain perception [##REF##33707975##11##] (Table ##TAB##1##2##). Healthcare-seeking behavior, too, may influence the presentation of arthritis. Patients with less access to rheumatology care may tend to downplay their concerns. The same applies to individuals who cannot afford quality care. One differential diagnosis of arthritis robustus is leprosy, which can present with deforming arthritis and lack of symptoms due to sensory impairment [##UREF##1##12##]. It may be possible that arthritis associated with sensory neuropathy, e.g., syphilis, diabetes, and acromegaly, can mimic arthritis robustus. Endemic diseases, such as nutritional and toxic neuropathy, may impact the perception of normal vs. abnormal, delaying their recognition of symptoms and signs [##UREF##2##13##, ####REF##24037045##14##, ##REF##32915310##15####32915310##15##]. Thus, arthritis robustus can be explained by the biopsychosocial model of health [##UREF##3##16##] (Table ##TAB##1##2##).</p>", "<p>Arthritis robustus has important clinical implications. Delayed recognition of the disease may lead to the progression of the disease and the occurrence of complications, including deformities. It may also result in delayed identification of comorbid conditions, such as autoimmune disorders, osteopenia, and sarcopenia, potentially leading to further morbidity. An early diagnosis allows the timely institution of disease-modifying anti-rheumatic drugs (DMARDs), and appropriate lifestyle changes, which may optimize long-term health. The construct of arthritis robustus also allows healthcare professionals to individualize therapy in a person-centered manner [##UREF##3##16##]. Arthritis robustus patients, for treatment adherence, require motivation, encouraging a proactive approach to managing their condition and controlling disease activity.</p>" ]

[ "<title>Conclusions</title>", "<p>The current case of arthritis robustus has associated tenosynovitis, which is rare. Apart from highlighting the existence of this syndrome, this discussion also underscores the need to spread awareness about this variant of RA while understanding its clinical presentation, differential diagnosis, and management.</p>" ]

[ "<p>Rheumatoid arthritis (RA) commonly presents as a chronic additive symmetric inflammatory polyarthritis involving the small and large joints. Rarely do patients present with few or no clinical symptoms, despite apparent signs of inflammation. This condition, known as arthritis robustus, typically occurs in elderly males who are manual laborers with an active lifestyle. It is essential to diagnose arthritis robustus and start treatment promptly to avoid the development of deformities and other complications in the future.</p>" ]

[ "<title>Case presentation</title>", "<p>A 45-year-old male was referred to the rheumatology clinic for complaints of swelling in both wrists for a six-month duration. The swelling persisted without progressing, and there was no associated joint pain. However, the patient experienced 30 minutes of early morning stiffness in the wrists and small joints of the hands. There was no associated history of fever, back pain, skin lesions, diarrhea, or urethral discharge. He worked as a milkman and was a former smoker who smoked 20 hand-rolled cigarettes (bidi) daily for ~ 10 years. Past medical and surgical history was unremarkable. On examination, swelling with fluctuation was present on both wrists and the left third proximal interphalangeal (PIP) joint, associated with warmth but without tenderness. A firm, non-tender swelling of size 1 x 1 cm was present over the left distal ulna (Figure ##FIG##0##1##). Radiographs of the bilateral wrist confirmed joint space narrowing with erosions but without significant periarticular osteopenia suggesting long-standing inflammatory disease.</p>", "<p>Investigations were remarkable for high rheumatoid factor (RF &gt;90 IU/mL), anti-cyclic citrullinated peptide (CCP) antibodies (&gt;80 IU/mL), and serum C-reactive protein (CRP) (8.2 mg/L) values (Table ##TAB##0##1##). Musculoskeletal ultrasonography revealed tenosynovitis of the left extensor carpi ulnaris tendon and cortical irregularity in the phalanges. The patient fulfilled the 2010 ACR/EULAR classification criteria with a DAS28-CRP score of 3.25, suggesting moderate disease activity.</p>", "<p>The patient was counseled and educated regarding this condition and was initiated on methotrexate, bridge therapy with low-dose prednisolone, and nonsteroidal anti-inflammatory drugs (NSAIDs) for RA. He was provided with physical and occupational therapy. The dose of prednisolone was tapered on follow-up and discontinued over a period of two months. The patient did not report a need for analgesics on follow-up visits. At the third-month follow-up visit, the patient’s swollen joint count was reduced to zero with disease activity measure DAS28-CRP 1.64 (suggestive of remission).</p>" ]

[ "<p>Vijay Karthik Bhogaraju and Arnav Kalra equally contributed to this work and should be considered co-first authors.</p>" ]

[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Both wrists of the patient</title><p>Note the left 3rd PIP swelling and left wrist tenosynovitis.</p><p>PIP: proximal interphalangeal</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Investigations at presentation</title><p>ESR: erythrocyte sedimentation rate; TLC: total leucocyte count; DLC: differential leucocyte count; CRP: C-reactive protein; ALT: alanine transaminase; AST: aspartate transferase; ALP: alkaline phosphatase; GGT: gamma-glutamyl transferase; CCP: cyclic citrullinated peptide; HCV: hepatitis C virus</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nTest\n</td><td rowspan=\"1\" colspan=\"1\">\nValue\n</td><td rowspan=\"1\" colspan=\"1\">\nTest\n</td><td rowspan=\"1\" colspan=\"1\">\nValue\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nHb(g/dL)\n</td><td rowspan=\"1\" colspan=\"1\">\n12.7\n</td><td rowspan=\"1\" colspan=\"1\">\nS. Uric acid\n</td><td rowspan=\"1\" colspan=\"1\">\n6.7 mg/dL\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nTLC per mm³\n</td><td rowspan=\"1\" colspan=\"1\">\n10,590/mm³\n</td><td rowspan=\"1\" colspan=\"1\">\nESR\n</td><td rowspan=\"1\" colspan=\"1\">\n14 mm in 1st hour\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nDLC (%)\n</td><td rowspan=\"1\" colspan=\"1\">\nN 69%, L 24%, M 6%, E 1%\n</td><td rowspan=\"1\" colspan=\"1\">\nS. CRP\n</td><td rowspan=\"1\" colspan=\"1\">\n8.2 mg/L \n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nPlatelet count/mm³\n</td><td rowspan=\"1\" colspan=\"1\">\n3,35,000\n</td><td rowspan=\"1\" colspan=\"1\">\nRheumatoid factor (RF)\n</td><td rowspan=\"1\" colspan=\"1\">\n&gt;90 IU/mL (&lt;14 )\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nS. Bilirubin (T/D) mg/dL\n</td><td rowspan=\"1\" colspan=\"1\">\n0.30/0.18\n</td><td rowspan=\"1\" colspan=\"1\">\nAnti-CCP\n</td><td rowspan=\"1\" colspan=\"1\">\n&gt;80 IU/mL (&lt;5 )\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nALT/AST(IU/L)\n</td><td rowspan=\"1\" colspan=\"1\">\n22.1/ 26.6\n</td><td rowspan=\"1\" colspan=\"1\">\nAnti-HIV-1 &amp; -2\n</td><td rowspan=\"1\" colspan=\"1\">\nNon-reactive\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nALP/GGT (IU/L)\n</td><td rowspan=\"1\" colspan=\"1\">\n337.3/ 60.4\n</td><td rowspan=\"1\" colspan=\"1\">\nHBsAg\n</td><td rowspan=\"1\" colspan=\"1\">\nNon-reactive\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nS. Total protein/ S. Albumin g/dL\n</td><td rowspan=\"1\" colspan=\"1\">\n8.02/4.04\n</td><td rowspan=\"1\" colspan=\"1\">\nAnti-HCV\n</td><td rowspan=\"1\" colspan=\"1\">\nNon-reactive\n</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>Potential explanations for arthritis robustus based upon the biopsychosocial model of health</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Biomedical</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Related to rheumatoid arthritis: lack of non-inflammatory pain pathways</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Associated sensory neuropathy, e.g., leprosy, syphilis, diabetes, nutritional deficiency, alcoholism</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Higher pain threshold</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Psychosocial</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Independent personality</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Delayed healthcare-seeking behavior</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Acceptance of joint-related symptoms due to the ubiquitous prevalence</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050583-i01\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>The symbiotic relationship between host and microorganisms is extremely complicated, and it is important to better understand these cellular interactions [##REF##36807754##1##]. Humans are colonized by the microbiota containing bacteria, archaea, fungi, and viruses; different body surfaces contain distinct members. Internal organs, such as the heart, once thought to be sterile, harbor organ-specific microorganisms [##REF##35255932##2##]. Blood from healthy individuals has diverse microbiota [##REF##26865079##3##], and specific bacteria even exist in immune cells [##REF##33853683##4##]. The colon is home to the most diverse microbiota composed of 100 trillion bacteria containing about 25 times the number of genes as the Homo sapiens genome [##UREF##0##5##]. The gut microbiota plays an important role in crosstalk between gut microbes and host cells as microbial metabolites can cross the intestinal wall and enter the bloodstream. Over 200 microbial metabolites have been identified in local and distal systems of the body [##REF##36732606##6##]. The gut is also home to about 70-80% of human immune cells [##REF##33803407##7##], and the symbiotic relationships between the microbiota and cells of the immune system are important in numerous medical disorders [##REF##34055983##8##].</p>", "<p>It is important to better understand how bacteria and immune cells coexist in the gut. It is commonly assumed that mixing bacteria with human cells in culture would result in the bacteria overgrowing the cells. We assumed that the oxygen in cell culture media would inhibit the growth or kill the anaerobic bacteria, allowing cellular reactions to be measured. Experiments adding a commercial mixture of probiotic anaerobic bacteria to peripheral blood mononuclear cells (PBMC) in culture with atmosphere air, which contains about 18% oxygen, did not show gross signs of bacterial contamination, and the growth rates were similar to control cultures at 24 hours. Here, we demonstrate that the four probiotics induced small amounts of cytokines/chemokines in PBMC, and cultures containing phytohemagglutinin (PHA) showed higher cytokine concentrations. There is a lot to learn from these mixed cultures, as the combination of PHA and probiotic bacteria resulted in significantly higher concentrations of the pro-inflammatory cytokine IL-1β, whereas the combination of PHA and bacteria significantly decreased the production of the chemokine MCP-1.</p>" ]

[ "<title>Materials and methods</title>", "<p>Study design</p>", "<p>The study involved four commercial anaerobic probiotic bacterial products: Neuro Byome (NB), Metabo Byome (MB), Male/Female Byome (M/F), and Immuno Byome (IB) were obtained from Nutri-Biome (Ogden, Utah). The four probiotics were made by reconstituting 18 freeze-dried (Gram - and Gram +) anaerobic bacteria and growing them on individual agar plates (Table ##TAB##0##1##). The agar plates with bacteria were incubated under strict anaerobic conditions (90% nitrogen, 5% carbon dioxide, 5% hydrogen at 37 °C) in a Bactron EZ Anaerobic Chamber (Sheldon Manufacturing, Cornelius, OR, USA).</p>", "<p>Bacteria were scrapped off the plates after one to five days of incubation depending on the growth of the particular strain. The bacteria were diluted in phosphate-buffered saline (PBS) to obtain an absorbance reading of about 1.0 at 600 nm on an Agilent 453E spectrophotometer (Santa Clara, CA, US). Equal numbers of each bacterium strain were combined to make the four probiotic products contain 4-6X 10⁸ bacteria/ml. Each strain was characterized by DNA typing with strain-specific PCR primers (Table ##TAB##1##2##).</p>", "<p>Human PBMCs from an individual donor were purchased from Cellular Technology Limited (CTL) (Shaker Heights, Ohio, USA). PBMCs contain a mixture of immune cells including T-cells, B-cells, NK cells, monocytes (blood macrophages), and dendritic cells. The complex mixture of immune cells in systemic PBMC makes it possible to study many immune interactions in test tubes. The PBMCs were stored in the vapor phase of liquid nitrogen until the day of use. PBMC culture reagents were obtained from CTL and company protocols were followed. A vial of PBMCs contains about 1 x10⁷ cells, which were plated at 2 x10⁵ cells/well in 200 μL of CTL media in flat-bottom 96-well plates.</p>", "<p>Ten microliters containing 4-6 x 10⁶ of the four probiotic bacteria products were added to each well in quadruplicate and allowed to incubate overnight with PBMC cultures under room air containing about 18% oxygen. At 24 hours, there were no signs of contamination or bacterial overgrowth (no cytotoxic granules or anomalous morphologies). To compare the amount of inflammation introduced by these anaerobic bacteria, PBMC control cultures were inflamed with 10 μg/ml PHA. PHA is a well-known mitogen that binds to toll-like receptor 2 (TLR2) on T-cells and monocytes and causes the production of high levels of inflammatory cytokines, which are associated with inflammation [##REF##30642137##24##].</p>", "<p>The HIEC-6 normal small intestine epithelial cell line obtained from American Type Culture Collection (ATCC) #CCL-3266 was cultured in Minimum Essential Media (MEM) with Glutamax^TM supplemented with 10 ng/ml EGF and 5% fetal bovine serum (FBS). HIEC-6 cells were plated at a density of 10,000 cells per well in 200 μl of the appropriate media in flat bottom 96-well culture plates in quadruplicate, which reached about 80% confluency in 48 hrs. Ten microliters containing 4-6 x 10⁶ of the four probiotic bacteria were added to each well and allowed to incubate overnight at 37 °C in room air containing about 18% oxygen.</p>", "<p>Viability assay</p>", "<p>The XTT assay (Biotium, Fremont, CA, USA) was used to evaluate cell viability at the end of culture experiments. XTT is a colorimetric detection assay utilizing tetrazolium dye to measure cell viability by enzymatic activity in the mitochondria of living cells. After the appropriate incubation time, 100 μl of the culture media from each well was removed for cytokine and chemokine analysis leaving 100 μl, to which 25 μl of the XTT reagent was added. The XTT plates were incubated at 37 °C for 120 minutes and the absorbance was recorded using a Tecan Genios plate reader (Mannedorf, Switzerland) that detects the absorption maximum (492 nm) of XTT.</p>", "<p>One hundred microliters of PBMC culture supernatants (absent of cells) of the quadruplicate wells were combined for a total of 400 μl and frozen. The cell culture supernatants were sent on dry ice to Quansys Bioscience (Logan, Utah, USA) for the determination of cytokines and chemokine concentrations using an enzyme-linked immunosorbent assay (ELISA) chemiluminescent immunoassay.</p>", "<p>Cytokines</p>", "<p>Pro-inflammatory cytokines include interleukin-6 (IL-6), interleukin-1β (IL-1β), granulocyte-macrophage colony-stimulating factor (GMCSF), and tumor necrosis factor-alpha (TNFα). Interleukin-8 (IL-8) and monocyte chemoattractant protein (MCP-1) are chemokines that attract innate immune cells, especially granulocytes to areas of inflammation. The 15-cytokine multiplex assay done at Quansys measures all cytokines at the same time by highly sensitive chemiluminescence.</p>", "<p>Statistical analysis</p>", "<p>One-way analysis of variance (ANOVA) followed by Dunnett’s multiple comparisons test was performed using GraphPad Prism version 10.0.0 for Mac (GraphPad Software, Boston, MA, USA, <ext-link xlink:href=\"https://www.graphpad.com\" ext-link-type=\"uri\">www.graphpad.com</ext-link>).</p>" ]

[ "<title>Results</title>", "<p>It was our decision to study cellular viability and cytokine production after incubating PBMC with probiotic mixtures of anaerobic bacteria. With 70-80% of immune cells residing in the gut, it is important to understand the intricate interactions between the local microbiota and immune cells. First, there was no sign of contamination or bacterial overgrowth in any of the PBMC or HIEC-6 cultures. Adding the four probiotic samples containing different anaerobic bacteria stimulated the PBMC to produce up to a couple of thousand picograms/ml of the various cytokines/chemokines (Table ##TAB##2##3##).</p>", "<p>Adding PHA to the PBMC cultures stimulated the cells to produce higher amounts of the various cytokines and combining PHA and anaerobic bacteria produced even higher levels of various cytokines. However, the PHA bacteria combination induced significantly lower levels of MCP-1 compared to PHA alone (Table ##TAB##3##4##). Adding bacteria to the HIEC-6 cultures did not induce the production of pro-inflammatory cytokines or chemokines (data not shown).</p>" ]

[ "<title>Discussion</title>", "<p>There was a minor difference in cell viability in the control cultures with no bacteria compared to the cultures with four probiotic mixtures as measured by the XTT assay (Figures ##FIG##0##1A##, ##FIG##0##1B##). This is not surprising, as bacteria have molecules on their surfaces that bind to pattern recognition receptors (PRR) on immune cells even if the bacteria are dead.</p>", "<p>These interactions of PRR on host cells and pathogen-associated molecular patterns (PAMPs) on bacteria are well-recognized and PAMPs exist in non-pathogenic bacteria [##REF##35478496##25##]. PBMCs have toll-like receptors (TLRs) and C-type lectin receptors, which recognize bacterial PAMPs. Additionally, it is well known that cytokines are produced by microbe-immune cell interactions [##REF##25879288##26##]. Cytokines are important chemical messengers that affect many cellular functions such as inflammation, cellular activation, and cellular proliferation [##REF##31958792##27##].</p>", "<p>The innate immune system has several defense mechanisms to detect and respond to invading microorganisms [##REF##25879288##26##]. One important response is the induction of inflammation to eliminate the invading microorganism. However, one must remember that there are trillions of bacteria living in our bodies. The coexistence of the countless strains of bacteria living in us cannot elicit full-blown inflammation like certain pathogenic infections. Therefore, inflammation must be carefully controlled or regulated to prevent excess tissue damage [##REF##22156412##28##,##UREF##6##29##]. The inflammatory cytokine (IL-1β) production was significantly higher in PHA plus bacteria compared to the PHA control (Figure ##FIG##1##2A##), whereas the combination of PHA and bacteria significantly decreased the chemokine MCP-1 (Figure ##FIG##1##2B##).</p>", "<p>Although this research strongly suggests that anaerobic bacteria do not overgrow PBMC when cultured overnight in media containing oxygen, there are many unanswered questions. For example, in the same experiment, the mixture of the four anaerobic bacteria with PHA resulted in 4-7-fold production of IL-1β (Figure ##FIG##0##1A##) and conversely a 2-4-fold decrease in the production of MCP-1 (Figure ##FIG##1##2B##) over PHA alone.</p>", "<p>It was decided to look at another non-immune cell to determine if the anaerobic bacteria affected growth patterns. The epithelial cell (HIEC-6) is a normal cell line that resides in the vicinity of the bulk of intestinal bacteria. The HIEC-6 growth patterns with the four anaerobic probiotic bacteria are not significantly different from control cultures without bacteria (Figure ##FIG##0##1B##) and no evidence of contamination was noted under careful microscopic examination. This is further evidence that the anaerobic bacteria do not rapidly expand in oxygen-containing media.</p>", "<p>The ability to examine anaerobic bacteria in culture with living cells in overnight culture suggests numerous experimental possibilities. The PBMC used in these experiments contained a mixture of white cells, and it is unclear which cells are responding to the bacteria. Purified monocytes, T-cells, B-cells, dendritic cells, etc. should be examined to determine anaerobic bacterial effects.</p>", "<p>Also, PBMC or white cells from individuals with specific diseases could be examined for cytokine responses. Single anaerobic bacteria should be examined to determine specific cytokine effects. Culture experiments longer than overnight may show differences and there may be stimulation of other cytokines or growth factors not measured in our current assays. Gene expression experiments could be useful, in certain experiments, to determine bacterial effects on a particular cell line.</p>", "<p>In our hands, the mixing of anaerobic bacteria with human cells in culture showed some interesting results. Limitations in new areas of research become apparent when different laboratories repeat similar experiments. Our research presented here evaluated four mixtures of probiotics of anaerobic bacteria. The first limitation of this work is there is the limited amount of data generated from this novel approach of combining anaerobic bacteria with cells in culture. The second limitation is we only looked at a small number of cytokines. Thirdly, we do not know which individual bacteria in the four mixtures is causing the cytokine effect, suggesting that cell culture experiments should be done using individual anaerobic bacteria. The last limitation is we only examined a small number of bacteria compared to the thousands of anaerobic bacteria that exist in the intestines.</p>" ]

[ "<title>Conclusions</title>", "<p>The data presented here clearly suggests that anaerobic bacteria do not grow rapidly in oxygen-containing conditions in the PBMC cell culture experiments. The presence of anaerobic bacteria in PBMC cultures stimulates a weak pro-inflammatory response in several cytokines. The addition of PHA and PHA plus anaerobic bacteria results in a more robust cytokine response. In the same experiments, PHA plus anaerobic bacteria significantly inhibits the chemokine MCP-1 response. The growth patterns of the normal cell line HIEC-6 are not strongly affected by the four anaerobic bacteria mixtures, nor was there a robust cytokine response.</p>" ]

[ "<p>In the last couple of decades, much progress has been made in studying bacteria living in humans. However, there is much more to learn about bacteria immune cell interactions. Here, we show that anaerobic bacteria do not grow when cultured overnight with human cells under atmospheric air. Air contains about 18% oxygen, which inhibits the growth of these bacteria while supporting the cultivation of human cells. The bacteria cultured with human peripheral blood mononuclear cells (PBMCs) inflamed with phytohemagglutinin (PHA) greatly increased the production of proinflammatory cytokines like tumor necrosis factor-alpha (TNFα) while inhibiting the production of monocyte chemoattractant protein-1 (MCP-1), an important chemokine.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Cell viability</title><p>A) XTT assay of PBMC after overnight culture. Control samples did not contain any bacteria. The cell viability of quadruplicate wells contained 2 x10⁵ PBMC/well in 200 µl of CLT media and 10 µml of bacteria (NB 6.2x10⁶, MB 6.0x10⁶, M/F 4.1x10⁶ and IB 5.8x10⁶) were examined after overnight culture. B) XTT assay of HIEC-6 after culture. Control samples did not contain any bacteria. Cell viability of cultures to which 10 µl of bacteria (NB 6.2x10⁶, MB 6.0x10⁶, M/F 4.1x10⁶, and IB 5.8x10⁶) were added to quadruplicate wells and examined after overnight culture.</p><p>*p &lt; 0.05 compared to the control samples.</p><p>PBMC: peripheral blood mononuclear cell</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>The production of cytokines and chemokines in PBMC cultures by the combination of bacteria and PHA. </title><p>A) IL-1β production is significantly stimulated and B) MCP-1 production is significantly inhibited. ****p &lt; 0.0001, ***p &lt; 0.001, **p &lt; 0.01 compared to the PHA control.</p><p>PBMC: peripheral blood mononuclear cell; PHA: phytohemagglutinin</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Anaerobic bacteria strains in four probiotic products</title><p>Gram + and gram- are listed as + or -. The particular agar used to grow the specific strain is shown in the second column.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Neuro Byome (NB)</td><td rowspan=\"1\" colspan=\"1\">Agar</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Faecalibacterium duncaniae (+)</td><td rowspan=\"1\" colspan=\"1\">Wilkins-Chalgren Agar</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Bacteroides fragilis (-)</td><td rowspan=\"1\" colspan=\"1\">Tryptic Soy Agar + 5% Sheep Blood</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Parabacteroides distasonis (-)</td><td rowspan=\"1\" colspan=\"1\">Wilkins-Chalgren Agar</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Agathobaculum butyriciproducens (+)</td><td rowspan=\"1\" colspan=\"1\">Wilkins-Chalgren Agar</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Metabo Byome (MB)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Akkermansia muciniphila (-)</td><td rowspan=\"1\" colspan=\"1\">Brain Heart Infusion Agar + 0.3% Porcine Mucin (Type II)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Eubacterium rectale (+)</td><td rowspan=\"1\" colspan=\"1\">Wilkins-Chalgren Agar</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Butyricicoccus pullicaecorum (+)</td><td rowspan=\"1\" colspan=\"1\">Wilkins-Chalgren Agar</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Anaerobutyricum hallii (+)</td><td rowspan=\"1\" colspan=\"1\">Wilkins-Chalgren Agar</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Dorea longicatena (+)</td><td rowspan=\"1\" colspan=\"1\">Wilkins-Chalgren Agar</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Christensenella minuta (-)</td><td rowspan=\"1\" colspan=\"1\">Wilkins-Chalgren Agar</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Male/Female Byome (MF)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Eubacterium limosum (+)</td><td rowspan=\"1\" colspan=\"1\">Tryptic Soy Agar + 5% Sheep Blood</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Bacteroides thetaiotaomicron (-)</td><td rowspan=\"1\" colspan=\"1\">Tryptic Soy Agar + 5% Sheep Blood</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Bacteroides uniformis (-)</td><td rowspan=\"1\" colspan=\"1\">Tryptic Soy Agar + 5% Sheep Blood</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Immuno Byome (Immuno)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Bacteroides ovatus (-)</td><td rowspan=\"1\" colspan=\"1\">Tryptic Soy Agar + 5% Sheep Blood</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Bacteroides uniformis (-)</td><td rowspan=\"1\" colspan=\"1\">Tryptic Soy Agar + 5% Sheep Blood</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Clostridium symbiosum (+)</td><td rowspan=\"1\" colspan=\"1\">Wilkins-Chalgren Agar</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Collinsella aerofaciens (+)</td><td rowspan=\"1\" colspan=\"1\">MRS Agar</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Roseburia hominis (+)</td><td rowspan=\"1\" colspan=\"1\">Wilkins-Chalgren Agar</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Anaerostipes caccae (+)</td><td rowspan=\"1\" colspan=\"1\">Wilkins-Chalgren Agar</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>Bacterial strain PCR primers</title><p>Individual bacteria strains were characterized by DNA typing with strain-specific polymerase chain reaction (PCR) primers. The forward and reverse primers are shown in columns 2 and 3, respectively. Published references for the strain-specific primers are shown in column 4.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td colspan=\"3\" rowspan=\"1\">Strain-Specific PCR Primers (5’--3’)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Neuro Byome (NB)</td><td rowspan=\"1\" colspan=\"1\">Forward</td><td rowspan=\"1\" colspan=\"1\">Reverse</td><td rowspan=\"1\" colspan=\"1\">Reference</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">F. duncaniae (+)</td><td rowspan=\"1\" colspan=\"1\">TCATCACGCCCAGATTGTCC</td><td rowspan=\"1\" colspan=\"1\">GGCGAGTATGTCCAGTTCGT</td><td rowspan=\"1\" colspan=\"1\">[##REF##22708584##9##]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">B. fragilis (-)</td><td rowspan=\"1\" colspan=\"1\">GTACACACCGCCCGT</td><td rowspan=\"1\" colspan=\"1\">AATTTAGAACCAATGAACG</td><td rowspan=\"1\" colspan=\"1\">[##REF##12757940##10##]</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">P. distasonis (-)</td><td rowspan=\"1\" colspan=\"1\">AATACCGCATGAAGCAGG</td><td rowspan=\"1\" colspan=\"1\">GACACGTCCCGCACTTTA</td><td rowspan=\"1\" colspan=\"1\">[##REF##7538270##11##]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">A. butyriciproducens (+)</td><td rowspan=\"1\" colspan=\"1\">GATCACTCTAGCCGGACTGC</td><td rowspan=\"1\" colspan=\"1\">GTTAGGCTACGGACTTCGGG</td><td rowspan=\"1\" colspan=\"1\">[##REF##22708584##9##]</td></tr><tr style=\"background-color:#ccc\"><td colspan=\"4\" rowspan=\"1\">Metabo Byome (MB)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">A. muciniphila (-)</td><td rowspan=\"1\" colspan=\"1\">CAGCACGTGAAGGTGGGGAC</td><td rowspan=\"1\" colspan=\"1\">CCTTGCGGTTGGCTTCAGAT</td><td rowspan=\"1\" colspan=\"1\">[##REF##27841267##12##]</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">E. rectale (+)</td><td rowspan=\"1\" colspan=\"1\">AAGGGAAGCAAAGCTGTGAA</td><td rowspan=\"1\" colspan=\"1\">CGGTTAGGTCACTGGCTTC</td><td rowspan=\"1\" colspan=\"1\">[##UREF##1##13##]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">B. pullicaecorum (+)</td><td rowspan=\"1\" colspan=\"1\">CGAGCAGGCAAACGACAA</td><td rowspan=\"1\" colspan=\"1\">CCAGGTCTTGGTACCGTCC</td><td rowspan=\"1\" colspan=\"1\">[##REF##34361916##14##]</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">A. hallii* (+)</td><td rowspan=\"1\" colspan=\"1\">TAATCGGTGCTTTCCTTCG</td><td rowspan=\"1\" colspan=\"1\">CAGCCTTACCTGCTGGCTAC</td><td rowspan=\"1\" colspan=\"1\">[##REF##20921388##15##]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">D. longicatena (+)</td><td rowspan=\"1\" colspan=\"1\">CGCATAAGACCACGTACC</td><td rowspan=\"1\" colspan=\"1\">TGATAGAAGTTTACATACCGAAAT</td><td rowspan=\"1\" colspan=\"1\">[##REF##22708584##9##]</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">C. minuta (-)</td><td rowspan=\"1\" colspan=\"1\">GTAATACGTAGGGAGCAAGC</td><td rowspan=\"1\" colspan=\"1\">CCCTCTCCTGTACTCAAGTC</td><td rowspan=\"1\" colspan=\"1\">[##REF##31190659##16##]</td></tr><tr><td colspan=\"4\" rowspan=\"1\">Male/Female Byome (MF)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">E. limosum (+)</td><td rowspan=\"1\" colspan=\"1\">GGCTTGCTGGACAAATACTG</td><td rowspan=\"1\" colspan=\"1\">CTAGGCTCGTCAGAGGATG</td><td rowspan=\"1\" colspan=\"1\">[##UREF##2##17##]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">B. thetaiotaomicron (-)</td><td rowspan=\"1\" colspan=\"1\">AACAGGTGGAAGCTGCGGA</td><td rowspan=\"1\" colspan=\"1\">AGCCTCCAACCGCATCAA</td><td rowspan=\"1\" colspan=\"1\">[##UREF##3##18##]</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">B. uniformis (-)</td><td rowspan=\"1\" colspan=\"1\">TATGCAACCAAGCTGATGAACGAAG</td><td rowspan=\"1\" colspan=\"1\">AGAGGTTGGCCACGATGTTGATAC</td><td rowspan=\"1\" colspan=\"1\">[##REF##32093252##19##]</td></tr><tr><td colspan=\"4\" rowspan=\"1\">Immuno Byome (Immuno)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">B. ovatus (-)</td><td rowspan=\"1\" colspan=\"1\">GTACACACCGCCCGT</td><td rowspan=\"1\" colspan=\"1\">AATATTGCATACTCGAACAC</td><td rowspan=\"1\" colspan=\"1\">[##REF##22708584##9##]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">B. uniformis (-)</td><td rowspan=\"1\" colspan=\"1\">TATGCAACCAAGCTGATGAACGAAG</td><td rowspan=\"1\" colspan=\"1\">AGAGGTTGGCCACGATGTTGATAC</td><td rowspan=\"1\" colspan=\"1\">[##REF##32093252##19##]</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">C. symbiosum (+)</td><td rowspan=\"1\" colspan=\"1\">GTGAGATGATGTGCCAGGC</td><td rowspan=\"1\" colspan=\"1\">TACCGGTTGCTTCGTCGATT</td><td rowspan=\"1\" colspan=\"1\">[##REF##29033369##20##]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">C. aerofaciens (+)</td><td rowspan=\"1\" colspan=\"1\">CCCGACGGGAGGGGAT</td><td rowspan=\"1\" colspan=\"1\">CTTCTGCAGGTACAGTCTTGA</td><td rowspan=\"1\" colspan=\"1\">[##REF##17631127##21##]</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">R. hominis (+)</td><td rowspan=\"1\" colspan=\"1\">GCACTTTAATTGATTTCTTCG</td><td rowspan=\"1\" colspan=\"1\">TCTTAGTCAGGTACCGTCATT</td><td rowspan=\"1\" colspan=\"1\">[##UREF##4##22##]</td></tr><tr><td rowspan=\"1\" colspan=\"1\">A. caccae (+)</td><td rowspan=\"1\" colspan=\"1\">GTTTTCGGATGGATTTCCTATAT</td><td rowspan=\"1\" colspan=\"1\">CTTTTCACACTGAATCATGCGATT</td><td rowspan=\"1\" colspan=\"1\">[##UREF##5##23##]</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB3\"><label>Table 3</label><caption><title>Cytokine determinations (pg/ml) in PBMC supernatant (bacteria only)</title><p>Measurements of cytokines and chemokines in PBMC. Control: PBMC only in media. The addition of the four probiotic bacteria mixtures to PBMC stimulated cytokine production. Quadruplicate culture wells contained 2x10⁵ PBMC/well in 200 µl of CLT media and 10 µl of bacteria (NB 6.2x10⁶, MB 6.0x10⁶, M/F 4.1x10⁶ and IB 5.8x10⁶).</p><p>PBMC: peripheral blood mononuclear cell; GMCSF: granulocyte-macrophage colony-stimulating factor; IL: interleukin; TNFα: tumor necrosis factor-alpha; MCP-1: monocyte chemoattractant protein-1</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">GMCSF</td><td rowspan=\"1\" colspan=\"1\">IL-1β</td><td rowspan=\"1\" colspan=\"1\">IL-6</td><td rowspan=\"1\" colspan=\"1\">TNFα</td><td rowspan=\"1\" colspan=\"1\">IL-8</td><td rowspan=\"1\" colspan=\"1\">MCP-1</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Control</td><td rowspan=\"1\" colspan=\"1\">6.5±0.8</td><td rowspan=\"1\" colspan=\"1\">15±0.9</td><td rowspan=\"1\" colspan=\"1\">9.7±0.6</td><td rowspan=\"1\" colspan=\"1\">27±0.6</td><td rowspan=\"1\" colspan=\"1\">509±28</td><td rowspan=\"1\" colspan=\"1\">80±11</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">NB</td><td rowspan=\"1\" colspan=\"1\">676±78</td><td rowspan=\"1\" colspan=\"1\">84±4.6 **</td><td rowspan=\"1\" colspan=\"1\">726±32 **</td><td rowspan=\"1\" colspan=\"1\">657±23 ***</td><td rowspan=\"1\" colspan=\"1\">6530±421 **</td><td rowspan=\"1\" colspan=\"1\">86±0.9</td></tr><tr><td rowspan=\"1\" colspan=\"1\">MB</td><td rowspan=\"1\" colspan=\"1\">458±39</td><td rowspan=\"1\" colspan=\"1\">57±4.6 *</td><td rowspan=\"1\" colspan=\"1\">1172±50 **</td><td rowspan=\"1\" colspan=\"1\">470±34 **</td><td rowspan=\"1\" colspan=\"1\">7335±128 ***</td><td rowspan=\"1\" colspan=\"1\">555±25 **</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">M/F</td><td rowspan=\"1\" colspan=\"1\">1844±91 *</td><td rowspan=\"1\" colspan=\"1\">283±6.8 ***</td><td rowspan=\"1\" colspan=\"1\">1851±47 ***</td><td rowspan=\"1\" colspan=\"1\">796±22 ***</td><td rowspan=\"1\" colspan=\"1\">5442±398 **</td><td rowspan=\"1\" colspan=\"1\">168±3.7 **</td></tr><tr><td rowspan=\"1\" colspan=\"1\">IB</td><td rowspan=\"1\" colspan=\"1\">1033±31 *</td><td rowspan=\"1\" colspan=\"1\">97±4.2 **</td><td rowspan=\"1\" colspan=\"1\">1514±52 ***</td><td rowspan=\"1\" colspan=\"1\">681±28 **</td><td rowspan=\"1\" colspan=\"1\">6208±238 **</td><td rowspan=\"1\" colspan=\"1\">427±1.4 ***</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB4\"><label>Table 4</label><caption><title>Cytokine determinations (pg/ml) in PBMC supernatants (PHA plus bacteria)</title><p>PHA controls contained 10 µg/ml of PHA without bacteria. The four probiotic bacteria samples containing 10 mg/ml PHA were compared to PHA-only cultures for statistical purposes; the mean and standard deviation of each parameter are shown. Samples that induced inhibition of the chemokines IL-8 and MCP-1 production are in italics. One-way ANOVA followed by Dunnett’s multiple comparisons test was performed.</p><p>****p &lt; 0.0001, ***p &lt; 0.001, **p &lt; 0.01 compared to the PHA control.</p><p>PBMC: peripheral blood mononuclear cell; PHA: phytohemagglutinin; IL: interleukin; MCP-1: monocyte chemoattractant protein-1; ANOVA: analysis of variance</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">GMCSF</td><td rowspan=\"1\" colspan=\"1\">IL-1β</td><td rowspan=\"1\" colspan=\"1\">IL-6</td><td rowspan=\"1\" colspan=\"1\">TNFα</td><td rowspan=\"1\" colspan=\"1\">IL-8</td><td rowspan=\"1\" colspan=\"1\">MCP-1</td></tr><tr><td rowspan=\"1\" colspan=\"1\">PHA Control</td><td rowspan=\"1\" colspan=\"1\">6800±230</td><td rowspan=\"1\" colspan=\"1\">859±22</td><td rowspan=\"1\" colspan=\"1\">20274±1400</td><td rowspan=\"1\" colspan=\"1\">7103±88</td><td rowspan=\"1\" colspan=\"1\">68016±720</td><td rowspan=\"1\" colspan=\"1\">11406±104</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">NB + PHA</td><td rowspan=\"1\" colspan=\"1\">12854±262 **</td><td rowspan=\"1\" colspan=\"1\">4212±163 **</td><td rowspan=\"1\" colspan=\"1\">25651±3891</td><td rowspan=\"1\" colspan=\"1\">7162±112</td><td rowspan=\"1\" colspan=\"1\">72432±3152</td><td rowspan=\"1\" colspan=\"1\">4034±108 ***</td></tr><tr><td rowspan=\"1\" colspan=\"1\">MB + PHA</td><td rowspan=\"1\" colspan=\"1\">8649±36 *</td><td rowspan=\"1\" colspan=\"1\">4347±11 ****</td><td rowspan=\"1\" colspan=\"1\">22986±834</td><td rowspan=\"1\" colspan=\"1\">8181±156 *</td><td rowspan=\"1\" colspan=\"1\">96049±2469 **</td><td rowspan=\"1\" colspan=\"1\">5661±127 ***</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">M/F + PHA</td><td rowspan=\"1\" colspan=\"1\">12440±79 **</td><td rowspan=\"1\" colspan=\"1\">5471±52 ***</td><td rowspan=\"1\" colspan=\"1\">24970±946 *</td><td rowspan=\"1\" colspan=\"1\">10092±226 **</td><td rowspan=\"1\" colspan=\"1\">62157±917 **</td><td rowspan=\"1\" colspan=\"1\">4245±12 ***</td></tr><tr><td rowspan=\"1\" colspan=\"1\">IB + PHA</td><td rowspan=\"1\" colspan=\"1\">14375±282 **</td><td rowspan=\"1\" colspan=\"1\">7182±156 ***</td><td rowspan=\"1\" colspan=\"1\">36508±15410</td><td rowspan=\"1\" colspan=\"1\">9349±225 *</td><td rowspan=\"1\" colspan=\"1\">77977±4066</td><td rowspan=\"1\" colspan=\"1\">3193±22 ***</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Anthony R. Torres, Shayne Morris</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Anthony R. Torres, Shayne Morris, Michael Benson, Craig Wilkinson, Rachael Lyon</p><p><bold>Drafting of the manuscript:</bold>  Anthony R. Torres</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Anthony R. Torres, Shayne Morris, Michael Benson, Craig Wilkinson, Rachael Lyon</p><p><bold>Supervision:</bold>  Anthony R. Torres</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Animal Ethics</title><fn fn-type=\"other\"><p><bold>Animal subjects:</bold> All authors have confirmed that this study did not involve animal subjects or tissue.</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>Dr. Anthony R. Torres and Dr. Shayne Morris have applied for a provisional patent. The other authors have contributed to the research under direction of Drs. Torres and Morris. However, they were not involved in the invention of mixing PBMC cells with anaerobic bacteria.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050586-i01\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050586-i02\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Bisphosphonate (BP) is a drug similar to pyrophosphate that has been prescribed since 1960 to treat various bone diseases [##REF##28875143##1##, ####REF##26096687##2##, ##UREF##0##3##, ##REF##35300956##4####35300956##4##]. It has been shown that BP causes the inhibition of osteoclasts, resulting in the reduction of bone resorption and bone remodelling, which leads to osteonecrosis of the jaw (ONJ), later known as BP-related osteonecrosis of the jaw (BRONJ) [##REF##25234529##5##,##UREF##1##6##]. It was first reported in 2003 by Marx [##REF##12966493##7##], and subsequent cases have been extensively reported in the scientific literature, leaving a significant impact on quality of life and substantial morbidity [##REF##17785387##8##, ####REF##20310042##9##, ##REF##20561470##10##, ##REF##31207196##11##, ##REF##28729804##12####28729804##12##]. It appears as a bone exposed for eight weeks or more without any history of radiation therapy [##REF##35300956##4##,##REF##19686928##13##]. In 2014, the American Association of Oral and Maxillofacial Surgeons (AAOMS) modified the old terminology to medication-related osteonecrosis of the jaw (MRONJ) because of cases of ONJ related with other antiresorptive medications [##REF##25234529##5##].</p>", "<p>The etiology of MRONJ is not entirely understood, but there are several suggested risk factors which include the length of time a patient has been on BP therapy, method of administration, age of the patient, history of dentoalveolar surgery, use of corticosteroids, and presence of systemic disease like diabetes mellitus [##REF##35300956##4##,##REF##31256803##14##]. There has been a gradual increase in the occurrence of complications associated with the use of these drugs. The understanding of the mechanisms behind MRONJ is still lacking, and various hypotheses have been proposed to explain why MRONJ explicitly affects the jaws. The hypotheses proposed involve various factors that may contribute to the observed effects. These factors include the excessive suppression of bone resorption, changes in bone remodelling processes, ongoing microtrauma, inhibition of angiogenesis, vitamin D deficiency, suppression of acquired or innate immunity, presence of infection or inflammation, and potential toxicity of soft tissue blood pressure [##REF##35300956##4##].</p>", "<p>In MRONJ, bone resorption and remodelling decrease as osteoclast differentiation and function are inhibited and apoptosis increases. In all skeletal bones, osteoclasts play a crucial role in bone remodelling and healing. Conversely, ONJ occurs in the mandible 73% of the time and in the maxilla 22.5% of the time [##REF##17919538##15##,##REF##30402414##16##]. This phenomenon may be explained by the higher remodelling rate of the jaws compared to other skeletal bones. ONJ has been proven to be triggered by infection and inflammation in numerous clinical studies. Biopsy samples of necrotic bone taken from patients with ONJ have been found to contain bacteria, specifically <italic>Actinomyces</italic> spp. [##REF##35300956##4##].</p>", "<p>MRONJ is mostly a drug-related disease, with risk factors including dosage, administration method, duration, and therapeutic use. Other risk factors for this disease include surgical operations, such as tooth extraction; individuals with comorbidities, such as diabetes; and concurrent use of corticosteroids. Some low-risk cancer patients are given treatment for noncancer conditions such as osteoporosis, osteopenia, and Paget's disease [##UREF##2##17##].</p>", "<p>It has been found that cancer-free individuals without having any additional risk who receive oral antiresorptive for less than four years have a relatively low risk of developing this disease. Low-risk patients can receive dental treatment without any modifications. However, cancer patients face a significant risk in developing multiple myeloma and bone metastases [##REF##27114946##18##,##REF##15459427##19##].</p>", "<p>The treatment of MRONJ involves a comprehensive approach that includes prevention, ongoing cancer care, preservation of bone health, and enhancing the quality of life for patients. Strategies involve taking proactive measures to prevent MRONJ, ensuring that individuals on specific therapies can continue their oncologic treatments without interruption, and prioritizing bone health to minimize the risk of fractures. In addition, patient education plays a crucial role in empowering individuals to actively participate in their care. Pain management, infection control, and preventive measures to stop the progression of lesions in the jaw are essential for improving comfort and minimizing potential complications [##REF##35300956##4##].</p>", "<p>Dentists have a significant impact on preventing BRONJ and MRONJ by offering preventive care and prioritizing preventive treatment before starting BP [##REF##35300956##4##,##REF##31207196##11##,##REF##25843818##20##,##REF##25757408##21##]. Therefore, dentists and physicians must possess sufficient knowledge about identifying potential complications and the appropriate treatment for patients who are at risk of MRONJ [##REF##25757408##21##].</p>", "<p>Guidelines for patients getting BPs on staging and treatment approaches were released by the AAOMS. These guidelines' primary goal was to give physicians a foundational understanding of BPs, MRONJ/BRONJ clinical characteristics and risk factors, and, most importantly, how to treat and prevent MRONJ/BRONJ. Regretfully, investigations have revealed that dentists have shown very poor knowledge about the care of patients receiving BP therapy, even in spite of these guidelines [##REF##25234529##5##,##UREF##1##6##].</p>", "<p>There have been limited studies exploring dental students' knowledge and awareness about MRONJ. However, no study has been conducted thus far to investigate their understanding of drugs associated with it. Therefore, the objective of this study was to analyse and assess knowledge about MRONJ among dental students and practitioners in the central region of Saudi Arabia.</p>" ]

[ "<title>Materials and methods</title>", "<p>An observational cross-sectional study was planned to collect data from dental students and dentists in the central region of Saudi Arabia. To collect information from participants, a valid and reliable questionnaire was used [##REF##33603544##22##] using a convenient non-probability sampling method during the period from October to December 2022. The Epi Info software (Centers for Disease Control and Prevention, Atlanta, Georgia, United States) was used to calculate the sample size, assuming a 50% incidence rate with a margin of error of 5% and a 95% level of confidence. The minimum sample required was 384; because of time constraints, we were able to collect responses from 250 participants. This study was approved by the Committee of Research Ethics of Qassim University (approval number: 21-12-03).</p>", "<p>Inclusion/exclusion criteria </p>", "<p>Dental students, graduates, and dental practitioners were included, whereas the general public, medical practitioners, and medical students were excluded from participation. </p>", "<p>Data collection method </p>", "<p>This study utilized a survey divided into five components (see Appendices section). There were six items in the first section of the questionnaire pertaining to demographic information, namely, age, gender, college affiliation (graduated from or currently enrolled in), years of professional experience, and highest educational degree attained. The second component consisted of five items designed to assess participants' general knowledge of antiresorptive medications. The third component assessed participants' understanding of the therapeutic applications of antiresorptive and antiangiogenic drugs. In the fourth component, participants were assessed on their knowledge of the correct definition and the associated risk factors. The fifth section addressed the dental management of patients taking BPs.</p>", "<p>Statistical analysis </p>", "<p>The data were analysed using IBM SPSS Statistics for Windows, Version 22.0 (Released 2013; IBM Corp., Armonk, New York, United States). Age, gender, marital status, and educational background were represented as frequencies and percentages. A chi-squared test was applied to determine the association between dentists' and students' knowledge related to ONJ. Statistical significance was determined by a p-value of less than 0.05.</p>" ]

[ "<title>Results</title>", "<p>A total of 250 participants were enrolled in the study. Of them, 128 (51.2%) were women, and 122 (48.8%) were men. Marital status revealed that most participants were single (198 or 79.2%) and 47 (18.8%) were married. Most participants (149 or 59.6%) were between the ages of 18 and 25, 82 (32.8%) were between the ages of 26 and 35, 13 (5.2%) were between the ages of 36 and 45, and only six (2.4%) were between the ages of 46 and 55. The colleges of dentistry at Qassim University, King Saud University, and Riyadh Elm University had 59 (23.6%), 55 (22%), and 39 (15.6%) participants, respectively. Additionally, 114 (45.6%) were students, whereas 136 (54.4%) were dentists, including dental interns, general practitioners, and specialists. Only 28 (11.2%) held a postgraduate degree (master's or PhD), as shown in Table ##TAB##0##1##. </p>", "<p>The general knowledge of antiresorptive/antiangiogenic medications revealed that most of the dentists (119 or 87.5%) knew about BP drugs as compared to students (78 or 68.4%), with a significant difference found among them (p&lt;0.05). Almost all of the dentists (121 or 89%) and about 81 (71.1%) students thought it was important to ask patients about their usage of antiresorptive/antiangiogenic medications, with a significant difference found between them (p=0.05). It was observed that the university was the primary source of information for both the dentists (97 or 71.3%) and students (70 or 61.4%). Regarding obtaining knowledge via variable additional sources (such as scientific journals and medical meetings), the dentists' group had a higher tendency to obtain knowledge than the students' group (p=0.136). Most of the dentists (117 or 86%) and 69 (60.5%) students believed BPs can lead to ONJ, with a significant difference between them (p&lt;0.05). Furthermore, most dentists (115 or 84.6%) and only 79 (23.7%) students thought that patients should be checked by a dentist before starting intravenous BP treatment, with a significant difference among them (p=0.05), as shown in Table ##TAB##1##2##. </p>", "<p>The study found that there was a general lack of knowledge regarding the therapeutic uses of antiresorptive and antiangiogenic medications in both dentists and students. Importantly, there were no significant differences between the two groups in terms of their knowledge (p=0.552). The data reveal that bone metastasis is the most commonly recognized therapeutic use of antiresorptive therapy among students, accounting for 25 (21.9%) of the responses. Furthermore, dentists primarily associate antiresorptive therapy with treating osteopenia and osteoporosis, which accounted for 28 (20.6%) of the responses. Interestingly, 62 (54.4%) students and 42 (30.9%) dentists could not identify BPs' active principle or commercial name. Out of all the listed BP medications, alendronate (Fosamax, Merck &amp; Co., Rahway, New Jersey, United States) was the most recognized, followed by zoledronate (Zometa, Novartis, Basel, Switzerland), with a significant difference between them (p&lt;0.05). Most of the dentists (77 or 56.6%) and students (73 or 64%) did not know that any other medications could lead to ONJ, with an insignificant difference among them (p=0.288), as shown in Table ##TAB##2##3##. </p>", "<p>Regarding knowledge of the correct definition of ONJ, only a small proportion of dentists (30 or 22.1%) and students (25 or 21.9%) knew the correct definition of MRONJ according to the AAOMS, but an insignificant difference was observed among them (p=0.779). Regarding the risk factors of MRONJ, tobacco was the most recognized by 28 (20.6%) dentists and 19 (16.7%) students, with an insignificant association among them (p=0.409) as shown in Table ##TAB##3##4##. </p>", "<p>Regarding the level of knowledge about the dental management of patients receiving BP therapy, most dentists (80 or 58.8%) and 58 students (50.9%) did not think invasive dental treatment could be performed safely on patients during intravenous BP therapy. In comparison, 32 (23.5%) dentists and 10 (8.8%) students thought that patients on intravenous BP therapy could possibly undergo invasive dental procedures without risk, with a significant difference between them (p&lt;0.05). Conversely, there was an insignificant difference observed between dentists and students that patients who are on oral BP therapy for a duration of less than four years and in the absence of any risk factors could safely undergo invasive dental treatment (p=0.186). Additionally, 47 (34.6%) dentists and 39 (34.2%) students recognized that taking oral BP therapy less than four years will make invasive dental treatment unsafe for such patients, with an insignificant difference observed between dentists and students (p=0.851). In addition, 42 (30.9%) dentists and 20 (17.5%) students indicated that for patients who are on oral BP therapy for more than four years, invasive dental treatment could be performed safely, with a significant difference observed between dentists and students (p=0.048). Most of the dentists (111 or 81.6%) and 91 (79.8%) students wanted to learn more about the ONJ; however, an insignificant association was noticed between them (p=0.913), as shown in Table ##TAB##4##5##.</p>" ]

[ "<title>Discussion</title>", "<p>MRONJ is a significant and debilitating adverse medication reaction observed in individuals undergoing prolonged treatment with antiresorptive or antiangiogenic drugs, primarily impacting the mandible more commonly than the maxilla. Having a sufficient understanding of MRONJ is essential to enhance treatment results and mitigate the problems linked to these drugs. Based on the available information, a scant amount of research has investigated the extent of knowledge of MRONJ among dental healthcare providers and dentistry students.</p>", "<p>This study was conducted to evaluate the knowledge of dentists and students regarding MRONJ to improve patient care. In the sample of 250 participants, this study found that most of the dentists knew about BP drugs (87.5%) and this is in alignment with previous research conducted in Saudi Arabia. However, both studies by Almousa et al. and Al-Eid et al. revealed a comparatively lower level of knowledge among dentists, with percentages of 66.5% and 60.8%, respectively. Furthermore, comparable findings were observed in previous studies conducted among dental professionals, indicating that 70% of dentists were aware of MRONJ. In a separate investigation, it was discovered that 83.3% of dental professionals and 99% of students reported possessing knowledge of BPs [##REF##33603544##22##,##UREF##3##23##]. In addition, similar results were found in Al-Maweri et al.'s study conducted among dentists, which showed that 70% knew about MRONJ [##REF##32032970##24##]. Other studies found 83.3% of dentists and 99% of students declared to know BPs [##REF##27114946##18##,##REF##32016175##25##].</p>", "<p>University teaching was the primary source of knowledge attained by dentists (71.3%) and students (61.4%). In general, the group of dentists tended to gain knowledge from many external sources, including the media, scientific journals, and professional gatherings, in contrast to the student group. One possible explanation for this phenomenon is that the dentist group may have a higher likelihood of seeing patients at risk of MRONJ and actively engaging in continuing medical education (CME) programs. The data indicate 89% of dentists consider it necessary to inquire about patients' use of antiresorptive/antiangiogenic medications, in contrast to 71.1% of students. The overall knowledge of dentists and students with regard to the therapeutic uses of antiresorptive and antiangiogenic medications was low. The most common therapeutic use recognized by students was bone metastasis (21.9%), whereas osteopenia and osteoporosis (20.6%) were also recognized by dentists.</p>", "<p>In the current study, it was observed that a significant portion of the participants lacked knowledge about the specific antiresorptive medications despite the inclusion of both the generic and brand names of these medications. The study conducted by de Lima et al. and Almousa et al. among dentists and dental students revealed similar results, indicating that most (86%) participants could not identify the commercial brand names of BP medication [##REF##25843818##20##,##REF##33603544##22##].</p>", "<p>Regarding BP side effects, a study revealed that alendronate (Fosamax) and zoledronate (Zometa) can cause osteonecrosis, whereas a large number of dentists (56.6%) and students (64%) were unaware that other medications might induce ONJ. The difference in knowledge between these two groups was statistically insignificant. Rosella et al. and Almousa et al. have reported similar findings regarding the impact of well-known BP medications [##REF##33603544##22##,##REF##29279671##26##].</p>", "<p>The process of identifying medications is crucial to minimizing the potential of providing care without fully understanding the associated risks.</p>", "<p>Regarding knowledge of the precise definition of ONJ, only a small proportion of dentists (22.1%) and students (21.9%) possessed this knowledge. The AAOMS defines MRONJ as the presence of exposed bone or bone that can be probed through a fistula in the maxillofacial region that persists for more than eight weeks in patients who have been treated with antiresorptive or antiangiogenic agents, without a history of radiation therapy to the jaws or evident metastatic disease in the jaws [##REF##35300956##4##]. The results demonstrate similarity with Almousa et al.'s, Al-Eid et al.'s, and Al-Maweri et al.'s studies, emphasizing the limited understanding of the clinical characteristics of MRONJ [##REF##33603544##22##, ####UREF##3##23##, ##REF##32032970##24####32032970##24##]. On the contrary, Spanish dentists and dental students exhibited more significant levels of knowledge because of their greater familiarity with the accurate definition of this disease, as reported by López-Jornet et al. [##REF##20663005##27##]. Lack of understanding about the definition can lead to delayed detection or unwarranted treatments, thereby heightening the likelihood of more serious complications.</p>", "<p>The participants' responses regarding the risk factors were inadequate because less than 50% of them correctly identified the risk factors. According to the data, a significant percentage of dentists (58.8%) and students (50.9%) hold the belief that invasive dental procedures may not be safe for patients undergoing intravenous BP therapy. The data indicate that a higher percentage of dentists (23.5%) compared to students (8.8%) believed that the task could be carried out without any risks. The findings from our study are consistent with the results of Almousa et al.'s study [##REF##33603544##22##].</p>", "<p>Overall, the findings indicate a lack of awareness about patient management, resulting in deferring necessary treatment when the risk is low while attempting high-risk treatments without taking the appropriate precautions.</p>", "<p>The present study acknowledges several limitations. The sample size was relatively small and limited to specific locations in Saudi Arabia. The outcomes reported may lack representativeness for dentists nationwide. Future studies should include larger sample sizes and broaden their sampling to include other regions in Saudi Arabia.</p>" ]

[ "<title>Conclusions</title>", "<p>The implications of the findings in the present study warrant increased emphasis on the importance of educating students and dentists about this disease. It is highly recommended to attend continuing education courses that focus on treating and preventing this ailment in patients undergoing BP therapy.</p>" ]

[ "<p>Background: Bisphosphonates (BPs) are often used in treating benign and malignant disorders. Medication-related osteonecrosis of the jaw (MRONJ) is a significant problem that arises from the long-term use of BPs.</p>", "<p>Objective: In this study, we assessed the knowledge of students and dentists about MRONJ in the central region of Saudi Arabia.</p>", "<p>Methods: A cross-sectional study was conducted to collect information from dental students and practitioners from the central region of Saudi Arabia. A valid, reliable, and structured questionnaire was used to gather data using a non-probability convenient sampling technique. IBM SPSS Statistics for Windows, Version 22.0 (Released 2013; IBM Corp., Armonk, New York, United States) was used to analyse the data. The descriptive data were expressed as frequencies and percentages to evaluate the association between dentists and students concerning overall knowledge related to osteonecrosis of the jaw, and a chi-squared test was applied.</p>", "<p>Results: In total, 250 individuals completed the questionnaire. The general knowledge of antiresorptive/antiangiogenic medications showed that most dentists (87.5%) and students (68.4%) knew about BP medications. A general lack of understanding about the therapeutic uses of antiangiogenic and antiresorptive medications was demonstrated by the participants. A significant proportion of dentists (58.8%) and students (50.9%) were not convinced that invasive dental procedures can be safely performed on patients receiving intravenous BP therapy. A significant proportion of the participants in the sample were unclear of the principal diseases that antiresorptive and antiangiogenic medications target. A mere 22% of respondents were aware of the accurate definition of medications-related MRONJ.</p>", "<p>Conclusion: There is insufficient knowledge about MRONJ among students and practitioners. Therefore, these findings suggest increased emphasis should be placed on educating dentists and students about this condition to ensure patients receive the best possible care.</p>" ]

[]

[ "<title>Appendices</title>" ]

[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Questionnaire (Section 1) </title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>Questionnaire (Section 2) </title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG3\"><label>Figure 3</label><caption><title>Questionnaire (Section 3) </title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG4\"><label>Figure 4</label><caption><title>Questionnaire (Section 4) </title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG5\"><label>Figure 5</label><caption><title>Questionnaire (Section 5) </title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Frequency distribution of demographic details of participants (n=250)</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td colspan=\"2\" rowspan=\"1\">Variable</td><td rowspan=\"1\" colspan=\"1\">% (n)</td></tr><tr><td rowspan=\"4\" colspan=\"1\">Age (years)</td><td rowspan=\"1\" colspan=\"1\">18-25</td><td rowspan=\"1\" colspan=\"1\">59.6% (149)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">26-35</td><td rowspan=\"1\" colspan=\"1\">32.8% (82)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">36-45</td><td rowspan=\"1\" colspan=\"1\">5.2% (13)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">46-55</td><td rowspan=\"1\" colspan=\"1\">2.4% (6)</td></tr><tr><td rowspan=\"2\" colspan=\"1\">Gender</td><td rowspan=\"1\" colspan=\"1\">Male</td><td rowspan=\"1\" colspan=\"1\">48.8% (122)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Female</td><td rowspan=\"1\" colspan=\"1\">51.2% (128)</td></tr><tr><td rowspan=\"3\" colspan=\"1\">Marital status</td><td rowspan=\"1\" colspan=\"1\">Married</td><td rowspan=\"1\" colspan=\"1\">18.8% (47)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Single</td><td rowspan=\"1\" colspan=\"1\">79.2% (198)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Separated</td><td rowspan=\"1\" colspan=\"1\">2% (5)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"6\" colspan=\"1\">University</td><td rowspan=\"1\" colspan=\"1\">Qassim University</td><td rowspan=\"1\" colspan=\"1\">23.6% (59)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">King Saud University</td><td rowspan=\"1\" colspan=\"1\">22% (55)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">King Saud bin Abdulaziz University for Health Sciences</td><td rowspan=\"1\" colspan=\"1\">6% (15)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Prince Sultan bin Abdulaziz University</td><td rowspan=\"1\" colspan=\"1\">8.4% (21)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Riyadh Elm University</td><td rowspan=\"1\" colspan=\"1\">15.6% (39)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Others</td><td rowspan=\"1\" colspan=\"1\">24.4% (61)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">Educational level</td><td rowspan=\"1\" colspan=\"1\">Student</td><td rowspan=\"1\" colspan=\"1\">45.6% (114)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Graduate</td><td rowspan=\"1\" colspan=\"1\">54.4% (136)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"7\" colspan=\"1\">Enrolled year in college</td><td rowspan=\"1\" colspan=\"1\">First year</td><td rowspan=\"1\" colspan=\"1\">7.2% (18)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Second year</td><td rowspan=\"1\" colspan=\"1\">7.6% (19)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Third year</td><td rowspan=\"1\" colspan=\"1\">6% (15)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Fourth year</td><td rowspan=\"1\" colspan=\"1\">5.2% (13)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Fifth year</td><td rowspan=\"1\" colspan=\"1\">14.8% (37)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Intern</td><td rowspan=\"1\" colspan=\"1\">4.8% (12)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Graduated</td><td rowspan=\"1\" colspan=\"1\">54.4% (136)</td></tr><tr><td rowspan=\"5\" colspan=\"1\">Highest degree obtained</td><td rowspan=\"1\" colspan=\"1\">Student</td><td rowspan=\"1\" colspan=\"1\">45.6% (114)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Bachelor</td><td rowspan=\"1\" colspan=\"1\">39.2% (98)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Masters</td><td rowspan=\"1\" colspan=\"1\">5.6% (14)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">PhD</td><td rowspan=\"1\" colspan=\"1\">5.6% (14)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Others</td><td rowspan=\"1\" colspan=\"1\">4% (10)</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>General knowledge of bisphosphonates</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Variables</td><td rowspan=\"1\" colspan=\"1\">Student (n=114) % (n)</td><td rowspan=\"1\" colspan=\"1\">Dentists (n=136) % (n)</td><td rowspan=\"1\" colspan=\"1\">p-value</td></tr><tr><td colspan=\"4\" rowspan=\"1\">Do you know bisphosphonate drugs?</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Yes</td><td rowspan=\"1\" colspan=\"1\">68.4% (78)</td><td rowspan=\"1\" colspan=\"1\">87.5% (119)</td><td rowspan=\"2\" colspan=\"1\">&lt;0.05</td></tr><tr><td rowspan=\"1\" colspan=\"1\">No</td><td rowspan=\"1\" colspan=\"1\">31.6% (36)</td><td rowspan=\"1\" colspan=\"1\">12.5% (17)</td></tr><tr style=\"background-color:#ccc\"><td colspan=\"4\" rowspan=\"1\">Is it important to ask patients about use of bisphosphonate medications?</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Yes</td><td rowspan=\"1\" colspan=\"1\">71.1% (81)</td><td rowspan=\"1\" colspan=\"1\">89% (121)</td><td rowspan=\"3\" colspan=\"1\">0.05</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">No</td><td rowspan=\"1\" colspan=\"1\">2.6% (3)</td><td rowspan=\"1\" colspan=\"1\">1.5% (2)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">I don't know</td><td rowspan=\"1\" colspan=\"1\">26.3% (30)</td><td rowspan=\"1\" colspan=\"1\">9.6% (13)</td></tr><tr style=\"background-color:#ccc\"><td colspan=\"4\" rowspan=\"1\">Where have you heard about bisphosphonate medications?</td></tr><tr><td rowspan=\"1\" colspan=\"1\">University</td><td rowspan=\"1\" colspan=\"1\">61.4% (70)</td><td rowspan=\"1\" colspan=\"1\">71.3% (97)</td><td rowspan=\"5\" colspan=\"1\">0.136</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Mass media</td><td rowspan=\"1\" colspan=\"1\">7% (8)</td><td rowspan=\"1\" colspan=\"1\">1.5% (1)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Scientific journals</td><td rowspan=\"1\" colspan=\"1\">9.6% (11)</td><td rowspan=\"1\" colspan=\"1\">10.3% (14)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Medical meetings</td><td rowspan=\"1\" colspan=\"1\">5.3% (6)</td><td rowspan=\"1\" colspan=\"1\">5.1% (7)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Never heard</td><td rowspan=\"1\" colspan=\"1\">16.7% (19)</td><td rowspan=\"1\" colspan=\"1\">11.8% (16)</td></tr><tr style=\"background-color:#ccc\"><td colspan=\"4\" rowspan=\"1\">Can bisphosphonates lead to osteonecrosis of the jaw?  </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Yes</td><td rowspan=\"1\" colspan=\"1\">60.5% (69)</td><td rowspan=\"1\" colspan=\"1\">86% (117)</td><td rowspan=\"3\" colspan=\"1\">&lt;0.05</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">No</td><td rowspan=\"1\" colspan=\"1\">11.4% (13)</td><td rowspan=\"1\" colspan=\"1\">5.1% (7)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">I don't know</td><td rowspan=\"1\" colspan=\"1\">28.1% (32)</td><td rowspan=\"1\" colspan=\"1\">8.8% (12)</td></tr><tr style=\"background-color:#ccc\"><td colspan=\"4\" rowspan=\"1\">Examination of patient before starting an IV bisphosphonate treatment</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Yes</td><td rowspan=\"1\" colspan=\"1\">23.7% (79)</td><td rowspan=\"1\" colspan=\"1\">84.6% (115)</td><td rowspan=\"3\" colspan=\"1\">0.05</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">No</td><td rowspan=\"1\" colspan=\"1\">7% (8)</td><td rowspan=\"1\" colspan=\"1\">1.5% (2)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">I don't know</td><td rowspan=\"1\" colspan=\"1\">69.3% (27)</td><td rowspan=\"1\" colspan=\"1\">14% (19)</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB3\"><label>Table 3</label><caption><title>Knowledge and therapeutic uses of medications</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Variables</td><td rowspan=\"1\" colspan=\"1\">Student (n=114) % (n)</td><td rowspan=\"1\" colspan=\"1\">Dentists (n=136) % (n)</td><td rowspan=\"1\" colspan=\"1\">p-value</td></tr><tr><td colspan=\"4\" rowspan=\"1\">What are the pathology target of a bisphosphonate therapy?</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Bone metastases</td><td rowspan=\"1\" colspan=\"1\">21.9% (25)</td><td rowspan=\"1\" colspan=\"1\">14% (19)</td><td rowspan=\"10\" colspan=\"1\">0.552</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Chondroblastoma</td><td rowspan=\"1\" colspan=\"1\">2.6% (3)</td><td rowspan=\"1\" colspan=\"1\">1.5% (2)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Hypercalcemia of malignancy</td><td rowspan=\"1\" colspan=\"1\">2.6% (3)</td><td rowspan=\"1\" colspan=\"1\">2.2% (3)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Multiple myeloma</td><td rowspan=\"1\" colspan=\"1\">7% (8)</td><td rowspan=\"1\" colspan=\"1\">8.1% (11)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Osteogenesis imperfecta</td><td rowspan=\"1\" colspan=\"1\">5.3% (6)</td><td rowspan=\"1\" colspan=\"1\">4.4% (6)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Osteomyelitis</td><td rowspan=\"1\" colspan=\"1\">13.2% (15)</td><td rowspan=\"1\" colspan=\"1\">14% (19)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Osteopenia and osteoporosis</td><td rowspan=\"1\" colspan=\"1\">14.9% (17)</td><td rowspan=\"1\" colspan=\"1\">20.6% (28)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Osteopetrosis</td><td rowspan=\"1\" colspan=\"1\">12.3% (14)</td><td rowspan=\"1\" colspan=\"1\">8.1% (11)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Paget's disease of bone</td><td rowspan=\"1\" colspan=\"1\">8.8% (10)</td><td rowspan=\"1\" colspan=\"1\">11.8% (16)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">All of the above</td><td rowspan=\"1\" colspan=\"1\">11.4% (13)</td><td rowspan=\"1\" colspan=\"1\">15.4% (21)</td></tr><tr style=\"background-color:#ccc\"><td colspan=\"4\" rowspan=\"1\">Do you know the active principle and commercial name of bisphosphonates?</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Alendronate (Fosamax, Merck &amp; Co., Rahway, New Jersey, United States)</td><td rowspan=\"1\" colspan=\"1\">17.5% (20)</td><td rowspan=\"1\" colspan=\"1\">22.8% (31)</td><td rowspan=\"8\" colspan=\"1\">&lt;0.05</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ibandronate (Boniva, Roche, Basel, Switzerland)</td><td rowspan=\"1\" colspan=\"1\">10.5% (12)</td><td rowspan=\"1\" colspan=\"1\">5.1% (7)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Neridronate (Nerixia, Abiogen Pharma, Pisa, Italy)</td><td rowspan=\"1\" colspan=\"1\">4.4% (5)</td><td rowspan=\"1\" colspan=\"1\">2.9% (4)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Pamidronate (Aredia, Pfizer, New York, New York, United States)</td><td rowspan=\"1\" colspan=\"1\">3.5% (4)</td><td rowspan=\"1\" colspan=\"1\">2.2% (3)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Risedronate (Actonel and Atelvia, Greenstone Pharma, North America, United States)</td><td rowspan=\"1\" colspan=\"1\">7% (8)</td><td rowspan=\"1\" colspan=\"1\">11.8% (16)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Tiludronate (Skelid, Sanofi, Paris, France)</td><td rowspan=\"1\" colspan=\"1\">0% (0)</td><td rowspan=\"1\" colspan=\"1\">1.5% (2)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Zoledronate (Zometa, Novartis, Basel, Switzerland)</td><td rowspan=\"1\" colspan=\"1\">2.6% (3)</td><td rowspan=\"1\" colspan=\"1\">22.8% (31)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">None</td><td rowspan=\"1\" colspan=\"1\">54.4% (62)</td><td rowspan=\"1\" colspan=\"1\">30.9% (42)</td></tr><tr><td colspan=\"4\" rowspan=\"1\">Do you know any other medications involved in the osteonecrosis of the jaw?</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Bevacizumab (Avastin, Roche, Basel, Switzerland)</td><td rowspan=\"1\" colspan=\"1\">7% (8)</td><td rowspan=\"1\" colspan=\"1\">10.3% (14)</td><td rowspan=\"6\" colspan=\"1\">0.288</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Denosumab (Xgeva and Prolia, Amgen, Thousand Oaks, California, United States)</td><td rowspan=\"1\" colspan=\"1\">13.2% (15)</td><td rowspan=\"1\" colspan=\"1\">20.6% (28)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Sirolimus (Rapamune, Pfizer, New York, New York, United States)</td><td rowspan=\"1\" colspan=\"1\">2.6% (3)</td><td rowspan=\"1\" colspan=\"1\">4.4% (6)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Sorafenib (Nexavar, Bayer, Berlin, Germany)</td><td rowspan=\"1\" colspan=\"1\">6.1% (7)</td><td rowspan=\"1\" colspan=\"1\">5.1% (7)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Sunitinib (Sutent, Pfizer, New York, New York, United States)</td><td rowspan=\"1\" colspan=\"1\">7% (8)</td><td rowspan=\"1\" colspan=\"1\">2.9% (4)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">I don't know any non-bisphosphonate drug</td><td rowspan=\"1\" colspan=\"1\">64% (73)</td><td rowspan=\"1\" colspan=\"1\">56.6% (77)</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB4\"><label>Table 4</label><caption><title>Knowledge about risk factors and correct definition of MRONJ</title><p>MRONJ: medication-related osteonecrosis of the jaw</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Variables</td><td rowspan=\"1\" colspan=\"1\">Student (n=114) % (n)</td><td rowspan=\"1\" colspan=\"1\">Dentists (n=136) % (n)</td><td rowspan=\"1\" colspan=\"1\">p-value</td></tr><tr><td colspan=\"4\" rowspan=\"1\">Risk factors of osteonecrosis of the jaw</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Alcohol</td><td rowspan=\"1\" colspan=\"1\">8.8% (10)</td><td rowspan=\"1\" colspan=\"1\">11% (15)</td><td rowspan=\"13\" colspan=\"1\">0.409</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Antibiotic therapy</td><td rowspan=\"1\" colspan=\"1\">10.5% (12)</td><td rowspan=\"1\" colspan=\"1\">2.9% (4)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Arterial hypertension</td><td rowspan=\"1\" colspan=\"1\">1.8% (2)</td><td rowspan=\"1\" colspan=\"1\">0.7% (1)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Hyperlipidemia</td><td rowspan=\"1\" colspan=\"1\">2.6% (3)</td><td rowspan=\"1\" colspan=\"1\">2.2% (3)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Length of therapy</td><td rowspan=\"1\" colspan=\"1\">1.8% (2)</td><td rowspan=\"1\" colspan=\"1\">5.9% (8)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Microtrauma</td><td rowspan=\"1\" colspan=\"1\">7.9% (9)</td><td rowspan=\"1\" colspan=\"1\">7.4% (10)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Steroid therapy</td><td rowspan=\"1\" colspan=\"1\">1.8% (2)</td><td rowspan=\"1\" colspan=\"1\">2.9% (4)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Tobacco</td><td rowspan=\"1\" colspan=\"1\">16.7% (19)</td><td rowspan=\"1\" colspan=\"1\">20.6% (28)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Tobacco, alcohol, length of therapy, steroid therapy</td><td rowspan=\"1\" colspan=\"1\">17.5% (20)</td><td rowspan=\"1\" colspan=\"1\">14% (19)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Total amount of drug administered</td><td rowspan=\"1\" colspan=\"1\">1.8% (2)</td><td rowspan=\"1\" colspan=\"1\">0.7% (1)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Way of administration</td><td rowspan=\"1\" colspan=\"1\">7% (8)</td><td rowspan=\"1\" colspan=\"1\">5.1% (7)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Way of administration, length of therapy, steroid therapy</td><td rowspan=\"1\" colspan=\"1\">3.5% (4)</td><td rowspan=\"1\" colspan=\"1\">5.1% (7)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">All of the above</td><td rowspan=\"1\" colspan=\"1\">18.4% (21)</td><td rowspan=\"1\" colspan=\"1\">21.3% (29)</td></tr><tr><td colspan=\"4\" rowspan=\"1\">Percentage of correct MRONJ definition</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Bone exposed to the maxillofacial region that is visible through an intraoral or extraoral fistula or has persisted for at least eight weeks in patients who are currently or have been treated with antiresorptive or antiangiogenic agents and who have no history of radiation therapy or metastatic disease in the jaws</td><td rowspan=\"1\" colspan=\"1\">21.9% (25)</td><td rowspan=\"1\" colspan=\"1\">22.1% (30)  </td><td rowspan=\"1\" colspan=\"1\">0.779</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB5\"><label>Table 5</label><caption><title>Knowledge about the dental management of patients on bisphosphonate therapy</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Variables</td><td rowspan=\"1\" colspan=\"1\">Student (n=114) % (n)</td><td rowspan=\"1\" colspan=\"1\">Dentists (n=136) % (n)</td><td rowspan=\"1\" colspan=\"1\">p-value</td></tr><tr><td colspan=\"4\" rowspan=\"1\">Can intravenous bisphosphonate patients receive invasive dental procedures?</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Yes</td><td rowspan=\"1\" colspan=\"1\">8.8% (10)</td><td rowspan=\"1\" colspan=\"1\">23.5% (32)</td><td rowspan=\"2\" colspan=\"1\">&lt;0.05</td></tr><tr><td rowspan=\"1\" colspan=\"1\">No</td><td rowspan=\"1\" colspan=\"1\">50.9% (58)</td><td rowspan=\"1\" colspan=\"1\">58.8% (80)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">I don't know</td><td rowspan=\"1\" colspan=\"1\">40.4% (46)</td><td rowspan=\"1\" colspan=\"1\">17.6% (24)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td colspan=\"4\" rowspan=\"1\">Are invasive dental procedures safe for patients taking oral bisphosphonates for less than four years without risk factors?</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Yes</td><td rowspan=\"1\" colspan=\"1\">20.2% (23)</td><td rowspan=\"1\" colspan=\"1\">29.4% (40)</td><td rowspan=\"3\" colspan=\"1\">0.186</td></tr><tr><td rowspan=\"1\" colspan=\"1\">No</td><td rowspan=\"1\" colspan=\"1\">31.6% (36)</td><td rowspan=\"1\" colspan=\"1\">31.6% (43)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">I don't know</td><td rowspan=\"1\" colspan=\"1\">48.2% (55)</td><td rowspan=\"1\" colspan=\"1\">39% (53)</td></tr><tr><td colspan=\"4\" rowspan=\"1\">Are invasive dental treatments safe for people with risk factors who have been taking oral bisphosphonates for &lt;4 years?</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Yes</td><td rowspan=\"1\" colspan=\"1\">20.2% (23)</td><td rowspan=\"1\" colspan=\"1\">22.8% (31)</td><td rowspan=\"3\" colspan=\"1\">0.851</td></tr><tr><td rowspan=\"1\" colspan=\"1\">No</td><td rowspan=\"1\" colspan=\"1\">34.2% (39)</td><td rowspan=\"1\" colspan=\"1\">34.6% (47)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">I don't know</td><td rowspan=\"1\" colspan=\"1\">45.6% (52)</td><td rowspan=\"1\" colspan=\"1\">42.6% (58)</td></tr><tr><td colspan=\"4\" rowspan=\"1\">Can oral bisphosphonate users over four years safely undergo invasive dental procedures?</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Yes</td><td rowspan=\"1\" colspan=\"1\">17.5% (20)</td><td rowspan=\"1\" colspan=\"1\">30.9% (42)</td><td rowspan=\"3\" colspan=\"1\">0.048</td></tr><tr><td rowspan=\"1\" colspan=\"1\">No</td><td rowspan=\"1\" colspan=\"1\">32.5% (37)</td><td rowspan=\"1\" colspan=\"1\">27.9% (38)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">I don't know</td><td rowspan=\"1\" colspan=\"1\">60% (57)</td><td rowspan=\"1\" colspan=\"1\">41.2% (56)</td></tr><tr><td colspan=\"4\" rowspan=\"1\">Do you want to learn more about jaw osteonecrosis?</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Yes</td><td rowspan=\"1\" colspan=\"1\">79.8% (91)</td><td rowspan=\"1\" colspan=\"1\">81.6% (111)</td><td rowspan=\"3\" colspan=\"1\">0.931</td></tr><tr><td rowspan=\"1\" colspan=\"1\">No</td><td rowspan=\"1\" colspan=\"1\">6.1% (7)</td><td rowspan=\"1\" colspan=\"1\">5.9% (8)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">I don't know</td><td rowspan=\"1\" colspan=\"1\">14% (16)</td><td rowspan=\"1\" colspan=\"1\">12.5% (17)</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Meshal M. Alghofaili, Syed Fareed Mohsin, Rayan Khaled Almazyad</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Meshal M. Alghofaili, Nada Mohammed Nahari, Tamim S. Alkhalifah, Mohammed K. Alsaegh</p><p><bold>Drafting of the manuscript:</bold>  Meshal M. Alghofaili, Syed Fareed Mohsin, Tamim S. Alkhalifah, Mohammed K. Alsaegh</p><p><bold>Supervision:</bold>  Syed Fareed Mohsin</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Nada Mohammed Nahari, Rayan Khaled Almazyad</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study. Committee of Research Ethics of Qassim University issued approval 21-12-03</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Animal Ethics</title><fn fn-type=\"other\"><p><bold>Animal subjects:</bold> All authors have confirmed that this study did not involve animal subjects or tissue.</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Leukemia is a malignancy of the bone marrow that arises from the abnormal proliferation and differentiation of hematopoietic stem cells. This results in the accumulation of immature or abnormal blood cells in the marrow and peripheral blood [##REF##27585213##1##]. Ocular manifestations of leukemia can impair vision. Of all leukemic ocular involvements, leukemic retinopathy is the most common, occurring in up to 50% of patients [##REF##2778482##2##, ####REF##14573980##3##, ##REF##30513404##4####30513404##4##]. Further classification divides leukemic retinopathy into primary and secondary retinopathy. Primary retinopathy is characterized by direct retinal infiltration of cancerous leukocytes [##REF##15002029##5##,##REF##32029770##6##]. Secondary retinopathy is a sequela of leukemic hematological abnormalities, including thrombocytopenia, anemia, and hyperviscosity [##REF##15002029##5##].</p>" ]

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[ "<title>Discussion</title>", "<p>Leukemia is a systemic hematological disease, with retinal involvement as the most common ocular manifestation [##REF##2778482##2##, ####REF##14573980##3##, ##REF##30513404##4####30513404##4##]. Leukemic retinopathy may arise from direct infiltration of cancerous leukocytes. It can also be a consequence of leukemia-induced hematologic abnormalities, which manifest as intraretinal hemorrhages (e.g., dot-blot hemorrhages, flame hemorrhages, Roth spots), preretinal hemorrhages, and cotton-wool spots [##REF##15002029##5##,##REF##32029770##6##]. Besides CEL, leukemic retinopathy has been noted in other leukemias such as acute lymphoblastic leukemia (ALL), acute myeloid leukemia (AML), chronic myeloid leukemia (CML), and adult T-cell leukemia [##REF##27585213##1##]. In this case, leukemic retinopathy was one of the first presenting signs of CEL, which led to our patient receiving a more immediate treatment. Regular follow-up with oncology specialists and ophthalmologists is necessary to monitor the patient’s condition and treatment efficacy.</p>", "<p>The presence of ocular involvement in leukemia indicates aggressive systemic disease and portends a poor prognosis [##REF##1477038##7##,##REF##8886591##8##]. Ohkoshi and Tsiaras reported that the five-year survival rate was significantly lower in leukemia patients with leukemic retinopathy on presentation than in those without ophthalmic involvement (21.4% vs. 45.7%) [##REF##1477038##7##]. This is due to a higher likelihood of central nervous system (CNS) involvement in patients exhibiting ophthalmic manifestations of leukemic retinopathy, which is a poor prognostic factor [##REF##1477038##7##]. Abu el-Asrar et al. prospectively evaluated the prognostic importance of retinopathy in adult and pediatric leukemia patients, reporting that the three-month mortality rate of patients with cotton-wool spots is eight times higher than in patients without these retinal lesions. Cotton-wool spots are a product of occluded precapillary arterioles and resultant retinal ischemia, which signify a disease state that is clinically and hematologically active [##REF##8886591##8##]. Therefore, the presence of any retinal hemorrhage or cotton-wool spot in a patient with no apparent systemic cause should prompt physicians to order a complete blood count, including WBC differential, to rule out leukemia and other hematologic irregularities [##REF##34252423##9##].</p>", "<p>Once the diagnosis of leukemia has been established, treatment of leukemic retinopathy involves treating the underlying cause with systemic chemotherapy [##REF##32029770##6##]. Imatinib, a BCR-ABL tyrosine kinase inhibitor, has shown promising systemic treatment of different hematological diseases, including myeloproliferative neoplasms with eosinophilia that have evidence of PDGFRA rearrangement. Allopurinol, a xanthine oxidase inhibitor, is often added to imatinib for prophylaxis against tumor lysis syndrome [##REF##8886591##8##]. Treatment regimens with imatinib have not only shown leukemic improvement, but cases have shown resolution of retinal hemorrhages and retinal infiltrates on fundus photography as soon as the one-month follow-up [##REF##25390438##10##,##REF##12614767##11##].</p>", "<p>After induction chemotherapy, physicians may consider additional therapies, such as hydroxyurea and leukapheresis, for leukemic treatment. Hydroxyurea given orally at a dose of 50-100 mg/kg daily can reduce the absolute WBC count by 50-80% percent within 48 hours [##REF##268956##12##]. Leukapheresis, the direct removal of WBCs from circulation, is another adjunct therapy and has been shown to improve VA in patients with retinal involvement [##REF##6585220##13##]. External radiation therapy may be indicated for cases of optic nerve and/or orbital involvement, however, it should be used sparingly due to the risk of radiation-induced retinopathy and cataracts [##REF##8886591##8##].</p>", "<p>In most cases, treatment of leukemia by systemic chemotherapy or radiation resolves primary and secondary leukemic retinopathy within the first two months [##REF##12614767##11##,##UREF##0##14##,##REF##18303160##15##]. If vitreoretinal leukemic infiltration, which can manifest as vitreous cell clumping or yellow-white subretinal infiltrates, persists despite systemic therapy, chemotherapeutic agents, such as methotrexate, can be injected intravitreally [##REF##18303160##15##]. This approach may reduce systemic chemo-drug toxicity when considering additional systemic chemotherapy [##REF##8886591##8##].</p>", "<p>Sequelae of untreated leukemic retinopathy include choroidal neovascularization and tractional retinal detachments. In patients with choroidal neovascularization, intravitreal anti-vascular endothelial growth factor (VEGF) agents may be used. If persistent vitreoretinal hemorrhages, vitreomacular traction, or retinal detachments arise in the setting of leukemic retinopathy, a pars plana vitrectomy is indicated [##REF##34252423##9##].</p>" ]

[ "<title>Conclusions</title>", "<p>In cases of unexplained retinal hemorrhages, a high index of suspicion for blood dyscrasias should warrant hematologic evaluation. This patient's visual complaints resulted in a workup that led to the diagnosis of eosinophilic leukemia, with prompt treatment allowing for a favorable prognosis. Ophthalmologists should thus be alert to retinal presentations of leukemia, as a comprehensive eye exam may lead to timely diagnosis and early intervention.</p>" ]

[ "<p>Leukemia is a systemic malignancy that can compromise various physiological functions, including vision. We report a case of a 37-year-old male presenting with worsening bilateral central vision loss, fatigue, shortness of breath, and ankle edema. Ophthalmic examination revealed extensive retinal hemorrhages, Roth spots, and subhyaloid hemorrhages, consistent with leukemic retinopathy. Further hematologic workup confirmed chronic eosinophilic leukemia. The patient showed systemic and visual improvement after prompt treatment with imatinib. This case highlights the importance of ophthalmological assessment in diagnosing leukemia, as ocular manifestations may often be the first sign of hematological disease.</p>" ]

[ "<title>Case presentation</title>", "<p>A 37-year-old Hispanic male presented with a two-day history of progressively worsening central vision in both eyes. The patient’s past medical history was significant only for diet-controlled hyperlipidemia. There was no past ocular history and no relevant family history. The patient worked as a forklift operator and denied alcohol and illicit drug use. Upon review of systems, the patient revealed that he had been having fatigue, shortness of breath on exertion, and bilateral ankle swelling for two weeks. He denied fever, chills, and recent weight loss.</p>", "<p>The best corrected visual acuity (BCVA) was counting fingers at 3 feet bilaterally. Pupils, intraocular pressure, confrontational visual fields, and motility were within normal limits bilaterally. Ishihara color plates were 0/8 in the right eye (OD) and 2/8 in the left eye (OS). Slit lamp examination was significant for bilateral conjunctival pallor. Dilated fundoscopic examination revealed Roth spots, macular edema, perivascular cotton-wool spots, extensive intra-retinal and pre-retinal hemorrhages, and chronic subhyaloid hemorrhages bilaterally (Figure ##FIG##0##1##). Vitreous was clear and optic nerves were sharp, pink, and without evidence of infiltration bilaterally.</p>", "<p>Differential diagnoses included infectious, inflammatory, and neoplastic etiologies. A hematologic workup revealed a high white blood cell (WBC) count (425 k/mm³) with elevated eosinophils and myelocytes (42%), as well as anemia (hemoglobin 6.2 g/dL, hematocrit 16.8%, mean corpuscular volume (MCV) 112 fL) and thrombocytopenia (15 k/mm³). Hematologic markers concerning for tumor lysis syndrome included hypocalcemia (8.2 mg/dL) and elevated lactate dehydrogenase (1063 U/L) while potassium levels were within normal limits (4.0 mEq/L). Infectious workup, blood cultures, fungal workup, and viral panels were negative. Bone marrow biopsy showed markedly increased eosinophils without an increase in blasts, consistent with chronic eosinophilic leukemia (CEL). The aspirate smears revealed a markedly increased eosinophilic component including mature segmented forms and precursors (42%). Immunochemistry showed an atypical myeloid population expressing CD13, CD33, CD11b, CD11c, CD9, and CD38. Flow cytometry was negative for HLA-DR, CD15, CD16, CD64, CD14, and immature markers (CD34, CD117). Polymerase chain reaction (PCR) testing came back negative for BCR-ABL and demonstrated a CHIC2 gene deletion, indicating a favorable prognosis with tyrosine kinase inhibitor therapy. PCR also revealed a PDGFRA gene rearrangement.</p>", "<p>The patient was admitted to the oncology service for treatment with leukapheresis, granulocyte colony-stimulating factor (G-CSF) injections, hydroxyurea, and imatinib, with allopurinol added for tumor lysis syndrome prophylaxis. He was given infection prophylaxis for seven days: acyclovir 400 mg PO BID, fluconazole 200 mg PO once a day, and ciprofloxacin 500 mg PO BID. The inpatient treatment regimen for CEL consisted of the following: two rounds of leukapheresis, several blood and platelet transfusions, one dose of G-CSF, allopurinol (300 mg PO once a day), hydroxyurea (500 mg q12 hours for 4 days), and imatinib (unspecified starting dose lowered to 100 mg PO once a day due to leukopenia). Upon discharge two weeks later, maintenance therapy of daily 100 mg imatinib led to normalized WBC count and resolution of fatigue and dyspnea. BCVA improved to 20/30 OD and 20/25 OS at the seven-month follow-up. Macular edema and retinal hemorrhage resolved after treatment when the patient was seen at his seven-month follow-up (Figures ##FIG##1##2##, ##FIG##2##3##), with residual foveal exudate present in the right (Figures ##FIG##1##2A##, ##FIG##2##3A##) and left (Figure ##FIG##2##3B##) eyes.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Color fundus photographs showing evidence of leukemic retinopathy</title><p>Color fundus photography of patient’s right (A) and left (B) eyes at initial presentation depicting Roth spots (solid white arrows), as well as extensive intraretinal (dashed white arrows) and chronic subhyaloid hemorrhages (white outlines).</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>Color fundus photographs after treatment of leukemic retinopathy</title><p>Color fundus photography at the seven-month follow-up depicting the resolution of hemorrhages in the right (A) and left (B) eyes. There is also the presence of right eye foveal exudate (solid white arrow). There is no evidence of left eye foveal exudate on fundus photography.</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG3\"><label>Figure 3</label><caption><title>Optical coherence tomography after treatment of leukemic retinopathy</title><p>Optical coherence tomography (OCT) of the macula at the seven-month follow-up further depicts foveal exudate in the right eye (A, solid white arrow) and minimal residual foveal exudate in the left eye (B, solid white arrow). There is no evidence of intraretinal or subretinal fluid in either eye.</p></caption></fig>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Emanuel Mordechaev, Tatyana Beketova, Brian Murillo, Max D. Schlesinger</p><p><bold>Drafting of the manuscript:</bold>  Emanuel Mordechaev, Tatyana Beketova, Brian Murillo, Max D. Schlesinger</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Emanuel Mordechaev, Tatyana Beketova, Brian Murillo, Max D. Schlesinger</p><p><bold>Concept and design:</bold>  Tatyana Beketova</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>The term \"musculoskeletal disorders\" (MSDs) refers to a group of periarticular conditions that affect the musculoskeletal system and primarily cause functional discomfort and everyday pain [##REF##36589714##1##,##UREF##0##2##]. Musculoskeletal disorders (MSDs) are more prevalent among schoolteachers, who are required to spend long periods of time standing, sitting, and engaging in repetitive tasks, such as grading papers or typing on a computer [##UREF##1##3##].</p>", "<p>These disorders can affect the teacher's physical health, causing pain, discomfort, and movement limitations, ultimately impacting their ability to perform their job effectively [##REF##36463148##4##]. In addition to physical discomfort, MSDs can significantly impact a teacher's mental health and job performance. Teachers with MSDs may experience increased stress and anxiety, reduced job satisfaction, and difficulty meeting their job demands [##UREF##2##5##]. Moreover, MSDs can lead to absenteeism and decreased productivity, ultimately impacting student learning and achievement [##REF##36589714##1##,##REF##23970968##6##].</p>", "<p>In Saudi Arabia, a study conducted in 2020 by Alqahtani et al. [##REF##32363025##7##] surveyed 261 high school teachers in Saudi Arabia and found that 73% of the respondents reported experiencing MSDs in at least one body part, with the highest prevalence being in the lower back (47.5%), neck (42.5%), and shoulders (33.3%). The study also found that female teachers reported a higher prevalence of MSDs compared to male teachers. They reported decreased work efficiency (73.2%), increased absenteeism (69.5%), and decreased job satisfaction (68.2%) [##REF##32363025##7##]. In another study done in Saudi Arabia, 79.2% reported experiencing MSDs in at least one body part, with the highest prevalence being in the neck (68.3%) and lower back (59.4%). This study reported decreased productivity (68.3%), increased absenteeism (57.4%), and decreased job satisfaction (46.5%) [##UREF##3##8##].</p>", "<p>The high prevalence of MSDs among teachers in Saudi Arabia indicates a need for further studies identifying what teachers are struggling with so that ergonomic and other interventions can be identified to help reduce this burden among teachers.</p>", "<p>To our knowledge, there are no studies exploring MSDs among schoolteachers in Buraydah City, and therefore, there are no data on the prevalence of MSDs among Buraydah City school teachers, their risk factors, or their impacts on performance. This study aimed to explore the prevalence and associated risk factors of MSD among teachers in Buraydah City, Saudi Arabia. The findings of this study could guide measures and strategies such as ergonomic interventions, physical activity programs, and education and awareness campaigns to prevent and manage MSDs among schoolteachers in Buraydah City and Saudi Arabia in general.</p>" ]

[ "<title>Materials and methods</title>", "<p>Study design and settings</p>", "<p>An analytic cross-sectional study was conducted for three months, from April 1 to June 30, 2023, in all schools in Buraydah City, Saudi Arabia, targeting all school teachers and other school workers in Buraydah City.</p>", "<p>Sample size</p>", "<p>The minimum sample size (n) was calculated as follows: </p>", "<p>n=Z²xPxQ/D²</p>", "<p>Where: </p>", "<p>n: Calculated sample size</p>", "<p>Z: The z-score for a 95% confidence level = 1.96. </p>", "<p>P: 50%, assumed proportion of participants for maximum sample size calculation</p>", "<p>Q: (1 - P) = 50%.</p>", "<p>D: The margin of error = 0.05.</p>", "<p>n = (1.96)²x 0.5x0.5/(0.05)² = 384</p>", "<p>The minimum calculated sample size to achieve a precision of ±5% with a 95% confidence interval was 384 teachers. To compensate for possible inaccurate responses and the erroneous completeness of questionnaires, we recruited 400 teachers.</p>", "<p>Sampling technique</p>", "<p>A multistage random sampling technique was used to select participants. All schools in Buraydah city were divided into four clusters based on their location in the city (East, West, North, and South). Teachers from each cluster were further divided into two strata based on their gender (Male and Female). Finally, 50 teachers were selected by systematic random sampling from each subcluster, making 400 participants in total.</p>", "<p>Data collection instrument and procedure</p>", "<p>We used a validated, self-administered questionnaire with four parts. The first part of the questionnaire had questions regarding socio-demographics, such as age, gender, marital status, duration of experience in teaching, reported weight and height, medications or physical therapy for MSD, and any other diseases. The second part inquired about MSDs using the Arabic version of the standardized Nordic Musculoskeletal Disorder Questionnaire [##UREF##4##9##,##REF##15676628##10##]. The third part inquired about the working conditions, how many standing hours per day, how many lectures, and what postures they took when teaching. The fourth part inquired about the effect of MSDs on daily life activities and work, absenteeism, and sick leaves.</p>", "<p>The investigators visited schools, sought permission from authorities, and attended teachers’ morning staff meetings to recruit them. Before data collection, participants were given all information about the study, including the study aims and objectives, and invited to participate voluntarily.</p>", "<p>Statistical analysis</p>", "<p>Both descriptive and inferential statistical analyses of the data were carried out. Simple frequencies and percentages of the sociodemographic characteristics and other categorical variables were calculated and tabulated. Percentages were also calculated for multiple-answer questions. For continuous variables, median and IQR were reported as central tendency and dispersion measures, respectively. To find any significant association between categorical variables, Fischer’s exact test was applied and interpreted. For continuous variables, the Kruskal-Wallis test was used to compare medians. Furthermore, to predict factors causing MSD symptoms, a binary logistic regression model with multiple predictors was created. The results of the model were presented as adjusted odds ratios (AOR). Statistical significance was established at a p-value of 0.05 or less with a 95% confidence interval. All the statistical calculations were performed using IBM Corp. Released 2020. IBM SPSS Statistics for Windows, Version 27.0. Armonk, NY: IBM Corp.</p>", "<p>Ethical considerations<italic> </italic>\n</p>", "<p>This study was approved by the ethics committee of Qassim province (Ref. No.: H-04-Q-001). Written consent was requested from participants before data collection. The study investigators requested approval from competent authorities and permission from the selected schools. There was no disclosure of the information obtained in this study to the hospital, legal or financial authorities, or anyone outside the study. The questionnaire collected anonymous information; no identifying data was collected, and participants had the right to withdraw. There was no disclosure of the information obtained in this study to anyone else outside of the study.</p>" ]

[ "<title>Results</title>", "<p>As indicated in Table ##TAB##0##1##, the total study participants were 787; among them, 648 were teachers, and the remaining 139 were other people working in school. The median age was 43 years, and the median years of experience were 16. The gender distribution shows 65.1% females and 34.9% males. Most participants (89.2%) were married. School level distribution was 29.9% high school, 22.2% intermediate, and 47.9% primary school. Non-smokers account for 94.4%. The BMI distribution includes 27.6% normal, 33.4% obese, 37.4% overweight, and 1.7% underweight participants. Regular exercise was undertaken by 41.7%, while 58.3% did not. Most (61%) had no chronic diseases, while 14.3% had osteoarthritis.</p>", "<p>When asked about experiencing troubles such as aches, pains, discomfort, or numbness in the past 12 months, most (78.4%) of the other school staff and 83.3% of teachers responded positively. Among other school staff, 71.9% reported being prevented from carrying out normal activities due to these troubles, compared to teachers (73.6%). In the past 12 months, 55.4% of other staff and 59% of teachers had sought medical consultation, and within the last seven days, 83.5% of other staff and 79.9% of teachers reported still having the symptoms (Table ##TAB##1##2##). </p>", "<p>The most prevalent musculoskeletal problem was lower back discomfort (46%), followed by neck pain (38%). Notably, shoulder, ankle, and foot troubles share a prevalence of 26% (Figure ##FIG##0##1##). Among the other school staff, those who experienced MSD in the past 12 months reported a median of two days absent from work due to muscle or joint pain, compared to 0 days for those who did not experience such pain. Among teachers, those who experienced MSD in the past 12 months reported a median of three days of absence, while those without such pain reported zero days. The most used treatments were topical analgesics (40%), massage (36%), and oral analgesics (30%) (Figure ##FIG##1##2##).</p>", "<p>The teachers who considered changing their jobs showed a significantly higher percentage of pain compared to those who did not consider changing their jobs (p&lt;0.001) (Table ##TAB##2##3##).</p>", "<p>Table ##TAB##3##4## shows that teachers who reported being prevented from normal activities, who reported aches, pain, discomfort, numbness, and due to MSD, and who sought physician consultation for MSD in the past 12 months were significantly more likely to suffer from major depressive disorders than those who did not report such problems (p&lt;0.001, p&lt;0.007, and p&lt;0.018, respectively). Teachers with MSD symptoms within the last seven days were significantly more likely to have a major depressive disorder (p&lt;0.001) (Table ##TAB##3##4##).</p>", "<p>The chi-square test showed that MSD prevalence significantly increases with age (p&lt;0.001). Females had a higher prevalence of MSD (67.0%) compared to males (33.0%) (p&lt;0.001). Working hours, including fixed rest times (sitting), significantly affect MSD prevalence (p = 0.002), and years of job experience were significantly associated with MSD prevalence (p=0.041) (Table ##TAB##4##5##).</p>", "<p>A multiple regression model was created to evaluate the risk factors associated with MSD (Table ##TAB##5##6##). Age showed a significant association with MSD (aOR: 1.070, 95%CI: 1.009-1.136, p=0.025); each increase in age is associated with a 7% increase in the odds of experiencing pain. Females experience higher MSD compared to males (aOR: 2.581, 95%CI: 1.617-4121, p &lt; 0.001). </p>", "<p>Regarding impacts of MSD and its associated factors, females had 2.906 times higher odds of experiencing disability due to MSD compared to males (p &lt; 0.001) (Table ##TAB##6##7##).</p>" ]

[ "<title>Discussion</title>", "<p>The study's findings shed light on the significant prevalence of MSD among teachers and other school staff and would also help educational institutions and policymakers take measures to promote safe and healthy working conditions for teachers to prevent the development of MSDs and improve their work performance. The gender distribution displayed a majority of females (65.1%), highlighting the gender composition in the teaching profession and in teachers with MSD reported by other studies [##REF##36589714##1##]. The prevalence of MSD, as indicated by the self-reported symptoms, such as pain and discomfort, varied across body regions. Neck and lower back pain were particularly common, affecting 38% and 46% of participants, respectively. These results are comparable to other Saudi Arabian studies from Abba (59.2%) [##UREF##5##11##], Dammam (63.8%) [##REF##23970968##6##], and a national-based survey (66.9%) [##REF##25674106##12##]. In contrast, a Japanese study showed a lower prevalence of back pain (20.6%) [##REF##22087739##13##]. Notably, the prevalence of MSD was high across multiple regions, underscoring the need for comprehensive interventions.</p>", "<p>Our findings showed that most participants in both groups experienced symptoms such as aches, pains, discomfort, or numbness in the past 12 months. Among the other school staff, 78.4% reported such symptoms, while the teachers had a slightly higher prevalence at 83.3%, similar to the findings of other studies [##UREF##1##3##,##REF##22087739##13##,##REF##22398261##14##]. Furthermore, the study examined whether these symptoms prevented participants from carrying out normal activities. It was observed that 73.6% of the teachers were affected, and 59.0% of the teachers had seen a physician for their condition in the past 12 months. These findings align with another study conducted in Cairo, Egypt [##UREF##6##15##]. </p>", "<p>The findings from this study underscore the substantial impact of musculoskeletal disorder (MSD) on both work performance and individuals' job considerations. Among teachers, the median number of absent days for those with MSD was three days, while those without such pain reported none. This significant difference (p&lt;0.001) further substantiates the impact of MSD on work attendance. A similar study conducted in Italy found that there was a greater intention to call off work and leave the job among those diagnosed with MSD [##REF##29788945##16##], aligning with another study conducted in Qassam, Saudi Arabia, that also showed a significant relationship between absenteeism and pain [##UREF##7##17##]. Moreover, the association between job considerations and MSD among teachers was also significant (p&lt;0.001). A higher percentage (94.3%) of participants contemplating job changes reported experiencing pain, highlighting the strong link between MSD and the inclination to explore alternative job options. Similar findings were reported by previous studies [##REF##22087739##13##,##REF##35342755##18##]. Altogether, these findings emphasize the multifaceted influence of MSD, encompassing both absenteeism and job-related decision-making, underscoring the importance of holistic interventions to mitigate its effects and enhance overall workplace well-being. Age exhibits a positive and significant association with MSD (p=0.025). This finding underscores the influence of age on the likelihood of experiencing pain and emphasizes the need for age-sensitive interventions. Similar findings were reported by other studies conducted in Saudi Arabia [##REF##23970968##6##,##REF##35342755##18##]. However, another study conducted in Saudi Arabia presents a contrasting result, showing no association of pain with age [##REF##30736782##19##]. Gender emerges as a significant predictor, with females reporting higher MSD rates compared to their male counterparts. The substantial difference highlights the gender-based disparities in MSD experiences. These findings are comparable to the findings of other studies conducted in Saudi Arabia and Cairo [##UREF##6##15##,##REF##35342755##18##]. Targeted interventions could enhance pain management strategies. Marital status, level of school, smoking habits, BMI categories, and fixed rest times were not significantly associated with MSD. These results suggest that these factors might not be associated with MSD. However, some of our study's findings contradict the previously conducted studies, which show that weight is significantly associated with MSD [##REF##23970968##6##,##UREF##7##17##]. While BMI categories do not independently predict MSD, the lack of regular exercise marginally increases the odds of experiencing pain. Although not statistically significant, further studies might focus on this as a potential intervention. Interestingly, the presence of major depressive disorder significantly correlates with a higher MSD prevalence (P&lt;0.001). This association signifies the intricate interplay between mental and physical well-being, emphasizing the need for holistic approaches to address both conditions simultaneously. Teachers with major depressive disorders reported more MSD-related symptoms, activity limitations, and medical consultations. This is in contrast to the results of a study that shows no correlation between depression and MSD [##REF##30808335##20##]. However, MSD severity's association with MDD was not significant, underscoring the complex interplay between mental and physical health in educators. Age and years of experience, though showing a slight trend, do not exhibit a significant correlation with disability, aligning with another previous study indicating that these factors might not strongly influence such outcomes [##UREF##7##17##]. Gender emerges as a significant determinant, with females facing higher disability rates. This could be attributed to biological differences, differing job roles, or varied coping mechanisms. Participants with major depressive disorders exhibited significantly higher odds of experiencing disability due to MSD, emphasizing the need for comprehensive health assessments and integrated care approaches. Similar findings were also reported by the study, showing the negative impact of MSD on quality of life among elementary school teachers [##REF##26327160##21##].</p>", "<p>Some limitations of this study include its cross-sectional design, which is unable to establish causal relationships between risk factors and MSD. The reliance on self-reported data for MSD might introduce recall bias or social desirability bias, affecting the accuracy of reported pain levels and associated factors. Finally, the study's findings might not be easily generalizable to teachers in other regions or countries due to cultural, organizational, or educational system differences.</p>" ]

[ "<title>Conclusions</title>", "<p>This study showed a high prevalence of MSD among teachers, highlighting the importance of addressing this issue for the well-being of educators. The study identifies key risk factors associated with MSD, including age and gender. MDD was also found to influence MSD. These findings emphasize the need for targeted interventions to alleviate pain and promote the overall health of teachers. Considering the high prevalence of MSD among teachers, implementing ergonomic interventions is crucial. Designing classrooms with adjustable furniture and promoting proper posture during teaching could reduce the strain on muscles and joints. Given the association between age and MSD, interventions should be tailored to different age groups. Younger teachers might benefit from preventive measures, while older teachers could benefit from pain management strategies. Recognizing the gender-based differences in MSD, design interventions that address the unique needs of male and female teachers. This might involve providing targeted exercises, workshops, or resources. Since major depressive disorder is linked to higher MSD, adopting an integrated approach to mental and physical health is essential. Collaboration between healthcare professionals specializing in both domains can yield comprehensive solutions.</p>" ]

[ "<p>Introduction: Musculoskeletal disorders (MSD) pose a significant challenge to the well-being and productivity of individuals and various occupational groups, including teachers. Among teachers, the prevalence of MSD has raised concerns globally, impacting their daily activities and overall quality of life. Buraidah and Saudi Arabia, like many other regions, face the implications of this issue. This study aimed to explore the prevalence and associated risk factors of MSD among teachers in Buraydah, providing valuable insights into the extent of the problem and potential areas for intervention.</p>", "<p>Methodology: An analytic cross-sectional study was conducted for three months, from April 1 to June 30, 2023, using the Arabic version of the standardized Nordic Musculoskeletal Disorder Questionnaire. This study was conducted in all schools in Buraydah City, Saudi Arabia. The study population was all schoolteachers (including principals, vice principals, etc.) in Buraydah City. The study analyzed responses from 648 teachers and 139 school workers using statistical tests, including chi-square tests and logistic regression models.</p>", "<p>Results: The results indicated a notable prevalence of MSD among teachers, with a significant association found between age, gender, and major depressive disorder (MDD) and MSD. The study reveals that females are at higher risk of MSD compared to males, emphasizing the need for gender-specific interventions. Moreover, the presence of MDD is identified as a significant contributor to MSD among teachers. However, certain demographic and lifestyle factors, such as marital status, level of school, smoking habits, and fixed rest times, do not show significant associations with MSD. Although age and years of experience are correlated, only age is found to significantly contribute to MSD. Regular exercise and BMI also do not emerge as significant contributors, although a lack of exercise shows a marginal impact.</p>", "<p>Conclusion: This study's findings have implications for educational institutions and policymakers, highlighting the need for tailored interventions to address MSD among teachers. It underscores the importance of ergonomic interventions, gender-sensitive approaches, and mental health support.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Prevalence of pain in different body parts</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>Treatment taken by teachers</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Sociodemographic characteristics of study participants</title><p>IQR: Interquartile range, N: Frequency, %: Percentage</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td colspan=\"2\" rowspan=\"2\">\n \n</td><td colspan=\"2\" rowspan=\"1\">\nTotal (N=787)\n</td><td colspan=\"2\" rowspan=\"1\">\nTeacher (N=648)\n</td><td colspan=\"2\" rowspan=\"1\">\nOther (N=139)\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nMedian\n</td><td rowspan=\"1\" colspan=\"1\">\nIQR (Q₃- Q₁)\n</td><td rowspan=\"1\" colspan=\"1\">\nMedian\n</td><td rowspan=\"1\" colspan=\"1\">\nIQR (Q₃- Q₁)\n</td><td rowspan=\"1\" colspan=\"1\">\nMedian\n</td><td rowspan=\"1\" colspan=\"1\">\nIQR (Q₃- Q₁)\n</td></tr><tr style=\"background-color:#ccc\"><td colspan=\"2\" rowspan=\"1\">\nAge\n</td><td rowspan=\"1\" colspan=\"1\">\n43\n</td><td rowspan=\"1\" colspan=\"1\">\n8 (47-39)\n</td><td rowspan=\"1\" colspan=\"1\">\n43\n</td><td rowspan=\"1\" colspan=\"1\">\n7 (47-40)\n</td><td rowspan=\"1\" colspan=\"1\">\n43\n</td><td rowspan=\"1\" colspan=\"1\">\n5 (47-38)\n</td></tr><tr><td colspan=\"2\" rowspan=\"1\">\nYears of experience\n</td><td rowspan=\"1\" colspan=\"1\">\n16\n</td><td rowspan=\"1\" colspan=\"1\">\n11 (22-11)\n</td><td rowspan=\"1\" colspan=\"1\">\n17\n</td><td rowspan=\"1\" colspan=\"1\">\n10 (22-12)\n</td><td rowspan=\"1\" colspan=\"1\">\n14\n</td><td rowspan=\"1\" colspan=\"1\">\n14 (22-8)\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\nCount\n</td><td rowspan=\"1\" colspan=\"1\">\nColumn N %\n</td><td rowspan=\"1\" colspan=\"1\">\nCount\n</td><td rowspan=\"1\" colspan=\"1\">\nColumn N %\n</td><td rowspan=\"1\" colspan=\"1\">\nCount\n</td><td rowspan=\"1\" colspan=\"1\">\nColumn N %\n</td></tr><tr><td rowspan=\"2\" colspan=\"1\">\nGender\n</td><td rowspan=\"1\" colspan=\"1\">\nFemale\n</td><td rowspan=\"1\" colspan=\"1\">\n512\n</td><td rowspan=\"1\" colspan=\"1\">\n65.1%\n</td><td rowspan=\"1\" colspan=\"1\">\n413\n</td><td rowspan=\"1\" colspan=\"1\">\n63.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n99\n</td><td rowspan=\"1\" colspan=\"1\">\n71.2%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nMale\n</td><td rowspan=\"1\" colspan=\"1\">\n275\n</td><td rowspan=\"1\" colspan=\"1\">\n34.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n235\n</td><td rowspan=\"1\" colspan=\"1\">\n36.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n40\n</td><td rowspan=\"1\" colspan=\"1\">\n28.8%\n</td></tr><tr><td rowspan=\"4\" colspan=\"1\">\nMarital status\n</td><td rowspan=\"1\" colspan=\"1\">\nDivorced\n</td><td rowspan=\"1\" colspan=\"1\">\n30\n</td><td rowspan=\"1\" colspan=\"1\">\n3.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n22\n</td><td rowspan=\"1\" colspan=\"1\">\n3.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n8\n</td><td rowspan=\"1\" colspan=\"1\">\n5.8%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nMarried\n</td><td rowspan=\"1\" colspan=\"1\">\n702\n</td><td rowspan=\"1\" colspan=\"1\">\n89.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n588\n</td><td rowspan=\"1\" colspan=\"1\">\n90.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n114\n</td><td rowspan=\"1\" colspan=\"1\">\n82.0%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nSingle\n</td><td rowspan=\"1\" colspan=\"1\">\n45\n</td><td rowspan=\"1\" colspan=\"1\">\n5.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n31\n</td><td rowspan=\"1\" colspan=\"1\">\n4.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n14\n</td><td rowspan=\"1\" colspan=\"1\">\n10.1%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nWindow\n</td><td rowspan=\"1\" colspan=\"1\">\n10\n</td><td rowspan=\"1\" colspan=\"1\">\n1.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n7\n</td><td rowspan=\"1\" colspan=\"1\">\n1.1%\n</td><td rowspan=\"1\" colspan=\"1\">\n3\n</td><td rowspan=\"1\" colspan=\"1\">\n2.2%\n</td></tr><tr><td rowspan=\"3\" colspan=\"1\">\nLevel of school you are working in\n</td><td rowspan=\"1\" colspan=\"1\">\nHigh school\n</td><td rowspan=\"1\" colspan=\"1\">\n235\n</td><td rowspan=\"1\" colspan=\"1\">\n29.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n181\n</td><td rowspan=\"1\" colspan=\"1\">\n27.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n54\n</td><td rowspan=\"1\" colspan=\"1\">\n38.8%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nIntermediate school\n</td><td rowspan=\"1\" colspan=\"1\">\n175\n</td><td rowspan=\"1\" colspan=\"1\">\n22.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n153\n</td><td rowspan=\"1\" colspan=\"1\">\n23.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n22\n</td><td rowspan=\"1\" colspan=\"1\">\n15.8%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nprimary school\n</td><td rowspan=\"1\" colspan=\"1\">\n377\n</td><td rowspan=\"1\" colspan=\"1\">\n47.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n314\n</td><td rowspan=\"1\" colspan=\"1\">\n48.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n63\n</td><td rowspan=\"1\" colspan=\"1\">\n45.3%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nSmoker\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n743\n</td><td rowspan=\"1\" colspan=\"1\">\n94.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n614\n</td><td rowspan=\"1\" colspan=\"1\">\n94.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n129\n</td><td rowspan=\"1\" colspan=\"1\">\n92.8%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n44\n</td><td rowspan=\"1\" colspan=\"1\">\n5.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n34\n</td><td rowspan=\"1\" colspan=\"1\">\n5.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n10\n</td><td rowspan=\"1\" colspan=\"1\">\n7.2%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">\nBMI\n</td><td rowspan=\"1\" colspan=\"1\">\nNormal\n</td><td rowspan=\"1\" colspan=\"1\">\n217\n</td><td rowspan=\"1\" colspan=\"1\">\n27.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n182\n</td><td rowspan=\"1\" colspan=\"1\">\n28.1%\n</td><td rowspan=\"1\" colspan=\"1\">\n35\n</td><td rowspan=\"1\" colspan=\"1\">\n25.2%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nObese\n</td><td rowspan=\"1\" colspan=\"1\">\n263\n</td><td rowspan=\"1\" colspan=\"1\">\n33.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n210\n</td><td rowspan=\"1\" colspan=\"1\">\n32.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n53\n</td><td rowspan=\"1\" colspan=\"1\">\n38.1%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nOverweight\n</td><td rowspan=\"1\" colspan=\"1\">\n294\n</td><td rowspan=\"1\" colspan=\"1\">\n37.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n248\n</td><td rowspan=\"1\" colspan=\"1\">\n38.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n46\n</td><td rowspan=\"1\" colspan=\"1\">\n33.1%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nUnderweight\n</td><td rowspan=\"1\" colspan=\"1\">\n13\n</td><td rowspan=\"1\" colspan=\"1\">\n1.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n8\n</td><td rowspan=\"1\" colspan=\"1\">\n1.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n5\n</td><td rowspan=\"1\" colspan=\"1\">\n3.6%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nRegular exercise (at least 30min 5 time/week)\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n459\n</td><td rowspan=\"1\" colspan=\"1\">\n58.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n385\n</td><td rowspan=\"1\" colspan=\"1\">\n59.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n74\n</td><td rowspan=\"1\" colspan=\"1\">\n53.2%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n328\n</td><td rowspan=\"1\" colspan=\"1\">\n41.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n263\n</td><td rowspan=\"1\" colspan=\"1\">\n40.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n65\n</td><td rowspan=\"1\" colspan=\"1\">\n46.8%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"8\" colspan=\"1\">\nChronic disease\n</td><td rowspan=\"1\" colspan=\"1\">\nNone\n</td><td rowspan=\"1\" colspan=\"1\">\n482\n</td><td rowspan=\"1\" colspan=\"1\">\n61%\n</td><td rowspan=\"1\" colspan=\"1\">\n394\n</td><td rowspan=\"1\" colspan=\"1\">\n60%\n</td><td rowspan=\"1\" colspan=\"1\">\n88\n</td><td rowspan=\"1\" colspan=\"1\">\n80.7%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nHypertension\n</td><td rowspan=\"1\" colspan=\"1\">\n55\n</td><td rowspan=\"1\" colspan=\"1\">\n6.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n47\n</td><td rowspan=\"1\" colspan=\"1\">\n8.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n8\n</td><td rowspan=\"1\" colspan=\"1\">\n7.3%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nAsthma\n</td><td rowspan=\"1\" colspan=\"1\">\n41\n</td><td rowspan=\"1\" colspan=\"1\">\n5.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n33\n</td><td rowspan=\"1\" colspan=\"1\">\n5.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n8\n</td><td rowspan=\"1\" colspan=\"1\">\n5.5%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nOsteoarthritis\n</td><td rowspan=\"1\" colspan=\"1\">\n113\n</td><td rowspan=\"1\" colspan=\"1\">\n14.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n102\n</td><td rowspan=\"1\" colspan=\"1\">\n15.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n11\n</td><td rowspan=\"1\" colspan=\"1\">\n7.9%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\ndiabetes\n</td><td rowspan=\"1\" colspan=\"1\">\n81\n</td><td rowspan=\"1\" colspan=\"1\">\n10%\n</td><td rowspan=\"1\" colspan=\"1\">\n61\n</td><td rowspan=\"1\" colspan=\"1\">\n9.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n20\n</td><td rowspan=\"1\" colspan=\"1\">\n14%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nVertebral disc disease\n</td><td rowspan=\"1\" colspan=\"1\">\n12\n</td><td rowspan=\"1\" colspan=\"1\">\n1.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n5\n</td><td rowspan=\"1\" colspan=\"1\">\n0.07%\n</td><td rowspan=\"1\" colspan=\"1\">\n7\n</td><td rowspan=\"1\" colspan=\"1\">\n5%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nHypothyroidism\n</td><td rowspan=\"1\" colspan=\"1\">\n20\n</td><td rowspan=\"1\" colspan=\"1\">\n2.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n16\n</td><td rowspan=\"1\" colspan=\"1\">\n2.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n4\n</td><td rowspan=\"1\" colspan=\"1\">\n2.8%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nOther\n</td><td rowspan=\"1\" colspan=\"1\">\n28\n</td><td rowspan=\"1\" colspan=\"1\">\n3.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n28\n</td><td rowspan=\"1\" colspan=\"1\">\n4.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n0\n</td><td rowspan=\"1\" colspan=\"1\">\n0%\n</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>Prevalence, impact, and severity of MSD in teachers and others working in school</title><p>N: Frequency, %: Percentage</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td colspan=\"2\" rowspan=\"2\">\n \n</td><td colspan=\"2\" rowspan=\"1\">\nOther (n=139)\n</td><td colspan=\"2\" rowspan=\"1\">\nTeacher (n=648)\n</td><td rowspan=\"2\" colspan=\"1\">\np-value\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nN\n</td><td rowspan=\"1\" colspan=\"1\">\n%\n</td><td rowspan=\"1\" colspan=\"1\">\nN\n</td><td rowspan=\"1\" colspan=\"1\">\n %\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nIn the past 12 months, have you had MSD problems (such as ache, pain, discomfort, numbness)\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n30\n</td><td rowspan=\"1\" colspan=\"1\">\n21.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n108\n</td><td rowspan=\"1\" colspan=\"1\">\n16.7%\n</td><td rowspan=\"2\" colspan=\"1\">\n0.167\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n109\n</td><td rowspan=\"1\" colspan=\"1\">\n78.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n540\n</td><td rowspan=\"1\" colspan=\"1\">\n83.3%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nIn the past 12 months, had you ‏been prevented from carrying out normal activities (e.g., job, housework, hobbies) because of this trouble\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n39\n</td><td rowspan=\"1\" colspan=\"1\">\n28.1%\n</td><td rowspan=\"1\" colspan=\"1\">\n171\n</td><td rowspan=\"1\" colspan=\"1\">\n26.4%\n</td><td rowspan=\"2\" colspan=\"1\">\n0.687\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n100\n</td><td rowspan=\"1\" colspan=\"1\">\n71.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n477\n</td><td rowspan=\"1\" colspan=\"1\">\n73.6%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nIn the past 12 months, have you seen a physician for this condition\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n62\n</td><td rowspan=\"1\" colspan=\"1\">\n44.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n266\n</td><td rowspan=\"1\" colspan=\"1\">\n41.0%\n</td><td rowspan=\"2\" colspan=\"1\">\n0.440\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n77\n</td><td rowspan=\"1\" colspan=\"1\">\n55.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n382\n</td><td rowspan=\"1\" colspan=\"1\">\n59.0%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nIn the past seven days, did you suffer from any of the previous symptoms (other than sports injuries)\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n23\n</td><td rowspan=\"1\" colspan=\"1\">\n16.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n130\n</td><td rowspan=\"1\" colspan=\"1\">\n20.1%\n</td><td rowspan=\"2\" colspan=\"1\">\n0.342\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n116\n</td><td rowspan=\"1\" colspan=\"1\">\n83.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n518\n</td><td rowspan=\"1\" colspan=\"1\">\n79.9%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"8\" colspan=\"1\">\nTreatment taken in last 12 months\n</td><td rowspan=\"1\" colspan=\"1\">\nNone\n</td><td rowspan=\"1\" colspan=\"1\">\n25\n</td><td rowspan=\"1\" colspan=\"1\">\n18%\n</td><td rowspan=\"1\" colspan=\"1\">\n132\n</td><td rowspan=\"1\" colspan=\"1\">\n20%\n</td><td rowspan=\"8\" colspan=\"1\">\n0.567\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nRest\n</td><td rowspan=\"1\" colspan=\"1\">\n2\n</td><td rowspan=\"1\" colspan=\"1\">\n1.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n8\n</td><td rowspan=\"1\" colspan=\"1\">\n1.5%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nSalt and water\n</td><td rowspan=\"1\" colspan=\"1\">\n1\n</td><td rowspan=\"1\" colspan=\"1\">\n.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n3\n</td><td rowspan=\"1\" colspan=\"1\">\n.5%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nMessage\n</td><td rowspan=\"1\" colspan=\"1\">\n41\n</td><td rowspan=\"1\" colspan=\"1\">\n29%\n</td><td rowspan=\"1\" colspan=\"1\">\n217\n</td><td rowspan=\"1\" colspan=\"1\">\n33%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nCompresses\n</td><td rowspan=\"1\" colspan=\"1\">\n21\n</td><td rowspan=\"1\" colspan=\"1\">\n15%\n</td><td rowspan=\"1\" colspan=\"1\">\n109\n</td><td rowspan=\"1\" colspan=\"1\">\n16%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nPhysiotherapy\n</td><td rowspan=\"1\" colspan=\"1\">\n22\n</td><td rowspan=\"1\" colspan=\"1\">\n16%\n</td><td rowspan=\"1\" colspan=\"1\">\n117\n</td><td rowspan=\"1\" colspan=\"1\">\n18%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nOral analgesics\n</td><td rowspan=\"1\" colspan=\"1\">\n37\n</td><td rowspan=\"1\" colspan=\"1\">\n27%\n</td><td rowspan=\"1\" colspan=\"1\">\n196\n</td><td rowspan=\"1\" colspan=\"1\">\n30%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nTopical analgesics\n</td><td rowspan=\"1\" colspan=\"1\">\n50\n</td><td rowspan=\"1\" colspan=\"1\">\n35%\n</td><td rowspan=\"1\" colspan=\"1\">\n264\n</td><td rowspan=\"1\" colspan=\"1\">\n40%\n</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB3\"><label>Table 3</label><caption><title>Association of MSD with the performance of teachers and others working in school </title><p>IQR: Interquartile range, N: Frequency, %: Percentage, MSD: Musculoskeletal disorder</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td colspan=\"4\" rowspan=\"3\">\n \n</td><td colspan=\"4\" rowspan=\"1\">\nMSD in past 12 months\n</td><td rowspan=\"3\" colspan=\"1\">\np-value\n</td></tr><tr><td colspan=\"2\" rowspan=\"1\">\nNo\n</td><td colspan=\"2\" rowspan=\"1\">\nYes\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nMedian N\n</td><td rowspan=\"1\" colspan=\"1\">\nIQR (Q₃- Q₁) %\n</td><td rowspan=\"1\" colspan=\"1\">\nMedian N\n</td><td rowspan=\"1\" colspan=\"1\">\nIQR (Q₃- Q₁) %\n</td></tr><tr><td rowspan=\"6\" colspan=\"1\">\n \n</td><td rowspan=\"3\" colspan=\"1\">\nOther\n</td><td colspan=\"2\" rowspan=\"1\">\nDuring the past 12 months, how many days have you been absent from work due to muscle or joint pain other than sports injuries?\n</td><td rowspan=\"1\" colspan=\"1\">\n0\n</td><td rowspan=\"1\" colspan=\"1\">\n0\n</td><td rowspan=\"1\" colspan=\"1\">\n2\n</td><td rowspan=\"1\" colspan=\"1\">\n5 (5-0)\n</td><td rowspan=\"1\" colspan=\"1\">\n&lt;0.001\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nHave you thought about changing your job due to muscle and joint pain?\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n28\n</td><td rowspan=\"1\" colspan=\"1\">\n93.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n95\n</td><td rowspan=\"1\" colspan=\"1\">\n87.2%\n</td><td rowspan=\"2\" colspan=\"1\">\n.348\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n2\n</td><td rowspan=\"1\" colspan=\"1\">\n6.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n14\n</td><td rowspan=\"1\" colspan=\"1\">\n12.8%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">\nTeacher\n</td><td colspan=\"2\" rowspan=\"1\">\nDuring the past 12 months, how many days have you been absent from work due to muscle or joint pain other than sports injuries?\n</td><td rowspan=\"1\" colspan=\"1\">\n0\n</td><td rowspan=\"1\" colspan=\"1\">\n1 (1-0)\n</td><td rowspan=\"1\" colspan=\"1\">\n0\n</td><td rowspan=\"1\" colspan=\"1\">\n3 (3-0)\n</td><td rowspan=\"1\" colspan=\"1\">\n&lt;0.001\n</td></tr><tr><td rowspan=\"2\" colspan=\"1\">\nHave you thought about changing your job due to muscle and joint pain?\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n100\n</td><td rowspan=\"1\" colspan=\"1\">\n92.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n401\n</td><td rowspan=\"1\" colspan=\"1\">\n74.3%\n</td><td rowspan=\"2\" colspan=\"1\">\n&lt;0.001\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n8\n</td><td rowspan=\"1\" colspan=\"1\">\n7.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n139\n</td><td rowspan=\"1\" colspan=\"1\">\n25.7%\n</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB4\"><label>Table 4</label><caption><title>Association of major depression with MSD among teachers</title><p>N: Frequency, %: Percentage</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td colspan=\"2\" rowspan=\"3\">\n \n</td><td colspan=\"4\" rowspan=\"1\">\nMajor depressive disorder\n</td><td rowspan=\"3\" colspan=\"1\">\np-value\n</td></tr><tr><td colspan=\"2\" rowspan=\"1\">\nNo\n</td><td colspan=\"2\" rowspan=\"1\">\nYes\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nN\n</td><td rowspan=\"1\" colspan=\"1\">\n%\n</td><td rowspan=\"1\" colspan=\"1\">\nN\n</td><td rowspan=\"1\" colspan=\"1\">\n%\n</td></tr><tr><td rowspan=\"2\" colspan=\"1\">\nIn the past 12 months had you trouble (such as ache, pain, discomfort, numbness)\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n96\n</td><td rowspan=\"1\" colspan=\"1\">\n18.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n12\n</td><td rowspan=\"1\" colspan=\"1\">\n9.0%\n</td><td rowspan=\"2\" colspan=\"1\">\n0.007\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n418\n</td><td rowspan=\"1\" colspan=\"1\">\n81.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n122\n</td><td rowspan=\"1\" colspan=\"1\">\n91.0%\n</td></tr><tr><td rowspan=\"2\" colspan=\"1\">\nIn the past 12 months have you ‏been prevented from carrying out normal activities (e.g., job, housework, hobbies) because of this trouble\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n155\n</td><td rowspan=\"1\" colspan=\"1\">\n30.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n16\n</td><td rowspan=\"1\" colspan=\"1\">\n11.9%\n</td><td rowspan=\"2\" colspan=\"1\">\n0.001\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n359\n</td><td rowspan=\"1\" colspan=\"1\">\n69.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n118\n</td><td rowspan=\"1\" colspan=\"1\">\n88.1%\n</td></tr><tr><td rowspan=\"2\" colspan=\"1\">\nIn the past 12 months have you seen a physician for this condition of MSD\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n223\n</td><td rowspan=\"1\" colspan=\"1\">\n43.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n43\n</td><td rowspan=\"1\" colspan=\"1\">\n32.1%\n</td><td rowspan=\"2\" colspan=\"1\">\n0.018\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n291\n</td><td rowspan=\"1\" colspan=\"1\">\n56.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n91\n</td><td rowspan=\"1\" colspan=\"1\">\n67.9%\n</td></tr><tr><td rowspan=\"2\" colspan=\"1\">\nIn the past seven days did you suffer from any of the previous symptoms (other than sports injuries)\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n118\n</td><td rowspan=\"1\" colspan=\"1\">\n23.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n12\n</td><td rowspan=\"1\" colspan=\"1\">\n9.0%\n</td><td rowspan=\"2\" colspan=\"1\">\n0.001\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n396\n</td><td rowspan=\"1\" colspan=\"1\">\n77.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n122\n</td><td rowspan=\"1\" colspan=\"1\">\n91.0%\n</td></tr><tr><td rowspan=\"2\" colspan=\"1\">\nSeverity of disease\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n241\n</td><td rowspan=\"1\" colspan=\"1\">\n46.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n69\n</td><td rowspan=\"1\" colspan=\"1\">\n51.5%\n</td><td rowspan=\"2\" colspan=\"1\">\n0.342\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n273\n</td><td rowspan=\"1\" colspan=\"1\">\n53.1%\n</td><td rowspan=\"1\" colspan=\"1\">\n65\n</td><td rowspan=\"1\" colspan=\"1\">\n48.5%\n</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB5\"><label>Table 5</label><caption><title>Association of sociodemographic factors with MSD among teachers</title><p>IQR: Interquartile range, N: Frequency, %: Percentage, MSD: Musculoskeletal disorder</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td colspan=\"2\" rowspan=\"3\">\n \n</td><td colspan=\"4\" rowspan=\"1\">\nAny MSD in past 12 months\n</td><td rowspan=\"3\" colspan=\"1\">\np-value\n</td></tr><tr><td colspan=\"2\" rowspan=\"1\">\nNo\n</td><td colspan=\"2\" rowspan=\"1\">\nYes\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nCount/Median\n</td><td rowspan=\"1\" colspan=\"1\">\n%/IQR\n</td><td rowspan=\"1\" colspan=\"1\">\nCount/Median\n</td><td rowspan=\"1\" colspan=\"1\">\n%/IQR\n</td></tr><tr><td colspan=\"2\" rowspan=\"1\">\nAge\n</td><td rowspan=\"1\" colspan=\"1\">\n40\n</td><td rowspan=\"1\" colspan=\"1\">\n8 (45-37)\n</td><td rowspan=\"1\" colspan=\"1\">\n44\n</td><td rowspan=\"1\" colspan=\"1\">\n7 (47-40)\n</td><td rowspan=\"1\" colspan=\"1\">\n0.001\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nGender\n</td><td rowspan=\"1\" colspan=\"1\">\nFemale\n</td><td rowspan=\"1\" colspan=\"1\">\n51\n</td><td rowspan=\"1\" colspan=\"1\">\n47.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n362\n</td><td rowspan=\"1\" colspan=\"1\">\n67.0%\n</td><td rowspan=\"2\" colspan=\"1\">\n0.001\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nMale\n</td><td rowspan=\"1\" colspan=\"1\">\n57\n</td><td rowspan=\"1\" colspan=\"1\">\n52.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n178\n</td><td rowspan=\"1\" colspan=\"1\">\n33.0%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">\nMarital status\n</td><td rowspan=\"1\" colspan=\"1\">\nDivorced\n</td><td rowspan=\"1\" colspan=\"1\">\n3\n</td><td rowspan=\"1\" colspan=\"1\">\n2.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n19\n</td><td rowspan=\"1\" colspan=\"1\">\n3.5%\n</td><td rowspan=\"4\" colspan=\"1\">\n0.069\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nMarried\n</td><td rowspan=\"1\" colspan=\"1\">\n95\n</td><td rowspan=\"1\" colspan=\"1\">\n88.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n493\n</td><td rowspan=\"1\" colspan=\"1\">\n91.3%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nSingle\n</td><td rowspan=\"1\" colspan=\"1\">\n10\n</td><td rowspan=\"1\" colspan=\"1\">\n9.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n21\n</td><td rowspan=\"1\" colspan=\"1\">\n3.9%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nWindow\n</td><td rowspan=\"1\" colspan=\"1\">\n0\n</td><td rowspan=\"1\" colspan=\"1\">\n0.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n7\n</td><td rowspan=\"1\" colspan=\"1\">\n1.3%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">\nLevel of school you are working in\n</td><td rowspan=\"1\" colspan=\"1\">\nHigh school\n</td><td rowspan=\"1\" colspan=\"1\">\n28\n</td><td rowspan=\"1\" colspan=\"1\">\n25.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n153\n</td><td rowspan=\"1\" colspan=\"1\">\n28.3%\n</td><td rowspan=\"3\" colspan=\"1\">\n0.859\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nIntermediate school\n</td><td rowspan=\"1\" colspan=\"1\">\n27\n</td><td rowspan=\"1\" colspan=\"1\">\n25.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n126\n</td><td rowspan=\"1\" colspan=\"1\">\n23.3%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nPrimary school\n</td><td rowspan=\"1\" colspan=\"1\">\n53\n</td><td rowspan=\"1\" colspan=\"1\">\n49.1%\n</td><td rowspan=\"1\" colspan=\"1\">\n261\n</td><td rowspan=\"1\" colspan=\"1\">\n48.3%\n</td></tr><tr><td colspan=\"2\" rowspan=\"1\">\nYears of job experience\n</td><td rowspan=\"1\" colspan=\"1\">\n15\n</td><td rowspan=\"1\" colspan=\"1\">\n11(21-10)\n</td><td rowspan=\"1\" colspan=\"1\">\n17\n</td><td rowspan=\"1\" colspan=\"1\">\n10(22-12)\n</td><td rowspan=\"1\" colspan=\"1\">\n0.041\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nSmoker\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n101\n</td><td rowspan=\"1\" colspan=\"1\">\n93.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n513\n</td><td rowspan=\"1\" colspan=\"1\">\n95.0%\n</td><td rowspan=\"2\" colspan=\"1\">\n0.528\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n7\n</td><td rowspan=\"1\" colspan=\"1\">\n6.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n27\n</td><td rowspan=\"1\" colspan=\"1\">\n5.0%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">\nBMI\n</td><td rowspan=\"1\" colspan=\"1\">\nNormal\n</td><td rowspan=\"1\" colspan=\"1\">\n37\n</td><td rowspan=\"1\" colspan=\"1\">\n34.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n145\n</td><td rowspan=\"1\" colspan=\"1\">\n26.9%\n</td><td rowspan=\"4\" colspan=\"1\">\n0.124\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nObese\n</td><td rowspan=\"1\" colspan=\"1\">\n27\n</td><td rowspan=\"1\" colspan=\"1\">\n25.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n183\n</td><td rowspan=\"1\" colspan=\"1\">\n33.9%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nOver weight\n</td><td rowspan=\"1\" colspan=\"1\">\n44\n</td><td rowspan=\"1\" colspan=\"1\">\n40.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n204\n</td><td rowspan=\"1\" colspan=\"1\">\n37.8%\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nUnder weight\n</td><td rowspan=\"1\" colspan=\"1\">\n0\n</td><td rowspan=\"1\" colspan=\"1\">\n0.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n8\n</td><td rowspan=\"1\" colspan=\"1\">\n1.5%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nRegular exercise? (at least 30 min. 5 time/week)\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n58\n</td><td rowspan=\"1\" colspan=\"1\">\n53.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n327\n</td><td rowspan=\"1\" colspan=\"1\">\n60.6%\n</td><td rowspan=\"2\" colspan=\"1\">\n0.186\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n50\n</td><td rowspan=\"1\" colspan=\"1\">\n46.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n213\n</td><td rowspan=\"1\" colspan=\"1\">\n39.4%\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nDo your working hours include fixed, frequent and regular rest times (sitting)?\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n41\n</td><td rowspan=\"1\" colspan=\"1\">\n38.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n126\n</td><td rowspan=\"1\" colspan=\"1\">\n23.3%\n</td><td rowspan=\"2\" colspan=\"1\">\n0.002\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nyes\n</td><td rowspan=\"1\" colspan=\"1\">\n67\n</td><td rowspan=\"1\" colspan=\"1\">\n62.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n414\n</td><td rowspan=\"1\" colspan=\"1\">\n76.7%\n</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB6\"><label>Table 6</label><caption><title>Risk factors for MSD among teachers</title><p>AOR: Adjusted odds ratio, C.I.: Confidence interval, IQR: Interquartile range, N: Frequency, %: Percentage</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td colspan=\"2\" rowspan=\"3\">\n \n</td><td colspan=\"4\" rowspan=\"1\">\nPain in last seven days\n</td><td rowspan=\"3\" colspan=\"1\">\np-value\n</td><td rowspan=\"3\" colspan=\"1\">\nAOR\n</td><td colspan=\"2\" rowspan=\"1\">\n95% C.I. for AOR\n</td></tr><tr><td colspan=\"2\" rowspan=\"1\">\nNo\n</td><td colspan=\"2\" rowspan=\"1\">\nYes\n</td><td rowspan=\"2\" colspan=\"1\">\nLower\n</td><td rowspan=\"2\" colspan=\"1\">\nUpper\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nCount/Median\n</td><td rowspan=\"1\" colspan=\"1\">\n%/ IQR\n</td><td rowspan=\"1\" colspan=\"1\">\nCount/Median\n</td><td rowspan=\"1\" colspan=\"1\">\n%/ IQR\n</td></tr><tr><td colspan=\"2\" rowspan=\"1\">\nAge\n</td><td rowspan=\"1\" colspan=\"1\">\n40\n</td><td rowspan=\"1\" colspan=\"1\">\n45-35\n</td><td rowspan=\"1\" colspan=\"1\">\n44\n</td><td rowspan=\"1\" colspan=\"1\">\n47-40\n</td><td rowspan=\"1\" colspan=\"1\">\n0.025\n</td><td rowspan=\"1\" colspan=\"1\">\n1.070\n</td><td rowspan=\"1\" colspan=\"1\">\n1.009\n</td><td rowspan=\"1\" colspan=\"1\">\n1.136\n</td></tr><tr style=\"background-color:#ccc\"><td colspan=\"2\" rowspan=\"1\">\nYears of experience\n</td><td rowspan=\"1\" colspan=\"1\">\n14\n</td><td rowspan=\"1\" colspan=\"1\">\n21-8\n</td><td rowspan=\"1\" colspan=\"1\">\n17\n</td><td rowspan=\"1\" colspan=\"1\">\n22-12\n</td><td rowspan=\"1\" colspan=\"1\">\n0.958\n</td><td rowspan=\"1\" colspan=\"1\">\n1.001\n</td><td rowspan=\"1\" colspan=\"1\">\n0.951\n</td><td rowspan=\"1\" colspan=\"1\">\n1.054\n</td></tr><tr><td rowspan=\"2\" colspan=\"1\">\nGender\n</td><td rowspan=\"1\" colspan=\"1\">\nFemale\n</td><td rowspan=\"1\" colspan=\"1\">\n56\n</td><td rowspan=\"1\" colspan=\"1\">\n43.1%\n</td><td rowspan=\"1\" colspan=\"1\">\n357\n</td><td rowspan=\"1\" colspan=\"1\">\n68.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n&lt;0.001\n</td><td rowspan=\"1\" colspan=\"1\">\n2.581\n</td><td rowspan=\"1\" colspan=\"1\">\n1.617\n</td><td rowspan=\"1\" colspan=\"1\">\n4.121\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nMale\n</td><td rowspan=\"1\" colspan=\"1\">\n74\n</td><td rowspan=\"1\" colspan=\"1\">\n56.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n161\n</td><td rowspan=\"1\" colspan=\"1\">\n31.1%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr><tr><td rowspan=\"4\" colspan=\"1\">\nMarital status\n</td><td rowspan=\"1\" colspan=\"1\">\ndivorced\n</td><td rowspan=\"1\" colspan=\"1\">\n4\n</td><td rowspan=\"1\" colspan=\"1\">\n3.1%\n</td><td rowspan=\"1\" colspan=\"1\">\n18\n</td><td rowspan=\"1\" colspan=\"1\">\n3.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.652\n</td><td rowspan=\"1\" colspan=\"1\">\n1.761\n</td><td rowspan=\"1\" colspan=\"1\">\n0.151\n</td><td rowspan=\"1\" colspan=\"1\">\n20.562\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nMarried\n</td><td rowspan=\"1\" colspan=\"1\">\n112\n</td><td rowspan=\"1\" colspan=\"1\">\n86.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n476\n</td><td rowspan=\"1\" colspan=\"1\">\n91.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.489\n</td><td rowspan=\"1\" colspan=\"1\">\n2.163\n</td><td rowspan=\"1\" colspan=\"1\">\n0.244\n</td><td rowspan=\"1\" colspan=\"1\">\n19.209\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nSingle\n</td><td rowspan=\"1\" colspan=\"1\">\n13\n</td><td rowspan=\"1\" colspan=\"1\">\n10.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n18\n</td><td rowspan=\"1\" colspan=\"1\">\n3.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.811\n</td><td rowspan=\"1\" colspan=\"1\">\n1.333\n</td><td rowspan=\"1\" colspan=\"1\">\n0.126\n</td><td rowspan=\"1\" colspan=\"1\">\n14.085\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nWindow\n</td><td rowspan=\"1\" colspan=\"1\">\n1\n</td><td rowspan=\"1\" colspan=\"1\">\n0.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n6\n</td><td rowspan=\"1\" colspan=\"1\">\n1.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr><tr><td rowspan=\"3\" colspan=\"1\">\nLevel of school you are working in\n</td><td rowspan=\"1\" colspan=\"1\">\nHigh school\n</td><td rowspan=\"1\" colspan=\"1\">\n38\n</td><td rowspan=\"1\" colspan=\"1\">\n29.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n143\n</td><td rowspan=\"1\" colspan=\"1\">\n27.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.834\n</td><td rowspan=\"1\" colspan=\"1\">\n0.948\n</td><td rowspan=\"1\" colspan=\"1\">\n0.948\n</td><td rowspan=\"1\" colspan=\"1\">\n0.577\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nIntermediate school\n</td><td rowspan=\"1\" colspan=\"1\">\n34\n</td><td rowspan=\"1\" colspan=\"1\">\n26.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n119\n</td><td rowspan=\"1\" colspan=\"1\">\n23.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.549\n</td><td rowspan=\"1\" colspan=\"1\">\n1.181\n</td><td rowspan=\"1\" colspan=\"1\">\n1.181\n</td><td rowspan=\"1\" colspan=\"1\">\n0.686\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nPrimary school\n</td><td rowspan=\"1\" colspan=\"1\">\n58\n</td><td rowspan=\"1\" colspan=\"1\">\n44.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n256\n</td><td rowspan=\"1\" colspan=\"1\">\n49.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nSmoker\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n120\n</td><td rowspan=\"1\" colspan=\"1\">\n92.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n494\n</td><td rowspan=\"1\" colspan=\"1\">\n95.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.346\n</td><td rowspan=\"1\" colspan=\"1\">\n0.664\n</td><td rowspan=\"1\" colspan=\"1\">\n0.283\n</td><td rowspan=\"1\" colspan=\"1\">\n1.558\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nyes\n</td><td rowspan=\"1\" colspan=\"1\">\n10\n</td><td rowspan=\"1\" colspan=\"1\">\n7.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n24\n</td><td rowspan=\"1\" colspan=\"1\">\n4.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">\nBMI\n</td><td rowspan=\"1\" colspan=\"1\">\nNormal\n</td><td rowspan=\"1\" colspan=\"1\">\n41\n</td><td rowspan=\"1\" colspan=\"1\">\n31.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n141\n</td><td rowspan=\"1\" colspan=\"1\">\n27.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.535\n</td><td rowspan=\"1\" colspan=\"1\">\n1.613\n</td><td rowspan=\"1\" colspan=\"1\">\n0.033\n</td><td rowspan=\"1\" colspan=\"1\">\n3.116\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nObese\n</td><td rowspan=\"1\" colspan=\"1\">\n29\n</td><td rowspan=\"1\" colspan=\"1\">\n22.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n181\n</td><td rowspan=\"1\" colspan=\"1\">\n34.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.681\n</td><td rowspan=\"1\" colspan=\"1\">\n1.375\n</td><td rowspan=\"1\" colspan=\"1\">\n0.037\n</td><td rowspan=\"1\" colspan=\"1\">\n3.642\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nOverweight\n</td><td rowspan=\"1\" colspan=\"1\">\n59\n</td><td rowspan=\"1\" colspan=\"1\">\n45.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n189\n</td><td rowspan=\"1\" colspan=\"1\">\n36.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.653\n</td><td rowspan=\"1\" colspan=\"1\">\n1.413\n</td><td rowspan=\"1\" colspan=\"1\">\n0.023\n</td><td rowspan=\"1\" colspan=\"1\">\n2.157\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nUnderweight\n</td><td rowspan=\"1\" colspan=\"1\">\n1\n</td><td rowspan=\"1\" colspan=\"1\">\n0.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n7\n</td><td rowspan=\"1\" colspan=\"1\">\n1.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nRegular exercise? (at least 30min 5 time/week)\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n67\n</td><td rowspan=\"1\" colspan=\"1\">\n51.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n318\n</td><td rowspan=\"1\" colspan=\"1\">\n61.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.382\n</td><td rowspan=\"1\" colspan=\"1\">\n1.214\n</td><td rowspan=\"1\" colspan=\"1\">\n0.786\n</td><td rowspan=\"1\" colspan=\"1\">\n1.874\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n63\n</td><td rowspan=\"1\" colspan=\"1\">\n48.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n200\n</td><td rowspan=\"1\" colspan=\"1\">\n38.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nDo your working hours include fixed, frequent and regular rest times (sitting)?\n</td><td rowspan=\"1\" colspan=\"1\">\nNo\n</td><td rowspan=\"1\" colspan=\"1\">\n35\n</td><td rowspan=\"1\" colspan=\"1\">\n26.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n132\n</td><td rowspan=\"1\" colspan=\"1\">\n25.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.392\n</td><td rowspan=\"1\" colspan=\"1\">\n1.239\n</td><td rowspan=\"1\" colspan=\"1\">\n0.786\n</td><td rowspan=\"1\" colspan=\"1\">\n1.874\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n95\n</td><td rowspan=\"1\" colspan=\"1\">\n73.1%\n</td><td rowspan=\"1\" colspan=\"1\">\n386\n</td><td rowspan=\"1\" colspan=\"1\">\n74.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB7\"><label>Table 7</label><caption><title>Impact of MSD and its association with various factors</title><p>AOR: Adjusted odds ratio, C.I.: Confidence interval, IQR: Interquartile range, N: Frequency, %: Percentage, MSD: Musculoskeletal disorder</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td colspan=\"2\" rowspan=\"3\">\n \n</td><td colspan=\"4\" rowspan=\"1\">\nDisability due to MSD\n</td><td rowspan=\"3\" colspan=\"1\">\np-value\n</td><td rowspan=\"3\" colspan=\"1\">\nAOR  \n</td><td colspan=\"2\" rowspan=\"1\">\n95% C.I.\n</td></tr><tr><td colspan=\"2\" rowspan=\"1\">\nNo\n</td><td colspan=\"2\" rowspan=\"1\">\nYes\n</td><td rowspan=\"2\" colspan=\"1\">\nLower\n</td><td rowspan=\"2\" colspan=\"1\">\nUpper\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nN/Median\n</td><td rowspan=\"1\" colspan=\"1\">\n%/ IQR\n</td><td rowspan=\"1\" colspan=\"1\">\nN/Median\n</td><td rowspan=\"1\" colspan=\"1\">\n%/ IQR\n</td></tr><tr><td colspan=\"2\" rowspan=\"1\">\nAge\n</td><td rowspan=\"1\" colspan=\"1\">\n40\n</td><td rowspan=\"1\" colspan=\"1\">\n(45-37)\n</td><td rowspan=\"1\" colspan=\"1\">\n44\n</td><td rowspan=\"1\" colspan=\"1\">\n48-40\n</td><td rowspan=\"1\" colspan=\"1\">\n1.110\n</td><td rowspan=\"1\" colspan=\"1\">\n1.048\n</td><td rowspan=\"1\" colspan=\"1\">\n1.175\n</td><td rowspan=\"1\" colspan=\"1\">\n1.110\n</td></tr><tr style=\"background-color:#ccc\"><td colspan=\"2\" rowspan=\"1\">\nYears of experience\n</td><td rowspan=\"1\" colspan=\"1\">\n15\n</td><td rowspan=\"1\" colspan=\"1\">\n(20-10)\n</td><td rowspan=\"1\" colspan=\"1\">\n17\n</td><td rowspan=\"1\" colspan=\"1\">\n22-12\n</td><td rowspan=\"1\" colspan=\"1\">\n0.084\n</td><td rowspan=\"1\" colspan=\"1\">\n0.957\n</td><td rowspan=\"1\" colspan=\"1\">\n0.911\n</td><td rowspan=\"1\" colspan=\"1\">\n1.006\n</td></tr><tr><td rowspan=\"2\" colspan=\"1\">\nGender\n</td><td rowspan=\"1\" colspan=\"1\">\nfemale\n</td><td rowspan=\"1\" colspan=\"1\">\n76\n</td><td rowspan=\"1\" colspan=\"1\">\n44.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n337\n</td><td rowspan=\"1\" colspan=\"1\">\n70.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n&lt;0.001\n</td><td rowspan=\"1\" colspan=\"1\">\n2.906\n</td><td rowspan=\"1\" colspan=\"1\">\n1.886\n</td><td rowspan=\"1\" colspan=\"1\">\n4.477\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nmale\n</td><td rowspan=\"1\" colspan=\"1\">\n95\n</td><td rowspan=\"1\" colspan=\"1\">\n55.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n140\n</td><td rowspan=\"1\" colspan=\"1\">\n29.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr><tr><td rowspan=\"4\" colspan=\"1\">\nMarital status\n</td><td rowspan=\"1\" colspan=\"1\">\ndivorced\n</td><td rowspan=\"1\" colspan=\"1\">\n7\n</td><td rowspan=\"1\" colspan=\"1\">\n4.1%\n</td><td rowspan=\"1\" colspan=\"1\">\n15\n</td><td rowspan=\"1\" colspan=\"1\">\n3.1%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.457\n</td><td rowspan=\"1\" colspan=\"1\">\n0.266\n</td><td rowspan=\"1\" colspan=\"1\">\n0.008\n</td><td rowspan=\"1\" colspan=\"1\">\n8.712\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nmarried\n</td><td rowspan=\"1\" colspan=\"1\">\n146\n</td><td rowspan=\"1\" colspan=\"1\">\n85.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n442\n</td><td rowspan=\"1\" colspan=\"1\">\n92.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.893\n</td><td rowspan=\"1\" colspan=\"1\">\n0.813\n</td><td rowspan=\"1\" colspan=\"1\">\n0.039\n</td><td rowspan=\"1\" colspan=\"1\">\n16.743\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nsingle\n</td><td rowspan=\"1\" colspan=\"1\">\n18\n</td><td rowspan=\"1\" colspan=\"1\">\n10.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n13\n</td><td rowspan=\"1\" colspan=\"1\">\n2.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.613\n</td><td rowspan=\"1\" colspan=\"1\">\n0.437\n</td><td rowspan=\"1\" colspan=\"1\">\n0.018\n</td><td rowspan=\"1\" colspan=\"1\">\n10.800\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nwindow\n</td><td rowspan=\"1\" colspan=\"1\">\n0\n</td><td rowspan=\"1\" colspan=\"1\">\n0.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n7\n</td><td rowspan=\"1\" colspan=\"1\">\n1.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr><tr><td rowspan=\"3\" colspan=\"1\">\nLevel of school you are working in\n</td><td rowspan=\"1\" colspan=\"1\">\nHigh school\n</td><td rowspan=\"1\" colspan=\"1\">\n49\n</td><td rowspan=\"1\" colspan=\"1\">\n28.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n132\n</td><td rowspan=\"1\" colspan=\"1\">\n27.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.735\n</td><td rowspan=\"1\" colspan=\"1\">\n0.924\n</td><td rowspan=\"1\" colspan=\"1\">\n0.584\n</td><td rowspan=\"1\" colspan=\"1\">\n1.462\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nIntermediate school\n</td><td rowspan=\"1\" colspan=\"1\">\n45\n</td><td rowspan=\"1\" colspan=\"1\">\n26.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n108\n</td><td rowspan=\"1\" colspan=\"1\">\n22.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.574\n</td><td rowspan=\"1\" colspan=\"1\">\n1.156\n</td><td rowspan=\"1\" colspan=\"1\">\n.698\n</td><td rowspan=\"1\" colspan=\"1\">\n1.913\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nprimary school\n</td><td rowspan=\"1\" colspan=\"1\">\n77\n</td><td rowspan=\"1\" colspan=\"1\">\n45.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n237\n</td><td rowspan=\"1\" colspan=\"1\">\n49.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nSmoker\n</td><td rowspan=\"1\" colspan=\"1\">\nno\n</td><td rowspan=\"1\" colspan=\"1\">\n161\n</td><td rowspan=\"1\" colspan=\"1\">\n94.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n453\n</td><td rowspan=\"1\" colspan=\"1\">\n95.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.035\n</td><td rowspan=\"1\" colspan=\"1\">\n0.399\n</td><td rowspan=\"1\" colspan=\"1\">\n0.170\n</td><td rowspan=\"1\" colspan=\"1\">\n0.935\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nyes\n</td><td rowspan=\"1\" colspan=\"1\">\n10\n</td><td rowspan=\"1\" colspan=\"1\">\n5.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n24\n</td><td rowspan=\"1\" colspan=\"1\">\n5.0%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">\nBMI\n</td><td rowspan=\"1\" colspan=\"1\">\nNormal\n</td><td rowspan=\"1\" colspan=\"1\">\n63\n</td><td rowspan=\"1\" colspan=\"1\">\n36.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n119\n</td><td rowspan=\"1\" colspan=\"1\">\n24.9%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.074\n</td><td rowspan=\"1\" colspan=\"1\">\n0.113\n</td><td rowspan=\"1\" colspan=\"1\">\n0.010\n</td><td rowspan=\"1\" colspan=\"1\">\n1.238\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nObese\n</td><td rowspan=\"1\" colspan=\"1\">\n44\n</td><td rowspan=\"1\" colspan=\"1\">\n25.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n166\n</td><td rowspan=\"1\" colspan=\"1\">\n34.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.137\n</td><td rowspan=\"1\" colspan=\"1\">\n0.161\n</td><td rowspan=\"1\" colspan=\"1\">\n0.014\n</td><td rowspan=\"1\" colspan=\"1\">\n1.788\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nOverweight\n</td><td rowspan=\"1\" colspan=\"1\">\n63\n</td><td rowspan=\"1\" colspan=\"1\">\n36.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n185\n</td><td rowspan=\"1\" colspan=\"1\">\n38.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.130\n</td><td rowspan=\"1\" colspan=\"1\">\n0.157\n</td><td rowspan=\"1\" colspan=\"1\">\n0.014\n</td><td rowspan=\"1\" colspan=\"1\">\n1.722\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nUnderweight\n</td><td rowspan=\"1\" colspan=\"1\">\n1\n</td><td rowspan=\"1\" colspan=\"1\">\n0.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n7\n</td><td rowspan=\"1\" colspan=\"1\">\n1.5%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nRegular exercise? (at least 30 min. 5 time/week)\n</td><td rowspan=\"1\" colspan=\"1\">\nno\n</td><td rowspan=\"1\" colspan=\"1\">\n93\n</td><td rowspan=\"1\" colspan=\"1\">\n54.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n292\n</td><td rowspan=\"1\" colspan=\"1\">\n61.2%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.743\n</td><td rowspan=\"1\" colspan=\"1\">\n1.070\n</td><td rowspan=\"1\" colspan=\"1\">\n0.713\n</td><td rowspan=\"1\" colspan=\"1\">\n1.606\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n78\n</td><td rowspan=\"1\" colspan=\"1\">\n45.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n185\n</td><td rowspan=\"1\" colspan=\"1\">\n38.8%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nDo your working hours include fixed, frequent, and regular rest times (sitting)?\n</td><td rowspan=\"1\" colspan=\"1\">\nno\n</td><td rowspan=\"1\" colspan=\"1\">\n49\n</td><td rowspan=\"1\" colspan=\"1\">\n28.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n118\n</td><td rowspan=\"1\" colspan=\"1\">\n24.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n0.828\n</td><td rowspan=\"1\" colspan=\"1\">\n0.952\n</td><td rowspan=\"1\" colspan=\"1\">\n0.610\n</td><td rowspan=\"1\" colspan=\"1\">\n1.486\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nyes\n</td><td rowspan=\"1\" colspan=\"1\">\n122\n</td><td rowspan=\"1\" colspan=\"1\">\n71.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n359\n</td><td rowspan=\"1\" colspan=\"1\">\n75.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">\nMajor depressive disorder\n</td><td rowspan=\"1\" colspan=\"1\">\nN0\n</td><td rowspan=\"1\" colspan=\"1\">\n155\n</td><td rowspan=\"1\" colspan=\"1\">\n90.6%\n</td><td rowspan=\"1\" colspan=\"1\">\n359\n</td><td rowspan=\"1\" colspan=\"1\">\n75.3%\n</td><td rowspan=\"1\" colspan=\"1\">\n&lt;0.001\n</td><td rowspan=\"1\" colspan=\"1\">\n0.316\n</td><td rowspan=\"1\" colspan=\"1\">\n0.175\n</td><td rowspan=\"1\" colspan=\"1\">\n0.571\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nYes\n</td><td rowspan=\"1\" colspan=\"1\">\n16\n</td><td rowspan=\"1\" colspan=\"1\">\n9.4%\n</td><td rowspan=\"1\" colspan=\"1\">\n118\n</td><td rowspan=\"1\" colspan=\"1\">\n24.7%\n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td><td rowspan=\"1\" colspan=\"1\">\n \n</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Tameem A. Alhomaid, Seham Alharbi, Nahla J. Alghafes, Yasmeen A. Alfouzan, Raghad I. Alhumaidan, Farah Alassaf, Abdullah Aldhuwyan </p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Tameem A. Alhomaid, Seham Alharbi, Nahla J. Alghafes, Yasmeen A. Alfouzan, Raghad I. Alhumaidan, Farah Alassaf, Abdullah Aldhuwyan </p><p><bold>Drafting of the manuscript:</bold>  Tameem A. Alhomaid, Seham Alharbi, Nahla J. Alghafes, Yasmeen A. Alfouzan, Raghad I. Alhumaidan, Farah Alassaf, Abdullah Aldhuwyan </p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Tameem A. Alhomaid, Seham Alharbi, Nahla J. Alghafes, Yasmeen A. Alfouzan, Raghad I. Alhumaidan, Farah Alassaf, Abdullah Aldhuwyan </p><p><bold>Supervision:</bold>  Tameem A. Alhomaid</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study. Regional Research Ethics Committee, Qassim Province issued approval H-04-Q-001</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Animal Ethics</title><fn fn-type=\"other\"><p><bold>Animal subjects:</bold> All authors have confirmed that this study did not involve animal subjects or tissue.</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050584-i01\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050584-i02\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>INTRODUCTION</title>", "<p>Transporters are transmembrane proteins that mediate selective uptake or export of solutes, metabolites, ions and drugs across the plasma membrane (PM) or other organellar membranes. Secondary active transporters of the PM couple the transport of substrates against their concentration gradients with the transport of other solutes down their concentration gradients. All PM transporters, despite their structural, functional and evolutionary differences, operate via an alternating access model [##REF##5968307##1##], where substrates bind on transporters from one side of the PM and elicit conformational changes, which lead to the opening of the transporter on the other side of the membrane, thus enabling release of substrates [##REF##27023848##2##–##REF##27023848##4##]. Mechanistic variations of the alternating access model, known as the rocker-switch [##REF##23403214##5##], the rocking-bundle [##REF##19996368##6##] or the elevator-type transporters [##REF##32369548##7##], reflect major structural differences among specific transporters. Notably, however, in all three mechanistic models, transporters undergo significant structural changes from an outward-facing to an inward-facing conformation, and vice versa, via a series of substrate-occluded structures [##REF##27023848##4##].</p>", "<p>The Amino Acid-Polyamine-Organocation (APC) superfamily is one of the largest and highly ubiquitous families of secondary transporters [##REF##10931886##8##, ##REF##10931885##9##], including well-studied transporters of biomedical interests, as for example neurotransmitter transporters DAT (dopamine) or SERT (serotonin), or transporters mediating the uptake of nucleobase-related drugs (e.g. 5-fluorouracil or 5-flurocytosine). APC transporters are characterized by a 5+5 α-helical inverted repeat (known as the 5HIRT or LeuT fold) formed by ten continuous transmembrane segments (TMS1-10). The ten TMSs are arranged in two discrete domains, the so-called ‘hash’/scaffold domain (TMS3, TMS4, TMS8, TMS9) and the ‘bundle’/core domain (TMS1, TMS2, TMS6, TMS7). In this arrangement, TMS5 and TMS10 function as dynamic gates controlling access to and release of substrates/cations from a central binding site. The substrate binding site in APCs is made by residues located in TMS3, TMS6, TMS8 and TMS10, at the interface of the hash and bundle domains. Based on this fold, all APC transporters function via variations of the rocking-bundle model, where the outward-facing to inward-facing conformational change occurs by the relative motion between the bundle or hash motifs, which underlies substrate accessibility and release [##REF##16041361##10##–##REF##21368142##14##]. It has been suggested that substrate binding in the outward-facing conformation is assisted by the simultaneous binding of a positive charge ion (Na⁺ or H⁺), which elicits the conformational change of the protein towards the inward-facing conformation. Recent high-resolution structures further showed that water molecules shape and stabilize the substrate-binding site and affect the functioning of gates in the bacterial AdiC, APC-type, transporter [##REF##34461897##15##]. Notably also, the cytosolic N- and C-terminal regions of several APC transporters have been shown to be involved in intramolecular interactions that are critical for function, substrate specificity or transporter turnover [##REF##31213137##16##].</p>", "<p>Most APC transporters possess two ‘extra’ TMSs at their C-terminal part (e.g. TMS11 and TMS12), the role of which does not seem to be directly related to substrate transport catalysis. Although the majority of APC structures have been resolved as monomeric transporters, in some cases, it has been proposed that these two extra C-terminal TMSs might be critical for APC oligomerization. For example, the AdiC transporter has been crystallized as a stable dimer where the homodimer interface is formed by non-polar amino acids from TMS11 and TMS12 [##REF##34461897##15##]. In this case, however, the role of APC oligomerization remains dubious as each monomer seems to be a self-contained transporter [##REF##34461897##15##, ##REF##19578361##17##]. LeuT has also been crystalized as a dimer via TMS9 and TMS12, and possibly TMS11 [##REF##19066470##18##, ##REF##16798738##19##], but to our knowledge there is no information whether dimerization is necessary for transport activity. There is also evidence, via co-immunoprecipitation [##REF##10716733##20##], crosslinking studies [##REF##8925924##21##] and FRET experiments [##REF##12746456##22##, ##REF##19338744##23##], that DAT and SERT transporters oligomerize. Similar results supporting oligomerization have been reported for rGAT1 and glycine transporters [##REF##11071889##24##, ##REF##18252709##25##]. However, in the aforementioned cases, only in SERT, TMS11 and TMS12 have been shown to be implicated in oligomerization <italic>in vivo</italic> [##REF##14660642##26##].</p>", "<p>The APC superfamily includes the Nucleobase Cation Symporter 1 (NCS1) group, of which several fungal and plant transporters have been extensively studied at the genetic and functional level [##REF##22969088##27##–##REF##26773540##34##]. In particular, work performed in <italic>Aspergillus nidulans</italic>, a filamentous fungus that has developed to be a model system to study transporters [##REF##17784857##35##–##REF##31583839##37##], has unveiled important knowledge on the regulation of expression, subcellular trafficking, turnover, transport kinetics and substrate specificity of NCS1 transporters [##REF##22969088##27##–##REF##31306663##31##]. All <italic>A. nidulans</italic> NCS1s function as H⁺ symporters selective for uracil, cytosine, allantoin, uridine, thiamine or nicotinamide riboside and secondarily for uric acid and xanthine. Previous studies have modeled <italic>A. nidulans</italic> NCS1s using the homologous prokaryotic Mhp1 benzyl-hydantoin/Na⁺ transporter [##REF##16041361##10##] as a structural template, and assessed structure-function relationships via extensive mutational analyses. These studies have identified the substrate binding site and substrate translocation trajectory, and revealed important roles of the cytosolic N-and C-terminal segments in regulating endocytic turnover, transport kinetics and, surprisingly, substrate specificity of NCS1 transporters [##REF##22969088##27##–##REF##31306663##31##]. Noticeably, all characterized functional mutations in NCS1 map in specific TMSs of the 5+5 inverted repeat fold or in the cytosolic terminal regions of these transporters.</p>", "<p>Here, we systematically investigate the role of TMS11 and TMS12 in the most extensively studied NCS1 transporter of <italic>A. nidulans</italic>, namely the allantoin-uracil-uric acid/H⁺ FurE symporter. We show that two specific aromatic residues in TMS12 are essential for ER-exit and traffic to the PM, apparently via structural interactions with specific residues in the core domain (TMS1-10) that catalyzes transport. We subsequently provide genetic evidence that TMS11-12 is essential for oligomerization or/and partitioning of FurE in specific membrane microdomains to achieve ER-exit and traffic to the PM.</p>" ]

[ "<title>MATERIAL AND METHODS</title>", "<title>Media, strains and growth conditions</title>", "<p>Standard complete (CM) and minimal media (MM) for <italic>A. nidulans</italic> growth were used. Media and supplemented auxotrophies were used at the concentrations given in <ext-link xlink:href=\"http://www.fgsc.net\" ext-link-type=\"uri\">http://www.fgsc.net</ext-link> [##REF##20413916##49##]. Glucose 1% (w/v) was used as carbon source. 10 mM sodium nitrate (NO₃) or 10 mM ammonium tartrate were used as standard nitrogen sources. Allantoin, uric acid and 5FU were used at the following final concentrations: 5FU at 100 μM; uric acid and allantoin 0.5 mM. All media and chemical reagents were obtained from Sigma-Aldrich (Life Science Chemilab SA, Hellas) or AppliChem (Bioline Scientific SA, Hellas). A Δ<italic>furD::riboB</italic> Δ<italic>furA::riboB</italic> Δ<italic>fcyB::argB</italic> Δ<italic>azgA</italic> Δ<italic>uapA</italic> Δ<italic>uapC::AfpyrG</italic> Δ<italic>cntA::riboB pabaA1 pantoB100</italic> mutant strain, named Δ7, was the recipient strain in transformations with plasmids carrying <italic>furE</italic> alleles based on complementation of the pantothenic acid auxotrophy <italic>pantoB100</italic> and/or the <italic>pabaA1</italic> auxotrophy [##REF##24355588##50##]. A <italic>pabaA1</italic> (paraminobenzoic acid auxotrophy) is a wt control strain. <italic>A. nidulans</italic> protoplast isolation and transformation was performed as previously described [##REF##14607411##51##]. Growth tests were performed at 37°C for 48 h, at pH 6.8.</p>", "<title>Standard molecular biology manipulations and plasmid construction</title>", "<p>Genomic DNA extraction from <italic>A. nidulans</italic> was performed as described in FGSC (<ext-link xlink:href=\"http://www.fgsc.net\" ext-link-type=\"uri\">http://www.fgsc.net</ext-link>). Plasmids, prepared in <italic>E. coli</italic>, and DNA restriction or PCR fragments were purified from agarose 1% gels with the Nucleospin Plasmid Kit or Nucleospin ExtractII kit, according to the manufacturer's instructions (Macherey-Nagel, Lab Supplies Scientific SA, Hellas). Standard PCR reactions were performed using KAPATaq DNA polymerase (Kapa Biosystems). PCR products used for cloning, sequencing and re-introduction by transformation in <italic>A. nidulans</italic> were amplified by a high-fidelity KAPA HiFi HotStart Ready Mix (Kapa Biosystems) polymerase. DNA sequences were determined by VBC-Genomics (Vienna, Austria). Site directed mutagenesis was carried out according to the instructions accompanying the Quik-Change® Site-Directed Mutagenesis Kit (Agilent Technologies, Stratagene). The principal vector used for most <italic>A. nidulans</italic> mutants is a modified pGEM-T-easy vector carrying a version of the <italic>gpdA</italic> promoter, the <italic>trpC</italic> 3' termination region and the <italic>panB</italic> (and <italic>pabaA</italic> for co-transformations) selection marker. Mutations were constructed by oligonucleotide-directed mutagenesis or appropriate forward and reverse primers (see ##SUPPL##0##Supplementary Table S6##).</p>", "<title>Protein Model Construction</title>", "<p>The FurE modeled structure was constructed based on homology modeling using Prime 2018-4 (Schrödinger, LLC, New York, NY, 2018) on Maestro platform (Maestro, version 2018-4, Schrödinger, LLC, New York, NY, 2018). Mhp1 was used as query in the three conformations: Outward (2JLN), Occluded (4D1C), inward open (2X79), sharing with FurE a 35% similarity. The models shown here are presented with PyMOL 2.5 (<ext-link xlink:href=\"https://pymol.org\" ext-link-type=\"uri\">https://pymol.org</ext-link>).</p>", "<title>Molecular Dynamics (MD)</title>", "<p>Protein model construction and MD simulations are described in detail elsewhere [##REF##37544358##52##]. In brief, homology models of FurE were constructed based on Mhp1 crystal structures 2JLN, 4D1B, 2X79. Each model was inserted into a lipid bilayer using the CHARMM-GUI tool and the resulting system was solvated using the TIP3P water model with final NaCl concentration of 150 mM. Calculations were conducted using GROMACS software, version 2019.2 and CHARMM36m force field [##REF##23407358##53##, ##UREF##5##54##]. The protein orientation into the membrane was calculated using the PPM server (<ext-link xlink:href=\"http://amber.manchester.ac.uk\" ext-link-type=\"uri\">http://amber.manchester.ac.uk</ext-link>, [##UREF##6##55##]). The system was first minimized to obtain stable structures and then equilibrated for 20ns by gradually heating and releasing the restraints. The resulting equilibrated structures were then used as an initial condition for the production runs of 100ns at constant pressure of 1 atm and constant target temperature of 300K using Nose-Hoover thermostat and Parrinello-Rahman semi-isotropic pressure coupling.</p>", "<title>Transport assays</title>", "<p>Kinetic analysis of wt and mutant FurE was measured by estimating uptake rates of [³H]-uracil uptake (40 Ci mmol⁻¹, Moravek Biochemicals, CA, USA), as previously described [##REF##24355588##50##]. In brief, [³H]-uracil uptake was assayed in <italic>A. nidulans</italic> conidiospores germinating for 4 h at 37°C, at 140 rpm, in liquid MM, pH 6.8. Initial velocities were measured on 10⁷ conidiospores/100 μL by incubation with concentration of 0.75 μM of [³H]-uracil at 37°C. The time points when the initial velocities (rates) are measured is 1 or 2 min. All transport assays were carried out at least in two independent experiments and the measurements in triplicate. Results were analysed in GraphPad Prism software.</p>", "<title>Epifluorescence microscopy</title>", "<p>Samples for standard epifluorescence microscopy were prepared as previously described [##REF##20002879##56##]. In brief, sterile 35 mm l-dishes with a glass bottom (Ibidi, Germany) containing liquid minimal media supplemented with NaNO₃ and 1% glucose were inoculated from a spore solution and incubated for 16 h at 25°C. The images were obtained using an inverted Zeiss Axio Observer Z1 equipped with an Axio Cam HR R3 camera. Image processing and contrast adjustment were made using the ZEN 2012 software while further processing of the TIFF files was made using Adobe Photoshop CS3 software for brightness adjustment, rotation, alignment and annotation. The GFP-fluorescence intensity ratio (PM/cytosolic) was calculated using the ICY software [##UREF##7##57##]. The areas of the plasma membrane and the cytosol were manually highlighted and the intensity of GFP-fluorescence was measured. For nuclear staining, the DAPI dye was added to the growth medium in a final concentration of 0,002 mg/ml. The strains of interest were incubated with the dye for 20 minutes (25°C) and then washed with liquid minimal medium, before observation.</p>", "<title>Data Availability Statement</title>", "<p>Strains and plasmids are available upon request. The authors affirm that all data necessary for confirming the conclusions of the article are present within the article, figures, and tables.</p>" ]

[ "<title>RESULTS</title>", "<title>Most residues in TMS11 and TMS12 of FurE are not critical for PM localization and have a moderate role in transport</title>", "<p>Previous studies concerning NCS1 transporters failed to show a functional role of residues of the last two TMSs (TMS11-12) in transport kinetics or substrate specificity [##REF##25712422##28##,##REF##31306663##31##]. In FurE specifically, where we have employed several unbiased genetic screens to select functional mutants, we have never obtained any mutation located in TMS11 or TMS12 affecting FurE function. In line with this, the reported distinct crystal structures (outward-facing, substrate-occluded or inward-facing) of the Mhp1 bacterial NCS1 homologue strongly suggested that TMS11 and TMS12 do not participate in transport catalysis [##REF##20413494##13##, ##REF##21169684##38##–##REF##28726379##42##].</p>", "<p>In order to investigate whether the last two TMSs of FurE play any role in transport activity, substrate specificity, subcellular trafficking or turnover, we constructed strains expressing triple alanine (Ala) substitutions of residues predicted to form the helices of TMS11 and TMS12 (<bold>##FIG##0##Figure 1A##</bold>; for details see Materials and methods). The choice of Ala substitution is based on the logic that Ala residues conserve the hydrophobic nature of these transmembrane segments but replace specific amino acid side chains that might be important for function. The predicted structures of FurE, shown in <bold>##FIG##0##Figure 1B##</bold>, have been constructed via homology modeling with Mhp1. By comparing the outward-facing or occluded structures to the inward-facing conformation of FurE, what becomes immediately apparent is a significant distancing of TMS11-12 from the main body of the transporter (TMS1-10), also associated with a tilt of the cytosolic-facing half of TMS12, which is now exposed towards the lipid bilayer (marked in red In <bold>##FIG##0##Figure 1##</bold>). The significance of this observation becomes apparent later.</p>", "<p><bold>##FIG##1##Figure 2A##</bold> shows growth tests relevant to FurE function of all triple Ala mutants and control strains. As also shown previously [##REF##28978674##30##, ##REF##31306663##31##], wild-type (wt) FurE expressed via the <italic>gpdA</italic> promoter in a genetic background lacking all other major nucleobase transporters (e.g., Δ7) confers growth on allantoin or uric acid as sole nitrogen sources and leads to sensitivity to 5-fluorouracil (5FU). The ‘empty’ Δ7 isοgenic strain lacking FurE expression (negative control), cannot grow on allantoin or uric acid and is resistant to 5FU. Notice that uracil, although it is also a FurE substrate, is not used as a N source in <italic>A. nidulans</italic>. Most of triple mutations did not significantly affect FurE-dependent growth on allantoin or sensitivity to 5FU, resembling the growth phenotype of the strain expressing the wt FurE. Some mutants, mostly those concerning TMS12, showed reduced growth on uric acid, which is characteristic of lower transport capacity of FurE. Notably, two triple Ala replacements in TMS12, those affecting residues 472–474 and 484–486 led to total loss of FurE function. Substitutions of residues 448–450, which mark the end of TMS11, but also of 487–490 and 490–492 led to partial loss-of-function of the FurE transporter, evident through the growth defects exhibited by the respective strains on uric acid. Finally, the very last triple Ala mutant (496–498), which corresponds to the beginning of the cytosolic C-tail, also led to a drastic reduction in FurE function (reduced growth on allantoin, significant sensitivity to 5FU and loss of growth on uric acid).</p>", "<p>To investigate whether the loss or reduction of FurE function in specific mutants is due to problematic translocation to the PM, reduced protein stability, or defective transport activity per se, we took advantage of the fact the all FurE alleles made were functionally fused in their C-terminus to a GFP epitope. <bold>##FIG##1##Figure 2A##</bold> (middle panel) shows the <italic>in vivo</italic> subcellular localization of wt and mutant versions of FurE, as followed by widefield epifluorescence microscopy. The two TMS12 mutations leading to apparently total FurE loss-of-function (472–474 and 484–486) led to retention of the transporter in the ER (notice the prominent rings, typical of nuclear ER in fungi; see also ##SUPPL##0##Supplementary Figure S1##). Mutation 493–495, which led to nearly total loss of FurE function also showed prominent ER-retention. Also, substitutions 448–450 and 490–92 led to partial ER-retention of FurE, rationalizing the growth phenotypes of the corresponding strains. In all other cases, FurE mutant versions label the PM, septa and vacuoles, similar to a correctly folded wt FurE transporter [##REF##28978674##30##, ##REF##31306663##31##]. Quantification of the ratio of PM-associated to cytosolic FurE-GFP fluorescence shows that most functional mutants give a similar result with that obtained with wt FurE, suggesting that the levels of FurE expression is in these strains is comparable (<bold>##FIG##1##Figure 2A##</bold>, right panel and ##SUPPL##0##Supplementary Figure S2##). Only mutations concerning residues 448–450 and 490–492 showed ∼5-fold reduced quantity of PM-associated FurE relative to the wt control, concomitant with partial ER-retention of the transporter.</p>", "<p>We performed direct uptake assays of all mutants made to test whether the mutant versions of FurE conserve transport capacity for radiolabeled uracil (radiolabeled allantoin is not available and radiolabeled uric acid is unstable). <bold>##FIG##1##Figure 2B##</bold> shows that most mutants conserve detectable transport rates, albeit often at a reduced degree. Transport rates measured ranged from those similar to wt FurE (i.e., close to 100%, in 438–440, 466–468 and 487–489) to moderate (∼44–55%, in 441–443, 444–447, 469–471 and 496–498), low (∼15–30%, in 475–477, 478–480, 481–483) or extremely low (just detectable, &lt;10%, in 448–450, 490–492 and 493–495). As might have been expected, the 472–474 and 484–486 Ala triple mutants, which showed ER-retention of FurE, did not exhibit detectable FurE transport capacity. Thus, uptake assays are in good agreement with growth tests and microscopy, which showed that most mutations, except those retained in the ER, could confer normal or reduced FurE-mediated growth on allantoin or uric acid and were sensitive to 5FU. Notice that for most transporters of nitrogen sources (e.g., amino acids, purines, nitrate, urea, etc.) studied in our laboratory, analogous mutations allowing transport rates &gt;10%, of the respective wt transporter are sufficient to confer detectable growth, while mutations allowing transport rates &gt;25% of wt can grow normally.</p>", "<p>In summary, our results showed that most of the residues in TMS11 and TMS12 are not essential for proper folding and localization of FurE to the PM and are not absolutely essential for transport. The exception concerns three amino acid triplets, namely 472–474, 484–486 and to a lower degree 493–495, which seem to contain sequence-specific information essential for proper ER-exit and trafficking to the PM, and thus for function. Among these mutations, 472–474 (Ser-Trp-Leu) and 484-486 (Tyr-Tyr-Leu) include conserved aromatic residues, Trp473 and Tyr484, respectively. Especially Tyr484 is absolutely conserved in all Fur-like transporters, replaced by aliphatic hydrophobic acids (Met or Val) in the homologous Fcy-like subgroup of NCS1 transporters (see <bold>##FIG##0##Figure 1A##</bold>). Noticeably also, Tyr484 is one helix turn downstream from the tilt-point (Gly481) where the TMS12 ‘breaks’ during the transition from outward- or occluded to the inward-facing conformation (see <bold>##FIG##0##Figure 1B##</bold>). Thus, we decided to investigate the role of these two aromatic residues in more detail.</p>", "<title>Tyr484 is irreplaceable for proper ER-exit of FurE possibly due to interactions with specific residues in TMS3, TMS10 and TMS12</title>", "<p>We constructed single Phe, Ser or Met replacements of Tyr484, the only well conserved residue of the triplet 484–486. The rationale for constructing and analyzing the Y484M is based on the observation that Met is present in the Fcy-type subgroup of NCS1 transporters, which exhibits distinct and non-overlapping substrate specificity (see <bold>##FIG##0##Figure 1A##</bold>). All three mutations led to inability for growth on allantoin and uric acid and resistance to 5FU (<bold>##FIG##2##Figure 3B##</bold>, left panel). This was shown to be due to ER retention, similar to what was found with the triple Ala replacement of residues 484–486 (<bold>##FIG##2##Figure 3B##</bold>, right panel). In line with growth tests and microscopy, direct uptake studies of radiolabeled uracil showed that Phe, Ser or Met replacements of Tyr484 led to FurE loss of function (<bold>##FIG##2##Figure 3C##</bold>). Thus, neither the presence of hydroxyl group (Ser) nor of an aromatic ring (Phe) could functionally replace Tyr484. In conclusion, Tyr484 is shown to be irreplaceable for the proper function of FurE, apparently due its essential role in ER-exit and translocation of the transporter to the PM.</p>", "<p>Defective ER-exit could be due to improper intramolecular folding, defective interaction with annular lipids, or due to abortive interactions with Sec24, the main membrane cargo receptor, or specific ER chaperones [##REF##23419775##43##]. To investigate these possibilities, we tried to select genetic revertants suppressing the lack of growth on allantoin of strains expressing FurE-Y484F or FurE-Y484S. We failed to obtain any, after repeated trials. This might suggest that Tyr484 is involved in complex interactions, these being intramolecular or with lipids or other proteins.</p>", "<p>We tried to identify, by modeling, the location of Tyr484 in respect to other domains of the transporter or the lipid bilayer. In the outward and occluded conformation Tyr484 faces the interior of FurE and seems to be at interacting distances with Phe111 located in the beginning of TMS3, and with Asp406 located towards the last turn of TMS10 (<bold>##FIG##2##Figure 3A##</bold>). More specifically, Tyr484 is predicted to interact via a H-bond with Asp406 and through pi-pi stacking with Phe111. Interestingly, in the inward-facing conformation, as a consequence of a tilt of half of the TMS12 helix, the side chain of Tyr484 although still oriented to Asp406, also gains direct contacts with the lipid bilayer. Consequently, the interaction with Asp406 and Phe111 is weakened (<bold>##FIG##2##Figure 3A##</bold> and ##SUPPL##0##Supplementary Figure S3##). This local conformational change is probably related to the final step of the transport cycle. Noticeably also, Tyr484 is also in close distance, especially in the inward conformation, with a series of aromatic acids in the end of TMS12, namely Tyr485, Phe488, Phe489, Trp491 and Phe493, which come into contact with the lipid bilayer (##SUPPL##0##Supplementary Figure S3##).</p>", "<p>The above findings suggested that the essentiality of Tyr484 might be associated to dynamic interactions with specific residues in TMS3 and TMS10, but also with downstream aromatic residues at the end TMS12, close to the cytoplasmic interphase with the lipid bilayer. If so, we thought that mutations in the interacting amino acids of Ty484, namely residues Phe111 and Asp406, but also Phe488, Phe489, Trp491 and Phe493, might lead to functional defects similar to those of mutation in Tyr484. Noticeably, Phe111 and Asp406 are nearly absolutely conserved in NCS1 transporters, while the other aromatic residues are only partially conserved in eukaryotic homologues.</p>", "<p>To further investigate the idea of a network of functional interactions of Tyr484 with Phe111 and Asp406 we constructed and analysed strains carrying the following FurE mutations: F111A, F111Y, D406A, D406E, F111Y/D406E, F111Y/Y484F, D406E/Y484F and F111Y/D406E/Y484F. Notice that the double mutation F111Y/Y484F inverts the topology of the aromatic acids involved, while some of the other mutations involve very conservative changes (e.g., F111Y or D406E). Nearly all of these amino acid substitutions scored as loss-of-function mutations associated to retention of FurE in the ER, very similar to Y484F (<bold>##FIG##2##Figure 3B##</bold>). An exception was only F111Y, one of the most conservative changes, which did not affect trafficking to the PM, but still led to a small defect in activity, reflected as reduced growth on allantoin and no growth on uric acid. Direct uptake assays were in excellent agreement with growth tests and microscopy (<bold>##FIG##2##Figure 3C##</bold>). The similarity of defects in ER-exit and FurE activity caused by mutations in Tyr484, Phe111 and Asp406 was in good agreement with the network of interactions of these residues, as proposed via structural modeling. Finally, notice that Ala mutations in amino acids triplets in TMS12 that include Trp491 and Phe493 also showed partial (490-492) or significant (493-495) ER retention, resembling the effect of replacing Tyr484 (see <bold>##FIG##1##Figure 2A##</bold>). Altogether, it seems that a network of interactions between residues in TMS12 and TMS3 or/and TMS10 might indeed be necessary for proper folding of FurE, and thus essential for its proper exit of FurE and traffic to the PM.</p>", "<title>An aromatic residue at position 473 is necessary for FurE ER-exit</title>", "<p>We also constructed and analysed W473A, W473F and W473Y substitutions of the well-conserved Trp473 (<bold>##FIG##3##Figure 4A##;</bold> see also <bold>##FIG##0##Figure 1A##</bold>). W473A conferred a FurE-dependent growth defect, which similarly to the Tyr484 mutations was shown to be related to ER-retention (<bold>##FIG##3##Figure 4B##</bold>). In contrast, both W473F and W473Y substitutions were functional, conferring growth on allantoin or uric acid and sensitivity to 5FU, and also showing proper localization of FurE to the PM (<bold>##FIG##3##Figure 4B##</bold>). In fact, W473F seems to enhance the presence of FurE protein in the PM, as the respective mutant not only shows increased PM associated fluorescence, but also the fraction detected in cytosolic foci originating from endocytosis (vacuoles and endosomes) is relatively reduced when compared to the image of wt FurE. Direct uptake measurements of radiolabeled uracil were in agreement with growth tests and microscopy, as W473A showed no transport activity, while W473F and W473Y exhibited increased (3-fold) or similar to wt transport rates, respectively (see <bold>##FIG##3##Figure 4C##</bold>). Thus, it seems that the presence of an aromatic residue at 473, not necessarily a Trp, is sufficient for proper ER-exit and FurE function. Notice that an aromatic residue is conserved in all Fur-like homologues.</p>", "<p>Molecular Dynamics (MD) were employed to identify putative interactive residues or whether Trp473 faces the membrane lipids. This showed that Trp473 is exactly in the middle of the lipid bilayer plane and might interact with Ile87, Pro88 and principally Leu91 in TMS2, Ala400 and Val404 in TMS10, and less so with Tyr469 and Ile477 in TMS12 (<bold>##FIG##3##Figure 4A##</bold>). The sites of residues corresponding to Leu91, Ala400, Val404 and Ile477 are conserved as aliphatic residues in other FurE homologues transporters, while Tyr469 is conserved as Tyr or Phe in all Fur proteins. The predicted interactions with the aforementioned residues appear rather stable. Trp473 has also contacts with lipid chains as the indole moiety is oriented in several structures towards the lipid-protein interface. In the outward and inward structure there are lipid chains parallel to the aromatic ring, but in the inward structure TMS12 appears to be slightly tilted exposing Trp473 further to lipid contacts (##SUPPL##0##Supplementary Figure S4##).</p>", "<p>Based on the above observations we mutated Leu91 and Val404 into Ala or Phe residues. The functional analysis of mutations L91A, L91F, V404A and V404F showed that none of them affects FurE localization to the PM. However, the Phe substitutions led to reduced growth on allantoin and uric acid and increased resistance to 5FU, whereas the Ala substituted mutants behaved as wt FurE in growth tests (<bold>##FIG##3##Figure 4B##</bold>). Uracil uptake assays confirmed that Ala substitutions did not affect FurE transport capacity, while Phe substitution led to differentially reduced FurE activity (<bold>##FIG##3##Figure 4C##</bold>). Substitution Y469A, concerning another residue possibly interacting with Trp473, is included in the already analyzed triple Ala mutation of residues 469-471, which showed no FurE defect. Overall, unlike the case of Tyr484, we could not identify putative intramolecular partners of Trp473 critical for FurE trafficking to the PM. Thus, the molecular basis that underlies the essentiality of an aromatic residue at position 473 for proper FurE folding and ER-exit remains elusive, although it might be related to dynamic interactions of TMS12 with TMS2 and/or TMS10, and probably with the membrane lipids too, as judged by MD.</p>", "<title>Evidence for oligomerization or concentrative partitioning of FurE molecules at ER-exit sites</title>", "<p>Previous reports have suggested that APC transporters might dimerize or oligomerize via their TMS11 and TMS12 domains (see Introduction). To address this issue in FurE and try to better understand the role of these segments in ER-exit and transport activity, we co-expressed ER-retained mutant versions of FurE with wt FurE. Co-expression was achieved by co-transformation of the Δ7 strain with plasmids carrying different FurE alleles expressed via the <italic>gpdA</italic> promoter (for details see Materials and methods).</p>", "<p><bold>##FIG##4##Figure 5##</bold> shows results obtained with selected transformants co-expressing FurE-Y484F or FurE-W473A with wt FurE. In both cases, we analyzed transformants where the GFP epitope is tagged in either the ER-retained mutant or the wt FurE. This allowed us to address the effect on subcellular localization of the ER-retained version on wt FurE and vice versa. As already discussed, both FurE-Y484F or FurE-W473A cannot confer growth on allantoin and uric acid and cannot accumulate the toxic analogue 5FU, while the strain expressing wt FurE grows on allantoin and uric acid and is sensitive to 5FU. Strains co-expressing ER-retained FurE-Y484F or FurE-W473A with wt FurE showed intermediate growth phenotypes (<bold>##FIG##4##Figure 5##, left panel</bold>). That is, they exhibited reduced growth on allantoin, no growth on uric acid, and intermediate resistance to 5FU. Epifluorescence microscopy of the same strains rationalized the growth phenotypes obtained (<bold>##FIG##4##Figure 5##, right panel</bold>). The strains co-expressing the ER-retained versions and wt FurE tagged with GFP showed partial ER retention of wt FurE-GFP (see panels with FurE-GFP/FurE-Y484F and FurE-GFP/FurE-W473A). Strains co-expressing the ER-retained versions tagged with GFP and untagged wt FurE showed no PM labeling, similar to the original ER-retained mutants. In agreement with the growth tests, direct uptake assays with radiolabeled uracil confirmed that FurE transport rates are significantly reduced in the strains co-expressing ER-retained FurE-Y484F and FurE-W473A with wt FurE, (<bold>##FIG##4##Figure 5B##</bold>). These findings point to the idea that FurE molecules associate by oligomerization or partitioning into specific ER microdomains, early after their co-translational insertion into the ER, so that misfolded versions, such as FurE-Y484F or FurE-W473A, trap a fraction of wt FurE in aggregates incapable of ER exit.</p>", "<title>Truncation of TMS11-12 abolishes oligomerization or partitioning of FurE at ER-exit sites</title>", "<p>To investigate the role of TMS11-12 in ER-exit and PM localization of FurE, we constructed and functionally analysed truncated versions of the transporter possessing either the core domains TMS1-10 (FurE-TMS1-10) or just the last two TMSs (FurE-TMS11-12). Both constructs were fused with GFP, and both were expressed via the <italic>gpdA</italic> promoter (for details see Materials and methods). First, we tested whether these two truncated versions translocate to the PM and whether FurE-TMS1-10 is functional. <bold>##FIG##5##Figure 6A##</bold> shows that FurE-TMS1-10 is totally trapped in the ER, and consequently could not confer FurE-dependent growth on allantoin or uric acid or sensitivity to 5FU, whereas FurE-TMS11-12 seems to be rapidly sorted in vacuole for degradation. We then co-expressed GFP-untagged FurE-TMS1-10 with FurE-TMS11-12-GFP, and tested whether the two parts of FurE are somehow functionally reconstituted or their co-expression promotes translocation to the PM. We showed that ‘split’ FurE could not be functionally or cellularly reconstituted (not shown). We also tried to employ Bifluorescence (BiF) assays using split YFP tags in all combinations (i.e., YFPn or YFPc N-terminally or C-terminally fused to the truncated parts of FurE), but still we did not obtain any evidence of reconstituted FurE parts (not shown). Subsequently, we co-expressed FurE-TMS1-10 with wt FurE and functionally analysed respective transformants. Results, summarized in <bold>##FIG##5##Figure 6A##</bold>, show no evidence for a dominant negative effect in respect to FurE function or localization, as seen when using non-truncated FurE versions.</p>", "<p>To obtain further evidence for concentrative oligomerization or partitioning of FurE in ERes (ER exit sites) and the role of TMS11-12, we also employed a distinct version of FurE, not related to TMS11-12 that shows tight retention in the ER, and consequently no transport activity. This is a FurE mutant where N-terminal residues 30–32 (Leu-Asp-Ser) have been replaced by alanines [##REF##31306663##31##] (named here FurE-Δ30-32). When full-length FurE-Δ30–32 was co-expressed with wt FurE, this led to synthetic phenotypes, basically a diminished transport capacity for uric acid and 5-fluorouracil (<bold>##FIG##5##Figure 6A##</bold>). However, when we used, in similar co-expression experiments, a truncated version of FurE-Δ30-32 lacking TMS11-12 (FurE-Δ30-32-TMS1-10), we ‘lost’ the dominant negative effect on growth (<bold>##FIG##5##Figure 6A##</bold>). In line with the above observations, direct uptake assays with radiolabeled uracil confirmed that FurE transport rates are significantly reduced in the strains co-expressing ER-retained FurE-Δ30-32 with wt FurE, while the uptake rate of strains co-expressing FurE-TMS1-10 or FurE-Δ30-32-TMS1-10 with wt FurE remained mostly unaffected (<bold>##FIG##5##Figure 6B##</bold>). Our findings strongly suggest that TMS11-12 are necessary for exit from the ER via their essential role in folding and oligomerization or partitioning of FurE in nascent ER exit sites.</p>" ]

[ "<title>DISCUSSION</title>", "<p>We showed that two specific aromatic residues in TMS12, namely Trp473 and Tyr484, are essential for ER-exit of FurE. In line with their functional importance, these residues are highly (Tyr484) or well (Trp473) conserved in FurE homologous proteins. Structural modelling and MD provided evidence that these residues, and especially Tyr484, might change their intramolecular topology relative to specific residues in other TMSs and aromatic residues at the end of TMS12, but also in respect to the lipid bilayer, during the transport cycle. This is a direct consequence of major topological changes that TMS12 undergoes during alteration from outward to inward conformation, as shown for Mhp1 [##REF##18927357##11##, ##REF##20413494##13##] and predicted by homology modelling in FurE here. More specifically, in the inward-facing conformation, TMS12 is shown to move away from the 5+5-fold core of the transporter and be directed towards annular lipids. Given the positioning of Trp473 in the middle of the bilayer and Tyr484 at the interphase of the transporter with lipids, the associated lack of proper ER-exit in the relative mutants suggests a structural defect in FurE folding due to modified intramolecular interactions and altered association with membrane lipids. In the case of Tyr484, we provided <italic>in silico</italic> evidence, supported by genetics, that this defect might be due to modified interactions of Tyr484 with two specific and highly conserved residues in TMS10 (Asp406) and TMS3 (Phe111), as well as, with downstream aromatic residues in TMS12 (mainly with Phe493).</p>", "<p>Previous mutations in the N-terminal part of TMS10 (named TMS10<bold>a</bold>) have been shown to affect the function or specificity of FurE, albeit without affecting ER-exit (e.g., mutations in M389) [##REF##25712422##28##]. Asp406, shown here to affect ER-exit, is located in the less flexible C-terminal part of TMS10 (named TMS10<bold>b</bold>). MD in Mhp1 has shown that the other half of TMS10 (TM10<bold>a</bold>) is dynamic gate able to occlude access to the major substrate binding site via local tilting in the middle of TMS10 [##REF##24952894##41##]. It is thus probable that interaction of Tyr484 (TMS12) with Asp406 in TMS10<bold>b</bold> is a structural interaction necessary for proper FurE folding, but this interaction seems to also affect gating and transport via stabilization of the neighbouring dynamic movements of TMS10a. In conclusion, we have identified a putative dynamic network of structural interactions necessary for proper FurE folding and thus for ER-exit.</p>", "<p>Our findings further show that specific TMS12 residues are involved principally in structural intramolecular interactions crucial for folding and proper ER-exit and localization to the PM, rather than being directly essential for substrate recognition and transport catalysis. Given the high conservation of residues involved in the network of interactions of TMS12, we predict that the ‘intramolecular chaperoning’ role TMS12 revealed herein might extent to all NCS1 or structurally similar APC-type transporters. In addition, the finding that the side-chain identity of most residues TMS12 and TMS11, except Trp473 and Tyr484, is little critical for ER-exit, further suggests that the packing of these helices with the 5+5 TMS core of the FurE transporter occurs via hydrophobic interactions of helical backbones, which are apparently strengthened by the specific interactions of involving Trp473 and Tyr484.</p>", "<p>To our opinion, an original finding of this work is the discovery that co-expression of wt and ER-retained FurE leads to a synthetic dominant negative phenotype, which can be best explained by FurE oligomerization or partitioning in common ERes, and that this phenomenon is dependent on TMS11 and TMS12. We exclude the possibility that overexpression of ER-retained versions of FurE in some transformants causes a general defect in cargo trafficking as we did not observe any growth defects in the relative strains in all growth media tested, except those revealing a defect in FurE activity (e.g., growth on allantoin or uric acid or sensitivity to 5FU; See ##SUPPL##0##Supplementary Figure S5##).</p>", "<p>The observed inter-molecular interactions of FurE versions might be explained by tight oligomerization, as is the case in other transporters (e.g., the UapA uric acid-xanthine transporter of <italic>A. nidulans</italic>; [##UREF##1##44##]). However, we failed to obtain any evidence of FurE oligomerization at the ER or the PM, using a BiF approach or blue native gel electrophoresis (not shown). This lack of evidence for <italic>in vivo</italic> oligomerization is also true for other NCS1 transporters and several APC transporters. An alternative explanation of the observed dominant negative or positive phenotypes is that FurE molecules soon after their co-translational insertion into the ER membrane partition laterally in common microdomains, which associate with ERes. This has as a consequence the concentrative packaging of distinct FurE versions in common COPII vesicles. Concentrative ER-exit of membrane cargoes has been previously reported, suggesting that a multimeric sorting ‘code’ drives selectivity in cargo sorting [##REF##12499351##45##].</p>", "<p>Why are FurE mutants, such as those including mutations Y484F or W473A, unable to exit the ER when expressed by themselves? We believe these are partially misfolded FurE versions that do not provide the proper multimeric structure to partition properly in ERes. Misfolding of these mutants seems only partial, as at least in the case of Y484F and W473A, the relative FurE versions can exhibit extremely low, but still measurable minimal transport, in some transformants (not shown). This hypothesis suggests that properly folded and misfolded FurE molecules are capable of self-associating and partitioning in common microdomains, probably prior to sorting to ERes. In this case, there might be an intrinsic propensity of self-association or loose oligomerization of FurE molecules immediately after their biogenesis. Alternatively, an ER-associated adaptor protein might exist that specifically recognizes nascent FurE molecules and promote their association into a common microdomain or in ERes. Such cargo-specific ER adaptors for specific membrane cargoes exist, with the <italic>Saccharomyces cerevisiae</italic> Erv14 being the most extensively studied [##UREF##2##46##–##UREF##4##48##]. Mutants unable to exit the ER might then be unable to be recognized efficiently by this adaptor, or association with this adaptor is defective for proper exit from ERes.</p>", "<p>An important finding related to the synthetic dominant negative phenotypes observed and the hypothesized association/oligomerization of FurE molecules in specific microdomains in the ER is the essential role of TMS11 and TMS12. We showed that these helices are essential for self-association of FurE molecules and/or partitioning to ERes. It is thus most reasonable to suggest that truncation of these helices leads to an unfolded FurE version, which unlike missense mutations in TMS3 (Phe111), TMS10 (Asp406), TMS12 (Trp473, Tyr484), or Δ30-32, cannot associate with co-expressed wt FurE molecules and thus fails to partition in ERes. This strengthens the notion that TMS11-12 function as an “intramolecular chaperone” essential for proper FurE folding, which in turn provides a structural code for FurE association and concentrative ER-exit the trafficking to the PM.</p>", "<p>As already discussed in the Introduction, there is <italic>in vitro</italic> and <italic>in vivo</italic> evidence that several members of the APC superfamily oligomerize, and in some cases oligomerization was shown to depend on TMS11 and TMS12. For example, in AdiC, which has been shown to form homodimers <italic>in vitro</italic> and in possibly <italic>in vivo</italic>, the homodimer interface is formed by non-polar amino acids from TMS11 and TMS12, where residues of TMS11 from one monomer interdigitate with residues of TMS12 from the other monomer. Further interactions between the two monomers are mediated by the loops between TMS2 and TMS3, the cytoplasmic ends of TMS2 and TMS3, the cytoplasmic halves of TMS10, and the C-termini. The latter embrace neighbouring monomers [##REF##34461897##15##]. However, given that monomers of AdiC have been reported to be transport active, the role of AdiC oligomerization remains unknown [##REF##34461897##15##, ##REF##19578361##17##]. Similarly, hSERT protomers have been shown to form oligomers via TMS11 and TMS12 [##REF##14660642##26##]. Oligomeric states have also been proposed for rGAT1 and glycine transporters [##REF##11071889##24##, ##REF##18252709##25##]. Finally, the LeuT dimeric structure has also been shown to involve interactions of TMS9 and TMS12, and possibly TMS11 [##REF##19066470##18##, ##REF##16798738##19##]. In conclusion, findings presented in this work, concerning the essential role of TMS11 and TMS12 in FurE folding, self-association and/or concentrative sorting to nascent ERes, might well extend to other NCS1 and similar transporters of the APC superfamily.</p>" ]

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[ "<p>Conflict of Interest: The authors declare no conflict of interest.</p>", "<p>Please cite this article as: Yiannis Pyrris, Georgia F. Papadaki, Emmanuel Mikros and George Diallinas (<bold>2024</bold>). The last two transmembrane helices in the APC-type FurE transporter act as an intramolecular chaperone essential for concentrative ER-exit. <bold>Microbial Cell</bold> 11: 1-15. doi: 10.15698/mic2024.01.811</p>", "<p>FurE is a H⁺ symporter specific for the cellular uptake of uric acid, allantoin, uracil, and toxic nucleobase analogues in the fungus <italic>Aspergillus</italic> nidulans. Being member of the NCS1 protein family, FurE is structurally related to the APC-superfamily of transporters. APC-type transporters are characterised by a 5+5 inverted repeat fold made of ten transmembrane segments (TMS1-10) and function through the rocking-bundle mechanism. Most APC-type transporters possess two extra C-terminal TMS segments (TMS11-12), the function of which remains elusive. Here we present a systematic mutational analysis of TMS11-12 of FurE and show that two specific aromatic residues in TMS12, Trp473 and Tyr484, are essential for ER-exit and trafficking to the plasma membrane (PM). Molecular modeling shows that Trp473 and Tyr484 might be essential through dynamic interactions with residues in TMS2 (Leu91), TMS3 (Phe111), TMS10 (Val404, Asp406) and other aromatic residues in TMS12. Genetic analysis confirms the essential role of Phe111, Asp406 and TMS12 aromatic residues in FurE ER-exit. We further show that co-expression of FurE-Y484F or FurE-W473A with wild-type FurE leads to a dominant negative phenotype, compatible with the concept that FurE molecules oligomerize or partition in specific microdomains to achieve concentrative ER-exit and traffic to the PM. Importantly, truncated FurE versions lacking TMS11-12 are unable to reproduce a negative effect on the trafficking of co-expressed wild-type FurE. Overall, we show that TMS11-12 acts as an intramolecular chaperone for proper FurE folding, which seems to provide a structural code for FurE partitioning in ER-exit sites.</p>" ]

[ "<title>SUPPLEMENTAL MATERIAL</title>", "<p>All supplemental data for this article are available online at <ext-link xlink:href=\"http://www.microbialcell.com/researcharticles/2023a-pyrris-microbial-cell/\" ext-link-type=\"uri\">www.microbialcell.com/researcharticles/2023a-pyrris-microbial-cell/</ext-link>.</p>" ]

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[ "<fig position=\"float\" id=\"fig1\"><label>Figure 1:</label><caption><title>FIGURE 1: TMS11-TMS12 form a distinct structural entity topologically distinct from the LeuT-fold.</title><p><bold>(A)</bold> Structural models of FurE in three conformations have been produced through homology modelling using the crystallographically resolved structures of its bacterial homolog Mhp1 as a template. TMS11 and TMS12 are highlighted in cyan and red, respectively. PyMOL 2.5 was used for structure depiction. For details on model construction see methods. <bold>(B)</bold> Multiple sequence alignments of a segment of representative NCS1 family transporters from fungi, plants and bacteria, including TMS11 and TMS12. Positions with &gt;50% identity are presented in boxes. For uniport IDs and taxonomy see ##SUPPL##0##Supplementary Table S7##.</p></caption></fig>", "<fig position=\"float\" id=\"fig2\"><label>Figure 2</label><caption><title>FIGURE 2: Functional analysis of triple Ala mutations in TMS11 and TMS12. (A) Left panel:</title><p>Growth tests of control strains and strains expressing GFP-tagged mutant FurE versions with triple alanine substitutions that cover the entire length of TMS11 and TMS12. WT is a standard <italic>A. nidulans</italic> wild-type strain. Δ7 is the negative control strain lacking all major nucleobase transporter and not expressing FurE. FurE is the positive control strain, which is a Δ7 strain functionally expressing FurE via the <italic>gpdA</italic> promoter. All mutant strains are isogenic to the positive control strain, also express-ing FurE versions from the <italic>gpdA</italic> promoter. Mutants are named by the number of residues substituted by Ala. Growth tests were performed on minimal media (MM) supplemented with ammonium (NH₄), uric acid (UA), allantoin (ALL) as nitrogen sources or on MM containing NO₃ and the toxic analogue 5-fluorouracil (5FU). <bold>Middle panel</bold>: Epifluorescence microscopy images of the same control and mutant strains. Scale bar is 5 μM. Right panel: PM/cytosol GFP fluorescence intensity ratios. 95% confidence intervals are pre-sented. For details see methods. <bold>(B)</bold> Ra-diolabeled uracil uptake rates, expressed as % of wt FurE rate, of strains expressing triple alanine substituted versions. For details see Materials and Methods.</p></caption></fig>", "<fig position=\"float\" id=\"fig3\"><label>Figure 3</label><caption><title>FIGURE 3: Y484 is irreplaceable for proper folding and ER-exit, mediating critical interactions between the LeuT-fold and TMS12.</title><p><bold>(A)</bold> Structural models of FurE in the inward- and outward-facing conformations highlighting important interactions in the Tyr484 neighbourhood. Notice the tilt of TMS12 between the two conformations that directs Tyr484 closer to Asp406 (TMS10) and Phe111 (TMS3) in the outward conformation. PyMOL 2.5 was used for structure depiction (see also ##SUPPL##0##Supplementary figure S2##). <bold>(B)</bold> Growth tests and epifluorescence microscopy of control strains and strains expressing GFP-tagged FurE mutations in Tyr484 (TMS12), Phe111 (TMS3) and Asp406 (TMS10). Growth tests were performed as described in ##FIG##1##figure 2A##. Scale bar for microscope images is 5 μM. <bold>(C)</bold> Radiolabeled uracil uptake rates, expressed as % of wt FurE rate, of strains expressing FurE mutant versions. For details see Materials and Methods.</p></caption></fig>", "<fig position=\"float\" id=\"fig4\"><label>Figure 4</label><caption><title>FIGURE 4: The aromatic character of W473 is critical for FurE folding.</title><p><bold>(A)</bold> Structural models of FurE in the inward- and outward-facing conformations depicting putative hydrophobic interactions of Trp473 (TMS12) with Val404, (TMS10), Leu91 (TMS2) and Tyr469 (TMS12). PyMOL 2.5 was used for structure presentation. <bold>(B)</bold> Growth tests and epifluorescence microscopy of controls and strains expressing GFP-tagged FurE mutations in Trp473, Leu91 and Val404. Scale bar for microscopy is 5 μM. <bold>(C)</bold> Radiolabeled uracil uptake rates, expressed as % of wt FurE rate, of strains expressing FurE mutant versions. For details see Materials and Methods.</p></caption></fig>", "<fig position=\"float\" id=\"fig5\"><label>Figure 5</label><caption><title>FIGURE 5: ER-retained mutants affect wild-type FurE localization, resulting in reduced transport activity.</title><p><bold>(A)</bold> Growth tests and epifluorescence microscopy of controls and strains co-expressing wild-type FurE with ER-retained FurE mutant versions. GFP-tagging is either on the wt FurE or in the ER-retained mutant. Notice the appearance of fluorescent signal associated with ER membranes (mostly as perinuclear rings) in the case were wt FurE-GFP is co-expressed with untagged Y484F or W473A. Scale bar corresponds to 5μM. <bold>(B)</bold> Radiolabeled uracil uptake rates, expressed as % of wt FurE rate, of strains expressing FurE mutant versions. For details see Materials and Methods.</p></caption></fig>", "<fig position=\"float\" id=\"fig6\"><label>Figure 6</label><caption><title>FIGURE 6: Truncated FurE versions containing the first 10 TMS are ER-retained and do not produce dominant negative effects when co-expressed with wild type FurE.</title><p><bold>(A)</bold> Growth tests and epifluorescence microscopy of controls, strains expressing truncated FurE versions con-sisting of TMS1-10 or TMS11-12, FurE molecules triple alanine substituted in positions 30–32 (Δ30–32) and strains co-expressing combinations of the above constructs with wild type FurE. In each strain one transporter version is GFP-tagged as noted in the strain name. Scale bar corresponds to 5 μM. <bold>(B)</bold> Radiolabeled uracil uptake rates, expressed as % of wt FurE rate, of strains co-expressing ER-retained FurE versions and wild-type FurE. For details see Materials and Methods.</p></caption></fig>" ]

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[ "<supplementary-material id=\"mic-11-001-s01\" position=\"float\" content-type=\"local-data\"></supplementary-material>" ]

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[ "<media xlink:href=\"mic-11-001-s01.pdf\" id=\"d64e846\" position=\"anchor\"><caption><p>Click here for supplemental data file.</p></caption></media>" ]

{ "acronym": [ "5FU", "APC", "BiF", "MD", "PM", "TMS", "wt" ], "definition": [ "– 5-fluorouracil,", "– amino acid-polyamine-oragnocation,", "– bifluorescence,", "– molecular dynamics,", "– plasma membrane,", "– transmembrane segment,", "– wild-type" ] }

[ "<title>Introduction</title>", "<p>Status epilepticus (SE) occurs within five minutes or more of continuous clinical and/or electrographic seizure activity or recurrent seizure activity without consciousness recovery between seizures. It is a common, life-threatening neurologic disorder with high morbidity and mortality rates [##REF##20738380##1##]. Its consequences can result in alterations of the neuronal network or death, and its incidence ranges from 10 to 40 cases per 100,000, with the rate of mortality ranging from 7.6 to 39% [##UREF##0##2##]. Rapid management is a priority to improve patient outcomes while etiologic investigation can be a challenge in people with SE. In Morocco, most people with epilepsy have no access to treatment, with traditional customs leading to very late patient management. Data on SE are lacking, and recognizing the burden of SE morbidity, we conducted a single-center retrospective study. The objective was to analyze the clinical characteristics of SE and its etiology. We also report mortality rates and risk factors.</p>" ]

[ "<title>Materials and methods</title>", "<p>Study design</p>", "<p>We conducted a single-center retrospective study at the A1 intensive care unit at the Hassan II University Hospital of Fez in Morocco. Three-year study data were obtained from January 2019 to December 2021. We included patients with SE according to defined inclusion and exclusion criteria.</p>", "<p>Inclusion and exclusion criteria for the study population</p>", "<p>Adult patients ( age &gt; 17 years) admitted to the intensive care unit with the diagnosis of SE were included in the study and data were obtained from medical records. Pregnant and lactating women, children, and patients with incomplete clinical data were excluded.</p>", "<p>Definition of variables</p>", "<p>All pre- and in-hospital records of SE patients were reviewed and data were collected using standardized forms. SE was defined according to the latest criteria of the International League Against Epilepsy (ILAE). SE involves a convulsive seizure lasting 5 minutes or more, nonconvulsive status with impaired consciousness lasting longer than 10 minutes, while refractory status epilepticus (RSE) is defined as persisting seizures after the failure of a sufficient benzodiazepine dose as a first-line treatment and an antiseizure medication (ASM) as a second-line treatment, irrespective of time [##REF##35503725##3##].</p>", "<p>Baseline variables</p>", "<p>These included gender, age, family history of epilepsy, SE type, SE duration, symptomatology, electroencephalogram (EEG), neuroimaging, cerebrospinal fluid results, etiology of seizure treatment, complications, length of hospital stay, and outcome.</p>", "<p>SE duration</p>", "<p>The end of SE was defined clinically as a cessation of seizure activity on EEG for patients with nonconvulsive status or under pharmacological sedation [##REF##35503725##3##], continuous EEG monitoring was not available, and an EEG examination was performed to confirm the cessation of SE.</p>", "<p>Neuroimaging</p>", "<p>This was performed in all cases on admission and was repeated according to clinical evolution.</p>", "<p>Etiologies of seizures</p>", "<p>The etiology was rated as unknown if no cause of SE could be identified.</p>", "<p>Management of SE</p>", "<p>According to our hospital’s management protocol, anti-seizure medication (ASM) was administered per the national guidelines, with the administration of benzodiazepines in the initial phase (diazepam or midazolam), followed by intravenous phenobarbital as a second line of treatment if SE did not resolve. Patients with refractory SE required sedation.</p>", "<p>Statistical analysis</p>", "<p>Data were analyzed using SPSS Statistics software version 22.0 (IBM Corp., Armonk, NY, USA). Frequencies were calculated using descriptive statistics, mean with standard deviation was used for continuous variables, and univariate comparisons of proportions were calculated using a chi-square test. P values &lt;0 .05 were considered statistically significant. Predictive factors of mortality were studied with a logistic regression model with multivariate analysis.</p>" ]

[ "<title>Results</title>", "<p>Overall, 82 patients with SE were admitted to the ICU from January 2019 to December 2021 and were included in the study.</p>", "<p>Patient characteristics</p>", "<p>Patients were aged 18 to 95 years, with a mean of 39.5 years, including 50 males (61%) and 32 females (39%). Seventy-two percent (72%) of patients (N: 59) presented with de-novo SE, 27.7% of patients (N: 23) had a history of epilepsy, of whom 40% (N: 9) were receiving regular antiepileptic drugs (AED) and 60% (N: 14) had poor therapeutic adherence. Sixteen patients (18%) had a history of brain injury, the majority of semiology was convulsive SE (93%, N: 77); and 7 patients had non-convulsive SE. EEG was only performed in 45 patients (54%). All patients underwent head CT scans, which showed abnormalities in 45 patients. MRI was performed on 48 patients, which showed abnormalities in 37 cases. Lumbar puncture was performed in 65 cases, and cerebrospinal fluid culture tested positive in 5 cases. Metabolic abnormalities notified at admission were hypernatremia, hyponatremia, and kidney failure.</p>", "<p>Causes of SE</p>", "<p>Epilepsy of unknown cause was the most common diagnosis (41.2%, N: 34). The cause of SE in 19 patients was a reduced seizure threshold. The most known etiology was acute/subacute cerebrovascular events (12 patients, 14.4%), primary tumors of the CNS (8 patients, 9.6%), metabolic abnormalities (8 patients, 9.6%), cerebral venous thrombosis (7 patients, 8%), and encephalitis (7 patients, 8%). The range of etiologies of super-refractory status epilepticus (SRSE) is illustrated in Figure ##FIG##0##1##.</p>", "<p>Drugs used to control SE</p>", "<p>All patients received benzodiazepines (diazepam or midazolam) as the first line of treatment, 96.4% of them received phenobarbital as the second line of treatment, and 65 patients had refractory SE requiring anesthesia.</p>", "<p>Patient characteristics are listed in Table ##TAB##0##1##.</p>", "<p>Outcomes</p>", "<p>Fifty-two patients developed at least one complication, the most common was a bloodstream infection, 31 patients (38%) died, mortality was attributed to cardiovascular and infectious complications in most cases, and the median days to mortality was 15.23 days.</p>", "<p>Mortality risk factors</p>", "<p>Acute Physiology and Chronic Health Evaluation (APACHE) II score ≥10 (p=0.0001), ischemic stroke (as an etiology of SE (80% vs 66.2%, P=0.048)), history of epilepsy(93% vs 66%, P =0.005), poor therapeutic adherence (100% vs 72%, P=0.001), cardiovascular complications (90% vs 43%, P= 0.0001), and presence of multiple complications (P=0.0001), pneumonia (96.7% vs 11.7%, P=0.0001), recurrence of SE (70.9% vs 58%, P=0.050) were variables significantly associated with mortality. However, there were no significant differences in mortality in relationship with the type of SE (for convulsive SE, P=0.419; refractory SE (P=0.173); or tracheal intubation, P=0.321 (Table ##TAB##1##2##).</p>" ]

[ "<title>Discussion</title>", "<p>Our study analyzed the clinical characteristics, etiologies, outcomes, and factors associated with mortality in patients with SE in a Moroccan center.</p>", "<p>Demographic and clinical characteristics</p>", "<p>The median age was 39.5 years, the incidence of SE was higher in elderly people in several studies, and 61% of patients were male, which is consistent with previous studies [##REF##27694014##4##, ####UREF##1##5####1##5##]. The rate of patients with a history of epilepsy in SE is up 40% to 50% in several studies [##REF##11422324##6##, ####REF##10980736##7####10980736##7##]. Only 27.7% of our patients had a history of epilepsy, illustrating the high rate of de-novo SE in our study, with the possibility of misdiagnosed epilepsy remaining abnormally high in our region [##UREF##2##8##]. Most SE was convulsive (93%, N: 76) in our study; only 7% of SE was non-convulsive, in consistency with other series [##REF##27694014##4##,##UREF##1##5##,##REF##10980736##7##]. Convulsive SE is usually easy to diagnose and EEG is not required for the initial diagnosis; however, the possibility of progressing to pauci-symptomatic epilepsy justifies daily EEG monitoring until recovery of consciousness or refractory SE [##UREF##2##8##, ####REF##9337142##9####9337142##9##]. In our context, EEG monitoring access is limited, leading to an underestimated incidence of nonconvulsive SE and an increase in the duration of sedation in refractory SE.</p>", "<p>Causes of SE</p>", "<p>In our study, we found a higher rate of unknown etiology (41.2%), considering that MRI was not always accessible in our context, particularly for unstable patients, followed by patients with acute symptomatic etiology; cerebrovascular events were found to be the main known cause (14.4%) in conformance with several studies [##REF##32331701##10##, ####REF##20890201##11####20890201##11##]. It is worth mentioning that, recently, autoimmune/paraneoplastic encephalitis has become one of the most known causes of SE [##REF##31260104##12##, ####REF##31105639##13####31105639##13##], probably undiagnosed and untreated in our study because anti-neuronal autoantibodies testing is not yet available nationally.</p>", "<p>Treatment</p>", "<p>Our study indicated that the use of benzodiazepines accounted for 96.4%, with phenobarbital as the second line to control seizure activity. Intravenous levetiracetam, phenytoin, and valproic acid recommended for second-line treatment are not available in our country. Sixty-five patients (79%) required anesthesia to be maintained for at least one to two days to have burst suppression. Midazolam and propofol were the main anesthetic drugs; a recent international cohort study proved that propofol and midazolam are equivalently efficacious for refractory SE [##UREF##3##14##]. We reserved ketamine in association with other anesthetics for SRSE. Publications on ketamine efficiency in SE are heterogeneous, but ketamine appears to have a promising outlook for refractory SE and SRSE. Larger randomized prospective studies should clarify its place in controlling seizures [##UREF##4##15##].</p>", "<p>Outcomes</p>", "<p>In contrast to several studies of SE that found mortality rates of approximately 10-20%, the outcome of patients with SE in our study is poor with 38% mortality; this could be explained by the lack of prehospital emergency care and specialized centers and limited access to EEG monitoring as well as limited diagnostic facility limitations. Several studies have shown that delay in treatment and a longer duration of SE contributed to poor clinical outcomes [##REF##29356811##16##, ####REF##11781422##17####11781422##17##].</p>", "<p>Mortality risk factors</p>", "<p>A history of epilepsy, ischemic stroke as an etiology, poor therapeutic adherence, the presence of complications, and recurrence of SE were associated with poor prognosis. This is compatible with the findings of other researchers, which identify also the age of the patient and duration of SE as risk factors for mortality [##REF##33530040##18##, ####REF##25915004##19##, ##REF##20697043##20####20697043##20##]. Also, studies have shown that in patients with ischemic stroke or those suffering from an anoxic brain injury, the occurrence of SE is identified as an independent factor of mortality [##REF##17201698##21##, ####REF##17636063##22####17636063##22##]. Immediate admission to the ICU of patients with a high risk of mortality should improve the prognosis of these patients.</p>" ]

[ "<title>Conclusions</title>", "<p>This study elucidated the clinical characteristics, etiologies, management, and outcomes of SE in our hospital. The patients in our study were young, with a high rate of de-novo SE. While cerebrovascular events are the most common cause of known etiology, the major diagnosis in our study was unknown etiology due in part to autoimmune encephalitis, most likely undiagnosed. The majority of SE patients in our study were managed with benzodiazepines and phenobarbital; patients who required anesthesia received midazolam or propofol. The mortality rate in patients with SE remained high in our study. Rapid determination of the causative etiology and initiation of therapy could decrease the mortality rate by improving prehospital emergency care and implementing elective ICU admission for patients at high risk. In our study, a history of epilepsy, ischemic stroke as an etiology, poor therapeutic adherence, presence of complications, and recurrence of SE were associated with a poor prognosis.</p>" ]

[ "<p>Background</p>", "<p>Status epilepticus (SE) is a common neurologic emergency with high rates of mortality and morbidity.</p>", "<p>Objective</p>", "<p>To analyze the clinical characteristics, causes, management, and outcomes of patients with SE in a tertiary care hospital in Morocco.</p>", "<p>Methods</p>", "<p>A retrospective study was conducted from January 2019 to December 2021, including all patients admitted to the medico-surgical general intensive care unit (ICU) with a diagnosis of SE. We recorded demographic characteristics, SE clinical history, management, causes, and discharge outcomes.</p>", "<p>Results</p>", "<p>Overall, 82 patients with SE were included, the median age was 39.5 years (18-95), 61% of the patients were male, the majority of semiology was convulsive SE (93%, N: 77), epilepsy of unknown cause was the most common diagnosis (41.2%, N: 34), and the most known etiology was acute/subacute cerebrovascular events (12 patients, 14.4%). All patients received benzodiazepines, 96.4% of them received phenobarbital as a second line of treatment, 65 patients required anesthesia, 52 patients developed one complication at least - the most common complication being systemic infection, and the mortality rate was noted to be 38% among patients with SE (N: 31). In this study, the factors associated with mortality were ischemic stroke (as an etiology of SE (p=0.048), history of epilepsy (p=0.005), poor therapeutic adherence (p=0.001), cardiovascular complications, presence of multiple complications (p=0.0001), pneumonia (p=0.0001), and the recurrence of SE (p=0.050).</p>", "<p>Conclusions</p>", "<p>We provide a single-center retrospective analysis of admissions in SE and note that mortality among SE patients is high in our settings. Improving prehospital emergency care and implementing elective ICU admission for patients at high risk could improve the mortality rate.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Proportion of status epilepticus etiologies</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Patient characteristics</title><p>SE: status epilepticus</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Items</td><td rowspan=\"1\" colspan=\"1\">Cases</td><td rowspan=\"1\" colspan=\"1\">Ratio</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Sex (Male/Female)</td><td rowspan=\"1\" colspan=\"1\">50/32</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">History of epilepsy</td><td rowspan=\"1\" colspan=\"1\">23</td><td rowspan=\"1\" colspan=\"1\">27.7%</td></tr><tr><td colspan=\"3\" rowspan=\"1\">Semiology</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">   Convulsive SE</td><td rowspan=\"1\" colspan=\"1\">77</td><td rowspan=\"1\" colspan=\"1\">93%</td></tr><tr><td rowspan=\"1\" colspan=\"1\">   Non convulsive SE</td><td rowspan=\"1\" colspan=\"1\">5</td><td rowspan=\"1\" colspan=\"1\">6%</td></tr><tr style=\"background-color:#ccc\"><td colspan=\"3\" rowspan=\"1\">Etiology</td></tr><tr><td rowspan=\"1\" colspan=\"1\">   Unknown</td><td rowspan=\"1\" colspan=\"1\">12</td><td rowspan=\"1\" colspan=\"1\">14.4%</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">   Vascular</td><td rowspan=\"1\" colspan=\"1\">8</td><td rowspan=\"1\" colspan=\"1\">9.6%</td></tr><tr><td rowspan=\"1\" colspan=\"1\">   Tumoral</td><td rowspan=\"1\" colspan=\"1\">8</td><td rowspan=\"1\" colspan=\"1\">9.6%</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">  Metabolic</td><td rowspan=\"1\" colspan=\"1\">6</td><td rowspan=\"1\" colspan=\"1\">8%</td></tr><tr><td rowspan=\"1\" colspan=\"1\">  Infectious</td><td rowspan=\"1\" colspan=\"1\">6</td><td rowspan=\"1\" colspan=\"1\">8%</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">  Cerebral venous thrombosis</td><td rowspan=\"1\" colspan=\"1\">6</td><td rowspan=\"1\" colspan=\"1\">8%</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Refractory SE</td><td rowspan=\"1\" colspan=\"1\">65</td><td rowspan=\"1\" colspan=\"1\">79%</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Acute complication</td><td rowspan=\"1\" colspan=\"1\">52</td><td rowspan=\"1\" colspan=\"1\">63%</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Mortality</td><td rowspan=\"1\" colspan=\"1\">31</td><td rowspan=\"1\" colspan=\"1\">38%</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>Risk factors related to mortality</title><p>SE: status epilepticus</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Death (N =31 ) 38%</td><td rowspan=\"1\" colspan=\"1\">Recovery (N = 51) 62%</td><td rowspan=\"1\" colspan=\"1\">P</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ischemic stroke</td><td rowspan=\"1\" colspan=\"1\">25 (80%)</td><td rowspan=\"1\" colspan=\"1\">49 (66.2%)</td><td rowspan=\"1\" colspan=\"1\">0.048</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">History of epilepsy</td><td rowspan=\"1\" colspan=\"1\">29 (93%)</td><td rowspan=\"1\" colspan=\"1\">34 (66%)</td><td rowspan=\"1\" colspan=\"1\">0.005</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Poor therapeutic adherence</td><td rowspan=\"1\" colspan=\"1\">31 (100%)</td><td rowspan=\"1\" colspan=\"1\">37 (72%)</td><td rowspan=\"1\" colspan=\"1\">0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Cardiovascular complication</td><td rowspan=\"1\" colspan=\"1\">28 (90%)</td><td rowspan=\"1\" colspan=\"1\">22 (43%)</td><td rowspan=\"1\" colspan=\"1\">0.0001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Pneumonia</td><td rowspan=\"1\" colspan=\"1\">30 (69.7%)</td><td rowspan=\"1\" colspan=\"1\">6 (11.7%)</td><td rowspan=\"1\" colspan=\"1\">0.0001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Recurrence of SE</td><td rowspan=\"1\" colspan=\"1\">22 (70.9%)</td><td rowspan=\"1\" colspan=\"1\">30 (58%)</td><td rowspan=\"1\" colspan=\"1\">0.05</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Ibrahim Bechri, Abdelkrarim Shimi, Ali Derkaoui, Mohammed Khatouf</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Ibrahim Bechri, Abdelkrarim Shimi, Ali Derkaoui, Mohammed Khatouf</p><p><bold>Drafting of the manuscript:</bold>  Ibrahim Bechri, Abdelkrarim Shimi, Ali Derkaoui, Mohammed Khatouf</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Ibrahim Bechri, Abdelkrarim Shimi, Ali Derkaoui, Mohammed Khatouf</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study. Hassan II University Hospital Ethics Committee issued approval 16/23</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Animal Ethics</title><fn fn-type=\"other\"><p><bold>Animal subjects:</bold> All authors have confirmed that this study did not involve animal subjects or tissue.</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050591-i01\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction and background</title>", "<p>Psoriasis is a chronic inflammatory skin illness characterized by a fast accumulation of skin cells, culminating in thick, scaly plaques [##REF##30909615##1##]. It can cause substantial physical and psychological anguish and impacts millions of individuals globally [##UREF##0##2##,##REF##16756666##3##]. Psoriasis’s precise etiology remains uncertain; nevertheless, it is hypothesized to be the result of an intricate interplay between environmental and genetic influences [##REF##15199399##4##,##REF##29387591##5##].</p>", "<p>Psoriasis has considerable variation in prevalence across distinct populations, with global estimates spanning from 0.1% to 3% [##REF##34977058##6##]. Although it has the potential to manifest at any stage of life, it is most frequently observed in early adulthood [##REF##28058392##7##]. In addition to nails, psoriasis can impact the scalp, elbows, and knees, among other body areas [##REF##34001566##8##].</p>", "<p>Biological drugs have significantly transformed the landscape of psoriasis treatment through the provision of focused therapy, which effectively mitigates inflammation and regulates the overproduction of skin cells. Having demonstrated exceptional effectiveness in clinical studies, they have received approval for the treatment of psoriasis. Risankizumab, secukinumab, guselkumab, adalimumab, certolizumab, etanercept, ustekinumab, brodalumab, ixekizumab, tildrakizumab, infliximab, methotrexate, briakinumab, golimumab, and adalimumab are some examples of biological medicines frequently used in the management of psoriasis [##REF##33521024##9##,##REF##32022825##10##].</p>", "<p>Biological therapies are advised for the treatment of psoriatic disease in all six domains of the disease [##REF##26749174##11##]. The primary aim in the therapy of psoriasis is to establish a comprehensive, safe, and efficacious treatment regimen that addresses all of its manifestations [##UREF##1##12##]. Nevertheless, the attainment of this objective is complicated by the diversity of the manifestations. Recent developments in our understanding of the disease's pathogenesis have prompted substantial research and approval of various modes of action, including TNFi (INFLIXIMAB, etanercept, golimumab, certolizumab, and adalimumab); IL-17i (secukinumab, ixekizumab, and brodalumab); and IL-12 and/or IL23i (ustekinumab, guselkumab, Risankizumab, and tildrakizumab).</p>", "<p>These pharmaceuticals function via distinct methods of action. As an illustration, ixekizumab, Risankizumab, secukinumab, and guselkumab selectively target interleukin-17A (IL-17A), a protein that is pivotal in the inflammatory mechanism underlying psoriasis [##REF##32022825##10##]. Through the inhibition of IL-17A, these pharmaceutical agents aid in the mitigation of inflammation and amelioration of symptoms [##REF##32022825##10##]. Additional biological drugs, including infliximab, adalimumab, certolizumab, and etanercept, selectively interact with tumor necrosis factor-alpha (TNF-alpha), a molecule that is implicated in the immune response associated with psoriasis [##REF##29518978##13##]. By suppressing TNF-alpha, these drugs aid in illness management and inflammation relief [##REF##29518978##13##].</p>", "<p>Ustekinumab selectively inhibits interleukin-12 (IL-12) and interleukin-23 (IL-23), cytokines that play a role in the psoriasis immune response [##REF##20421912##14##]. Through the inhibition of IL-12 and IL-23, ustekinumab aids in inflammation reduction and immune system regulation [##REF##20421912##14##]. Alternative pharmaceuticals, including methotrexate, briakinumab, golimumab, and ADA (adalimumab), operate by means of distinct pathways and selectively target distinct immune system components in order to elicit therapeutic responses [##UREF##2##15##].</p>", "<p>Notwithstanding the accessibility of these biological drugs, rigorous evaluations that juxtapose their safety and efficacy profiles are necessary. These studies offer significant insights regarding the relative efficacy of various treatments and serve as a reference for clinical decision-making. Nevertheless, the number of systematic reviews and comparisons of these biological medicines for the treatment of psoriasis is insufficient.</p>", "<p>As a result, by a systematic evaluation of the existing literature concerning the efficacy of several biological medicines for the treatment of psoriasis, this study seeks to fill this knowledge gap. The objective of this study is to conduct a thorough investigation of the relative efficacy of these drugs in alleviating psoriasis symptoms, decreasing inflammation, and increasing the quality of life for patients by synthesizing the existing evidence. The outcomes of this research results will augment the existing body of knowledge regarding the management of psoriasis and provide guidance to medical practitioners in the process of prescribing the most suitable biological therapy for their clients.</p>" ]

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[ "<title>Conclusions</title>", "<p>The systematic review assessed the performance of several biological drugs in the treatment of psoriasis offers significant insights into the treatment's success. Adalimumab, certolizumab, secukinumab, ustekinumab, and methotrexate had moderate efficacy, whereas guselkumab, ixekizumab, Risankizumab, and briakinumab appeared as exceptionally successful alternatives. Clinicians can utilize these findings as a guide for determining which treatment is most suitable for specific patients. When making treatment options, it is essential to evaluate patient characteristics, treatment objectives, and potential adverse effects. Additional investigation into the long-term effects of various drugs and comparative analyses of their efficacy is necessary in order to advance our knowledge of psoriasis care.</p>" ]

[ "<p>Psoriasis is a chronic inflammatory skin illness that has the potential to manifest at any stage of life, it is most frequently observed in early adulthood. Biological drugs have significantly transformed the landscape of psoriasis treatment through the provision of focused therapy, which effectively mitigates inflammation and regulates the overproduction of skin cells. Notwithstanding the accessibility of these biological drugs, rigorous evaluations that juxtapose their safety and efficacy profiles are necessary. The objective of this study is to conduct a thorough investigation of the relative efficacy of these drugs in alleviating psoriasis symptoms and increasing the quality of life for patients by synthesizing the existing evidence. A comprehensive review was conducted to evaluate and compare the safety and effectiveness of different biochemical medicines utilized in the management of psoriasis. In accordance with the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) recommendations, the review process was conducted among the available studies. A search was conducted across electronic databases, such as Web of Science, PubMed, and Embase, utilizing a combination of keywords and Mesh phrases pertaining to psoriasis, biological medications, and particular names of pharmaceuticals.</p>", "<p>In total, 475 studies were ascertained by the preliminary search of the database. After eliminating duplicate research, 358 distinct studies remained. After meticulous screening of titles and abstracts against the predefined inclusion criteria, 281 papers were deemed ineligible and thus excluded. For final inclusion, the whole texts of the remaining 77 studies were evaluated. Forty additional papers were removed during the full-text evaluation for a variety of reasons, including improper research design, or insufficient outcome data. Finally, 37 studies were included in this systematic review since they satisfied all inclusion criteria. The results of the current systematic review showed that all biological medications showed high efficacy in the treatment of skin psoriasis compared with placebo based on the clinical assessment outcomes using different tools such as PASI.</p>" ]

[ "<title>Review</title>", "<p>Methodology</p>", "<p>Study Design</p>", "<p>A comprehensive review was undertaken to evaluate and compare the effectiveness of diverse biological medicines utilised in the management of psoriasis. In accordance with the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) recommendations, the review process was conducted with integrity and transparency.</p>", "<p>Search Strategy</p>", "<p>A thorough examination of the literature was undertaken in order to locate pertinent studies. A search was conducted throughout electronic databases, such as Web of Science, PubMed, and Embase. The search strategy involved a meticulous combination of keywords and MeSH phrases related to psoriasis, biological medications, and specific pharmaceuticals. The keywords used in this research including “Psoriasis”, “biological treatment”, “TNF-alpha inhibitor”, “IL-12/IL-23 Inhibitors”, ” IL-17 Inhibitors”, “IL-23 Inhibitors”, “IL-23/IL-17 Inhibitors”, “Risankizumab”, “secukinumab”, “guselkumab”, “adalimumab”, “certolizumab”, “etanercept”, “ustekinumab”, “brodalumab”, “ixekizumab”, “tildrakizumab”, “Infliximab”, “methotrexate”, “briakinumab”, “golimumab”, “bimekizumab” and “ADA”. The search methodology was modified in accordance with the specific criteria of every database.</p>", "<p>Study Selection Criteria</p>", "<p>The following inclusion criteria were applied to identify eligible studies.</p>", "<p>Participants: Studies involving patients diagnosed with psoriasis, including both plaque psoriasis.</p>", "<p>Intervention: Randomized controlled trials (RCTs) evaluating the use of biological medications (Risankizumab, secukinumab, guselkumab, adalimumab, certolizumab, etanercept, ustekinumab, brodalumab, ixekizumab, tildrakizumab, infliximab, methotrexate, briakinumab, golimumab, bimekizumab and ADA) for the treatment of psoriasis.</p>", "<p>Comparator: Studies comparing the efficacy of different biological medications or comparing biological medications with placebo or other standard treatments.</p>", "<p>Outcome measures: Studies reporting outcomes related to disease severity, such as Psoriasis Area and Severity Index (PASI) scores.</p>", "<p>Study Design</p>", "<p>Only RCTs were included in this review.</p>", "<p>Study Selection Process</p>", "<p>Five independent reviewers independently screened the titles and abstracts of the identified studies to assess their eligibility based on the inclusion criteria. Full-text articles of potentially eligible studies were obtained and assessed for final inclusion. Any discrepancies or disagreements between reviewers were resolved through discussion or consultation with a sixth or seventh reviewer if necessary.</p>", "<p>Data Extraction</p>", "<p>Data extraction was performed independently by five reviewers using a standardized data extraction form. The following information was extracted from each included study: study characteristics (authors, year of publication, study design), participant characteristics (sample size, demographics), intervention details (type of biological medication, dosage, duration of treatment), comparator details, outcome measures, and results.</p>", "<p>Results</p>", "<p>In total, 475 studies were ascertained by the preliminary search of the database. After eliminating duplicate research, 358 distinct studies remained. After meticulous screening of titles and abstracts against the predefined inclusion criteria, 281 papers were deemed ineligible and thus excluded. For final inclusion, the whole texts of the remaining 77 studies were evaluated. Forty additional papers were removed during the full-text evaluation for a variety of reasons, including improper research design, irrelevant intervention, or insufficient outcome data. Finally, 37 studies were included in this systematic review since they satisfied all inclusion criteria. A detailed flowchart with the results of the literature review is shown in Figure ##FIG##0##1##.</p>", "<p>For the skin domain, results between 10 and 16 weeks were considered which is reported in all studies. The systematic review included a total of 37 RCTs focusing on the outcomes of the GRAPPA domain. These trials investigated various drugs and dosages and assessed multiple outcomes related to psoriasis. The review encompassed studies conducted between 2005 and 2021, with sample sizes ranging from 78 to 1,881 participants. The drugs evaluated in the trials included Risankizumab, secukinumab, guselkumab, adalimumab, certolizumab, etanercept, ustekinumab, brodalumab, ixekizumab, tildrakizumab, infliximab, methotrexate, briakinumab, golimumab, and ADA (adalimumab). The primary outcomes assessed in the included studies were categorized into several domains. These domains included measures of disease severity such as PASI (Psoriasis Area and Severity Index) scores, ACR (American College of Rheumatology) scores for arthritis, and assessments for dactylitis, enthesitis, and nail involvement (Table ##TAB##0##1##).</p>", "<p>The study included a total of 30,023 participants, with approximately two-thirds of them being males (19,929 individuals, accounting for 66.37% of the total sample). The mean age of the participants was 45.76 years, with a standard deviation of 2.10. The age range varied from 40.1 to 53.3 years. The average duration of plaque psoriasis among the participants was 16.47 years, with a standard deviation of 4.04. The minimum duration reported was 2.8 years, while the maximum duration reached 21 years. The body surface area affected by psoriasis had a mean value of 26.87%, with a standard deviation of 5.06. The range for body surface area varied from 12% to 41.6%.</p>", "<p>Among the participants, 15,507 individuals had their sPSA (severity of psoriasis) categorized. Of these, 8,773 patients (56.5%) were classified as having severe psoriasis (category 3), 6,671 patients (43.0%) were categorized as having very severe psoriasis (category 4 or above), and 63 patients (0.5%) had milder cases of psoriasis (below category 3).</p>", "<p>The study also investigated the use of different medications for psoriasis treatment. The sample sizes ranged from 277 patients for briakinumab to 3,709 patients for secukinumab. The mean ages were generally similar across medications, ranging from 41.8 years for methotrexate to 48.65 years for Risankizumab. The percentage of male patients was also comparable, varying from 57.3% for infliximab to 75.7% for guselkumab. The number of studies analyzed per medication ranged from one trial for tildrakizumab, golimumab, and briakinumab to eight studies for etanercept and secukinumab. Placebo arms accounted for the largest pooled sample size of 5,542 patients across 27 studies (Table ##TAB##1##2##).</p>", "<p>The results of the study revealed significant differences between the medications and placebo, as well as variations among the different medications themselves. Among the various medications, guselkumab demonstrated the highest efficacy, with PASI75, PASI90, and PASI100 improvement rates of 89.63%, 72.7%, and 48.47% respectively. Following closely behind was ixekizumab, exhibiting impressive improvement rates of 81.33%, 71.53%, and 37.83% respectively for PASI75, PASI90, and PASI100. Risankizumab and briakinumab also showed notable efficacy, with PASI scores of 84.5%, 66.83%, and 34.8% for Risankizumab, and 83.14%, 65.94%, and 34.62% for briakinumab. Adalimumab, certolizumab, secukinumab, ustekinumab, and methotrexate also exhibited moderate effectiveness in improving psoriasis symptoms, although with varying degrees. These medications demonstrated PASI improvement rates ranging from 50% to 73.8% for PASI75, 37.65% to 50.73% for PASI90, and 13.15% to 24.28% for PASI100. On the other hand, etanercept, golimumab (50 mg dosage), and TIL 100 mg showed relatively lower efficacy compared to other medications. Etanercept resulted in PASI75, PASI90, and PASI100 improvement rates of 43.24%, 17%, and 4.48%, respectively, while golimumab and TIL 100 mg exhibited results of 40.3%, 20.8%, and 62.5%, 36.9%, 13.15%, respectively. Comparing the medications to the placebo group, all the biological medications showed significantly higher improvement rates across the PASI scores. The placebo group had minimal improvements, with PASI75, PASI90, and PASI100 rates of 5.76%, 1.8%, and 0.45% respectively (Table ##TAB##2##3##).</p>", "<p>Discussion</p>", "<p>From among the several drugs that were evaluated, guselkumab consistently exhibited the most notable rates of improvement across all three PASI categories: 89.63%, 72.7%, and 48.47% for PASI75, PASI90, and PASI100, respectively. Consistent with prior research, the effectiveness of Guselkumab in the treatment of psoriasis certifies its status as a very successful therapeutic alternative [##REF##29470778##53##, ####UREF##3##54##, ##REF##36549381##55####36549381##55##]. Ixekizumab had noteworthy effectiveness as well, as seen by improvement rates of 81.33%, 71.53%, and 37.83%, respectively, on the PASI75, PASI90, and PASI100, respectively. The findings of this study provide further support for the notion that ixekizumab is an effective treatment for psoriasis [##REF##29387609##56##].</p>", "<p>Both briakinumab and Risankizumab exhibited significant efficacy, as evidenced by the considerable rates of improvement observed in the PASI scores. Briakinumab demonstrated PASI scores of 34.62%, 63.14%, and 66.94%, whereas Risankizumab demonstrated PASI75, PASI90, and PASI100 scores of 84.5%, 66.83%, and 34.8%, respectively. The results underscore the efficacy of these pharmaceuticals in mitigating the symptoms associated with psoriasis.</p>", "<p>Our results are similar to many previous studies that showed that the new biologic medicines, including Risankizumab [##REF##30097359##26##], guselkumab [##REF##31402114##20##,##REF##28057360##31##,##REF##31809827##57##,##REF##29799960##58##], ixekizumab [##REF##29969700##59##, ####REF##28917383##60##, ##REF##28635026##61##, ##REF##28551073##62####28551073##62##], and brodalumab [##REF##26422722##36##,##REF##31175909##63##], have proven high efficacy in patients with moderate-to-severe psoriasis. In the respective clinical trials, approximately 70%-80% of patients attained a reduction in the Psoriasis Area Severity Index (PASI) score of 90% or above within 16 weeks of therapy initiation (PASI 90) [##REF##32121574##64##]. At 52 weeks, the proportion decreased to between 80% and 90%. PASI 100 was between 50% and 60% at 52 weeks. The biologic medicines exhibited significant efficacy [##REF##32121574##64##].</p>", "<p>Adalimumab, certolizumab, secukinumab, ustekinumab, and methotrexate were among the additional drugs that exhibited a modest degree of effectiveness in ameliorating symptoms associated with psoriasis. The observed variations in improvement rates among various drugs underscore the significance of taking into account the unique qualities and preferences of each patient when determining the most suitable course of treatment. It is noteworthy that although the improvement rates of these drugs may be comparatively lower than those of ixekizumab and guselkumab, they nonetheless provide substantial advantages in the management of psoriasis.</p>", "<p>Conversely, the effectiveness of etanercept, golimumab (at a dosage of 50 mg), and TIL 100 mg was comparatively diminished in comparison to the aforementioned drugs. Etanercept induced the following percentage improvements in PASI75, PASI90, and PASI100: 43.24%, 17%, and 4.48%, respectively. The administration of 50 mg of golimumab resulted in 40.3% and 20.8% improvement rates for PASI75 and PASI90, respectively. Similarly, 100 mg of TIL produced improvement rates of 62.5%, 36.9%, and 13.15% for PASI75, PASI90, and PASI100, respectively. Patients whose responses to these drugs are inadequate may benefit more from alternate treatment modalities, according to these results.</p>", "<p>Brodalumab demonstrated a significantly higher level of efficacy compared to secukinumab, ustekinumab, and etanercept, as evidenced by four 52-week RCTs. Similarly, secukinumab demonstrated more efficacy than ustekinumab, and both agents beat etanercept. The results obtained from thirteen supplementary trials and four additional therapeutic interventions (ixekizumab, apremilast, infliximab, and brodalumab) demonstrated that brodalumab exhibited the highest efficacy, followed by ustekinumab, infliximab, and ixekizumab. It was expected that etanercept would have the least lasting effect. At week 52, brodalumab was associated with a higher likelihood of prolonged PASI response, including complete clearance, in comparison to comparable medications. Furthermore, Sawyer et al. [##UREF##4##65##] did a network meta-analysis comprising 34,816 patients and 77 studies. The effectiveness of brodalumab, ixekizumab, secukinumab, guselkumab, and Risankizumab in the treatment of plaque psoriasis was shown to be superior to that of ustekinumab, tildrakizumab, all TNF-α inhibitors, non-biologic systemic medicines, as demonstrated by the researchers. Furthermore, it was observed that brodalumab, ixekizumab, and Risankizumab exhibited greater efficacy than secukinumab, however not by a substantial margin compared to guselkumab. In terms of PASI 90 and PASI 100 response, brodalumab, ixekizumab, guselkumab, and Risankizumab shown the most substantial improvements. According to a meta-analysis of 140 studies conducted by Shidian et al. [##UREF##5##66##], the percentage of patients attained by ixekizumab, secukinumab, bimekizumab, brodalumab, Risankizumab, and guselkumab with PASI 90 demonstrated that these agents were more effective than ustekinumab, adalimumab, certolizumab, and etanercept. Furthermore, adalimumab and ustekinumab had a higher degree of efficacy compared to certolizumab and etanercept. A comparison between the biological drugs and the placebo group provides more evidence of the biological therapies' better efficacy. The improvement rates of all biological drugs assessed on the PASI were found to be significantly greater in comparison to the placebo group. This underscores the significance of regarding these drugs as the benchmark for the management of psoriasis.</p>", "<p>An optimal treatment regimen for a patient with psoriasis should consist of a solitary medication that exhibits efficacy across all indications. The study's exhaustive literature review offers significant insights into the effectiveness of several biological drugs in the treatment of psoriasis. Consistent with other investigations, the results validate the concept that briakinumab, ixekizumab, guselkumab, and ixekizumab are exceedingly efficacious therapeutic alternatives. Furthermore, methotrexate, adalimumab, certolizumab, secukinumab, and ustekinumab exhibit a moderate degree of efficacy in the management of symptoms associated with psoriasis.</p>", "<p>By giving a complete review of the efficacy of various drugs, the results of this study contribute to the current body of knowledge on psoriasis treatment. However, a few limitations should be taken into account. The study initially utilized data obtained from randomised controlled trials, which might not comprehensively represent treatment outcomes in the real world. A more positive response to treatment may be observed in the controlled trial setting as opposed to ordinary clinical practice.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>The PRISMA figure showing the steps to choose the studies for systematic review</title><p>PRISMA: Preferred Reporting Items for Systematic Reviews and Meta-Analyses</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title> RCT included in the systematic review focusing on the outcomes of GRAPPA domain</title><p>RCT: Randomized Controlled Trial, GRAPPA: Group for Research and Assessment of Psoriasis and Psoriatic Arthritis, PASI: Psoriasis Area and Severity Index Score, ACR: American College of Rheumatology score for arthritis.</p><p>The data have been represented as a year of publication, number of participants (N), name of the drug used, dosage of the used drug (mg) and outcomes based on Psoriasis Area and Severity Index Score (PASI).</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Author</td><td rowspan=\"1\" colspan=\"1\">Year</td><td rowspan=\"1\" colspan=\"1\">Number</td><td rowspan=\"1\" colspan=\"1\">Drug</td><td rowspan=\"1\" colspan=\"1\">Dosage</td><td rowspan=\"1\" colspan=\"1\">Outcomes</td></tr><tr><td rowspan=\"2\" colspan=\"1\">1</td><td rowspan=\"2\" colspan=\"1\">Warren et al. [##REF##32594522##16##]</td><td rowspan=\"2\" colspan=\"1\">2021</td><td rowspan=\"2\" colspan=\"1\">327</td><td rowspan=\"1\" colspan=\"1\">Risankizumab</td><td rowspan=\"1\" colspan=\"1\">150 mg</td><td rowspan=\"2\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">300 mg</td></tr><tr><td rowspan=\"2\" colspan=\"1\">2</td><td rowspan=\"2\" colspan=\"1\">Ferris et al. [##REF##30887876##17##]</td><td rowspan=\"2\" colspan=\"1\">2020</td><td rowspan=\"2\" colspan=\"1\">78</td><td rowspan=\"1\" colspan=\"1\">Guselkumab</td><td rowspan=\"2\" colspan=\"1\">100 mg</td><td rowspan=\"2\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr><td rowspan=\"2\" colspan=\"1\">3</td><td rowspan=\"2\" colspan=\"1\">McInnes et al. [##REF##32386593##18##]</td><td rowspan=\"2\" colspan=\"1\">2020</td><td rowspan=\"2\" colspan=\"1\">853</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">300 mg</td><td rowspan=\"1\" colspan=\"1\">ACR20, ACR50, ACR70, PASI75</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">40 mg</td><td rowspan=\"1\" colspan=\"1\">PASI90, PASI100, dactylitis assessment, enthesitis assessment</td></tr><tr><td rowspan=\"2\" colspan=\"1\">4</td><td rowspan=\"2\" colspan=\"1\">Mease et al. [##REF##32178766##19##]</td><td rowspan=\"2\" colspan=\"1\">2020</td><td rowspan=\"2\" colspan=\"1\">741</td><td rowspan=\"1\" colspan=\"1\">Guselkumab</td><td rowspan=\"2\" colspan=\"1\">100 mg</td><td rowspan=\"1\" colspan=\"1\">ACR20, ACR50, ACR70, PASI75</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">PASI90, PASI100, dactylitis assessment, enthesitis assessment</td></tr><tr><td rowspan=\"2\" colspan=\"1\">5</td><td rowspan=\"2\" colspan=\"1\">Reich et al. [##REF##31402114##20##]</td><td rowspan=\"2\" colspan=\"1\">2019</td><td rowspan=\"2\" colspan=\"1\">1048</td><td rowspan=\"1\" colspan=\"1\">Guselkumab</td><td rowspan=\"1\" colspan=\"1\">100 mg</td><td rowspan=\"2\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">300 mg</td></tr><tr><td rowspan=\"3\" colspan=\"1\">6</td><td rowspan=\"3\" colspan=\"1\">Ohtsuki et al. [##REF##31237727##21##]</td><td rowspan=\"3\" colspan=\"1\">2019</td><td rowspan=\"3\" colspan=\"1\">171</td><td rowspan=\"1\" colspan=\"1\">Risankizumab</td><td rowspan=\"1\" colspan=\"1\">75 mg</td><td rowspan=\"3\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Risankizumab</td><td rowspan=\"2\" colspan=\"1\">150 mg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">7</td><td rowspan=\"2\" colspan=\"1\">Reich et al. [##REF##31280967##22##]</td><td rowspan=\"2\" colspan=\"1\">2019</td><td rowspan=\"2\" colspan=\"1\">605</td><td rowspan=\"1\" colspan=\"1\">Risankizumab</td><td rowspan=\"1\" colspan=\"1\">150 mg</td><td rowspan=\"2\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">40 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">8</td><td rowspan=\"3\" colspan=\"1\">Mease et al. [##REF##29550766##23##]</td><td rowspan=\"3\" colspan=\"1\">2018</td><td rowspan=\"3\" colspan=\"1\">774</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">150 mg</td><td rowspan=\"1\" colspan=\"1\">ACR20, ACR50, ACR70</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">300 mg</td><td rowspan=\"1\" colspan=\"1\">PASI75, PASI100, dactylitis assessment</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"6\" colspan=\"1\">9</td><td rowspan=\"3\" colspan=\"1\">Gottlieb et al. [##REF##29660421##24##] (CIMPASI-1)</td><td rowspan=\"3\" colspan=\"1\">2018</td><td rowspan=\"3\" colspan=\"1\">234</td><td rowspan=\"1\" colspan=\"1\">Certolizumab</td><td rowspan=\"1\" colspan=\"1\">200 mg</td><td rowspan=\"3\" colspan=\"1\">PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Certolizumab</td><td rowspan=\"2\" colspan=\"1\">400 mg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Gottlieb et al. [##REF##29660421##24##] (CIMPASI-2)</td><td rowspan=\"3\" colspan=\"1\">2018</td><td rowspan=\"3\" colspan=\"1\">227</td><td rowspan=\"1\" colspan=\"1\">Certolizumab</td><td rowspan=\"1\" colspan=\"1\">200 mg</td><td rowspan=\"3\" colspan=\"1\">PASI90, PASI100</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Certolizumab</td><td rowspan=\"2\" colspan=\"1\">400 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr><td rowspan=\"4\" colspan=\"1\">10</td><td rowspan=\"4\" colspan=\"1\">Lebwohl et al. [##REF##29660425##25##]</td><td rowspan=\"4\" colspan=\"1\">2018</td><td rowspan=\"4\" colspan=\"1\">559</td><td rowspan=\"1\" colspan=\"1\">Certolizumab</td><td rowspan=\"1\" colspan=\"1\">200 mg</td><td rowspan=\"4\" colspan=\"1\">PASI75, PASI90</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Certolizumab</td><td rowspan=\"1\" colspan=\"1\">400 mg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"2\" colspan=\"1\">50 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr><td rowspan=\"6\" colspan=\"1\">11</td><td rowspan=\"3\" colspan=\"1\">Gordon et al. [##REF##30097359##26##] (UltIMMa-1)</td><td rowspan=\"3\" colspan=\"1\">2018</td><td rowspan=\"3\" colspan=\"1\">506</td><td rowspan=\"1\" colspan=\"1\">Risankizumab</td><td rowspan=\"1\" colspan=\"1\">150 mg</td><td rowspan=\"3\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"2\" colspan=\"1\">45/90 mg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Gordon et al. [##REF##30097359##26##] (UltIMMa-2)</td><td rowspan=\"3\" colspan=\"1\">2018</td><td rowspan=\"3\" colspan=\"1\">491</td><td rowspan=\"1\" colspan=\"1\">Risankizumab</td><td rowspan=\"1\" colspan=\"1\">150 mg</td><td rowspan=\"3\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"2\" colspan=\"1\">45/90 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr><td rowspan=\"3\" colspan=\"1\">12</td><td rowspan=\"3\" colspan=\"1\">Reich et al. [##REF##30367462##27##]</td><td rowspan=\"3\" colspan=\"1\">2018</td><td rowspan=\"3\" colspan=\"1\">198</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">150 mg</td><td rowspan=\"1\" colspan=\"1\">PASI75, PASI90</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">300 mg</td><td rowspan=\"1\" colspan=\"1\">PASI100, nail assessment</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">13</td><td rowspan=\"2\" colspan=\"1\">Bagel et al. [##REF##30334147##28##]</td><td rowspan=\"2\" colspan=\"1\">2018</td><td rowspan=\"2\" colspan=\"1\">1102</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">300 mg</td><td rowspan=\"2\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">45/90 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"7\" colspan=\"1\">14</td><td rowspan=\"3\" colspan=\"1\">Reich et al. [##REF##28596043##29##] (reSURFACE 1)</td><td rowspan=\"3\" colspan=\"1\">2017</td><td rowspan=\"3\" colspan=\"1\">772</td><td rowspan=\"1\" colspan=\"1\">Tildrakizumab</td><td rowspan=\"1\" colspan=\"1\">100 mg</td><td rowspan=\"3\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Tildrakizumab</td><td rowspan=\"2\" colspan=\"1\">200 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr><td rowspan=\"4\" colspan=\"1\">Reich et al. [##REF##28596043##29##] (reSURFACE 2)</td><td rowspan=\"4\" colspan=\"1\">2017</td><td rowspan=\"4\" colspan=\"1\">1,090</td><td rowspan=\"1\" colspan=\"1\">Tildrakizumab</td><td rowspan=\"1\" colspan=\"1\">100 mg</td><td rowspan=\"4\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Tildrakizumab</td><td rowspan=\"1\" colspan=\"1\">200 mg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"2\" colspan=\"1\">50 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr><td rowspan=\"2\" colspan=\"1\">15</td><td rowspan=\"2\" colspan=\"1\">Reich et al. [##REF##28542874##30##]</td><td rowspan=\"2\" colspan=\"1\">2017</td><td rowspan=\"2\" colspan=\"1\">302</td><td rowspan=\"1\" colspan=\"1\">Ixekizumab</td><td rowspan=\"1\" colspan=\"1\">80 mg</td><td rowspan=\"2\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">45/90 mg</td></tr><tr><td rowspan=\"3\" colspan=\"1\">16</td><td rowspan=\"3\" colspan=\"1\">Blauvelt et al. [##REF##28057360##31##]</td><td rowspan=\"3\" colspan=\"1\">2017</td><td rowspan=\"3\" colspan=\"1\">837</td><td rowspan=\"1\" colspan=\"1\">Guselkumab</td><td rowspan=\"1\" colspan=\"1\">100 mg</td><td rowspan=\"3\" colspan=\"1\">PASI75, PASI90, PASI100, nail assessment</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"2\" colspan=\"1\">40 mg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">17</td><td rowspan=\"4\" colspan=\"1\">Mease et al. [##REF##27553214##32##]</td><td rowspan=\"4\" colspan=\"1\">2017</td><td rowspan=\"4\" colspan=\"1\">417</td><td rowspan=\"1\" colspan=\"1\">Ixekizumab</td><td rowspan=\"1\" colspan=\"1\">80 mg 2 w</td><td rowspan=\"1\" colspan=\"1\">ACR20, ACR50, PASI75</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ixekizumab</td><td rowspan=\"1\" colspan=\"1\">80 mg 4 w</td><td rowspan=\"1\" colspan=\"1\">PASI90, PASI100, dactylitis assessment</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"2\" colspan=\"1\">40 mg</td><td rowspan=\"2\" colspan=\"1\">enthesitis assessment, nail assessment</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">18</td><td rowspan=\"4\" colspan=\"1\">Gordon et al. [##REF##27299809##33##]</td><td rowspan=\"4\" colspan=\"1\">2016</td><td rowspan=\"4\" colspan=\"1\">1,346</td><td rowspan=\"1\" colspan=\"1\">IXE</td><td rowspan=\"1\" colspan=\"1\">80 mg 2 w</td><td rowspan=\"4\" colspan=\"1\">PASI100, PASI90, PASI75</td></tr><tr><td rowspan=\"1\" colspan=\"1\">IXE</td><td rowspan=\"1\" colspan=\"1\">80 mg 4 w</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">ETN</td><td rowspan=\"1\" colspan=\"1\">50 mg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">PLB</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">19</td><td rowspan=\"3\" colspan=\"1\">Papp et al. [##REF##26914406##34##]</td><td rowspan=\"3\" colspan=\"1\">2016</td><td rowspan=\"3\" colspan=\"1\">661</td><td rowspan=\"1\" colspan=\"1\">Brodalumab</td><td rowspan=\"1\" colspan=\"1\">140 mg</td><td rowspan=\"3\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Brodalumab</td><td rowspan=\"2\" colspan=\"1\">210 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr><td rowspan=\"8\" colspan=\"1\">20</td><td rowspan=\"4\" colspan=\"1\">Griffiths et al. [##REF##26072109##35##] (UNCOVER-2 desgin)</td><td rowspan=\"4\" colspan=\"1\">2015</td><td rowspan=\"4\" colspan=\"1\">1,224</td><td rowspan=\"1\" colspan=\"1\">Ixekizumab</td><td rowspan=\"1\" colspan=\"1\">80 mg 2 w</td><td rowspan=\"4\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ixekizumab</td><td rowspan=\"1\" colspan=\"1\">80 mg 4 w</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"2\" colspan=\"1\">50 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr><td rowspan=\"4\" colspan=\"1\">Griffiths et al. [##REF##26072109##35##] (UNCOVER-3 design)</td><td rowspan=\"4\" colspan=\"1\">2015</td><td rowspan=\"4\" colspan=\"1\">1,346</td><td rowspan=\"1\" colspan=\"1\">Ixekizumab</td><td rowspan=\"1\" colspan=\"1\">80 mg 2 w</td><td rowspan=\"4\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ixekizumab</td><td rowspan=\"1\" colspan=\"1\">80 mg 4 w</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"2\" colspan=\"1\">50 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr><td rowspan=\"8\" colspan=\"1\">21</td><td rowspan=\"4\" colspan=\"1\">Lebwohl et al. [##REF##26422722##36##] (AMAGINE-2 )</td><td rowspan=\"4\" colspan=\"1\">2015</td><td rowspan=\"4\" colspan=\"1\">1,831</td><td rowspan=\"1\" colspan=\"1\">Brodalumab</td><td rowspan=\"1\" colspan=\"1\">140 mg</td><td rowspan=\"4\" colspan=\"1\">PASI75, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Brodalumab</td><td rowspan=\"1\" colspan=\"1\">210 mg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"2\" colspan=\"1\">45/95 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr><td rowspan=\"4\" colspan=\"1\">Lebwohl et al. [##REF##26422722##36##] (AMAGINE-3)</td><td rowspan=\"4\" colspan=\"1\">2015</td><td rowspan=\"4\" colspan=\"1\">1,881</td><td rowspan=\"1\" colspan=\"1\">Brodalumab</td><td rowspan=\"1\" colspan=\"1\">140 mg</td><td rowspan=\"4\" colspan=\"1\">PASI75, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Brodalumab</td><td rowspan=\"1\" colspan=\"1\">210 mg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"2\" colspan=\"1\">45/95 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr><td rowspan=\"2\" colspan=\"1\">22</td><td rowspan=\"2\" colspan=\"1\">Thaçi et al [##REF##26092291##37##]</td><td rowspan=\"2\" colspan=\"1\">2015</td><td rowspan=\"2\" colspan=\"1\">676</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">300 mg</td><td rowspan=\"2\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">45/90 mg</td></tr><tr><td rowspan=\"7\" colspan=\"1\">23</td><td rowspan=\"3\" colspan=\"1\">Langley et al. [##REF##25007392##38##] (ERASURE study)</td><td rowspan=\"3\" colspan=\"1\">2014</td><td rowspan=\"3\" colspan=\"1\">738</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">150 mg</td><td rowspan=\"3\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">300 mg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">Langley et al. [##REF##25007392##38##] (FIXTURE study)</td><td rowspan=\"4\" colspan=\"1\">2014</td><td rowspan=\"4\" colspan=\"1\">1,306</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">150 mg</td><td rowspan=\"4\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">300 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">50 mg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">24</td><td rowspan=\"3\" colspan=\"1\">Mease et al. [##REF##23942868##39##]</td><td rowspan=\"3\" colspan=\"1\">2014</td><td rowspan=\"3\" colspan=\"1\">409</td><td rowspan=\"1\" colspan=\"1\">Certolizumab</td><td rowspan=\"1\" colspan=\"1\">200 mg</td><td rowspan=\"1\" colspan=\"1\">ACR20, ACR50, ACR70</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Certolizumab</td><td rowspan=\"1\" colspan=\"1\">400 mg</td><td rowspan=\"1\" colspan=\"1\">PASI50, PASI75, PASI90</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Dactylitis assessment, enthesitis assessment, nail assessment</td></tr><tr><td rowspan=\"2\" colspan=\"1\">25</td><td rowspan=\"2\" colspan=\"1\">Baranauskaite et al. [##REF##21994233##40##]</td><td rowspan=\"2\" colspan=\"1\">2012</td><td rowspan=\"2\" colspan=\"1\">115</td><td rowspan=\"1\" colspan=\"1\">Infliximab + Methotrexate</td><td rowspan=\"1\" colspan=\"1\">5 mg/kg</td><td rowspan=\"1\" colspan=\"1\">ACR20, ACR50, ACR70</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Methotrexate</td><td rowspan=\"1\" colspan=\"1\">15 mg</td><td rowspan=\"1\" colspan=\"1\">PASI75, dactylitis assessment, enthesitis assessment</td></tr><tr><td rowspan=\"2\" colspan=\"1\">26</td><td rowspan=\"2\" colspan=\"1\">Gottlieb et al. [##REF##22533447##41##]</td><td rowspan=\"2\" colspan=\"1\">2012</td><td rowspan=\"2\" colspan=\"1\">478</td><td rowspan=\"1\" colspan=\"1\">Methotrexate + Etanercept</td><td rowspan=\"1\" colspan=\"1\">15 mg + 50 mg</td><td rowspan=\"2\" colspan=\"1\">PASI50, PASI75, PASI90</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Etanercept + placebo</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"2\" colspan=\"1\">27</td><td rowspan=\"2\" colspan=\"1\">Barker et al. [##REF##21910713##42##]</td><td rowspan=\"2\" colspan=\"1\">2011</td><td rowspan=\"2\" colspan=\"1\">868</td><td rowspan=\"1\" colspan=\"1\">Infliximab</td><td rowspan=\"1\" colspan=\"1\">5 mg/kg</td><td rowspan=\"1\" colspan=\"1\">PASI50, PASI75, PASI90</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Methotrexate</td><td rowspan=\"1\" colspan=\"1\">15 mg</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">28</td><td rowspan=\"3\" colspan=\"1\">Gottlieb et al. [##REF##21574983##43##]</td><td rowspan=\"3\" colspan=\"1\">2011</td><td rowspan=\"3\" colspan=\"1\">347</td><td rowspan=\"1\" colspan=\"1\">Briakinumab</td><td rowspan=\"1\" colspan=\"1\">200 mg</td><td rowspan=\"3\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"2\" colspan=\"1\">50 mg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">29</td><td rowspan=\"3\" colspan=\"1\">Strober et al. [##REF##21574984##44##]</td><td rowspan=\"3\" colspan=\"1\">2011</td><td rowspan=\"3\" colspan=\"1\">350</td><td rowspan=\"1\" colspan=\"1\">Briakinumab</td><td rowspan=\"1\" colspan=\"1\">200 mg</td><td rowspan=\"3\" colspan=\"1\">PASI75, PASI90, PASI100</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"2\" colspan=\"1\">50 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td></tr><tr><td rowspan=\"3\" colspan=\"1\">30</td><td rowspan=\"3\" colspan=\"1\">Kavanaugh et al. [##REF##19333944##45##]</td><td rowspan=\"3\" colspan=\"1\">2009</td><td rowspan=\"3\" colspan=\"1\">405</td><td rowspan=\"1\" colspan=\"1\">Golimumab</td><td rowspan=\"1\" colspan=\"1\">50 mg</td><td rowspan=\"1\" colspan=\"1\">ACR20, ACR50, ACR70</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Golimumab</td><td rowspan=\"1\" colspan=\"1\">100 mg</td><td rowspan=\"1\" colspan=\"1\">PASI50, PASI75, PASI90</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">dactylitis assessment enthesitis assessment, nail assessment</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">31</td><td rowspan=\"3\" colspan=\"1\">Leonardi et al. [##REF##18486739##46##]</td><td rowspan=\"3\" colspan=\"1\">2008</td><td rowspan=\"3\" colspan=\"1\">766</td><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">45 mg</td><td rowspan=\"3\" colspan=\"1\">PASI50, PASI75, PASI90</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">90 mg</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"2\" colspan=\"1\">32</td><td rowspan=\"2\" colspan=\"1\">Menter et al. [##REF##17936411##47##]</td><td rowspan=\"2\" colspan=\"1\">2008</td><td rowspan=\"2\" colspan=\"1\">1212</td><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">40 mg</td><td rowspan=\"2\" colspan=\"1\">PASI90, PASI100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">PLB</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">33</td><td rowspan=\"3\" colspan=\"1\">Papp et al. [##REF##18486740##48##]  </td><td rowspan=\"3\" colspan=\"1\">2008</td><td rowspan=\"3\" colspan=\"1\">1,230</td><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">45 mg</td><td rowspan=\"3\" colspan=\"1\">PASI50, PASI75, PASI90</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">90 mg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">34</td><td rowspan=\"2\" colspan=\"1\">Tyring et al. [##REF##17576937##49##]</td><td rowspan=\"2\" colspan=\"1\">2007</td><td rowspan=\"2\" colspan=\"1\">618</td><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">50 mg</td><td rowspan=\"2\" colspan=\"1\">PASI50, PASI75, PASI90</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">35</td><td rowspan=\"2\" colspan=\"1\">Antoni et al. [##REF##15677701##50##]  </td><td rowspan=\"2\" colspan=\"1\">2005</td><td rowspan=\"2\" colspan=\"1\">200</td><td rowspan=\"1\" colspan=\"1\">Infliximab</td><td rowspan=\"1\" colspan=\"1\">5 mg/kg</td><td rowspan=\"1\" colspan=\"1\">ACR20, ACR50, ACR70, PASI50</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">PASI75, PASI90, PASI100, dactylitis assessment</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">36</td><td rowspan=\"2\" colspan=\"1\">Mease et al. [##REF##16200601##51##]</td><td rowspan=\"2\" colspan=\"1\">2005</td><td rowspan=\"2\" colspan=\"1\">313</td><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">40 mg</td><td rowspan=\"1\" colspan=\"1\">ACR20, ACR50, ACR70</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">PASI50, PASI75, PASI90</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">37</td><td rowspan=\"2\" colspan=\"1\">Reich et al. [##REF##16226614##52##]</td><td rowspan=\"2\" colspan=\"1\">2005</td><td rowspan=\"2\" colspan=\"1\">378</td><td rowspan=\"1\" colspan=\"1\">Infliximab</td><td rowspan=\"1\" colspan=\"1\">5 mg/kg</td><td rowspan=\"1\" colspan=\"1\">PASI50, PASI75</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">PASI90, nail assessment</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>General characteristics of the population and treatment</title><p>sPGA: Static Physicians Global Assessment, N: Number of participants, N%: Percentage of the participants from the total participants in the study, NA: Not Available.</p><p>The data have been represented as the name of the drug used, sample size, age (Year), gender (Male), average duration of plaque psoriasis (Year), sPGA category represents the score of psoriasis based on Static physician global assessment score which ranges from 0 (No signs of plaque psoriasis) to 4 (Dark, red erythematous psoriatic plaques), next to each score is the number of participants in the study and their percentage from the overall participants in that particular study, last column shows the percentage of the average of body area involved.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">Author</td><td rowspan=\"2\" colspan=\"1\">Drug</td><td rowspan=\"2\" colspan=\"1\">Sample size</td><td rowspan=\"2\" colspan=\"1\">Age (Year) Mean</td><td rowspan=\"2\" colspan=\"1\">Gender (Male) Number (N%)</td><td rowspan=\"2\" colspan=\"1\">Duration of plaque psoriasis (years) Mean</td><td colspan=\"3\" rowspan=\"1\">                             sPGA category, n (%)</td><td rowspan=\"2\" colspan=\"1\">Body surface area (%), mean</td></tr><tr><td rowspan=\"1\" colspan=\"1\">3  N (N%)  </td><td rowspan=\"1\" colspan=\"1\">4 N (N%)</td><td rowspan=\"1\" colspan=\"1\">&lt; 3 or missing. N (N%)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">Warren et al. [##REF##32594522##16##]</td><td rowspan=\"1\" colspan=\"1\">Risankizumab</td><td rowspan=\"1\" colspan=\"1\">164</td><td rowspan=\"1\" colspan=\"1\">47·3</td><td rowspan=\"1\" colspan=\"1\">112 (68·3)</td><td rowspan=\"1\" colspan=\"1\">18·6</td><td rowspan=\"1\" colspan=\"1\">140 (85·4)</td><td rowspan=\"1\" colspan=\"1\">24 (14·6)</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">23·8</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">163</td><td rowspan=\"1\" colspan=\"1\">46·8</td><td rowspan=\"1\" colspan=\"1\">101 (62·0)</td><td rowspan=\"1\" colspan=\"1\">17·4</td><td rowspan=\"1\" colspan=\"1\">137 (84·0)</td><td rowspan=\"1\" colspan=\"1\">25 (15·3)</td><td rowspan=\"1\" colspan=\"1\">1 (0·6)</td><td rowspan=\"1\" colspan=\"1\">26·0</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">Ferris et al. [##REF##30887876##17##]</td><td rowspan=\"1\" colspan=\"1\">Guselkumab</td><td rowspan=\"1\" colspan=\"1\">62</td><td rowspan=\"1\" colspan=\"1\">46.2</td><td rowspan=\"1\" colspan=\"1\">41 (66.1)</td><td rowspan=\"1\" colspan=\"1\">19.1</td><td rowspan=\"1\" colspan=\"1\">52 (83.9)</td><td rowspan=\"1\" colspan=\"1\">10 (16.1)</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">20.1</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">16</td><td rowspan=\"1\" colspan=\"1\">45.4</td><td rowspan=\"1\" colspan=\"1\">12 (75.0)</td><td rowspan=\"1\" colspan=\"1\">17.4</td><td rowspan=\"1\" colspan=\"1\">14 (87.5)</td><td rowspan=\"1\" colspan=\"1\">2 (12.5)</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">18.6</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">McInnes et al. [##REF##32386593##18##]</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">426</td><td rowspan=\"1\" colspan=\"1\">48·5</td><td rowspan=\"1\" colspan=\"1\">208 (49%)</td><td rowspan=\"1\" colspan=\"1\">5·1</td><td rowspan=\"1\" colspan=\"1\">215 (50%)</td><td rowspan=\"1\" colspan=\"1\">-</td><td rowspan=\"1\" colspan=\"1\">-</td><td rowspan=\"1\" colspan=\"1\">-</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">427</td><td rowspan=\"1\" colspan=\"1\">49·5</td><td rowspan=\"1\" colspan=\"1\">229 (54%)</td><td rowspan=\"1\" colspan=\"1\">5·7</td><td rowspan=\"1\" colspan=\"1\">202 (47%)</td><td rowspan=\"1\" colspan=\"1\">-</td><td rowspan=\"1\" colspan=\"1\">-</td><td rowspan=\"1\" colspan=\"1\">-</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">Mease et al. [##REF##32178766##19##]</td><td rowspan=\"1\" colspan=\"1\">Guselkumab</td><td rowspan=\"1\" colspan=\"1\">493</td><td rowspan=\"1\" colspan=\"1\">45.4</td><td rowspan=\"1\" colspan=\"1\">271 (54.9%)</td><td rowspan=\"1\" colspan=\"1\">5·5</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">18·2</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">246</td><td rowspan=\"1\" colspan=\"1\">46.3</td><td rowspan=\"1\" colspan=\"1\">117 (48%)</td><td rowspan=\"1\" colspan=\"1\">5·8</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">17·1%</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">Reich et al. [##REF##31402114##20##]</td><td rowspan=\"1\" colspan=\"1\">Guselkumab</td><td rowspan=\"1\" colspan=\"1\">534</td><td rowspan=\"1\" colspan=\"1\">46·3</td><td rowspan=\"1\" colspan=\"1\">365 (68%)</td><td rowspan=\"1\" colspan=\"1\">18·5</td><td rowspan=\"1\" colspan=\"1\">407 (76%)</td><td rowspan=\"1\" colspan=\"1\">127 (24%)</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">23·7</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">514</td><td rowspan=\"1\" colspan=\"1\">45·3</td><td rowspan=\"1\" colspan=\"1\">342 (67%)</td><td rowspan=\"1\" colspan=\"1\">18·3</td><td rowspan=\"1\" colspan=\"1\">391 (76%)</td><td rowspan=\"1\" colspan=\"1\">122 (24%)</td><td rowspan=\"1\" colspan=\"1\">1 (&lt;1%)</td><td rowspan=\"1\" colspan=\"1\">24·5</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Ohtsuki et al. [##REF##31237727##21##]</td><td rowspan=\"1\" colspan=\"1\">Risankizumab (75 mg)</td><td rowspan=\"1\" colspan=\"1\">58</td><td rowspan=\"1\" colspan=\"1\">51.5</td><td rowspan=\"1\" colspan=\"1\">48 (83)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">7 (12)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">41.6</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Risankizumab (150 mg)</td><td rowspan=\"1\" colspan=\"1\">55</td><td rowspan=\"1\" colspan=\"1\">53.3</td><td rowspan=\"1\" colspan=\"1\">50 (91)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">9 (16)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">40.5</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">58</td><td rowspan=\"1\" colspan=\"1\">50.9</td><td rowspan=\"1\" colspan=\"1\">45 (78)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">4 (7)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">33.2</td></tr><tr><td rowspan=\"2\" colspan=\"1\">Reich et al. [##REF##31280967##22##]</td><td rowspan=\"1\" colspan=\"1\">Risankizumab</td><td rowspan=\"1\" colspan=\"1\">301</td><td rowspan=\"1\" colspan=\"1\">45·3</td><td rowspan=\"1\" colspan=\"1\">210 (70%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">26·5</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">304</td><td rowspan=\"1\" colspan=\"1\">47·0</td><td rowspan=\"1\" colspan=\"1\">212 (70%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">25·5</td></tr><tr><td rowspan=\"3\" colspan=\"1\">Mease et al. [##REF##29550766##23##]</td><td rowspan=\"1\" colspan=\"1\">Secukinumab (300 mg)</td><td rowspan=\"1\" colspan=\"1\">222</td><td rowspan=\"1\" colspan=\"1\">48.9</td><td rowspan=\"1\" colspan=\"1\">108 (48.6)</td><td rowspan=\"1\" colspan=\"1\">6.7</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Secukinumab (150 mg)</td><td rowspan=\"1\" colspan=\"1\">220</td><td rowspan=\"1\" colspan=\"1\">48.4</td><td rowspan=\"1\" colspan=\"1\">111 (50.5)</td><td rowspan=\"1\" colspan=\"1\">6.7</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">332</td><td rowspan=\"1\" colspan=\"1\">49</td><td rowspan=\"1\" colspan=\"1\">161 (48.5)</td><td rowspan=\"1\" colspan=\"1\">6.6</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Gottlieb et al. [##REF##29660421##24##] (CIMPASI-1)</td><td rowspan=\"1\" colspan=\"1\">Certolizumab (200 mg)</td><td rowspan=\"1\" colspan=\"1\">95</td><td rowspan=\"1\" colspan=\"1\">44.5</td><td rowspan=\"1\" colspan=\"1\">67 (70.5)</td><td rowspan=\"1\" colspan=\"1\">16.6</td><td rowspan=\"1\" colspan=\"1\">62 (65.3)</td><td rowspan=\"1\" colspan=\"1\">33 (34.7)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">25.4</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Certolizumab (400)</td><td rowspan=\"1\" colspan=\"1\">88</td><td rowspan=\"1\" colspan=\"1\">43.6</td><td rowspan=\"1\" colspan=\"1\">60 (68.2)</td><td rowspan=\"1\" colspan=\"1\">18.4</td><td rowspan=\"1\" colspan=\"1\">65 (73.9)</td><td rowspan=\"1\" colspan=\"1\">23 (26.1)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">24.1</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">51</td><td rowspan=\"1\" colspan=\"1\">47.9</td><td rowspan=\"1\" colspan=\"1\">35 (68.6)</td><td rowspan=\"1\" colspan=\"1\">18.5</td><td rowspan=\"1\" colspan=\"1\">35 (68.6)</td><td rowspan=\"1\" colspan=\"1\">16 (31.4)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">26.1</td></tr><tr><td rowspan=\"3\" colspan=\"1\">Gottlieb et al. [##REF##29660421##24##] (CIMPASI-2)</td><td rowspan=\"1\" colspan=\"1\">Certolizumab (200 mg)</td><td rowspan=\"1\" colspan=\"1\">91</td><td rowspan=\"1\" colspan=\"1\">46.7</td><td rowspan=\"1\" colspan=\"1\">58 (63.7)</td><td rowspan=\"1\" colspan=\"1\">18.8</td><td rowspan=\"1\" colspan=\"1\">66 (72.5)</td><td rowspan=\"1\" colspan=\"1\">25 (27.5)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">21.4</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Certolizumab (400)</td><td rowspan=\"1\" colspan=\"1\">87</td><td rowspan=\"1\" colspan=\"1\">46.4</td><td rowspan=\"1\" colspan=\"1\">43 (49.4)</td><td rowspan=\"1\" colspan=\"1\">18.6</td><td rowspan=\"1\" colspan=\"1\">61 (70.1)</td><td rowspan=\"1\" colspan=\"1\">26 (29.9)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">23.1</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">49</td><td rowspan=\"1\" colspan=\"1\">43.3</td><td rowspan=\"1\" colspan=\"1\">26 (53.1)</td><td rowspan=\"1\" colspan=\"1\">15.4</td><td rowspan=\"1\" colspan=\"1\">37 (75.5)</td><td rowspan=\"1\" colspan=\"1\">12 (24.5)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">20</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">Lebwohl et al. [##REF##29660425##25##]</td><td rowspan=\"1\" colspan=\"1\">Certolizumab (200 mg)</td><td rowspan=\"1\" colspan=\"1\">165</td><td rowspan=\"1\" colspan=\"1\">46.7</td><td rowspan=\"1\" colspan=\"1\">113 (68.5)</td><td rowspan=\"1\" colspan=\"1\">19.5</td><td rowspan=\"1\" colspan=\"1\">114 (69.1)</td><td rowspan=\"1\" colspan=\"1\">51 (30.9)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">28.1</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Certolizumab (400 mg )</td><td rowspan=\"1\" colspan=\"1\">167</td><td rowspan=\"1\" colspan=\"1\">45.4</td><td rowspan=\"1\" colspan=\"1\">107 (64.1)</td><td rowspan=\"1\" colspan=\"1\">17.8</td><td rowspan=\"1\" colspan=\"1\">113 (67.7)</td><td rowspan=\"1\" colspan=\"1\">54 (32.3)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">27.6</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">170</td><td rowspan=\"1\" colspan=\"1\">44.6</td><td rowspan=\"1\" colspan=\"1\">127 (74.7)</td><td rowspan=\"1\" colspan=\"1\">17.4</td><td rowspan=\"1\" colspan=\"1\">115 (67.6)</td><td rowspan=\"1\" colspan=\"1\">55 (32.4)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">27.5</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">57</td><td rowspan=\"1\" colspan=\"1\">46.5</td><td rowspan=\"1\" colspan=\"1\">34 (59.6)</td><td rowspan=\"1\" colspan=\"1\">18.9</td><td rowspan=\"1\" colspan=\"1\">40 (70.2</td><td rowspan=\"1\" colspan=\"1\">17 (29.8)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">24.3</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Gordon et al. [##REF##30097359##26##] (UltIMMa-1)</td><td rowspan=\"1\" colspan=\"1\">Risankizumab</td><td rowspan=\"1\" colspan=\"1\">304</td><td rowspan=\"1\" colspan=\"1\">48.3</td><td rowspan=\"1\" colspan=\"1\">212 (70%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">256 (84%)</td><td rowspan=\"1\" colspan=\"1\">48 (16%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">26.2</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">100</td><td rowspan=\"1\" colspan=\"1\">46.5</td><td rowspan=\"1\" colspan=\"1\">70 (70%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">85 (85%)</td><td rowspan=\"1\" colspan=\"1\">15 (15%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">25.2</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">102</td><td rowspan=\"1\" colspan=\"1\">49.3</td><td rowspan=\"1\" colspan=\"1\">79 (77%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">86 (84%)</td><td rowspan=\"1\" colspan=\"1\">16 (16%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">27.9</td></tr><tr><td rowspan=\"3\" colspan=\"1\">Gordon et al. [##REF##30097359##26##] (UltIMMa-2)</td><td rowspan=\"1\" colspan=\"1\">Risankizumab</td><td rowspan=\"1\" colspan=\"1\">294</td><td rowspan=\"1\" colspan=\"1\">46.2</td><td rowspan=\"1\" colspan=\"1\">203 (69%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">228 (78%)</td><td rowspan=\"1\" colspan=\"1\">66 (22%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">26.2</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">99</td><td rowspan=\"1\" colspan=\"1\">48.6</td><td rowspan=\"1\" colspan=\"1\">66 (67%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">81 (82%)</td><td rowspan=\"1\" colspan=\"1\">18 (18%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">20.9</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">98</td><td rowspan=\"1\" colspan=\"1\">46.3</td><td rowspan=\"1\" colspan=\"1\">67 (68%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">77 (79%)</td><td rowspan=\"1\" colspan=\"1\">21 (21%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">23.9</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Reich et al. [##REF##30367462##27##]</td><td rowspan=\"1\" colspan=\"1\">Secukinumab (300 mg)</td><td rowspan=\"1\" colspan=\"1\">66</td><td rowspan=\"1\" colspan=\"1\">45.1</td><td rowspan=\"1\" colspan=\"1\">53 (80)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">28</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Secukinumab (150 mg)</td><td rowspan=\"1\" colspan=\"1\">67</td><td rowspan=\"1\" colspan=\"1\">43.5</td><td rowspan=\"1\" colspan=\"1\">55 (82)</td><td rowspan=\"1\" colspan=\"1\">20</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">26.4</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">65</td><td rowspan=\"1\" colspan=\"1\">43.6</td><td rowspan=\"1\" colspan=\"1\">52 (80)</td><td rowspan=\"1\" colspan=\"1\">17.4</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">25.8</td></tr><tr><td rowspan=\"2\" colspan=\"1\">Bagel et al. [##REF##30334147##28##]</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">550</td><td rowspan=\"1\" colspan=\"1\">45.4</td><td rowspan=\"1\" colspan=\"1\">356 (64.7)</td><td rowspan=\"1\" colspan=\"1\">16.8</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">29.2</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">552</td><td rowspan=\"1\" colspan=\"1\">45.3</td><td rowspan=\"1\" colspan=\"1\">376 (68.1)</td><td rowspan=\"1\" colspan=\"1\">17.3</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">29.5</td></tr><tr><td rowspan=\"3\" colspan=\"1\">Reich et al. [##REF##28596043##29##] (reSURFACE 1)</td><td rowspan=\"1\" colspan=\"1\">Tildrakizumab 200 mg</td><td rowspan=\"1\" colspan=\"1\">308</td><td rowspan=\"1\" colspan=\"1\">46·9</td><td rowspan=\"1\" colspan=\"1\">226 (73%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">30.9</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Tildrakizumab 100 mg</td><td rowspan=\"1\" colspan=\"1\">309</td><td rowspan=\"1\" colspan=\"1\">46·4</td><td rowspan=\"1\" colspan=\"1\">207 (67%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">29.7</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">155</td><td rowspan=\"1\" colspan=\"1\">47·9</td><td rowspan=\"1\" colspan=\"1\">100 (65%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">29.6</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">Reich et al. [##REF##28596043##29##] (reSURFACE 2)</td><td rowspan=\"1\" colspan=\"1\">Tildrakizumab 200 mg</td><td rowspan=\"1\" colspan=\"1\">314</td><td rowspan=\"1\" colspan=\"1\">44·6</td><td rowspan=\"1\" colspan=\"1\">225 (72%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">31.8</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Tildrakizumab 100 mg</td><td rowspan=\"1\" colspan=\"1\">307</td><td rowspan=\"1\" colspan=\"1\">44·6</td><td rowspan=\"1\" colspan=\"1\">220 (72%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">34.2</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">313</td><td rowspan=\"1\" colspan=\"1\">45·8</td><td rowspan=\"1\" colspan=\"1\">222 (71%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">31.6</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">156</td><td rowspan=\"1\" colspan=\"1\">46·4</td><td rowspan=\"1\" colspan=\"1\">112 (72%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">31.3</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">Reich et al. [##REF##28542874##30##]</td><td rowspan=\"1\" colspan=\"1\">Ixekizumab</td><td rowspan=\"1\" colspan=\"1\">136</td><td rowspan=\"1\" colspan=\"1\">42.7</td><td rowspan=\"1\" colspan=\"1\">90 (66·2)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">26.7</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">166</td><td rowspan=\"1\" colspan=\"1\">44·0</td><td rowspan=\"1\" colspan=\"1\">112 (67·5)</td><td rowspan=\"1\" colspan=\"1\">18.2</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">27.5</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Blauvelt et al. [##REF##28057360##31##]</td><td rowspan=\"1\" colspan=\"1\">Guselkumab</td><td rowspan=\"1\" colspan=\"1\">329</td><td rowspan=\"1\" colspan=\"1\">43.9</td><td rowspan=\"1\" colspan=\"1\">240 (72.9)</td><td rowspan=\"1\" colspan=\"1\">17.9</td><td rowspan=\"1\" colspan=\"1\">252 (76.6)</td><td rowspan=\"1\" colspan=\"1\">77 (23.4)</td><td rowspan=\"1\" colspan=\"1\">3 (0.9)</td><td rowspan=\"1\" colspan=\"1\">28.3</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">334</td><td rowspan=\"1\" colspan=\"1\">42.9</td><td rowspan=\"1\" colspan=\"1\">249 (74.6)</td><td rowspan=\"1\" colspan=\"1\">17</td><td rowspan=\"1\" colspan=\"1\">241 (72.2)</td><td rowspan=\"1\" colspan=\"1\">90 (26.9)</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">28.6</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">174</td><td rowspan=\"1\" colspan=\"1\">44.9</td><td rowspan=\"1\" colspan=\"1\">119 (68.4)</td><td rowspan=\"1\" colspan=\"1\">17.6</td><td rowspan=\"1\" colspan=\"1\">131 (75.3)</td><td rowspan=\"1\" colspan=\"1\">43 (24.7)</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">25.8</td></tr><tr><td rowspan=\"4\" colspan=\"1\">Mease et al. [##REF##27553214##32##]</td><td rowspan=\"1\" colspan=\"1\">Ixekizumab (once every 2 weeks)</td><td rowspan=\"1\" colspan=\"1\">103</td><td rowspan=\"1\" colspan=\"1\">49.8</td><td rowspan=\"1\" colspan=\"1\">48 (46.6)</td><td rowspan=\"1\" colspan=\"1\">17</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">12</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ixekizumab (once every 4 weeks)</td><td rowspan=\"1\" colspan=\"1\">107</td><td rowspan=\"1\" colspan=\"1\">49.1</td><td rowspan=\"1\" colspan=\"1\">45 (42.1)</td><td rowspan=\"1\" colspan=\"1\">16.5</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">15.1</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">101</td><td rowspan=\"1\" colspan=\"1\">48.6</td><td rowspan=\"1\" colspan=\"1\">51 (50.5)</td><td rowspan=\"1\" colspan=\"1\">15.7</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">14.8</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">106</td><td rowspan=\"1\" colspan=\"1\">50.6</td><td rowspan=\"1\" colspan=\"1\">48 (45.3)</td><td rowspan=\"1\" colspan=\"1\">16</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">14.4</td></tr><tr><td rowspan=\"4\" colspan=\"1\">Gordon et al. [##REF##27299809##33##]</td><td rowspan=\"1\" colspan=\"1\">Ixekizumab (once every 2 weeks)</td><td rowspan=\"1\" colspan=\"1\">386</td><td rowspan=\"1\" colspan=\"1\">46</td><td rowspan=\"1\" colspan=\"1\">254 (66.0)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">178 (46.2)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">28</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ixekizumab (once every 4 weeks)</td><td rowspan=\"1\" colspan=\"1\">385</td><td rowspan=\"1\" colspan=\"1\">46</td><td rowspan=\"1\" colspan=\"1\">258 (66.8)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">177 (46.2)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">28</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">382</td><td rowspan=\"1\" colspan=\"1\">46</td><td rowspan=\"1\" colspan=\"1\">269 (70.4)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">192 (50.3)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">28</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">PLB</td><td rowspan=\"1\" colspan=\"1\">193</td><td rowspan=\"1\" colspan=\"1\">46</td><td rowspan=\"1\" colspan=\"1\">137 (71.0)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">101 (52.3)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">29</td></tr><tr><td rowspan=\"3\" colspan=\"1\">Papp et al. [##REF##26914406##34##]</td><td rowspan=\"1\" colspan=\"1\">Brodalumab(140 mg)</td><td rowspan=\"1\" colspan=\"1\">219</td><td rowspan=\"1\" colspan=\"1\">46</td><td rowspan=\"1\" colspan=\"1\">162 (74)</td><td rowspan=\"1\" colspan=\"1\">19</td><td rowspan=\"1\" colspan=\"1\">129 (59)</td><td rowspan=\"1\" colspan=\"1\">94 (41)</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">27.4</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Brodalumab (210 mg)</td><td rowspan=\"1\" colspan=\"1\">222</td><td rowspan=\"1\" colspan=\"1\">46</td><td rowspan=\"1\" colspan=\"1\">161 (73)</td><td rowspan=\"1\" colspan=\"1\">20</td><td rowspan=\"1\" colspan=\"1\">121 (55)</td><td rowspan=\"1\" colspan=\"1\">97 (45)</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">25.1</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">220</td><td rowspan=\"1\" colspan=\"1\">47</td><td rowspan=\"1\" colspan=\"1\">161 (73)</td><td rowspan=\"1\" colspan=\"1\">21</td><td rowspan=\"1\" colspan=\"1\">114 (52)</td><td rowspan=\"1\" colspan=\"1\">106 (48)</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">26.9</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">Griffiths et al. [##REF##26072109##35##] (UNCOVER-2 desgin)</td><td rowspan=\"1\" colspan=\"1\">Ixekizumab (once every 2 weeks)</td><td rowspan=\"1\" colspan=\"1\">351</td><td rowspan=\"1\" colspan=\"1\">45</td><td rowspan=\"1\" colspan=\"1\">221 (63%)</td><td rowspan=\"1\" colspan=\"1\">19</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">173 (49%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">25</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ixekizumab (once every 4 weeks)</td><td rowspan=\"1\" colspan=\"1\">347</td><td rowspan=\"1\" colspan=\"1\">45</td><td rowspan=\"1\" colspan=\"1\">244 (70%)</td><td rowspan=\"1\" colspan=\"1\">19</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">181 (52%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">27</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">358</td><td rowspan=\"1\" colspan=\"1\">45</td><td rowspan=\"1\" colspan=\"1\">236 (66%)</td><td rowspan=\"1\" colspan=\"1\">19</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">172 (48%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">25</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">168</td><td rowspan=\"1\" colspan=\"1\">45</td><td rowspan=\"1\" colspan=\"1\">120 (71%)</td><td rowspan=\"1\" colspan=\"1\">19</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">82 (49%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">27</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">Griffiths et al. [##REF##26072109##35##] (UNCOVER-3 design)</td><td rowspan=\"1\" colspan=\"1\">Ixekizumab (once every 2 weeks)</td><td rowspan=\"1\" colspan=\"1\">385</td><td rowspan=\"1\" colspan=\"1\">46</td><td rowspan=\"1\" colspan=\"1\">254 (66%)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">178 (46%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">28</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ixekizumab (once every 4 weeks)</td><td rowspan=\"1\" colspan=\"1\">386</td><td rowspan=\"1\" colspan=\"1\">46</td><td rowspan=\"1\" colspan=\"1\">258 (67%)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">177 (46%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">28</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">382</td><td rowspan=\"1\" colspan=\"1\">46</td><td rowspan=\"1\" colspan=\"1\">269 (70%)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">192 (50%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">28</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">193</td><td rowspan=\"1\" colspan=\"1\">46</td><td rowspan=\"1\" colspan=\"1\">137 (71%)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">101 (52%)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">29</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">Lebwohl et al. [##REF##26422722##36##] (AMAGINE-2 )</td><td rowspan=\"1\" colspan=\"1\">Brodalumab(140 mg)</td><td rowspan=\"1\" colspan=\"1\">610</td><td rowspan=\"1\" colspan=\"1\">45</td><td rowspan=\"1\" colspan=\"1\">413 (68)</td><td rowspan=\"1\" colspan=\"1\">19</td><td rowspan=\"1\" colspan=\"1\">358 (59)</td><td rowspan=\"1\" colspan=\"1\">52</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">27</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Brodalumab (210 mg)</td><td rowspan=\"1\" colspan=\"1\">612</td><td rowspan=\"1\" colspan=\"1\">45</td><td rowspan=\"1\" colspan=\"1\">421 (69)</td><td rowspan=\"1\" colspan=\"1\">19</td><td rowspan=\"1\" colspan=\"1\">316 (52)</td><td rowspan=\"1\" colspan=\"1\">296 (48)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">26</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">300</td><td rowspan=\"1\" colspan=\"1\">45</td><td rowspan=\"1\" colspan=\"1\">205 (68)</td><td rowspan=\"1\" colspan=\"1\">19</td><td rowspan=\"1\" colspan=\"1\">153 (51)</td><td rowspan=\"1\" colspan=\"1\">147 (49)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">27</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">309</td><td rowspan=\"1\" colspan=\"1\">44</td><td rowspan=\"1\" colspan=\"1\">219 (71)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">167 (54)</td><td rowspan=\"1\" colspan=\"1\">142 (46)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">28</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">Lebwohl et al.  [##REF##26422722##36##] (AMAGINE-3)</td><td rowspan=\"1\" colspan=\"1\">Brodalumab</td><td rowspan=\"1\" colspan=\"1\">629</td><td rowspan=\"1\" colspan=\"1\">45</td><td rowspan=\"1\" colspan=\"1\">437 (70)</td><td rowspan=\"1\" colspan=\"1\">17</td><td rowspan=\"1\" colspan=\"1\">412 (66)</td><td rowspan=\"1\" colspan=\"1\">217 (34)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">29</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Brodalumab</td><td rowspan=\"1\" colspan=\"1\">624</td><td rowspan=\"1\" colspan=\"1\">45</td><td rowspan=\"1\" colspan=\"1\">431 (69)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">373 (60)</td><td rowspan=\"1\" colspan=\"1\">251 (40)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">28</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">313</td><td rowspan=\"1\" colspan=\"1\">45</td><td rowspan=\"1\" colspan=\"1\">212 (68)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">192 (61)</td><td rowspan=\"1\" colspan=\"1\">121 (39)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">28</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">315</td><td rowspan=\"1\" colspan=\"1\">44</td><td rowspan=\"1\" colspan=\"1\">208 (66)</td><td rowspan=\"1\" colspan=\"1\">18</td><td rowspan=\"1\" colspan=\"1\">192 (61)</td><td rowspan=\"1\" colspan=\"1\">123 (39)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">28</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">Thaçi et al. [##REF##26092291##37##]</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">337</td><td rowspan=\"1\" colspan=\"1\">45.2</td><td rowspan=\"1\" colspan=\"1\">229 (68.0)</td><td rowspan=\"1\" colspan=\"1\">19.6</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">130 (38.6)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">32.6</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ustekinumab</td><td rowspan=\"1\" colspan=\"1\">339</td><td rowspan=\"1\" colspan=\"1\">44.6</td><td rowspan=\"1\" colspan=\"1\">252 (74.3)</td><td rowspan=\"1\" colspan=\"1\">16.1</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">125 (36.9)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">32</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Langley et al. [##REF##25007392##38##] (ERASURE study)</td><td rowspan=\"1\" colspan=\"1\">Secukinumab (300 mg)</td><td rowspan=\"1\" colspan=\"1\">245</td><td rowspan=\"1\" colspan=\"1\">44.9</td><td rowspan=\"1\" colspan=\"1\">169 (69.0)</td><td rowspan=\"1\" colspan=\"1\">17.4</td><td rowspan=\"1\" colspan=\"1\">154 (62.9)</td><td rowspan=\"1\" colspan=\"1\">91 (37.1)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">32.8</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Secukinumab (150 mg)</td><td rowspan=\"1\" colspan=\"1\">245</td><td rowspan=\"1\" colspan=\"1\">44.9</td><td rowspan=\"1\" colspan=\"1\">168 (68.6)</td><td rowspan=\"1\" colspan=\"1\">17.5</td><td rowspan=\"1\" colspan=\"1\">161 (65.7)</td><td rowspan=\"1\" colspan=\"1\">84 (34.3)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">33.3</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">248</td><td rowspan=\"1\" colspan=\"1\">45.4</td><td rowspan=\"1\" colspan=\"1\">172 (69.4)</td><td rowspan=\"1\" colspan=\"1\">17.3</td><td rowspan=\"1\" colspan=\"1\">151 (60.9)</td><td rowspan=\"1\" colspan=\"1\">97 (39.1)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">29.7</td></tr><tr><td rowspan=\"4\" colspan=\"1\">Langley et al. [##REF##25007392##38##] (FIXTURE study)</td><td rowspan=\"1\" colspan=\"1\">Secukinumab (300 mg)</td><td rowspan=\"1\" colspan=\"1\">327</td><td rowspan=\"1\" colspan=\"1\">44.5</td><td rowspan=\"1\" colspan=\"1\">224 (68.5)</td><td rowspan=\"1\" colspan=\"1\">15.8</td><td rowspan=\"1\" colspan=\"1\">203 (62.1)</td><td rowspan=\"1\" colspan=\"1\">124 (37.9)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">34.3</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Secukinumab (150 mg)</td><td rowspan=\"1\" colspan=\"1\">327</td><td rowspan=\"1\" colspan=\"1\">45.4</td><td rowspan=\"1\" colspan=\"1\">236 (72.2)</td><td rowspan=\"1\" colspan=\"1\">17.3</td><td rowspan=\"1\" colspan=\"1\">206 (63 )</td><td rowspan=\"1\" colspan=\"1\">121 (37)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">34.5</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">326</td><td rowspan=\"1\" colspan=\"1\">43.8</td><td rowspan=\"1\" colspan=\"1\">232 (71.2)</td><td rowspan=\"1\" colspan=\"1\">16.4</td><td rowspan=\"1\" colspan=\"1\">195 (59.8)</td><td rowspan=\"1\" colspan=\"1\">131 (40.2)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">33.6</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">326</td><td rowspan=\"1\" colspan=\"1\">44.1</td><td rowspan=\"1\" colspan=\"1\">237 (72.7)</td><td rowspan=\"1\" colspan=\"1\">16.6</td><td rowspan=\"1\" colspan=\"1\">202 (62)</td><td rowspan=\"1\" colspan=\"1\">124 (38.0)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">35.2</td></tr><tr><td rowspan=\"3\" colspan=\"1\">Mease et al. [##REF##23942868##39##]</td><td rowspan=\"1\" colspan=\"1\">Certolizumab (200 mg Q2W)</td><td rowspan=\"1\" colspan=\"1\">138</td><td rowspan=\"1\" colspan=\"1\">48.2</td><td rowspan=\"1\" colspan=\"1\">64 (46.4)</td><td rowspan=\"1\" colspan=\"1\">9.6</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Certolizumab (400 mg Q4W)</td><td rowspan=\"1\" colspan=\"1\">135</td><td rowspan=\"1\" colspan=\"1\">47.1</td><td rowspan=\"1\" colspan=\"1\">62 (45.9)</td><td rowspan=\"1\" colspan=\"1\">8.1</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">136</td><td rowspan=\"1\" colspan=\"1\">47.3</td><td rowspan=\"1\" colspan=\"1\">57 (41.9)</td><td rowspan=\"1\" colspan=\"1\">7.9</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">Baranauskaite et al. [##REF##21994233##40##]</td><td rowspan=\"1\" colspan=\"1\">Infliximab + Methotrexate</td><td rowspan=\"1\" colspan=\"1\">56</td><td rowspan=\"1\" colspan=\"1\">40.1</td><td rowspan=\"1\" colspan=\"1\">27 (48.2)</td><td rowspan=\"1\" colspan=\"1\">2.8</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Methotrexate</td><td rowspan=\"1\" colspan=\"1\">54</td><td rowspan=\"1\" colspan=\"1\">42.3</td><td rowspan=\"1\" colspan=\"1\">33 (61.1)</td><td rowspan=\"1\" colspan=\"1\">3.7</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">Gottlieb et al. [##REF##22533447##41##]</td><td rowspan=\"1\" colspan=\"1\">Methotrexate + Etanercept</td><td rowspan=\"1\" colspan=\"1\">239</td><td rowspan=\"1\" colspan=\"1\">43</td><td rowspan=\"1\" colspan=\"1\">153 (64·0)</td><td rowspan=\"1\" colspan=\"1\">17.9</td><td rowspan=\"1\" colspan=\"1\">138 (57·7)</td><td rowspan=\"1\" colspan=\"1\">69 (28.9)</td><td rowspan=\"1\" colspan=\"1\">32 (13.4)</td><td rowspan=\"1\" colspan=\"1\">24.4</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Etanercept + placebo</td><td rowspan=\"1\" colspan=\"1\">239</td><td rowspan=\"1\" colspan=\"1\">45.2</td><td rowspan=\"1\" colspan=\"1\">167 (69·9)</td><td rowspan=\"1\" colspan=\"1\">16.9</td><td rowspan=\"1\" colspan=\"1\">139 (58·2)</td><td rowspan=\"1\" colspan=\"1\">74(31)</td><td rowspan=\"1\" colspan=\"1\">26(10.8)</td><td rowspan=\"1\" colspan=\"1\">24.2</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">Barker et al. [##REF##21910713##42##]</td><td rowspan=\"1\" colspan=\"1\">Infliximab</td><td rowspan=\"1\" colspan=\"1\">653</td><td rowspan=\"1\" colspan=\"1\">44.1</td><td rowspan=\"1\" colspan=\"1\">438 (67)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">31.9</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Methotrexate</td><td rowspan=\"1\" colspan=\"1\">215</td><td rowspan=\"1\" colspan=\"1\">41.9</td><td rowspan=\"1\" colspan=\"1\">148 (69)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">31</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Gottlieb et al. [##REF##21574983##43##]</td><td rowspan=\"1\" colspan=\"1\">Briakinumab</td><td rowspan=\"1\" colspan=\"1\">138</td><td rowspan=\"1\" colspan=\"1\">43.6</td><td rowspan=\"1\" colspan=\"1\">89 (64.5)</td><td rowspan=\"1\" colspan=\"1\">16.1</td><td rowspan=\"1\" colspan=\"1\">77 (55.8)</td><td rowspan=\"1\" colspan=\"1\">61 (44.2)</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">23.6</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">141</td><td rowspan=\"1\" colspan=\"1\">43.1</td><td rowspan=\"1\" colspan=\"1\">98 (69.5)</td><td rowspan=\"1\" colspan=\"1\">17</td><td rowspan=\"1\" colspan=\"1\">72 (51.1)</td><td rowspan=\"1\" colspan=\"1\">69 (48.9)</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">24.1</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">68</td><td rowspan=\"1\" colspan=\"1\">44</td><td rowspan=\"1\" colspan=\"1\">47 (69.1)</td><td rowspan=\"1\" colspan=\"1\">19.1</td><td rowspan=\"1\" colspan=\"1\">42 (61.8)</td><td rowspan=\"1\" colspan=\"1\">26 (38.2)</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">23.8</td></tr><tr><td rowspan=\"3\" colspan=\"1\">Strober et al. [##REF##21574984##44##]</td><td rowspan=\"1\" colspan=\"1\">Briakinumab</td><td rowspan=\"1\" colspan=\"1\">139</td><td rowspan=\"1\" colspan=\"1\">44.9</td><td rowspan=\"1\" colspan=\"1\">93 (66.9)</td><td rowspan=\"1\" colspan=\"1\">16.3</td><td rowspan=\"1\" colspan=\"1\">63 (45.3)</td><td rowspan=\"1\" colspan=\"1\">38 (44.7)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">24.9</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">139</td><td rowspan=\"1\" colspan=\"1\">45.2</td><td rowspan=\"1\" colspan=\"1\">85 (61.2)</td><td rowspan=\"1\" colspan=\"1\">15.2</td><td rowspan=\"1\" colspan=\"1\">69 (49.6)</td><td rowspan=\"1\" colspan=\"1\">70 (50.4)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">24.7</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">72</td><td rowspan=\"1\" colspan=\"1\">45</td><td rowspan=\"1\" colspan=\"1\">46 (63.9)</td><td rowspan=\"1\" colspan=\"1\">15.5</td><td rowspan=\"1\" colspan=\"1\">34 (47.2)</td><td rowspan=\"1\" colspan=\"1\">76 (52.8)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">22.1</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Kavanaugh et al. [##REF##19333944##45##]</td><td rowspan=\"1\" colspan=\"1\">Golimumab (50 mg)</td><td rowspan=\"1\" colspan=\"1\">146</td><td rowspan=\"1\" colspan=\"1\">45.7</td><td rowspan=\"1\" colspan=\"1\">89 (61)</td><td rowspan=\"1\" colspan=\"1\">7.2</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">16.2</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Golimumab (100 mg)</td><td rowspan=\"1\" colspan=\"1\">146</td><td rowspan=\"1\" colspan=\"1\">48.2</td><td rowspan=\"1\" colspan=\"1\">86 (59)</td><td rowspan=\"1\" colspan=\"1\">7.7</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">17.7</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">113</td><td rowspan=\"1\" colspan=\"1\">47</td><td rowspan=\"1\" colspan=\"1\">69 (61)</td><td rowspan=\"1\" colspan=\"1\">7.6</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">14.7</td></tr><tr><td rowspan=\"3\" colspan=\"1\">Leonardi et al. [##REF##18486739##46##]</td><td rowspan=\"1\" colspan=\"1\">Ustekinumab (45 mg)</td><td rowspan=\"1\" colspan=\"1\">255</td><td rowspan=\"1\" colspan=\"1\">44.8</td><td rowspan=\"1\" colspan=\"1\">175 (68·6%)</td><td rowspan=\"1\" colspan=\"1\">19.7</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">27.2</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ustekinumab (90 mg)</td><td rowspan=\"1\" colspan=\"1\">256</td><td rowspan=\"1\" colspan=\"1\">46.2</td><td rowspan=\"1\" colspan=\"1\">173 (67·6%)</td><td rowspan=\"1\" colspan=\"1\">19.6</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">25.2</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">255</td><td rowspan=\"1\" colspan=\"1\">44.8</td><td rowspan=\"1\" colspan=\"1\">183 (71·8%)</td><td rowspan=\"1\" colspan=\"1\">20.4</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">27.7</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"2\" colspan=\"1\">Menter et al. [##REF##17936411##47##]</td><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">814</td><td rowspan=\"1\" colspan=\"1\">44.1</td><td rowspan=\"1\" colspan=\"1\">546 (67.1)</td><td rowspan=\"1\" colspan=\"1\">18.1</td><td rowspan=\"1\" colspan=\"1\">417 (51.2)</td><td rowspan=\"1\" colspan=\"1\">397 (48.8)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">25.8</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">398</td><td rowspan=\"1\" colspan=\"1\">45.4</td><td rowspan=\"1\" colspan=\"1\">257 (64.6)</td><td rowspan=\"1\" colspan=\"1\">18.4</td><td rowspan=\"1\" colspan=\"1\">220 (55.3)</td><td rowspan=\"1\" colspan=\"1\">178 (44.7)</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">25.6</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Papp et al. [##REF##18486740##48##]</td><td rowspan=\"1\" colspan=\"1\">Ustekinumab (45 mg)</td><td rowspan=\"1\" colspan=\"1\">409</td><td rowspan=\"1\" colspan=\"1\">45.1</td><td rowspan=\"1\" colspan=\"1\">283 (69·2%)</td><td rowspan=\"1\" colspan=\"1\">19.3</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">25.9</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Ustekinumab(90 mg)</td><td rowspan=\"1\" colspan=\"1\">411</td><td rowspan=\"1\" colspan=\"1\">46.4</td><td rowspan=\"1\" colspan=\"1\">274 (66·7%)</td><td rowspan=\"1\" colspan=\"1\">20.3</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">27.1</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">410</td><td rowspan=\"1\" colspan=\"1\">47</td><td rowspan=\"1\" colspan=\"1\">283 (69·0%)</td><td rowspan=\"1\" colspan=\"1\">20.8</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">26.1</td></tr><tr><td rowspan=\"2\" colspan=\"1\">Tyring et al. [##REF##17576937##49##]</td><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">311</td><td rowspan=\"1\" colspan=\"1\">45.8</td><td rowspan=\"1\" colspan=\"1\">203 (65.3)</td><td rowspan=\"1\" colspan=\"1\">20.2</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">27.2</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">307</td><td rowspan=\"1\" colspan=\"1\">45.5</td><td rowspan=\"1\" colspan=\"1\">215 (70.0)</td><td rowspan=\"1\" colspan=\"1\">19.7</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">27.2</td></tr><tr><td rowspan=\"2\" colspan=\"1\">Antoni et al. [##REF##15677701##50##]</td><td rowspan=\"1\" colspan=\"1\">Infliximab</td><td rowspan=\"1\" colspan=\"1\">100</td><td rowspan=\"1\" colspan=\"1\">47.1</td><td rowspan=\"1\" colspan=\"1\">71 (71)</td><td rowspan=\"1\" colspan=\"1\">8.4</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">100</td><td rowspan=\"1\" colspan=\"1\">46.5</td><td rowspan=\"1\" colspan=\"1\">51 (51)</td><td rowspan=\"1\" colspan=\"1\">7.5</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr><tr><td rowspan=\"2\" colspan=\"1\">Mease et al. [##REF##16200601##51##]</td><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">151</td><td rowspan=\"1\" colspan=\"1\">48.6</td><td rowspan=\"1\" colspan=\"1\">85 (56.3)</td><td rowspan=\"1\" colspan=\"1\">9.8</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">162</td><td rowspan=\"1\" colspan=\"1\">49.2</td><td rowspan=\"1\" colspan=\"1\">89 (54.9)</td><td rowspan=\"1\" colspan=\"1\">9.2</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td></tr><tr><td rowspan=\"2\" colspan=\"1\">Reich et al. [##REF##16226614##52##]</td><td rowspan=\"1\" colspan=\"1\">Infliximab</td><td rowspan=\"1\" colspan=\"1\">301</td><td rowspan=\"1\" colspan=\"1\">42.6</td><td rowspan=\"1\" colspan=\"1\">207 (69)</td><td rowspan=\"1\" colspan=\"1\">19.1</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">34.1</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">77</td><td rowspan=\"1\" colspan=\"1\">43.8</td><td rowspan=\"1\" colspan=\"1\">61 (79)</td><td rowspan=\"1\" colspan=\"1\">17.3</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">33.5</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB3\"><label>Table 3</label><caption><title> sPASI Improvements in patients with psoriasis skin</title><p>sPASI: Simplified Psoriasis Severity Index, PASI: Psoriasis Area and Severity Index Score, N: Number of the participants, %: Percentage of the participants from the overall participants.</p><p>The data have been represented as weeks of treatment, name of the drug used, PASI100 (Completely clear skin), PASI90 (Clear to almost clear skin), and PASI75 (75% reduction of severity from the baseline)</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">PASI100</td><td rowspan=\"1\" colspan=\"1\">PASI90</td><td rowspan=\"1\" colspan=\"1\">PASI75</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Study</td><td rowspan=\"1\" colspan=\"1\">Weeks</td><td rowspan=\"1\" colspan=\"1\">Drug</td><td rowspan=\"1\" colspan=\"1\">n/total (%)</td><td rowspan=\"1\" colspan=\"1\">n/total (%)</td><td rowspan=\"1\" colspan=\"1\">n/total (%)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Warren et al. [##REF##32594522##16##]</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Risankizumab</td><td rowspan=\"1\" colspan=\"1\">44/164 (26.9)</td><td rowspan=\"1\" colspan=\"1\">74/164 (45.1)</td><td rowspan=\"1\" colspan=\"1\">92/164 (56.1)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">34/163 (20.9)</td><td rowspan=\"1\" colspan=\"1\">66/163 (40.5)</td><td rowspan=\"1\" colspan=\"1\">80/163 (49.1)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Ferris et al. [##REF##30887876##17##]</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Guselkumab</td><td rowspan=\"1\" colspan=\"1\">31/62 (50.0)</td><td rowspan=\"1\" colspan=\"1\">47/62 (75.8)</td><td rowspan=\"1\" colspan=\"1\">55/62 (88.7)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">0/16 (0)</td><td rowspan=\"1\" colspan=\"1\">0/16 (0)</td><td rowspan=\"1\" colspan=\"1\">0/16 (0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">McInnes et al. [##REF##32386593##18##]</td><td rowspan=\"1\" colspan=\"1\">52 weeks</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">99/215 (46)</td><td rowspan=\"1\" colspan=\"1\">140/215 (54)</td><td rowspan=\"1\" colspan=\"1\">170/215 (79)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">52 weeks</td><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">61/202 (30)</td><td rowspan=\"1\" colspan=\"1\">87/202 (43)</td><td rowspan=\"1\" colspan=\"1\">123/202 (61)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Mease et al. [##REF##32178766##19##]</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">132/283 (46.6)</td><td rowspan=\"1\" colspan=\"1\">158/283 (55.8)</td><td rowspan=\"1\" colspan=\"1\">195/238 (68.9)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">ixekizumab</td><td rowspan=\"1\" colspan=\"1\">170/283 (60.1)</td><td rowspan=\"1\" colspan=\"1\">203/283 (71.7)</td><td rowspan=\"1\" colspan=\"1\">227/283 (80.2)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Reich et al. [##REF##31402114##20##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">GUSELKUMAB</td><td rowspan=\"1\" colspan=\"1\">311/534 (58)</td><td rowspan=\"1\" colspan=\"1\">369/534 (69)</td><td rowspan=\"1\" colspan=\"1\">477/534 (89)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">249/514 (48)</td><td rowspan=\"1\" colspan=\"1\">391/514 (76)</td><td rowspan=\"1\" colspan=\"1\">471/514 (92)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"5\" colspan=\"1\">Ohtsuki et al. [##REF##31237727##21##]</td><td rowspan=\"2\" colspan=\"1\">16 weeks</td><td rowspan=\"2\" colspan=\"1\">Risankizumab75 mg</td><td rowspan=\"2\" colspan=\"1\">13/58 (22.4)</td><td rowspan=\"2\" colspan=\"1\">–</td><td rowspan=\"2\" colspan=\"1\">52/58 (89.8)  </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Risankizumab 150 mg</td><td rowspan=\"1\" colspan=\"1\">18/55 (32.7)</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">52/55 (94.5)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">0/0</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">5/58 (8.6)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Reich et al. [##REF##31280967##22##]</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Risankizumab 150 mg</td><td rowspan=\"1\" colspan=\"1\">120/301 (40)</td><td rowspan=\"1\" colspan=\"1\">218/301 (72)</td><td rowspan=\"1\" colspan=\"1\">237/301 (91)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">70/304 (23)</td><td rowspan=\"1\" colspan=\"1\">144/304 (47)</td><td rowspan=\"1\" colspan=\"1\">218/304 (72)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">Mease et al. [##REF##29550766##23##]</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">31/332 (9.3)</td><td rowspan=\"1\" colspan=\"1\">40/332 (12.3)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Secukinumab 150 mg</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">81/220 (36.8)</td><td rowspan=\"1\" colspan=\"1\">132/220 (60.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Secukinumab 300 mg</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">119/222 (53.6)</td><td rowspan=\"1\" colspan=\"1\">155/222 (70.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Gottlieb et al. [##REF##29660421##24##]</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">0/51 (0.0)</td><td rowspan=\"1\" colspan=\"1\">0/51 (0.0)</td><td rowspan=\"1\" colspan=\"1\">3/51 (6.5)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">certolizumab 200 mg</td><td rowspan=\"1\" colspan=\"1\">13/95 (13.7)</td><td rowspan=\"1\" colspan=\"1\">34/95 (35.8)</td><td rowspan=\"1\" colspan=\"1\">63/95 (66.3)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Gottlieb et al. [##REF##29660421##24##]</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">1/49 (1.8)</td><td rowspan=\"1\" colspan=\"1\">2/49 (2.2)</td><td rowspan=\"1\" colspan=\"1\">6/49 (11.6)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">certolizumab 200 mg</td><td rowspan=\"1\" colspan=\"1\">14/91 (15.4)</td><td rowspan=\"1\" colspan=\"1\">48/91 (52.6)</td><td rowspan=\"1\" colspan=\"1\">74/92 (81.4)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Lebwohl et al. [##REF##29660425##25##]</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">5/57 (0.0)</td><td rowspan=\"1\" colspan=\"1\">3/57 (5.3)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">certolizumab 200 mg</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">66/165 (40.0)</td><td rowspan=\"1\" colspan=\"1\">113/165 (68.5)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"4\" colspan=\"1\">Gordon et al. [##REF##30097359##26##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">0/102 (0.0)</td><td rowspan=\"1\" colspan=\"1\">2/102 (2.0)</td><td rowspan=\"1\" colspan=\"1\">10/102 (9.8)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">ustekinumab*</td><td rowspan=\"1\" colspan=\"1\">12/100 (12.0)</td><td rowspan=\"1\" colspan=\"1\">42/100 (42.0)</td><td rowspan=\"1\" colspan=\"1\">70/100 (70)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Risankizumab</td><td rowspan=\"1\" colspan=\"1\">109/304 (35.9)</td><td rowspan=\"1\" colspan=\"1\">229/304 (75.3)</td><td rowspan=\"1\" colspan=\"1\">264/304 (86.8)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"4\" colspan=\"1\">Gordon et al. [##REF##30097359##26##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">2/98 (2.0)</td><td rowspan=\"1\" colspan=\"1\">2/98 (2.0)</td><td rowspan=\"1\" colspan=\"1\">8/98 (8.1)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">ustekinumab</td><td rowspan=\"1\" colspan=\"1\">24/99 (24.2)</td><td rowspan=\"1\" colspan=\"1\">47/99 (47.5)</td><td rowspan=\"1\" colspan=\"1\">69/99 (69.7)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Risankizumab</td><td rowspan=\"1\" colspan=\"1\">149/294 (50.7)</td><td rowspan=\"1\" colspan=\"1\">220/294 (74.9)</td><td rowspan=\"1\" colspan=\"1\">261/294 (88.8)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Reich et al. [##REF##30367462##27##]</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">0/65 (0.0)</td><td rowspan=\"1\" colspan=\"1\">1/65 (1.5)</td><td rowspan=\"1\" colspan=\"1\">3/65 (4.6)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Secukinumab 300 mg</td><td rowspan=\"1\" colspan=\"1\">22/66 (33.3)</td><td rowspan=\"1\" colspan=\"1\">48/66 (72.7)</td><td rowspan=\"1\" colspan=\"1\">56/66 (84.8)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Bagel et al. [##REF##30334147##28##]</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Secukinumab</td><td rowspan=\"1\" colspan=\"1\">249/550 (45.3)</td><td rowspan=\"1\" colspan=\"1\">421/550 (76.6)</td><td rowspan=\"1\" colspan=\"1\">504/550 (91.7)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">ustekinumab</td><td rowspan=\"1\" colspan=\"1\">147/552 (26.7)</td><td rowspan=\"1\" colspan=\"1\">299/552 (54.1)</td><td rowspan=\"1\" colspan=\"1\">440/552 (79.8)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Reich et al. [##REF##28596043##29##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">2/154 (1.3)</td><td rowspan=\"1\" colspan=\"1\">4/154 (3.0)</td><td rowspan=\"1\" colspan=\"1\">9/154 (5.8)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">tildrakizumab 100 mg</td><td rowspan=\"1\" colspan=\"1\">43/309 (13.9)</td><td rowspan=\"1\" colspan=\"1\">107/309 (35.0)</td><td rowspan=\"1\" colspan=\"1\">197/309 (63.8)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"4\" colspan=\"1\">Reich et al. [##REF##28596043##29##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">0/156 (0.0)</td><td rowspan=\"1\" colspan=\"1\">2/156 (1.3)</td><td rowspan=\"1\" colspan=\"1\">9/156 (5.8)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">tildrakizumab 100 mg*</td><td rowspan=\"1\" colspan=\"1\">38/307 (12.4)</td><td rowspan=\"1\" colspan=\"1\">119/307 (38.8)</td><td rowspan=\"1\" colspan=\"1\">188/307 (61.2)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">15/313 (4.8)</td><td rowspan=\"1\" colspan=\"1\">67/313 (21.4)</td><td rowspan=\"1\" colspan=\"1\">151/313 (48.2)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Reich et al. [##REF##28542874##30##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">ixekizumab</td><td rowspan=\"1\" colspan=\"1\">49/136 (36.0)</td><td rowspan=\"1\" colspan=\"1\">99/136 (72.8)</td><td rowspan=\"1\" colspan=\"1\">120/136 (88.2)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">ustekinumab</td><td rowspan=\"1\" colspan=\"1\">24/166 (14.5)</td><td rowspan=\"1\" colspan=\"1\">70/166 (42, 2)</td><td rowspan=\"1\" colspan=\"1\">114/166 (68.7)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"4\" colspan=\"1\">Blauvelt et al. [##REF##28057360##31##]</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">1/174(0.6)</td><td rowspan=\"1\" colspan=\"1\">5/174 (2.9)</td><td rowspan=\"1\" colspan=\"1\">10/174 (5.7)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">GUSELKUMAB*</td><td rowspan=\"1\" colspan=\"1\">123/329 (37.4)</td><td rowspan=\"1\" colspan=\"1\">241/329 (73, 3)</td><td rowspan=\"1\" colspan=\"1\">300/329 (91.2)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">57/334 (17.4)</td><td rowspan=\"1\" colspan=\"1\">166/334 (49.7)</td><td rowspan=\"1\" colspan=\"1\">244/334 (73.1)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"4\" colspan=\"1\">Mease et al. [##REF##27553214##32##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">1/67 (1.5)</td><td rowspan=\"1\" colspan=\"1\">1/67 (1.5)</td><td rowspan=\"1\" colspan=\"1\">5/67 (7.5)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">ixekizumab Q4W*</td><td rowspan=\"1\" colspan=\"1\">23/73 (31.5)</td><td rowspan=\"1\" colspan=\"1\">38/73 (52.0)</td><td rowspan=\"1\" colspan=\"1\">55/73 (75.3)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">10/68 (14.7)</td><td rowspan=\"1\" colspan=\"1\">15/68 (22.1)</td><td rowspan=\"1\" colspan=\"1\">23/68 (33.8)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Gordon et al. [##REF##27299809##33##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">0/431 (0.0)</td><td rowspan=\"1\" colspan=\"1\">7/431 (1.7)</td><td rowspan=\"1\" colspan=\"1\">17/431 (3.9)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">ixekizumab Q4W</td><td rowspan=\"1\" colspan=\"1\">145/432 (33.6)</td><td rowspan=\"1\" colspan=\"1\">279/432 (64.6)</td><td rowspan=\"1\" colspan=\"1\">357/432 (82.6)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Papp et al. [##REF##26914406##34##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">1/220 (0.5)</td><td rowspan=\"1\" colspan=\"1\">2/220 (0.9)</td><td rowspan=\"1\" colspan=\"1\">6/220 (2.7)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">brodalumab</td><td rowspan=\"1\" colspan=\"1\">93/222 (41.9)</td><td rowspan=\"1\" colspan=\"1\">156/220 (70.9)</td><td rowspan=\"1\" colspan=\"1\">185/222 (83.3)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"4\" colspan=\"1\">Griffiths et al. [##REF##26072109##35##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">1/168 (0.6)</td><td rowspan=\"1\" colspan=\"1\">1/168 (0.6)</td><td rowspan=\"1\" colspan=\"1\">4/168 (2.4)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">ixekizumab Q4W*</td><td rowspan=\"1\" colspan=\"1\">107/347 (30.8)</td><td rowspan=\"1\" colspan=\"1\">267/347 (76.9)</td><td rowspan=\"1\" colspan=\"1\">269/347 (77.5)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">19/358 (5.3)</td><td rowspan=\"1\" colspan=\"1\">67/358 (18.7)</td><td rowspan=\"1\" colspan=\"1\">149/358 (41.6)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"4\" colspan=\"1\">Griffiths et al. [##REF##26072109##35##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">0/193 (0.0)</td><td rowspan=\"1\" colspan=\"1\">6/193 (3.1)</td><td rowspan=\"1\" colspan=\"1\">14/193 (7.2)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">ixekizumab*</td><td rowspan=\"1\" colspan=\"1\">135/386 (35.0)</td><td rowspan=\"1\" colspan=\"1\">352/386 (91.2)</td><td rowspan=\"1\" colspan=\"1\">325/386 (84.2)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">19/358 (5.3)</td><td rowspan=\"1\" colspan=\"1\">98/382 (25.6)</td><td rowspan=\"1\" colspan=\"1\">201/382 (52.6)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"4\" colspan=\"1\">Lebwohl et al. [##REF##26422722##36##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">2/309 (0.6)</td><td rowspan=\"1\" colspan=\"1\">12/309 (3.9)</td><td rowspan=\"1\" colspan=\"1\">25/309 (8.1)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">ustekinumab</td><td rowspan=\"1\" colspan=\"1\">65/300 (21.7)</td><td rowspan=\"1\" colspan=\"1\">141/300 (47.0)</td><td rowspan=\"1\" colspan=\"1\">210/300 (70.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">brodalumab 210 mg</td><td rowspan=\"1\" colspan=\"1\">272/612 (44.4)</td><td rowspan=\"1\" colspan=\"1\">428/612 (69.9)</td><td rowspan=\"1\" colspan=\"1\">528/612 (86.3)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"4\" colspan=\"1\">Lebwohl et al. [##REF##26422722##36##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">1/315 (0.3)</td><td rowspan=\"1\" colspan=\"1\">6/315 (1.9)</td><td rowspan=\"1\" colspan=\"1\">19/315 (6.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">ustekinumab</td><td rowspan=\"1\" colspan=\"1\">58/313 (18.5)</td><td rowspan=\"1\" colspan=\"1\">141/313 (45.0)</td><td rowspan=\"1\" colspan=\"1\">217/313 (69.3)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">brodalumab</td><td rowspan=\"1\" colspan=\"1\">229/624 (36.7)</td><td rowspan=\"1\" colspan=\"1\">430/624 (68.9)</td><td rowspan=\"1\" colspan=\"1\">531/624 (85.1)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Thaci et al. [##REF##26092291##37##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">SEC</td><td rowspan=\"1\" colspan=\"1\">148/334 (44.3)</td><td rowspan=\"1\" colspan=\"1\">264/334 (79.0)</td><td rowspan=\"1\" colspan=\"1\">311/334 (93.1)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">ustekinumab</td><td rowspan=\"1\" colspan=\"1\">130/334 (38.9)</td><td rowspan=\"1\" colspan=\"1\">277/334 (82.9)</td><td rowspan=\"1\" colspan=\"1\">277/334 (82.9)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Langley et al. [##REF##25007392##38##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">2/246 (0.8)</td><td rowspan=\"1\" colspan=\"1\">3/246 (1.2)</td><td rowspan=\"1\" colspan=\"1\">11/246 (4.5)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Secukinumab 300 mg</td><td rowspan=\"1\" colspan=\"1\">70/245 (26.6)</td><td rowspan=\"1\" colspan=\"1\">145/245 (59.2)</td><td rowspan=\"1\" colspan=\"1\">200/245 (81.6)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"4\" colspan=\"1\">Langley et al. [##REF##25007392##38##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">0/324 (0.0)</td><td rowspan=\"1\" colspan=\"1\">5/324 (1.5)</td><td rowspan=\"1\" colspan=\"1\">16/324 (4.9)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Secukinumab 300 mg</td><td rowspan=\"1\" colspan=\"1\">78/323 (24.1)</td><td rowspan=\"1\" colspan=\"1\">175/323 (54.1)</td><td rowspan=\"1\" colspan=\"1\">249/323 (77.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">14/323 (4.3)</td><td rowspan=\"1\" colspan=\"1\">67/323 (20.7)</td><td rowspan=\"1\" colspan=\"1\">142/323 (44.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Mease et al. [##REF##23942868##39##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">4/86 (4.7)</td><td rowspan=\"1\" colspan=\"1\">12/86 (13.9)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">certolizumab 200 mg</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">20/90 (22.2)</td><td rowspan=\"1\" colspan=\"1\">42/90 (46.7)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Baranauskaite et al. [##REF##21994233##40##]</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Methotrexate</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">19/35 (54.3%)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Methotrexate+ infliximab</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">33/34 (97.1%)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Gottlieb et al. [##REF##22533447##41##]</td><td rowspan=\"1\" colspan=\"1\">24 weeks</td><td rowspan=\"1\" colspan=\"1\">Methotrexate + etanercept</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">184/239 (77.3%)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">24 weeks</td><td rowspan=\"1\" colspan=\"1\">etanercept+ Placebo</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">144/239 (60·3%)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Barker et al. [##REF##21910713##42##]</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Methotrexate</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">41/216 (19.0)</td><td rowspan=\"1\" colspan=\"1\">90/216 (41.7)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">Infliximab</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">356/656 (54.2)</td><td rowspan=\"1\" colspan=\"1\">508/656 (77.4)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"4\" colspan=\"1\">Gottlieb et al. [##REF##21574983##43##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">0/68 (0.0)</td><td rowspan=\"1\" colspan=\"1\">1/68 (1.5)</td><td rowspan=\"1\" colspan=\"1\">5/68 (7.4)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">briakinumab</td><td rowspan=\"1\" colspan=\"1\">39/138 (28.3)</td><td rowspan=\"1\" colspan=\"1\">83/138 (60.0)</td><td rowspan=\"1\" colspan=\"1\">112/138 (81.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">5/141 (3.6)</td><td rowspan=\"1\" colspan=\"1\">18/141 (12.7)</td><td rowspan=\"1\" colspan=\"1\">78/141 (55.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"4\" colspan=\"1\">Strober et al. [##REF##21574984##44##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">0/72 (0.0)</td><td rowspan=\"1\" colspan=\"1\">3/72 (4.2)</td><td rowspan=\"1\" colspan=\"1\">5/72 (6.9)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">briakinumab</td><td rowspan=\"1\" colspan=\"1\">30/139 (21.9)</td><td rowspan=\"1\" colspan=\"1\">83/139 (60)</td><td rowspan=\"1\" colspan=\"1\">111/139 (80.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">5/139 (3.6)</td><td rowspan=\"1\" colspan=\"1\">18/139 (13.0)</td><td rowspan=\"1\" colspan=\"1\">40/139 (28.8)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Kavanaugh et al. [##REF##19333944##45##]</td><td rowspan=\"1\" colspan=\"1\">14 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">0/73 (0.0)</td><td rowspan=\"1\" colspan=\"1\">2/79 (2.5)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">14 weeks</td><td rowspan=\"1\" colspan=\"1\">golimumab 50 mg</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">22/106 (20.8)</td><td rowspan=\"1\" colspan=\"1\">44/109 (40.3)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Leonardi et al. [##REF##18486739##46##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">0/255 (0.0)</td><td rowspan=\"1\" colspan=\"1\">5/255 (2.0)</td><td rowspan=\"1\" colspan=\"1\">5/255 (2.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">ustekinumab 45 mg</td><td rowspan=\"1\" colspan=\"1\">32/255 (12.5)</td><td rowspan=\"1\" colspan=\"1\">106/255 (41.6)</td><td rowspan=\"1\" colspan=\"1\">171/255 (67.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Menter et al. [##REF##17936411##47##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">4/398 (1.0)</td><td rowspan=\"1\" colspan=\"1\">8/398 (2.0)</td><td rowspan=\"1\" colspan=\"1\">20/398 (5.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">114/814 (14.0)</td><td rowspan=\"1\" colspan=\"1\">301/814 (37.0)</td><td rowspan=\"1\" colspan=\"1\">554/814 (68.1)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Papp et al. [##REF##18486740##48##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">0/410 (0.0)</td><td rowspan=\"1\" colspan=\"1\">3/410 (0.7)</td><td rowspan=\"1\" colspan=\"1\">15/410 (3.7)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">ustekinumab 45 mg</td><td rowspan=\"1\" colspan=\"1\">74/409 (18.1)</td><td rowspan=\"1\" colspan=\"1\">173/409 (42.3)</td><td rowspan=\"1\" colspan=\"1\">273/409 (66.5)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Tyring et al. [##REF##17576937##49##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">1/292 (0.3)</td><td rowspan=\"1\" colspan=\"1\">5/292 (1.7)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Etanercept</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">21/305 (6.9)</td><td rowspan=\"1\" colspan=\"1\">47/305 (15.4)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Antoni et al. [##REF##15677701##50##]</td><td rowspan=\"1\" colspan=\"1\">14 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">0/87 (0.0)</td><td rowspan=\"1\" colspan=\"1\">1/87 (1.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">14 weeks</td><td rowspan=\"1\" colspan=\"1\">Infliximab</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">34/87 (41.0)</td><td rowspan=\"1\" colspan=\"1\">55/87 (64.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"3\" colspan=\"1\">Mease et al. [##REF##16200601##51##]</td><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">0/69 (0.0)</td><td rowspan=\"1\" colspan=\"1\">4/69 (5.8)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12 weeks</td><td rowspan=\"1\" colspan=\"1\">Adalimumab</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">30/69 (43.5)</td><td rowspan=\"1\" colspan=\"1\">49/69 (71.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"3\" colspan=\"1\">Reich et al. [##REF##16226614##52##]</td><td rowspan=\"1\" colspan=\"1\">10 weeks</td><td rowspan=\"1\" colspan=\"1\">Placebo</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">1/77 (1.0)</td><td rowspan=\"1\" colspan=\"1\">2/77 (3.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">10 weeks</td><td rowspan=\"1\" colspan=\"1\">Infliximab</td><td rowspan=\"1\" colspan=\"1\">–</td><td rowspan=\"1\" colspan=\"1\">172/301 (57.0)</td><td rowspan=\"1\" colspan=\"1\">242/301 (80.0)</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significance</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\">Significant</td><td rowspan=\"1\" colspan=\"1\"> </td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Sattam A. Alzahrani , Abdulaziz M. Aljuhni, Norah Y. Alqahtani, Naif A. Al Thaqfan, Sara A. Alwarwari, Rakan A. Alzabadin, Abdullah A. Alhajlah, Fajer M. Alzamil, Reema A. Alzehairi</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Sattam A. Alzahrani , Abdulaziz M. Aljuhni, Naif A. Al Thaqfan, Sara A. Alwarwari, Abdullah A. Alkharashi, Rakan A. Alzabadin, Abdullah A. Alhajlah, Fajer M. Alzamil</p><p><bold>Drafting of the manuscript:</bold>  Sattam A. Alzahrani , Abdulaziz M. Aljuhni, Norah Y. Alqahtani, Naif A. Al Thaqfan, Sara A. Alwarwari, Abdullah A. Alkharashi, Rakan A. Alzabadin, Abdullah A. Alhajlah, Fajer M. Alzamil, Reema A. Alzehairi</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Sattam A. Alzahrani , Abdulaziz M. Aljuhni, Norah Y. Alqahtani, Sara A. Alwarwari, Abdullah A. Alkharashi, Rakan A. Alzabadin, Abdullah A. Alhajlah, Fajer M. Alzamil</p><p><bold>Supervision:</bold>  Fajer M. Alzamil</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050588-i01\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Diverticulitis, an inflammatory condition affecting colonic diverticula, is a relatively common gastrointestinal ailment in the general population [##REF##26416187##1##,##UREF##0##2##]. Its severity is broad but commonly presents with signs of inflammation and left lower quadrant pain due to its propensity to affect the sigmoid colon [##REF##32218442##3##, ####REF##24430321##4##, ##REF##35977135##5####35977135##5##].</p>", "<p>Although the incidence of diverticular disease has seen an increase in younger populations [##REF##26416187##1##,##UREF##0##2##,##REF##19212172##6##,##REF##21876861##7##], the co-occurrence of diverticulitis and pregnancy is incredibly rare, with only around a dozen documented case reports and two observational studies [##REF##36438049##8##, ####REF##15565304##9##, ##REF##11383700##10##, ##REF##9950133##11####9950133##11##], and may often be overlooked when evaluating an acute abdomen in the obstetric population [##REF##26526440##12##]. Furthermore, one study reviewing a 20-year period of pregnancies within an obstetric department reported an incidence of one in 6000 pregnancies [##REF##21629625##13##], but these cases included small bowel diverticular diseases like Meckel’s diverticulitis. The convergence of these two distinct clinical scenarios can be particularly complex and requires careful consideration to optimize both maternal and fetal outcomes. As such, diverticulitis in pregnancy presents a unique and intricate medical challenge, demanding nuanced decision-making and a multidisciplinary approach to care regarding fetal and maternal outcomes. This was possible because the patient was managed in a large regional medical center. Additionally, the SCARE and CARE Checklist has been completed by the authors for this case report and attached as online supplementary material (Appendices).</p>" ]

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[ "<title>Discussion</title>", "<p>Early in training, diverticulitis is taught as a quintessential gastrointestinal diagnosis. However, when paired with other comorbid conditions and, in this case, several potential contraindications in management, it is important to realize the extent to which risk-benefit analysis plays a more instructive role than the general algorithms for treatment of a common and increasingly prevalent gastrointestinal disease.</p>", "<p>This case of diverticulitis in pregnancy is only one of a few cases, each with significant differences concerning both the mother’s health and the fetal status. It is these differences that provide both the strengths and limitations of this case report. It is unlikely a case with these exact circumstances will occur again. Therefore, a major limitation of this case report is that these diagnostic steps and treatment options were tailored for this specific case. However, the strength is that should a similar case present in the future, this along with several other cases will illuminate more patterns in better ways to care for diverticulitis in pregnancy. Furthermore, this case resulted in positive outcomes both for the newborn and the mother, which may inform future decisions regarding diagnostic and treatment considerations compared to other case reports.</p>", "<p>As mentioned earlier, the existing literature on diverticulitis during pregnancy is notably limited. Moreover, even within comparable instances of sigmoid diverticulitis, the complications and gestational age varied. A study by Kechagias et al. conducted a systematic review revealing all documented cases of colonic diverticulitis in pregnancy amounting to just 12 cases [##REF##36438049##8##]. In one case, the patient was at 33 weeks’ gestation and experienced only one episode of diverticular inflammation before undergoing delivery and subsequent resection due to small bowel obstruction [##REF##11383700##10##]. Furthermore, The American College of Obstetricians and Gynecologists (ACOG) published the ACOG committee opinion number 723, which is supported by the American College of Radiology and the American Institute of Ultrasound in Medicine [##UREF##1##14##]. The outcomes of this case report and similar literature provide evidence to support this committee's opinion.</p>" ]

[ "<title>Conclusions</title>", "<p>Although diagnosis and treatment of diverticular disease have been well documented and should serve as an initial guide when making initial considerations, management of complications of diverticulitis, such as abscesses and recurrent flares, should be managed with drainage when possible and frequent follow-up until delivery. Resection should be considered in patients where inflammation, pain, or other complications do not resolve in the postpartum period.</p>", "<p>The complex decision-making regarding induction of preterm fetuses for maternal beneficence. CT-guided drainage of the abscess was necessary to prevent further perforation and worsening of infection, possibly leading to peritoneal sepsis. In this case, risk-benefit truly takes on a new meaning and is more akin to which risk is the patient willing to take: fetal demise or a possible worsening of the diverticulitis until the sigmoid abscess completely perforates.</p>", "<p>Timing and management of recurrent diverticulitis in pregnancy necessitate further research to establish comprehensive guidelines tailored to these at-risk populations.</p>" ]

[ "<p>The co-occurrence of diverticulitis with pregnancy is incredibly rare and the management of recurrent complicated diverticulitis may not be feasible in a pregnant patient. Adding cases to the incredibly sparse literature may highlight similarities and create potential recommendations for at-risk populations. We present a case of a female at 18 weeks’ gestation who presented with left lower quadrant pain. The initial physical exam and clinical findings revealed severe abdominal tenderness without signs of generalized peritonitis, leukocytosis with predominant neutrophils, and fundal height with confirmatory ultrasonography of intrauterine pregnancy. The main diagnosis was diverticulitis complicated by an abscess and pregnancy, confirmed with CT imaging. The initial intervention was IV antibiotics and bowel rest; however, with each subsequent discharge, she returned to the emergency department with worsening symptoms. Eventually, at 28 weeks, she was escalated to IV meropenem, CT-guided drainage of the abscess, and preterm vaginal delivery at 28 weeks, with a lower anterior resection and diverting ileostomy at six weeks postpartum. She is currently being followed outpatient with improvement in pain, meaningful healthy weight gain, and a healthy newborn child. While management of diverticulitis is generally straightforward, severe presentations like this, even when compared to existing literature, suggest traditional notions of contraindications and risks may not fully apply. Timing and management of recurrent diverticulitis in pregnancy necessitate further research to establish comprehensive guidelines tailored to these at-risk populations.</p>" ]

[ "<title>Case presentation</title>", "<p>The patient is a 30-year-old Caucasian female, 18 weeks’ gestation, with a history of diverticulosis. The patient initially reported a history of left lower quadrant pain and nausea. The pain was described as sharp/stabbing without radiation, severe, and sudden for 16 hours. She was referred from an outside hospital and had already been started on IV antibiotics and pain control. Ultrasound and CT without contrast from the referring hospital revealed an intrauterine pregnancy at 18 weeks’ gestation and a left sigmoid diverticular abscess.</p>", "<p>On admission, the patient was alert but in acute distress with an ill appearance. Her vitals were temperature 100.1 °F (37.8 °C), heart rate 118, respiratory rate 26, blood pressure 140/77, and O2 99%. Mucous membranes dry, left lower quadrant (LLQ) abdominal tenderness, guarding, and hypoactive bowel sounds but without generalized peritoneal findings. Table ##TAB##0##1## reveals the initial abnormal tests and Table ##TAB##1##2## is a timeline of the relevant hospital encounters.</p>", "<p>Diagnostic assessment and therapeutic interventions</p>", "<p>She was ultimately diagnosed with diverticulitis complicated by a 3 cm sigmoid abscess, Figure ##FIG##0##1##, and a singleton 18-week fetus with no signs of subchorionic hemorrhage or other abnormalities. CT-guided drainage was not considered during this ED visit due to potential complications secondary to the proximity of the abscess to the placenta. She was placed on IV piperacillin-tazobactam, hydromorphone, and fluids and was discharged once her pain and infection were at manageable levels.</p>", "<p>However, over the subsequent weeks, she had multiple flares, as outlined in Table ##TAB##1##2##. Ultimately, the discussion for CT-guided drainage of the now-confirmed sigmoid abscess was discussed with the patient and her family as a preintervention step in managing her diverticulitis. Furthermore, given the lack of meaningful resolution of her diverticulitis, the option of a partial colectomy was discussed; however, this was once she was well into her postpartum period or in the event of an emergency following her CT-guided drainage at 20 weeks (Figures ##FIG##1##2##, ##FIG##2##3##). </p>", "<p>Once her diverticulitis flares were controlled with IV antibiotics, which eventually escalated to meropenem, and the baby was delivered and out of the Neonatal Intensive Care Unit (NICU), she underwent lower anterior resection of her colon at six weeks postpartum.</p>", "<p>A laparoscopic approach was performed, via a median infraumbilical incision. However, after a safe entrance to the abdomen was obtained, pelvic inspection revealed multiple loops of bowel scarred into the pelvis, initially suggested by CT imaging in Figure ##FIG##3##4##, thus making the operation unsafe for a laparoscopic approach. The median incision was enlarged, and the procedure was converted into an open approach. The multiple loops of small bowel were dissected from their attachments into the site of the pelvis abscess to mobilize them to expose the colon. Once the colon was mobilized, dissection into the pelvis revealed a significant hydroureter which prompted an intraoperative urological consult and stent placement. Following stent placement, further dissection revealed the extent of scar tissue encapsulating a majority of the rectum, vaginal, uterus, the entirety of the sigmoid colon, and a distal portion of the descending colon. Once the scar tissue was adequately dissected and the inflamed colon was separated from the healthy colon, an end-to-end anastomosis from the healthy proximal colon to the rectum was achieved. However, due to the extent of the inflammation and infection, this was considered high risk, and a diverting ileostomy was put in place along with a 19 French black drain.</p>", "<p>Pathology revealed 22.5 cm in length and 2.2 cm in diameter sigmoid and rectum with multiple diverticula and peri-diverticular inflammation, benign pseudocyst in subserosa with peri-cystic inflammation, negative for dysplasia/malignancy, and a single benign reactive lymph node that was negative for malignancy.</p>", "<p>The patient was admitted and placed on IV piperacillin-tazobactam 4.5 g 100 mL run in at 25 mL/hr with pain management rotating on an as-needed basis between ketorolac 30 mg IV and hydromorphone 0.5 mg IV, and oxycodone-acetaminophen 325 mg by mouth (PO) once oral feeding was tolerated.</p>", "<p>Postoperative monitoring confirmed the stability of the patient’s vital signs. Her general condition had improved, and the patient was discharged four days after surgery, afebrile. She had a stoma output of 200 cc, drain output of 100 cc of serosanguinous fluid, and urine output of 650 cc. Incisions were well healed. She was discharged on amoxicillin-clavulanate 125 mg twice daily PO for 10 days and oxycodone-acetaminophen 325 mg every six hours/as needed PO for seven days. She followed up three weeks later no longer in pain, her stoma functioning well, and regaining weight. Her child is healthy and doing well.</p>" ]

[ "<title>Appendices</title>" ]

[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Confirmatory CT Imaging without Contrast of Sigmoid Diverticular Abscess</title><p>CT scans used dose modulation, iterative reconstruction, and/or weight-based dosing to reduce radiation dose to as low as reasonably achievable.</p><p>Impression:</p><p>A: 2.8 x 3.9 cm fluid and gas collection within the rectal pouch of Douglas tracking along the right posterior lateral and superior uterus.</p><p>B: Sigmoid colonic inflammatory changes and ascites when compared to previous imaging. Prominent gas and stool-filled diverticulum noted extending along the sigmoid antimesenteric border. No gross pneumoperitoneum.</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>CT-Guided Drain Placement</title><p>Impression: CT-guided right 8 French transgluteal drain placement into rectouterine pouch fluid collection.</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG3\"><label>Figure 3</label><caption><title>Drain Visualized in Abscess</title><p>Impression: CT-guided right 8 French transgluteal drain placement into rectouterine pouch fluid collection.</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG4\"><label>Figure 4</label><caption><title>Abdominal and Pelvic CT with IV Contrast Imaging before Lower Anterior Resection</title><p>Impression:</p><p>A: Severe diffuse sigmoid colon wall thickening which could reflect colitis or diverticulitis, hyperdensity extrinsic to the sigmoid colon. Pelvic small bowel inflammatory changes with wall thickening, tethering, and areas of distension.</p><p>B: Small pelvic air-fluid collection suspicious for an interloop abscess on axial image on the right.</p><p>Diverticulitis may be classified as Hinchey/Kaiser Ib according to CT imaging impression.</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG5\"><label>Figure 5</label><caption><title>CARE Guideline</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG6\"><label>Figure 6</label><caption><title>SCARE Guideline</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG7\"><label>Figure 7</label><caption><title>Patient Perspective</title><p>The patient's perspective was paraphrased from follow-up encounters post-operatively through discussions with the attending colorectal surgeon.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Abnormal Results of Initial Diagnostic Laboratory Tests</title><p>CRP - C-reactive protein</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Test</td><td rowspan=\"1\" colspan=\"1\">Unit</td><td rowspan=\"1\" colspan=\"1\">Value</td><td rowspan=\"1\" colspan=\"1\">Normal range</td></tr><tr><td rowspan=\"1\" colspan=\"1\">White Blood Cells</td><td rowspan=\"1\" colspan=\"1\">1000/µl </td><td rowspan=\"1\" colspan=\"1\">14.9</td><td rowspan=\"1\" colspan=\"1\">3.5 - 10.8</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Absolute Neutrophils</td><td rowspan=\"1\" colspan=\"1\">1000/µl </td><td rowspan=\"1\" colspan=\"1\">13.6</td><td rowspan=\"1\" colspan=\"1\">1.8 - 7.7</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Percent Neutrophils</td><td rowspan=\"1\" colspan=\"1\">%</td><td rowspan=\"1\" colspan=\"1\">91</td><td rowspan=\"1\" colspan=\"1\">50 - 70</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Absolute Lymphocytes</td><td rowspan=\"1\" colspan=\"1\">1000/µl </td><td rowspan=\"1\" colspan=\"1\">0.8</td><td rowspan=\"1\" colspan=\"1\">1.0 - 4.8</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Percent Lymphocytes</td><td rowspan=\"1\" colspan=\"1\">%</td><td rowspan=\"1\" colspan=\"1\">5</td><td rowspan=\"1\" colspan=\"1\">18 - 42</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Blood urea nitrogen</td><td rowspan=\"1\" colspan=\"1\">mg/dL</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">7 - 20</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Co2</td><td rowspan=\"1\" colspan=\"1\">mmol/L</td><td rowspan=\"1\" colspan=\"1\">20</td><td rowspan=\"1\" colspan=\"1\">23 - 29</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Total calcium</td><td rowspan=\"1\" colspan=\"1\">mg/dL</td><td rowspan=\"1\" colspan=\"1\">8.4</td><td rowspan=\"1\" colspan=\"1\">8.8 - 10.6</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Protein</td><td rowspan=\"1\" colspan=\"1\">g/dL</td><td rowspan=\"1\" colspan=\"1\">5.8</td><td rowspan=\"1\" colspan=\"1\">6 - 8.3</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Albumin</td><td rowspan=\"1\" colspan=\"1\">g/dL</td><td rowspan=\"1\" colspan=\"1\">2.8</td><td rowspan=\"1\" colspan=\"1\">3.5 - 5.0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Quantitative CRP</td><td rowspan=\"1\" colspan=\"1\">mg/L</td><td rowspan=\"1\" colspan=\"1\">90.2</td><td rowspan=\"1\" colspan=\"1\">&lt; 3.0</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Procalcitonin </td><td rowspan=\"1\" colspan=\"1\">ng/mL</td><td rowspan=\"1\" colspan=\"1\">17.61</td><td rowspan=\"1\" colspan=\"1\">&lt; 0.1</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Hemoglobin </td><td rowspan=\"1\" colspan=\"1\">g/dL</td><td rowspan=\"1\" colspan=\"1\">9.2</td><td rowspan=\"1\" colspan=\"1\">12.0 - 16.0</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Hematocrit</td><td rowspan=\"1\" colspan=\"1\">%</td><td rowspan=\"1\" colspan=\"1\">27.4</td><td rowspan=\"1\" colspan=\"1\">36.0 - 46.0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Mean corpuscular volume</td><td rowspan=\"1\" colspan=\"1\">fL</td><td rowspan=\"1\" colspan=\"1\">61</td><td rowspan=\"1\" colspan=\"1\">80 - 100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Red cell distribution width</td><td rowspan=\"1\" colspan=\"1\">%</td><td rowspan=\"1\" colspan=\"1\">17.1</td><td rowspan=\"1\" colspan=\"1\">35.1 - 46.3</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>Timeline of Relevant Data from Episodes of Care</title><p>CT - Computer Tomography, IV - Intravenous, LLQ - Left Lower Quadrant, NGT- Nasogastric Tube, MRI - Magnetic Resonance Imaging, PO - Per Os, NICU - Neonatal Intensive Care Unit, LAR - Lower Anterior Resection</p><p> </p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Date</td><td rowspan=\"1\" colspan=\"1\">Key Events</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Day 1</td><td rowspan=\"1\" colspan=\"1\">1st Emergency department visit. Initial symptoms were left lower quadrant pain and nausea. Ultrasound revealed an 18-week fetus. CT without contrast suggested a left sigmoid diverticular abscess. The patient was started on IV antibiotics, pain control, and antiemetics.</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Day 2</td><td rowspan=\"1\" colspan=\"1\">Interventional radiology did not consider CT drainage due to the position of the abscess relative to the uterus and placenta. IV antibiotics treatment showed improvement. The patient was discharged on PO antibiotics.</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Day 3</td><td rowspan=\"1\" colspan=\"1\">2nd emergency department visit. Presenting symptoms were left lower quadrant pain but with new blood in stool. Repeat CT showed a 3 cm abscess. The patient started on IV antibiotics. This CT image and all subsequent CT images performed utilize dose modulation and/or weight-based dose reduction when appropriate to reduce radiation dose to the patient as low as reasonably achievable. </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Day 5</td><td rowspan=\"1\" colspan=\"1\">Flexible sigmoidoscopy was performed. Revealed small diverticuli with purulent fluid and mild erythema, rectosigmoid polyps, but otherwise normal. Polyps were removed.</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Day 7</td><td rowspan=\"1\" colspan=\"1\">Discharged on PO antibiotics with stable labs: minimally elevated WBCs. However, mild LLQ tenderness persisted.</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Day 16</td><td rowspan=\"1\" colspan=\"1\">3rd emergency department visit. The presenting symptom was lower abdominal pain and severe abdominal distention. Labs showed significant leukocytosis (20.7 1000/mL). MRI imaging suggested enteritis of the Jejunum. The patient was started on IV antibiotics.</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Day 17</td><td rowspan=\"1\" colspan=\"1\">The patient's infection continued to worsen with leukocytosis peaking at 24.5 1000/mL (neutrophils abs 21.3 1000/mL, neutrophils 87%), Na+ 132 mEq/L, CO2 16 mmol/L, Glucose 109 mg/dL. The patient was started on meropenem and an NGT was placed.</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Day 18</td><td rowspan=\"1\" colspan=\"1\">CT imaging suggested concern for rectal perforation with small bowel ileus compared to MRI. The patient's status mildly improved. CT-guided drainage was recommended but emergency surgery was discussed for potential complications.</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Day 19</td><td rowspan=\"1\" colspan=\"1\">Leukocytosis began to resolve. Proceeded with CT-guided drain placed with no complications. Leukocytosis continued to resolve, and the patient began to ambulate. Meropenem was continued for 2 weeks.</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Day 25</td><td rowspan=\"1\" colspan=\"1\">Discharged with PO antibiotics. The patient's diet was advanced, the pain was controlled, and the drain was in place.</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Day 31</td><td rowspan=\"1\" colspan=\"1\">1st office visit. The patient had no tenderness along the incision site, &lt;5cc drainage, mild purulent, only flush returns. The drain was removed. Left lower quadrant pain was not resolved but was manageable.</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Day 45</td><td rowspan=\"1\" colspan=\"1\">2nd office visit. MRI was performed two days prior with no significant changes. No tenderness, no drainage from previous drain site. Left lower quadrant pain is still present but manageable.</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Day 64</td><td rowspan=\"1\" colspan=\"1\">The patient delivered a 28-week neonate vaginally. The neonate was placed in the NICU.</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Day 88</td><td rowspan=\"1\" colspan=\"1\">4th emergency department visit. The patient has 6 day history of constipation and lower abdominal pain. The patient was given Miralax, had multiple bowel movements, and was discharged on PO antibiotics for diverticular flares.</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Day 112</td><td rowspan=\"1\" colspan=\"1\">3rd office visit. The patient continues to have abdominal pain and poor appetite. Options for meaningful resolution of diverticulitis discussed at length: Laparoscopic vs open LAR discussed. Her child was discharged from NICU and at patient's home.</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Day 134</td><td rowspan=\"1\" colspan=\"1\">Lower anterior resection was performed. Severe scarring and adhesions. Converted to an open procedure, ureteral stent incision and drainage of the pelvis, drain placement, and diverting ileostomy.</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Day 154</td><td rowspan=\"1\" colspan=\"1\">Post-operative visit. The patient was no longer in pain, and the stoma was functioning.</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Day 169</td><td rowspan=\"1\" colspan=\"1\">Post-operative visit 2. The patient was doing well, and regaining weight. The patient's incision has healed, the stoma was healthy, and the patient no longer possesses diverticular pain.</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Nathan K. Louie</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Nathan K. Louie, Bradley Champagne</p><p><bold>Drafting of the manuscript:</bold>  Nathan K. Louie</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Nathan K. Louie, Bradley Champagne</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>A tibial fracture is a common and serious lower extremity injury that is characterized by a break or crack in the shinbone. These fractures are frequently caused by high-energy trauma, but they can also result from overuse or underlying illnesses that weaken the bone [##REF##7744927##1##]. The age, sex, location, and lifestyle of an individual can all have an impact on the incidence of tibial fractures. Tibial fractures are among the most common long bone fractures, accounting for 10-15% of all adult fractures [##REF##16814787##2##]. An estimated 492,000 visits per year were attributed to tibial fractures in the United States alone [##REF##16424818##3##]. Since these fractures can seriously impair a person's mobility, prompt and appropriate care is frequently required for the best possible healing functional recovery and quality of life [##REF##16056075##4##]. According to the American Academy of Orthopaedic Surgeons, tibial fracture accounts for 85% of all lower extremity fractures, making them an important part of orthopaedic medicine. Because of the inherent instability and limited potential for bone healing, multiple bone fragments in a fracture, also referred to as a comminuted fracture, present additional challenges [##UREF##0##5##]. Traumatic tibial fractures can take many different forms and range in severity, rendering individuals immobile and requiring immediate medical care. Conservative measures and surgical interventions, such as internal or external fixation, are among the available treatment options [##REF##7744927##1##].</p>", "<p>Intense lower limb pain is one of the symptoms and indicators, and it frequently gets worse with movement or activities that require bearing weight on the affected limbs. Together with a visible deformity or angulation of the leg, there may also be pain and swelling near the fracture site. The afflicted limb might be unable to support much weight, and the area around the wound might be bruised or discoloured. A complex fracture may, in extreme circumstances, result in an open wound or bone protrusion. In more severe cases, tingling or numbness may also be caused by damaged blood vessels or nerve involvement [##REF##27026999##6##].</p>", "<p>Open reduction and internal fixation (ORIF) is a common surgical procedure used to stabilize and align fractured bones, particularly tibial plateau fractures, with the use of a plate and screws. For the treatment of proximal tibial fractures involving the tibial plateau, especially those that are displaced or comminuted, ORIF using a plate and screws is commonly used. Using an open reduction technique, the fractured bone is exposed, the fragments are realigned, and then they are stabilized with a metal plate and screws. In order to guarantee the best possible long-term function, the main objectives of this procedure are to promote early mobilization, improve fracture healing, and restore the tibia's articular integrity. The strength and stability of the fixation have been significantly improved by the use of locking screws and anatomically shaped plates, which has decreased the incidence of non-union and malunion in tibial fracture cases. To reduce the risk of complications like infection, implant failure, and delayed wound healing, it is essential to carefully evaluate the soft tissue condition, fracture pattern, and patient-specific factors [##UREF##1##7##].</p>", "<p>Plate osteosynthesis, internal fixation, and open reduction are frequently used in the treatment of tibial fractures and have shown promise in improving functional recovery and enabling early weight bearing. This is especially noticeable in cases of complex or extremely unstable fractures. Strict adherence to well-defined rehabilitation protocols and close patient monitoring after surgery is essential for optimizing the overall recovery process and reducing the risk of complications [##REF##16497997##8##]. Musculoskeletal disorders that are known to be difficult to treat, like complicated De Quervain's syndrome and lateral epicondylitis, are generally not treated with manual therapy. On the other hand, Mulligan's mobilization with movement (MWM)-based therapeutic approaches are becoming more well-known for their efficacy in these circumstances. In manual therapy treatment, a specific joint glide is maintained during a compromised function by applying a manual force in MWM. The impaired joint can move freely, pain-free, with this technique called MWM. In MWM, the force is typically applied perpendicular to the plane of movement or impaired action and parallel to the treatment plane. We do not yet know the precise mechanisms by which MWM enhances patient outcomes. MWM may be able to correct joint positional errors brought on by sprains or injuries. To completely comprehend this suggested mechanism of action, more study is necessary, though, especially regarding the controversy surrounding other spinal manipulation theories like chiropractic subluxation [##REF##16959529##9##].</p>", "<p>The significance of customizing treatment regimens to meet each patient's unique needs is emphasized by this case study. It describes a multidisciplinary strategy that was used to treat stiffness after a tibia fracture that had comminuted. There are various benefits to using Mulligan mobilization to treat knee stiffness. This method uses mild and painless mobilization techniques during targeted movements in an effort to improve joint mechanics, lessen pain, and improve functional outcomes [##UREF##2##10##]. Unlike conventional methods, Mulligan mobilization actively involves patients in their rehabilitation process. Better results and higher patient compliance are frequently the results of this proactive involvement [##UREF##3##11##].</p>" ]

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[ "<title>Discussion</title>", "<p>This case study features a 45-year-old male patient who had a proximal tibial fracture managed surgically with ORIF. After three months, the patient visited the orthopaedic department for follow-up with complaints of stiffness, pain and restricted ROM, for the same he was referred to physical therapy. Preventing subsequent problems, enhancing lower limb strength and ROM, reducing discomfort, and encouraging early weight-bearing were the main objectives of physiotherapy management. An extensive review of pertinent literature is conducted to determine the efficacy of Mulligan MWM approaches in the treatment of osteoarthritis in the knee [##REF##31324377##14##]. Changes in knee ROM, pain alleviation, and improved function are just a few of the outcomes that are included in the evaluation. In addition, a randomized clinical trial is being conducted to evaluate and compare Mulligan's mobilization techniques and ischemia compression as treatments for patellofemoral pain syndrome. In order to identify the most efficient method for treating this condition with manual therapy approaches, the study focuses on knee function, pain severity, and patient-reported outcomes [##REF##30583976##15##].</p>", "<p>According to Gaston et al., 20% of patients with tibial plateau fractures experience stiffening (a persistent knee flexion contracture of greater than 5º) a year following surgery [##REF##16129749##16##]. Flexion contractures, extension contractures, and combination contractures are the three categories into which Pujol et al. have classified the causes of post-traumatic knee stiffness. Fibrotic alteration of periarticular tissues and/or extensive intra-articular adhesions may be the cause of post-traumatic stiffness. In general, posterior adhesions and/or anterior impingement are the cause of flexion contractures. Conversely, posterior impingement or anterior adhesions cause extension contractures [##UREF##6##17##].</p>", "<p>A comprehensive method of fracture care management emphasizes the significance of comprehending the biomechanics of tibial fractures. It offers justification for surgical fracture therapy and assesses the effects of different fixation techniques, such as intramedullary nailing, plate osteosynthesis, and external fixation, on soft tissue injuries and fracture patterns [##REF##22744206##18##]. Tibial fracture-specific problems can be effectively addressed with Mulligan mobilization techniques. These techniques can address a variety of problems, such as the improvement of weight-bearing capacity, the restoration of normal gait patterns, and the correction of muscular imbalances that frequently result from immobilization. Following a fracture, patients can achieve a more thorough healing process, functional independence, and improved quality of life by concentrating on these areas [##UREF##7##19##]. According to Bhandari et al., two major advantages of operational treatment are enhanced fracture stability and early mobilization; the fact that our patient recovered well after surgery is evidence of the efficacy of these approaches [##REF##11701800##20##].</p>" ]

[ "<title>Conclusions</title>", "<p>The results of this case study on physiotherapeutic therapies for postoperative stiffness in tibial fractures treated surgically demonstrate the important role that specialized physiotherapy plays in fostering functioning and flexibility. The case's comprehensive physiotherapy approach addressed postoperative stiffness in an effective manner, which increased the ROM and functional recovery. Tibial fractures require not just precise surgery but also an individualized approach to postoperative therapy for optimal care. This highlights the necessity of tailored approaches to reduce postoperative stiffness and improve the overall functional outcome, underscoring the significance of including physiotherapeutic therapies in the overall treatment strategy for tibial fractures. In order to achieve the best results, the case report highlights the benefits of working together between surgical treatments and focused physical therapy.</p>" ]

[ "<p>This case study explains the complete care of a 45-year-old male patient who had a high-impact road injury that resulted in a displaced proximal tibial fracture. Substantial soft tissue damage was discovered during the initial assessment, requiring careful thought before undergoing surgery. A customized physiotherapy program was instituted after an incremental strategy involving open reduction and internal fixation. The patient made a satisfactory functional recovery, regaining nearly normal mobility and going back to daily activities within 12 weeks despite difficulties encountered during the rehabilitation phase, including temporary postoperative complications. The present study underscores the significance of a multidisciplinary approach involving Mulligan mobilization in the effective management of intricate proximal tibial fractures. It also underscores the importance of meticulous surgical intervention and organized rehabilitation protocols in enhancing patient outcomes and regaining functional abilities to improve patients' quality of life.</p>" ]

[ "<title>Case presentation</title>", "<p>A 45-year-old male patient reported he had been experiencing pain in his left knee joint over three months. He also reported stiffness in the same joint and difficulty walking. Three months prior, the patient underwent surgical management for a tibial fracture that was treated with ORIF. After seeing an orthopaedic department, the patient was given medication prescriptions and directed to the physiotherapy department for additional care. Table ##TAB##0##1## presents the current episodes' chronology.</p>", "<p>Clinical findings</p>", "<p>The patient was conscious, cooperative, and able to follow instructions and had a mesomorphic build. When the patient was first observed, the head was raised at a 30-degree angle while the patient was reclining in a supine position. While the patient was in the supine position, his posture was assessed. The ankle was in a plantarflexed position, the knee was flexed, and the left hip was extended. Upon examination of the left leg, the skin covering appeared tense, and there was generalized swelling observed over the upper portion of the leg and knee. A bandage was present on the anterior aspect of the left knee. Upon palpation, the local temperature was within normal limits. Tenderness was detected over the upper portion of the left leg, specifically over the tibial condyles. The dimensions of the bandage were measured to be 15 cm x 6 cm. The range of motion (ROM) at the knee joint is documented in Table ##TAB##1##2##.</p>", "<p>Clinical investigations</p>", "<p>Figure ##FIG##0##1## depicts a pre-operative X-ray. A wedge fracture of the medial and lateral tibial plateau is shown on preoperative radiographs, suggesting a type V Schatzker classification. Figure ##FIG##1##2## shows the X-ray taken after surgery. The radiograph in Figure ##FIG##1##2## illustrates how screws and plates are used to fix a type V Schatzker fracture internally. CT scan seen in Figure ##FIG##2##3## demonstrates comminuted bicondylar fracture of the tibia with tibial eminence fracture that is Schatzker classification type V.</p>", "<p>Treatment</p>", "<p>Physiotherapy Intervention</p>", "<p>The physiotherapeutic intervention lasted for a duration of three weeks, with a frequency of five sessions per week, each session lasting approximately 40 minutes.</p>", "<p>Week 1: In order to relieve pain, the patient was advised to have cryotherapy sessions thrice daily for 10 minutes focused on the left knee joint. In order to avoid oedema and lessen swelling, it was also advised that the patient elevate the afflicted limb. Gentle passive stretching exercises were given, with three 30-second holds, to improve quadriceps flexibility. In order to strengthen the glutes, hamstrings, and quadriceps, isometric exercises were also recommended. It was suggested to perform straight leg raises to increase general strength. To improve mobility, active ROM exercises were recommended. Additionally, it was advised to perform the vastus medialis oblique (VMO) strengthening exercise in a single set of 10 repetitions. Mulligan mobilization for the knee joint was administered, which entails ten separate sets of MWM, to ease pain and reduce stiffness.</p>", "<p>Week 2: The patient was instructed to perform three sets of light passive stretching, with a 30-second rest in between, in order to increase the flexibility of their quadriceps. Furthermore, perform isometric workouts targeting the quadriceps, hamstrings, and glutes. Leg raises done straight are one type of strengthening exercise. Active ROM exercises were done to increase mobility. He did dynamic squats and mini squats, two weight-bearing exercises, 10 times. In addition, there were three sets of 10 repetitions of VMO strengthening and Mulligan mobilization for the knee joint. To ease pain and lessen stiffness, MWM was employed for ten distinct sets in addition to proprioceptive training. It was advised to ice for 10 minutes three times a day.</p>", "<p>Week 3:<italic> </italic>To improve the strength of the quadricep muscles, a dynamic training program was employed, which involved performing ten repetitions in a single set with a one-kilogram weight. Furthermore, two sets of 10 repetitions of VMO strength training exercises were performed with resistance. Mulligan mobilization was applied to the knee joint, involving 15 separate sets. The quadriceps were further strengthened with 20 mini squats. Ten repetitions of the muscle energy technique were used to increase the muscle strength. Lastly, icing the affected area three times a day for 10 minutes each time can help with relief and recovery.</p>", "<p>Physiotherapists and other medical professionals use Mulligan mobilization, a manual therapy technique, to treat musculoskeletal pain and movement limitations [##UREF##4##12##]. This approach combines prolonged joint mobilization with active patient mobility. The main objectives are to improve joint mobility and relieve pain during particular movements [##UREF##2##10##]. On the other hand, traditional methods include the standard physiotherapy exercises and techniques used to treat musculoskeletal problems. To increase strength, flexibility, and general function, these methods could involve massage, stretches, exercises, and other modalities [##UREF##5##13##]. Figures ##FIG##3##4##-##FIG##4##5## show patients performing exercises.</p>", "<p>Follow-up and outcomes</p>", "<p>An interventional physical therapy process was initiated after thorough planning. A follow-up was performed once a week for four weeks. Telerehabilitation was used for routine post-discharge follow-up. Outcome measures are listed in Tables ##TAB##2##3##-##TAB##3##4##.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>The pre-operative radiograph of the left knee joint lateral and anteroposterior views</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>The post-operative radiograph of the left knee joint fixed with screws and plates</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG3\"><label>Figure 3</label><caption><title>CT scan of the left knee joint</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG4\"><label>Figure 4</label><caption><title>The patient performing active range of motion exercises.</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG5\"><label>Figure 5</label><caption><title>The patient performing dynamic quads.</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Timeline of current episodes</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Events</td><td rowspan=\"1\" colspan=\"1\">Dates</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Date of accident</td><td rowspan=\"1\" colspan=\"1\">July 03, 2023</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Date of operation</td><td rowspan=\"1\" colspan=\"1\">July 10, 2023</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Duration of physical therapy rehabilitation</td><td rowspan=\"1\" colspan=\"1\">Three weeks (October 21 to November 10, 2023)</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>The range of motion of the knee joint</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Knee joint</td><td rowspan=\"1\" colspan=\"1\">Right</td><td rowspan=\"1\" colspan=\"1\">Left</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Flexion</td><td rowspan=\"1\" colspan=\"1\">0-120^o\n</td><td rowspan=\"1\" colspan=\"1\">0-30^o\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Extension</td><td rowspan=\"1\" colspan=\"1\">120-0^o\n</td><td rowspan=\"1\" colspan=\"1\">30-0^o\n</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB3\"><label>Table 3</label><caption><title>Outcome measures before and after the treatment</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Outcome measures</td><td rowspan=\"1\" colspan=\"1\">On admission</td><td rowspan=\"1\" colspan=\"1\">On discharge</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Numerical pain rating scale</td><td rowspan=\"1\" colspan=\"1\">9/10</td><td rowspan=\"1\" colspan=\"1\">4/10</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Lower extremity functional scale</td><td rowspan=\"1\" colspan=\"1\">7/80</td><td rowspan=\"1\" colspan=\"1\">36/80</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB4\"><label>Table 4</label><caption><title>Before and after the assessment of range of motion</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> Joint</td><td rowspan=\"1\" colspan=\"1\">Movement</td><td rowspan=\"1\" colspan=\"1\">On admission</td><td rowspan=\"1\" colspan=\"1\">At discharge</td><td rowspan=\"1\" colspan=\"1\">On admission</td><td rowspan=\"1\" colspan=\"1\">At discharged</td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Left</td><td rowspan=\"1\" colspan=\"1\">Left</td><td rowspan=\"1\" colspan=\"1\">Right</td><td rowspan=\"1\" colspan=\"1\">Right</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> Hip</td><td rowspan=\"1\" colspan=\"1\">Extension</td><td rowspan=\"1\" colspan=\"1\">0-70</td><td rowspan=\"1\" colspan=\"1\">0-90</td><td rowspan=\"1\" colspan=\"1\">0-130</td><td rowspan=\"1\" colspan=\"1\">0-130</td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Flexion</td><td rowspan=\"1\" colspan=\"1\">0-15</td><td rowspan=\"1\" colspan=\"1\">0-25</td><td rowspan=\"1\" colspan=\"1\">0-30</td><td rowspan=\"1\" colspan=\"1\">0-30</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> Knee</td><td rowspan=\"1\" colspan=\"1\">Extension</td><td rowspan=\"1\" colspan=\"1\">0-10</td><td rowspan=\"1\" colspan=\"1\">0-60</td><td rowspan=\"1\" colspan=\"1\">0-120</td><td rowspan=\"1\" colspan=\"1\">0-120</td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Flexion</td><td rowspan=\"1\" colspan=\"1\">10-0</td><td rowspan=\"1\" colspan=\"1\">0-60</td><td rowspan=\"1\" colspan=\"1\">0-120</td><td rowspan=\"1\" colspan=\"1\">0-120</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> Ankle</td><td rowspan=\"1\" colspan=\"1\">Plantar-flexion</td><td rowspan=\"1\" colspan=\"1\">0-50</td><td rowspan=\"1\" colspan=\"1\">0-60</td><td rowspan=\"1\" colspan=\"1\">0-40</td><td rowspan=\"1\" colspan=\"1\">0-40</td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Dorsiflexion</td><td rowspan=\"1\" colspan=\"1\">0-16</td><td rowspan=\"1\" colspan=\"1\">0-25</td><td rowspan=\"1\" colspan=\"1\">0-25</td><td rowspan=\"1\" colspan=\"1\">0-25</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Krishnayani Shende, Nishigandha P. Deodhe, Grisha Ratnani, Khushi M. Gandhi</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Krishnayani Shende, Nishigandha P. Deodhe, Grisha Ratnani, Khushi M. Gandhi</p><p><bold>Drafting of the manuscript:</bold>  Krishnayani Shende, Nishigandha P. Deodhe, Grisha Ratnani, Khushi M. Gandhi</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Krishnayani Shende, Nishigandha P. Deodhe, Grisha Ratnani, Khushi M. Gandhi</p><p><bold>Supervision:</bold>  Krishnayani Shende, Nishigandha P. Deodhe, Grisha Ratnani, Khushi M. Gandhi</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050589-i01\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050589-i02\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050589-i03\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050589-i04\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050589-i05\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Simulation studies use computer-generated data to investigate research questions (##UREF##0##Beaujean, 2018##). Monte Carlo simulation is a commonly used procedure that uses repeated random number selections to solve modeling problems (##UREF##10##Gelfand and Smith, 1990##). It is especially useful when a statistical assumption (e.g., normality) is violated or in situations without theoretical distribution (##UREF##6##Fan, 2012##). The Monte Carlo simulation was introduced to psychometrics by ##UREF##19##Patz and Junker (1999a##,##UREF##20##b)##.</p>", "<p>Psychological researchers are often interested in determining the sampling distributions of test statistics, comparing parameter estimators (e.g., Cohen’s <italic>d</italic>), and comparing multiple statistics that perform the same function. In a Monte Carlo simulation context, a key factor is the design of the specific conditions to evaluate.</p>", "<p>Different simulation studies use different designs with a variety of conditions. This is because the study’s aims usually dictate the selection of the conditions. Suppose a Monte Carlo study has been designed to test the violation of a certain assumption (e.g., normality). Both the condition that the assumption has been met (e.g., normal distributed population) and the condition that the assumption has been violated (e.g., skewed distributed population) should be included in the study. Now, assume a different study has been designed to test the performance of a statistic (or several statistics) across different population distributions. In this case, multiple population distributions should be included in the study design. In general, “the major factors that may potentially affect the outcome of interest should be included” (##UREF##6##Fan, 2012##, p. 437).</p>", "<p>However, some recent studies overlooked the inclusion of the null distribution of statistics in the simulations. A null distribution of statistics represents a scenario with no estimated relationship between variables within a given sample (##REF##14596476##Hunter and May, 2003##; ##UREF##24##Spurrier, 2003##).</p>", "<p>In this study, we advocate for including a null distribution of statistic conditions in the Monte Carlo simulation when evaluating and comparing statistical measures. Furthermore, we suggest that the performance of a statistic should be assessed in light of commonly used cut-offs. Psychologists often employ informal tests in their research to compare the statistics values to a pre-determined cut-off to reach a binary decision. For example, an area under the curve (AUC) greater than 0.7 in ROC analysis is considered the minimum acceptable threshold (##REF##17375868##Streiner and Cairney, 2007##); A Root Mean Square Error of Approximation (RMSEA) of 0.08 is regarded as the upper limit for Structural Equation Model fitting (SEM, ##UREF##5##Fabrigar, 1999##). An internal consistency (e.g., Cronbach Alpha) greater than 0.7 is considered an acceptable level of reliability, according to ##UREF##26##Taber (2018)##. ##REF##27303333##Trizano-Hermosilla and Alvarado (2016)## have conducted a Monte Carlo simulation study with a focus on internal consistency performance evaluation. In this paper, we will utilize the influential study by ##REF##27303333##Trizano-Hermosilla and Alvarado (2016)## as a practical example to demonstrate the inclusion of a null distribution and the assessment of the statistic using commonly used cut-offs.</p>", "<p>This paper is organized into several sections. We review current practices regarding including null distribution in psychological Monte Carlo simulation studies and their associated limitations. Subsequently, we introduce a simulation design rooted in the confusion matrix as a proposed solution. The study conducted by ##REF##27303333##Trizano-Hermosilla and Alvarado (2016)## will be used as a practical example of this design. In conclusion, we engage in a comprehensive discussion about the design, supplemented by another illustrative sample.</p>" ]

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[ "<title>How will the missing null distribution of statistic conditions influence the result of a simulation study?</title>", "<p>In the following section, we will offer a general overview of the original study. We will specifically address the shortcomings of not including null distribution of the statistics conditions in their simulation design and propose enhancements through the methodology developed in this study.</p>", "<p>In the original study, ##REF##27303333##Trizano-Hermosilla and Alvarado (2016)## compared the performances of four internal consistency statistics: Cronbach’s Alpha, Omega (##UREF##15##McDonald, 1999##), GLB (Greatest Lower Bound, ##REF##20037639##Sijtsma, 2009##), and GLBa (Greatest Lower Bound algebraic, ##UREF##16##Moltner and Revelle, 2015##). They made a comparison of these statistics with various normal and nonnormal distributions and two kinds of inter-correlation between items: tau-equivalent and congeneric.</p>", "<p>The original study used Root mean square error (RMSE) and %bias as their criteria.</p>", "<p>where <italic>p̂</italic> refers to the observed statistics for each replication, <italic>p</italic> refers to the true value of statistics in the simulation population, and <italic>Nr</italic> refers to the number of replications. Larger absolute values in the RMSE and the %bias statistics indicate worse performance.</p>", "<p>Based on the RMSE and the %bias, the authors reported that Omega showed the best performance across most conditions included in their study. In other words, when comparing the difference between observed sample statistics and the associated true population parameter values, Omega showed the smallest discrepancies across most of the conditions. This led the authors to conclude that Omega should be recommended as the preferred index of internal consistency in psychological research. Specifically, the original study suggests that Omega should be preferred over Cronbach’s Alpha, which is the most widely used measure of internal consistency. Various studies across multiple disciplines shared the opinion with the original study that Omega rather than Alpha should be used as an internal consistency measurement method (##REF##28429633##Watkins, 2017##; ##REF##28557467##McNeish, 2018##; ##REF##32772525##Cortina et al., 2020##).</p>", "<p>Importantly, for our purposes, ##REF##27303333##Trizano-Hermosilla and Alvarado (2016)## original study included only simulation conditions in which there was an effect measured by the statistic (i.e., populations with internal consistency). Specifically, it only included conditions with an acceptable level of internal consistency between items in the questionnaires (i.e., a true internal consistency of 0.731 and 0.845) for the condition of 6 and 12 questionnaire lengths, respectively. As mentioned above, Alpha and Omega values of 0.7 or above are indicated as acceptable internal consistency in psychological research (##UREF##26##Taber, 2018##). Therefore, it included the null effect of some factors (e.g., no distribution error). However, it did not include a null distribution statistic condition, as we suggest here. According to ##UREF##27##Tang et al. (2014)##, internal consistency refers to the degree of inter-item correlations among items with factor saturation. Thus, to simulate a null distribution for these internal consistency statistics, one can independently assign random numbers to each item.</p>", "<p>As a result, we would argue that the conditions included in the original study are insufficient to support their conclusions. To elaborate, we propose a new hypothetical index, C, which is used to measure internal consistency, with 0.7 being set as an acceptable cut-off. C is a constant number that can be computed and observed across all the 1,000 simulated datasets. Suppose <italic>C</italic> is found to be 0.78 from each replicated sample, i.e., (##FORMU##2##3##)</p>", "<p>In other words, <italic>C</italic> is a dummy index without validity according to internal consistency estimation. However, based on the criteria employed in previous studies (i.e., RMSE and %bias), <italic>C</italic> has a similar level of error as the Omega index. Across conditions in the original study for length = 6 items, the population parameter of internal consistency is 0.731. This is based on Equations ##FORMU##0##(1)## and ##FORMU##1##(2)##, in which <italic>p</italic> is always 0.731, and <italic>p̂</italic> is always 0.78. As a result, the RMSE is 0.049 (##FORMU##3##4##)</p>", "<p>and the %bias is −4.9% (##FORMU##4##5##)</p>", "<p>in all conditions. Across conditions included in the original study length of 12 items, with similar calculations, RMSE is 0.065, and the %bias is 6.5%. These two results will remain consistent regardless of other factors, like the type of distribution. Therefore, it appears that in a number of conditions, this dummy index can provide similar or even superior performance to the genuine indices included in the original study. Importantly, this indicates that based only on the empirical evidence provided in the original study, we cannot distinguish Omega from this dummy index <italic>C</italic>. <italic>C</italic> is an extreme theoretical case, and a statistic with a consistent number cannot be applied. However, A dummy index similar to C with variations can be simulated easily. For example, can be simulated from a continuous uniform distribution [0.711, 0.751] and also cannot be distinguished from Omega with the simulation conditions and criteria used in the original study.</p>", "<p>To sum up, simulation studies often evaluate the performance of a statistic based on RMSE and %bias in Monte Carlo simulations, with a view to quantifying the distance between the sample estimates of an observed statistic and the true parameter values (i.e., TP) in the population. We agree that this approach can offer insights regarding the degree to which observed sample estimates are different from true population values. However, without the introduction of the null distribution of statistic conditions in simulation, researchers may reach incorrect or incomplete conclusions, as in the above example with the dummy <italic>C</italic> index.</p>", "<p>To address this problem, we introduce in this study a Monte Carlo design based on criteria commonly used in psychology and machine learning to evaluate models with categorical or binary results: the “confusion matrix” (##UREF##14##Marom et al., 2010##). In psychology, researchers typically use a confusion matrix to evaluate the performance of a categorization model in real psychological practice (i.e., ##REF##29330090##Ruuska et al., 2018##). A confusion matrix comprises four quadrants: True Positive (TP), False Positive (FP), True Negative (TN), and False Negative (FN). Their relationships are shown in ##TAB##0##Table 1##.</p>", "<p>We use the original study as an example to illustrate how to apply confusion matrix methodology to simulation studies in psychology, aiming at statistical comparison. The null distribution of the statistic in the original study is the condition where there is zero internal consistency between items in the population.</p>", "<p>We also include the interpretation of results that come from the design. Internal consistency is both continuous and binary (i.e., cut-off), exemplifying the problem adequately. The present study will keep the original study’s design unless alternative designs better target the research problem, or the original design is not applicable.</p>", "<title>Simulation Study 1: estimating the true negative</title>", "<p>As noted above, the original study does not include the null distribution of the internal consistency statistics. The original study only provides empirical evidence of TP. The following simulation aims to distinguish Omega from a dummy index like <italic>C</italic>. As suggested by several studies (##UREF##17##Moriasi et al., 2007##; ##UREF##28##Wang and Lu, 2018##), for the continuous variable, RMSE and %bias can provide more information than a percentage. Therefore, we propose TN can also use these two criteria. In the case that Omega is an efficacy statistic or index and that TN conditions should have an RMSE close to 0 and a %bias close to 0%, upon which a dummy statistic or index like <italic>C</italic> should have an RMSE close to 0.70 and a %bias close to 70% in the TN condition. Therefore, using TN to test whether a statistic or index is merely a “dummy” one is crucial, and its inclusion in the simulation represents an important step toward obtaining truly conclusive results.</p>", "<title>Design</title>", "<p>In the original study, the researchers simulated several factors, including sample size (250, 500, and 1,000) and item length (6 or 12). This study will also use the same design for these two factors. The original study also included the distribution of errors following ##UREF##12##Headrick’s (2002)## that were introduced to 2, 4, or all 6 items of the 6-item condition and to 2, 4, 6, 8, 10, or all 12 items of the 12-item condition. However, Headrick’s method (2002) was not introduced to our current study to ensure there is no internal consistency created from this method between items and results and also for simplicity.</p>", "<p>The original study included the tau-equivalent and congeneric models as simulation conditions. This aspect of the study does not apply to the condition of null distribution to internal consistency statistics. This is because there is no correlation between any two items in the null distribution population, regardless of its type. Therefore, this design is not included in our study. In summary, 2*3 = 6 conditions are included in the first simulation.</p>", "<p>The original study simulated all datasets in R (##UREF##21##R Core Team, 2021##) with R Studio (##UREF##22##Racine, 2012##). The present study will also use these platforms (for details, please see <xref rid=\"app1\" ref-type=\"app\">Appendix</xref>). For each condition, the design of the original study was replicated 10,000 times. The current study will use the same replication time with 10,000 across six conditions.</p>", "<p>Four kinds of internal consistency measurement indices were included in the original study: Alpha, Omega, GLB, and GLBa. As provided by the sample code in the original study, these functions were used in the original study to obtain the following results: omega$alpha for Alpha, omega$omega.tot for Omega, glb.fa$glb for GLB, and glb.algebraic$glb for GLBa. In addition, two packages were used in calculations in the original studies: Psych (##UREF##23##Revelle, 2015##) and GPArotation (##UREF##1##Bernaards and Jennrich, 2015##), which are also used in this study. Further, the Omega.total was used as the chosen index from Omega in the original study because Trizano-Hermosilla and Alvarado also reported and evaluated the performance of , and consequently, the present study will also make use of the same reliability index.</p>", "<p>To create a null distribution of internal consistency, we simulated the dataset from a standard normal distribution N(0,1) for each item across participants during the replications. Accordingly, each item and each participant’s response are totally independent, which ensures that the true covariances and factor loadings in the population are always zero. To check the validity of this design, we followed ##UREF##6##Fan (2012##, p. 436), who suggests, “We may do a quick data generation verification by generating a large sample.” We simulated a large dataset from N(0,1) and calculated four internal consistency indices, as they yielded results close to zero, which supported the simulated null distribution of statistics as accurate. This part of the code is provided separately. This study will also use RMSE and %bias as criteria, similar to the original study, to evaluate the performance of the statistics.</p>" ]

[ "<title>Discussion</title>", "<p>Our study, alongside the original study by ##REF##27303333##Trizano-Hermosilla and Alvarado (2016)##, presents a new Monte Carlo simulation design within the confusion matrix paradigm. We have proposed new conditions, guided by the perspective of the confusion matrix, that should be included in the evaluation of statistical simulation studies. Firstly, we will discuss the findings of internal consistency indices. Secondly, we will provide a summary of how to apply this novel confusion matrix design to simulation studies in statistics comparison. Thirdly, we will engage in a general discussion.</p>", "<title>Issues of internal consistency indices</title>", "<p>This study is not primarily focused on which kind of internal consistency indices should be used in psychological research. Therefore, the study has replicated the design of the original study (i.e., sample size and questionnaire length) when applicable to provide an example of how to apply this confusion matrix design. This does not imply that we see no space for improvement in the conditions included in the study. For instance, Likert scale variables should be included in the simulation as internal consistency indexes are usually applied to the Likert scale variables in psychological research (##UREF##3##Croasmun and Ostrom, 2011##). However, we have found additional empirical evidence that should be used as a reference for the performance of these statistics. Through this additional evidence, we have found that Omega and GLB indices do not perform well enough for small sample sizes under some conditions. Yet, our results do not imply that Alpha should necessarily be preferred over Omega. We admit that Alpha has shortcomings as an index for measuring internal consistency, which is boosted by the length of the questionnaire or prerequisites that are violated, as described in previous studies (##REF##28557467##McNeish, 2018##; ##UREF##11##Hayes and Counts, 2020##).</p>", "<p>However, we have found that under some conditions (e.g., sample size = 20, 30, or 40), Omega.total and GLB are boosted and thus become unreliable. Specifically, it is difficult to distinguish a population with random numbers from a population that has high internal consistency. Therefore, in these conditions (i.e., sample size &lt;40), we recommend that Omega.total and GLB be avoided in estimating the internal consistency, no matter what kind of performance Omega.total has when there is an acceptable level of the parameters in a given population. These suggestions are based on the results of this simulation study, which are limited by the study’s design.</p>", "<p>We simulated a null distribution for internal consistency, specifically using a normal distribution generated randomly for each item. This implies that all effects in the dataset are essentially noise. To our understanding, the reason why the Omega statistic tends to be inflated in small sample sizes is due to its value range being restricted to [0,1]. Consequently, any noise in the dataset disproportionately affects Omega positively. As suggested by ##REF##27334468##Okada (2017)##, the zero-winsorized method can create positive biases. Especially in conditions of small sample sizes, such biases can lead to inflated results, sometimes even exceeding the established cut-off (i.e., 0.7).</p>", "<p>Moreover, related Omega indices, such as Hierarchical Omega, should also be tested when researchers aim to measure the reliability of the general factor only. All these indices with these conditions should be tested through the TP and TN conditions, corresponding to FPm and FNm. Most importantly, all these conditions should be tested simultaneously in a simulation study to provide empirical evidence for applied researchers. Suppose the proposed design had been applied in the original study. A more conservative recommendation of Omega with a discussion of Omega’s limitations will be provided in the original study and studies influenced by the original study (##REF##28429633##Watkins, 2017##; ##REF##28557467##McNeish, 2018##; ##REF##32772525##Cortina et al., 2020##).</p>", "<title>Practical recommendations and steps when implementing a confusion matrix design through Monte Carlo simulation</title>", "<p>Step 1: Both conditions in which there is a certain relationship between variables and the condition in which the expected association is deemed as absent should be included in the simulation design (i.e., the null distribution of statistics), together with other relevant criteria such as sample size, distribution, and alike. These two kinds of conditions ought to be included as TP and TN, respectively.</p>", "<p>In simulating the null distribution of statistics, we advocate for consistently employing the method outlined in the APA guidelines (##UREF##6##Fan, 2012##). This approach ensures that the simulation design accurately represents a population with a null statistic distribution and assesses its impact on the observed sample statistics. Our findings confirm that it is possible to reconstruct an estimation by a normally distributed dataset in the absence of internal consistency across four reliability statistics, which have theoretical and practical implications that are related to the definition and calculation of what is considered to be a large sample. For instance, as described in Study 1, we calculated all four statistics (i.e., Omega, Alpha, GLB, and GLBa) with a large sample of 100,000 and a standard normal distribution, ensuring the inclusion of a null distribution of statistics since all the statistics are close to zero in such an extensive sample.</p>", "<p>Meanwhile, it’s important to acknowledge that there are various types of null distributions for a statistic. Although our simulation study only includes normal distributions, we encourage researchers to explore a broader range of nonnormal distributions. This expansion is crucial to estimating the robustness of statistics under a variety of True Negative (TN) conditions. When doing so, researchers should employ the checking method we mentioned earlier to ensure that the design excludes any relationship specific to the statistic being tested.</p>", "<p>Step 2: Suppose there is a commonly used cut-off or an acceptable level of a statistic with a continuous result. FPm ##FORMU##4##(5)## and FNm ##FORMU##8##(6)## should be measured in various conditions, such as those conditions commonly occurring in practice.</p>", "<p>We have already described the difficulty of practicing FN and FP directly in statistics used in psychology. Yet, we also admit that FNm is necessary but not sufficient to estimate FN. Analogically speaking, using FNm to replace FN and using FPm to replace FP would be like trying to measure whether an unknown number X is bigger than 1 to solve the question of whether X &gt; 2. If X ≤ 1, then X is definitely less than 2. However, if X &gt; 1, it does not necessarily mean X is greater than 2.</p>", "<p>The confusion matrix design also works in this way. Suppose a statistic can report a result above the cut-off or an acceptable level of a relationship between variables measured by this statistic when there is a null distribution of the statistics in this condition. In this case, it is also highly likely that the statistic will report a result above this cut-off when the population parameter is lower than the acceptable level. As a result, the statistics in this condition are not reliable. To estimate the possibility of this situation, we conducted another simulation study that used internal consistency levels of 0.3 and 0.5 as the true parameters of the population. The result is in ##TAB##4##Table 5##. According to our findings, the Omega is also boosted in the conditions tested as questionable by FPm. Therefore, FPm scores above 5% are reliable enough to ascertain when a statistic should be considered questionable. Some researchers might argue that this part of the simulation may also be included in our proposed confusion matrix design. Yet, for some statistics, it is not easy to find a present but not acceptable level of the statistic.</p>", "<p>Furthermore, our research identified two key relationships between True Negative (TN) and FPm. If a statistic shows poor performance in the TN condition, it is likely to also fare poorly in the FPm condition. This observation aligns with the rationale we discussed earlier. Additionally, we found that a positive bias in TN is correlated with an increased likelihood of simulation study results meeting the acceptable cut-off. Using the original study as an empirical example of True Positive (TP), we can reasonably infer that all four indices demonstrate robust performance in FNm. Thus, for statistics without a pre-established cut-off, we recommend using TN and TP as predictive references. A large absolute value in percentage bias and RMSE suggests that the statistical output is likely derived from population samples.</p>", "<title>Several research scenarios</title>", "<p>We have demonstrated a comprehensive example of applying this enhanced confusion matrix design in evaluating internal consistency indices. To further clarify, we propose that this design is versatile and can be applied to a broader range of tasks. Before delving into a general discussion, we will present three concise examples illustrating how the confusion matrix design can be implemented in other published simulation studies. In the first two studies, only TN conditions can be applied, as these studies do not have a common cut-off for their respective statistics (i.e., correlation coefficients and mediation correlation coefficients). However, for the third study, we will apply the full confusion matrix design, as it involves a cut-off for Root Mean Square Error of Approximation (RMSEA) in Structural Equation Modeling (SEM).</p>", "<p>##REF##35792742##Ventura-León et al. (2023)## executed a Monte Carlo simulation study focusing on correlation coefficients commonly used in psychology research. They examined various population correlation conditions, such as 0.12, 0.20, 0.31, and 0.50, under nonnormal distributions and distributions with outliers. Their findings indicated that the Winzorized Pearson correlation coefficient (##UREF##29##Wilcox, 2011##) performed the best within the simulated conditions they included. Based on the design of our study, we suggest that ##REF##35792742##Ventura-León et al. (2023)## should also consider including conditions with a null distribution of the statistics, specifically where the population correlation is zero that can be used as TN. The absence of TN in their study leaves a gap in empirical evidence regarding the performance of correlation coefficients under this condition. This omission poses a risk, as certain correlation coefficients may exhibit poor performance at the zero point, like the Eta square effect size (##UREF##18##Okada, 2013##) and the Omega statistics in our simulation.</p>", "<p>##REF##34992307##Sim et al. (2022)## conducted a Monte Carlo simulation study to estimate the necessary sample size for detecting mediation effects in various models. Their study included partial and full mediation conditions, providing the minimum sample size required to detect these effects. However, their design overlooked the inclusion of null distribution of mediation effects conditions, which are crucial for assessing the sample size needed to maintain a reasonable Type-I error level. This omission can bring significant problems. For instance, suppose a sample size requirement of 200 is found under some null distribution conditions to ensure the correct result is found in most replications. Then, the conclusions of ##REF##34992307##Sim et al. (2022)## might be called into question. They concluded that a sample size of 90 is sufficient to detect a mediation effect when the factor loading is 0.7 with a large effect size. Yet, this sample size level may not avoid the detection of a mediation effect in a population where no such effect exists. Including conditions with no mediation effect, as TN proposed in our study, is essential to test and validate the sample size requirements thoroughly.</p>", "<p>In the case of the studies by ##REF##35792742##Ventura-León et al. (2023)## and ##REF##34992307##Sim et al. (2022)##, the simulation conditions of FPm and FNm are not applicable, as these studies lack defined criteria for determining satisfactory levels of mediation effect or correlation coefficients. Next, we will examine another study by ##UREF##9##Gao et al. (2020)##, which focuses on the RMSEA in SEM. Our discussion will first detail the design of Gao, Shi, and Maydeu-Olivares’s study, followed by its shortcomings. We will then explore how the methodology of our study can be applied to theirs to address these limitations.</p>", "<p>##UREF##9##Gao et al. (2020)## used a Monte Carlo simulation study to examine the robustness of several RMSEA measurements. Their studies have included several robust RMSEA measurement methods and conditions with normal and nonnormal distributions. They found that RMSEA with mean and variance corrections is the most robust as it performs best across all conditions.</p>", "<p>From our perspective, the study conducted by ##UREF##9##Gao et al. (2020)## has shortcomings. One significant limitation is their failure to test the statistics under a null distribution condition, such as a simulated distribution in which items bear no relationship to the model. This omission means that they have not provided empirical evidence about the performance of these statistics in such a null condition. Therefore, it is essential to include TN conditions in their analysis. Additionally, they should test whether any RMSEA correction methods can yield results considered a good fit under null distribution conditions. This FPm design could be assessed using a cut-off of 0.08, as ##UREF##5##Fabrigar (1999)## suggested, across various conditions. If certain conditions reveal a good fit using an RMSEA correction method, then the performance of these statistics under these specific conditions becomes questionable. A similar approach could be applied to assess FNm.</p>" ]

[]

[ "<p>Edited by: Ioannis Tsaousis, National and Kapodistrian University of Athens, Greece</p>", "<p>Reviewed by: Jung Yeon Park, George Mason University, United States; Franca Crippa, University of Milano-Bicocca, Italy</p>", "<p>Monte Carlo simulation is a common method of providing empirical evidence to verify statistics used in psychological studies. A representative set of conditions should be included in simulation studies. However, several recently published Monte Carlo simulation studies have not included the conditions of the null distribution of the statistic in their evaluations or comparisons of statistics and, therefore, have drawn incorrect conclusions. This present study proposes a design based on a common statistic evaluation procedure in psychology and machine learning, using a confusion matrix with four cells: true positive, true negative, false negative modified, and false positive modified. To illustrate this design, we employ an influential Monte Carlo simulation study by Trizano-Hermosilla and Alvarado (2016), which concluded that the Omega-indexed internal consistency should be preferred over other alternatives. Our results show that Omega can report an acceptable level of internal consistency (i.e., &gt; 0.7) in a population with no relationship between every two items in some conditions, providing novel empirical evidence for comparing internal consistency indices.</p>" ]

[ "<title>The null distributions conditions included in the Monte Carlo simulation psychological studies</title>", "<p>We observed that the null distribution of statistics is generally included in existing Monte Carlo simulation studies in two ways: First, the null distribution of statistics is included to represent the condition that there is no true mean difference between two groups of scores and are usually referred to as conditions of null effect (e.g., ##UREF##4##Derrick et al., 2016##; ##UREF##2##Carter et al., 2019##; ##UREF##8##Fernández-Castilla et al., 2021##). This is consistent with the suggestion of the American Psychological Association (APA) guidelines. That is, researchers should include the null distribution of statistics (i.e., no mean difference between two groups; ##UREF##6##Fan, 2012##) in any simulation of effect to test and evaluate the potential threat of Type I error.</p>", "<p>Second, the researchers include the condition of a null distribution in factors in the simulation (e.g., ##UREF##13##Heggestad et al., 2015##). In ##REF##31318246##Greene et al.’s (2019)## study evaluating the bias of different kinds of fit indices, the authors manipulated (a) the strength of the cross-loadings between factors as 0, 0.1, 0.3, and 0.5, (b) the strength of the between-factor correlated residuals as 0, 0.1, 0.3, and 0.5, and (c) the strength of the within-factor correlated residuals as 0, 0.1, 0.3, and 0.5 in a model. In this sample, 0 represents the condition in which the relationship of cross-loadings or correlated residuals does not exist in the population of variables.</p>", "<p>In summary, researchers commonly include the null distribution of the statistic condition when estimating a statistic’s performance closely related to the mean difference. For example, when examining Cohen’s d in a Monte Carlo simulation study, researchers typically include a condition of no mean difference between two populations. Researchers also include the conditions of null distribution in factors in simulation studies for statistical comparison. However, psychological researchers may sometimes neglect to include the null distribution of the statistic in some other circumstances, such as in cases where the examined statistic does not have a close relationship with the mean difference.</p>", "<p>Returning to the fit indices study (##REF##31318246##Greene et al., 2019##) one paragraph above, the authors should include conditions that a null distribution in factors, such as no between-factor correlated residual, and the conditions with the null distribution of the statistic, such that some simulated samples should have no relationship with the proposed model (i.e., no model fitting). In our view, the failure to include conditions of null distribution weakens the conclusion of the simulation in the study. This may occur because some researchers have not considered the performance of the statistic in the condition that the dataset follows a null distribution of this statistic. (i.e., how will the fitting index perform on random data?), although other researchers recognize its importance. For instance, ##UREF##25##Stone (2000##, p. 64) points out: “In order to test statistically the fit of an item, it is then necessary to compare the statistic that is calculated with a null distribution.” Stone conducted a Monte Carlo simulation based on null distribution to compare goodness-of-fit test statistics in item response theory (IRT) models, and the results showed the superiority of the statistic he proposed. ##UREF##7##Fan and Sivo (2007)## and ##REF##36601254##Fisk et al. (2023)## examined the performance of fit indices in structural equation modeling (SEM) under conditions of model misspecification. This misspecification refers to discrepancies between the theoretical structure of the model and the simulated dataset.</p>", "<p>In summary, the null distribution of the statistic is widely included in NHST-related statistics. Yet, when evaluating a statistic that does not have a close relationship in NHST (e.g., fit indices), psychological researchers sometimes neglect the null distribution condition. This study demonstrates the importance of this issue using the example of an influential simulation study about the several common statistics of internal consistencies and will propose a new design based on a confusion matrix that always includes a test with null distribution in statistics and evaluates the statistics from these conditions. As our example, we have selected a study conducted by ##REF##27303333##Trizano-Hermosilla and Alvarado (2016)##, which we will henceforth refer to as the “original study” for convenience.</p>", "<title>Results</title>", "<p>Our results indicated that none of these indices performed as a dummy index. However, according to the criteria used in the original study, Omega (i.e., Omega.total) performed the worst in some TN conditions and never the best. In contrast, Alpha showed the best performance across all conditions. This is possibly because Omega, by definition, cannot be smaller than zero, implying that errors can only inflate its results. The full results of our simulation are displayed in ##TAB##1##Table 2##.</p>", "<p>It should be noted that this design is subject to a limitation. The performances of statistics, which are RMSE and %bias used in the original study, have different meanings when the true effects are different. For example, a 10%bias in a condition where the true effect is 0.731 can lead to a considerable number of studies establishing wrong predictions that view the internal consistency of a study as not acceptable because it could create a 95% distribution like [0.5, 0.9] and consequently yield wrong decisions based on these outputs. For instance, a researcher could consider a 0.6 measurement error of Alpha level in a study not acceptable. A 20%bias in the condition that the true effect is zero will not usually influence the decision-making process since it could create a 95% distribution like [0, 0.3]. In the scenario of a null distribution of internal consistency in the population, it does not matter whether the internal consistency is 0.1 or 0.2, as neither internal consistency score is acceptable in psychological studies.</p>", "<p>We propose adding two new designs to the Monte Carlo simulation in psychological statistic testing to overcome this limitation: FNm and FPm.</p>", "<title>Simulation Study 2: estimating the modified false positive and modified false negative</title>", "<p>First, it is essential to review the definition of FP and FN in the confusion matrix. As shown in ##TAB##0##Table 1##, an accurate definition of FP is the percentage of results that erroneously indicate that a particular condition or attribute (e.g., correlation between variables in the same test) is present, whereas FN is the percentage of results that erroneously indicate that a particular condition or attribute is absent.</p>", "<p>These two percentages can be used as criteria in binary outcomes. However, their usage with continuous results is problematic. FP and FN are originally designed for binary results (e.g., yes or no, acceptable or unacceptable). In computer science, the results tend to be clear and objective (i.e., an object is a dog or not a dog). However, this is not always the case in psychological science. The pre-determined cut-off used in psychology for binary conclusions is arbitrary. For instance, it is hard to justify why 0.69 is an unacceptable level of internal consistency while 0.70 is acceptable. This kind of binary thinking is often inappropriate in psychological research. It further implies that designing and measuring P (Internal consistency in simulation &lt;0.70| Population internal consistency = 0.71) or P (Internal consistency in simulation &gt;0.70| Population internal consistency = 0.69) becomes questionable since there is no substantive difference between an internal consistency of 0.69 and 0.70, in which P (X|Y) is a conditional probability, means the possibility of X in the condition of Y.</p>", "<p>In addition, as discussed above, it is also meaningless to measure the percentage of internal consistency and report a weak relationship among variables when the internal consistency in the population follows a null distribution (i.e., P (Internal consistency in simulation &gt; = 0.05| Population internal consistency = 0) and (Internal consistency in simulation &lt;= 0.05| Population internal consistency = 0)). Thus, internal consistency close to 0.1 is not acceptable in psychological research.</p>", "<p>Therefore, these FP and FN percentages have little practical meaning. However, there is a clear difference between the null distribution condition (e.g., Internal consistency = 0.0) and an acceptable level of relationship (e.g., Internal consistency = 0.7). Therefore, we propose two new metrics based on FP and FN, named FPm and FNm, and suggest a study similar to the original that additionally measures these metrics, in which FPm is the percentage that a statistic returns an acceptable level of statistics result when the statistic follows a null distribution in the population (##FORMU##8##6##), and FNm is the percentage that a statistic returns a null result statistic when, in fact, the parameter is at an acceptable level in the population (##FORMU##9##7##).</p>", "<p>The letter M in FPm and FNm stands for modification.</p>", "<p>According to the percentage that should be measured, the FPm in this study is (##FORMU##10##8##).</p>", "<p>The FNm in this study is (##FORMU##11##9##).</p>", "<p>RMSE is not applicable for this design. Yet, criteria are needed for the purpose of this new design. Hence, we propose two criteria:</p>", "<p>Ideally, FPm and FNm should be close to 0% across all conditions. Therefore, for comparison between statistics, the fewer the number of conditions having a number larger than zero, the better the statistic.</p>", "<p>In addition, suppose that FPm or FNm is larger than 5% in a certain condition. We suggest that the statistic should be deemed questionable in this condition and not used. This suggestion is based on the standard tolerable level of binary decision error. For instance, if a statistic shows an FPm of 0.3 when the sample size is 200, we would propose that this statistic is unreliable in this sample size condition because an acceptable level of relationship can be reported by this statistic even if the statistic in the population follows a null distribution. However, this statistic could be reliable with a sample size of 1,000, depending on TP, TN, FPm, and FNm values following this rationale. As a result, we suggest that extreme conditions in psychological research should be included in the simulation study to provide comprehensive results.</p>", "<title>Design</title>", "<p>At first, for both FPm and FNm, the following conditions were included in our study as the original study did: the four internal consistency indices and questionnaire lengths of 6 and 12 items. We included 250, 500, and 1,000 for sample size. In addition, small sample sizes of 20, 25, 30, 35, 40, 45, and 50 are included in this study to test whether there is any condition in psychological studies that these biases will influence TN results.</p>", "<p>In the evaluation of FPm, the datasets were simulated with the same population [i.e., N (0.1)] as in Study 1 to create the null distribution of statistics. In the evaluation of FNm, the datasets were simulated with the same method implemented in the original study. This makes the overall conditions 7*2 = 14. Both tau-equivalent and congeneric models are included. The population covariance matrixes are displayed in the code. All four statistics in the original study are included with questionnaire lengths of 6 and 12. Consequently, this makes the overall conditions 14 in FPm and 28 in FNm.</p>", "<title>Results</title>", "<p>The simulation results of FPm are displayed in ##TAB##2##Table 3##, while the results of FNm are displayed in ##TAB##3##Table 4##. As can be seen in ##TAB##2##Table 3##, <italic>based</italic> on the criteria we proposed, (1) Alpha performs best when there is a null distribution in the internal consistency, and (2) the acceptable level of results of Omega, GLB, and GLBa is questionable when the sample size is less than 30 to 40, depending on the questionnaire length. As can be seen in ##TAB##3##Table 4##, based on the criteria we proposed, all internal consistency indices showed good FNm. This suggests that, under the conditions of our study using the four indices, a result close to zero is highly unlikely to originate from a population with an acceptable level of internal consistency.</p>", "<title>General discussion</title>", "<p>This study introduces a novel simulation design based on a confusion matrix framework. As we propose, this innovative design is particularly suited for use in simulation studies that focus on comparing the performance of statistical methods under various conditions. To demonstrate its applicability, we have presented three potential scenarios and a detailed example illustrating the implementation of this design.</p>", "<p>It is somewhat surprising that researchers might overlook the fact that studies like the original one can only yield empirical evidence when the attribute under investigation reaches an acceptable level. Consider a hypothetical scenario where all populations in psychological research exhibit an acceptable level of a particular statistical parameter. In such a case, regardless of whether the original study violated any assumptions, there would be no necessity to develop statistics to verify the existence of an effect. Furthermore, it’s important to reiterate that APA guidelines advise researchers to include a null effect in any simulation of effect, specifically the absence of a mean difference between two groups (##UREF##6##Fan, 2012##). However, the rationale provided by the APA primarily aims to prevent Type-I errors, potentially leading researchers to mistakenly believe that the null distribution of statistics is only relevant for inferential statistics closely related to NHST. Our research findings suggest otherwise; different statistics may perform variably under different conditions. Identifying the most suitable statistic for these conditions requires including these conditions with an evaluation of the commonly used criteria.</p>", "<title>Data availability statement</title>", "<p>The original contributions presented in the study are included in the article/##SUPPL##0##Supplementary material##, further inquiries can be directed to the corresponding author.</p>", "<title>Author contributions</title>", "<p>YC: Data curation, Writing – original draft. PP-D: Writing – review &amp; editing. KP: Writing – review &amp; editing. JL: Writing – review &amp; editing.</p>" ]

[ "<title>Conflict of interest</title>", "<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>", "<title>Publisher’s note</title>", "<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>", "<title>Supplementary material</title>", "<p>The Supplementary material for this article can be found online at: <ext-link xlink:href=\"https://www.frontiersin.org/articles/10.3389/fpsyg.2023.1298534/full#supplementary-material\" ext-link-type=\"uri\">https://www.frontiersin.org/articles/10.3389/fpsyg.2023.1298534/full#supplementary-material</ext-link></p>", "<title>Appendix</title>", "<p>The R code that generated all the data and simulation results in this study is available in a separate file that is attached to the current submission to the journal Frontiers in Psychology. It is also available through the URL: <ext-link xlink:href=\"https://liqas.org/code-under-review/\" ext-link-type=\"uri\">https://liqas.org/code-under-review/</ext-link>. Researchers are encouraged to simulate and replicate the results for future research. This study was not preregistered.</p>" ]

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[ "<table-wrap position=\"float\" id=\"tab1\"><label>Table 1</label><caption><p>The elements of a confusion matrix.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th rowspan=\"1\" colspan=\"1\"/><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Predicted true result</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Predicted false result</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Actual true result</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TP</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">FN</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Actual false result</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">FP</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TN</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"tab2\"><label>Table 2</label><caption><p>Estimation of true negative in Study 1.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" valign=\"top\" colspan=\"2\" rowspan=\"1\"/><th align=\"center\" valign=\"top\" colspan=\"4\" rowspan=\"1\">%bias</th><th align=\"center\" valign=\"top\" colspan=\"4\" rowspan=\"1\">RMSE</th></tr><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Length</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">SS</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Alpha</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Omega</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">GLB</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">GLBa</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Alpha</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Omega</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">GLB</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">GLBa</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">250</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12.41%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">38.39%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">15.89%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">30.67%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">14.17%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">39.35%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">17.58%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">32.53%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">500</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">33.42%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">11.52%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">27.67%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">10.35%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">34.70%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12.87%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">29.73%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1,000</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6.42%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">28.99%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">8.23%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">24.77%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">7.42%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">30.63%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9.22%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">27.11%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">250</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">17.67%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">29.27%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">26.59%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">34.45%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">18.62%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">29.71%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">27.47%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">35.52%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">500</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12.76%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">23.38%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">19.13%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">30.34%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">13.58%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">23.87%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">19.87%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">31.45%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1,000</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9.28%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">19.15%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">13.78%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">27.04%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9.90%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">19.74%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">14.35%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">28.26%</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"tab3\"><label>Table 3</label><caption><p>Estimation of false positive modified in Study 2.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Length</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">SS</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Alpha</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Omega</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">GLB</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">GLBa</th></tr></thead><tbody><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">20</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.19%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>49.93%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>11.07%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>8.48%</bold>\n</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">30</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>16.55%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">1.73%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">1.85%</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">40</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>5.43%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.22%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.36%</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">50</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">1.63%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.07%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.16%</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">250</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">500</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">1,000</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">20</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.89%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>40.98%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>85.25%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>35.37%</bold>\n</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">30</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>5.03%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>42.15%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>11.59%</bold>\n</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">40</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.32%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>14.03%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">3.36%</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">50</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.02%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">3.70%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.98%</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">250</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">500</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">1,000</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"tab4\"><label>Table 4</label><caption><p>Estimation of false negative modified in Study 2.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">QL</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">SS</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Condition</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Alpha</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Omega</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">GLB</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">GLBa</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">20</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.28%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">20</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.34%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">20</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.01%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">20</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.01%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">30</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.09%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">30</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.15%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">30</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.03%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">30</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.01%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">40</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.03%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">40</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.03%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">40</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.01%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">40</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">50</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">50</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.03%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">50</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.02%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">50</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.02%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">250</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">250</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.01%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">250</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">250</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">500</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">500</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.01%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">500</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">500</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1,000</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1,000</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.06%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1,000</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1,000</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"tab5\"><label>Table 5</label><caption><p>Estimation of false positive method with unacceptable internal consistency level.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">QL</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">SS</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Condition</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">IL</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Alpha</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Omega</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">GLB</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">GLBa</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">20</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.3</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.75%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>56.45%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>35.83%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>34.83%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">25</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.3</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.20%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>39.08%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>23.99%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>25.93%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">30</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.3</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.12%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>27.14%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>16.02%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>19.77%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">35</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.3</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.02%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>18.08%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>10.25%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>14.81%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">40</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.3</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.01%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>12.68%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>6.28%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>10.77%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">45</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.3</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.02%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>8.89%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">4.12%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>8.35%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">50</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.3</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>6.63%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">2.60%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>6.68%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">250</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.3</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">500</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.3</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1,000</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TE</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.3</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">20</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.5</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">6.82%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>60.38%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>99.63%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>92.11%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">25</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.5</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">3.65%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>40.39%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>98.48%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>88.77%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">30</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.5</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">2.07%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>28.16%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>96.93%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>85.42%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">35</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.5</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">1.21%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>21.08%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>95.50%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>82.38%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">40</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.5</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">1.07%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>16.04%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>93.19%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>79.36%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">45</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.5</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.60%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>12.09%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>90.52%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>76.02%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">50</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.5</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.47%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>9.49%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>87.67%</bold>\n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>73.44%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">250</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.5</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">2.18%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">\n<bold>18.57%</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">500</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.5</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">7.78%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1,000</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CG</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.5</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.00%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">3.43%</td></tr></tbody></table></table-wrap>" ]

[ "<disp-formula id=\"EQ1\">\n<label>(1)</label>\n<mml:math id=\"M1\" overflow=\"scroll\"><mml:mi mathvariant=\"italic\">RMSE</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:msup><mml:mfrac><mml:mrow><mml:mo stretchy=\"true\">∑</mml:mo><mml:mrow><mml:mfenced open=\"(\" close=\")\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo>−</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:mfrac><mml:mn>2</mml:mn></mml:msup></mml:msqrt></mml:math>\n</disp-formula>", "<disp-formula id=\"EQ2\">\n<label>(2)</label>\n<mml:math id=\"M2\" overflow=\"scroll\"><mml:mo>%</mml:mo><mml:mi mathvariant=\"italic\">bias</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy=\"true\">∑</mml:mo><mml:mrow><mml:mfenced open=\"(\" close=\")\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo>−</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:mfrac><mml:mo>×</mml:mo><mml:mn>100</mml:mn><mml:mo>%</mml:mo></mml:math>\n</disp-formula>", "<disp-formula id=\"EQ3\">\n<label>(3)</label>\n<mml:math id=\"M3\" overflow=\"scroll\"><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mn>0.78</mml:mn><mml:mtext>.</mml:mtext></mml:math>\n</disp-formula>", "<disp-formula id=\"EQ4\">\n<label>(4)</label>\n<mml:math id=\"M4\" overflow=\"scroll\"><mml:mtext>RMSE</mml:mtext><mml:mo>=</mml:mo><mml:msqrt><mml:mfrac><mml:mrow><mml:msubsup><mml:mstyle displaystyle=\"true\"><mml:mo stretchy=\"true\">∑</mml:mo></mml:mstyle><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>000</mml:mn></mml:mrow></mml:msubsup><mml:msup><mml:mfenced open=\"(\" close=\")\"><mml:mrow><mml:mn>0.731</mml:mn><mml:mo>−</mml:mo><mml:mn>0.78</mml:mn></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>1000</mml:mn></mml:mfrac></mml:msqrt><mml:mo>=</mml:mo><mml:mn>0.049</mml:mn></mml:math>\n</disp-formula>", "<disp-formula id=\"EQ5\">\n<label>(5)</label>\n<mml:math id=\"M5\" overflow=\"scroll\"><mml:mo>%</mml:mo><mml:mtext>bias</mml:mtext><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mstyle displaystyle=\"true\"><mml:mo stretchy=\"true\">∑</mml:mo></mml:mstyle><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>000</mml:mn></mml:mrow></mml:msubsup><mml:mfenced open=\"(\" close=\")\"><mml:mrow><mml:mn>0.731</mml:mn><mml:mo>−</mml:mo><mml:mn>0.78</mml:mn></mml:mrow></mml:mfenced></mml:mrow><mml:mn>1000</mml:mn></mml:mfrac><mml:mo>×</mml:mo><mml:mn>100</mml:mn><mml:mo>%</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>4.9</mml:mn><mml:mo>%</mml:mo></mml:math></disp-formula>", "<inline-formula>\n<mml:math id=\"M6\" overflow=\"scroll\"><mml:mover accent=\"true\"><mml:mi>C</mml:mi><mml:mo stretchy=\"true\">˙</mml:mo></mml:mover></mml:math>\n</inline-formula>", "<inline-formula>\n<mml:math id=\"M7\" overflow=\"scroll\"><mml:mover accent=\"true\"><mml:mi>C</mml:mi><mml:mo stretchy=\"true\">˙</mml:mo></mml:mover></mml:math>\n</inline-formula>", "<inline-formula>\n<mml:math id=\"M8\" overflow=\"scroll\"><mml:msub><mml:mi>ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math>\n</inline-formula>", "<disp-formula id=\"EQ6\">\n<label>(6)</label>\n<mml:math id=\"M9\" overflow=\"scroll\"><mml:mrow><mml:mi mathvariant=\"normal\">FPm</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">P</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mtable columnalign=\"left\"><mml:mtr><mml:mtd><mml:mi mathvariant=\"normal\">Acceptable level of</mml:mi><mml:mspace width=\"thickmathspace\"/><mml:mi mathvariant=\"normal\">a</mml:mi><mml:mspace width=\"thickmathspace\"/><mml:mi mathvariant=\"normal\">statistic in </mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi mathvariant=\"normal\">simulation</mml:mi><mml:mspace width=\"thickmathspace\"/><mml:mi mathvariant=\"normal\">|</mml:mi><mml:mspace width=\"thickmathspace\"/><mml:mi mathvariant=\"normal\">Null distribution of statistic</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math>\n</disp-formula>", "<disp-formula id=\"EQ7\">\n<label>(7)</label>\n<mml:math id=\"M10\" overflow=\"scroll\"><mml:mrow><mml:mi mathvariant=\"normal\">FNm</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">P</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mtable columnalign=\"left\"><mml:mtr><mml:mtd><mml:mi mathvariant=\"normal\">Null distribution of statistic </mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi mathvariant=\"normal\">in simulation</mml:mi><mml:mspace width=\"thickmathspace\"/><mml:mi mathvariant=\"normal\">|</mml:mi><mml:mspace width=\"thickmathspace\"/><mml:mi mathvariant=\"normal\">Acceptable level </mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi mathvariant=\"normal\">parameter in population</mml:mi></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math>\n</disp-formula>", "<disp-formula id=\"EQ8\">\n<label>(8)</label>\n<mml:math id=\"M11\" overflow=\"scroll\"><mml:mrow><mml:mi mathvariant=\"normal\">P</mml:mi><mml:mspace width=\"thickmathspace\"/><mml:mrow><mml:mo>(</mml:mo><mml:mtable columnalign=\"left\"><mml:mtr><mml:mtd><mml:mi mathvariant=\"normal\">Internal consistency in simulation</mml:mi><mml:mo>&gt;</mml:mo><mml:mo>=</mml:mo><mml:mn>0.7</mml:mn><mml:mspace width=\"thickmathspace\"/><mml:mi mathvariant=\"normal\">|</mml:mi><mml:mspace width=\"thickmathspace\"/></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi mathvariant=\"normal\">Population internal consistency parameter</mml:mi><mml:mo>=</mml:mo><mml:mn>0.0</mml:mn></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math>\n</disp-formula>", "<disp-formula id=\"EQ9\">\n<label>(9)</label>\n<mml:math id=\"M12\" overflow=\"scroll\"><mml:mrow><mml:mi mathvariant=\"normal\">P</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mtable columnalign=\"left\"><mml:mtr><mml:mtd><mml:mi mathvariant=\"normal\">Internal consistency in simulation</mml:mi><mml:mo>&lt;</mml:mo><mml:mo>=</mml:mo><mml:mn>0.0</mml:mn><mml:mspace width=\"thickmathspace\"/><mml:mi mathvariant=\"normal\">|</mml:mi><mml:mspace width=\"thickmathspace\"/></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi mathvariant=\"normal\">Population internal consistency</mml:mi><mml:mo>=</mml:mo><mml:mn>0.70</mml:mn></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>\n</disp-formula>" ]

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[ "<supplementary-material id=\"SM1\" position=\"float\" content-type=\"local-data\"></supplementary-material>" ]

[ "<table-wrap-foot><p>True Positive (TP) is the proportion of results that correctly indicates the presence of a condition or characteristic in the population; True Negative (TN) is the proportion of results that correctly indicates the absence of a condition or characteristic in the population; False Positive (FP) is the proportion of results that erroneously indicates that a particular condition or attribute is present in the population, while False Negative (FN) is the proportion of results which erroneously indicates that a particular condition or attribute is absent in the population. In an ideal perfect model, TP and TN should be at 100%, and FN and FP should be at 0%.</p></table-wrap-foot>", "<table-wrap-foot><p>Length is the length of items; SS is the sample size, and RMSE is Root Mean Square Error without the degree of freedom adjustment.</p></table-wrap-foot>", "<table-wrap-foot><p>Length is the length of items; SS is the sample size. The percentage values are acceptable (i.e., adequate reliability) when the dataset follows a null distribution (i.e., zero reliability) in the population. Percentages in bold are the Percentages above 5%, which suggests the result of a specific statistic is questionable in this condition.</p></table-wrap-foot>", "<table-wrap-foot><p>QL is the length of items; SS is the sample size. TE is tau-equivalent model. CG is Congeneric model. Percentage values are failures that suggest statistics report that there is no internal consistency when in fact, there is an acceptable internal consistency in the population.</p></table-wrap-foot>", "<table-wrap-foot><p>QL is the length of the questionnaire or the item number in a questionnaire; SS is the sample size. TE is tau-equivalent model. CG is Congeneric model. IL is the population internal consistency parameter of Alpha. Percentage values are failures that suggest that a statistic report that internal consistency is above the cut-off when in fact, there is an internal consistency parameter that is considerably away from this cut-off.</p></table-wrap-foot>" ]

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[ "<media xlink:href=\"Data_Sheet_1.ZIP\"><caption><p>Click here for additional data file.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>The concept of autism has had a central role in cultural and clinical thinking throughout the twentieth and twenty-first centuries but has lacked a developed and representative phenomenology. Accounting for this will lead us to the heart of many current debates and dilemmas in this field. As a clinical concept, autism has historically been attended to and interpreted by clinicians and philosophers and, until recently, has very little representation of autistic voices themselves. This does not reflect the evolving preoccupations and understanding of the current time. In this paper, we argue that autism conceptualisation and clinical practise have reached a pivotal point, making a representative and systematic phenomenology approach possible. We offer suggestions as to how we can progress beyond this situation and the importance and value of so doing.<xref rid=\"fn0001\" ref-type=\"fn\">¹</xref></p>" ]

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[ "<title>Discussion</title>", "<p>This study contains and juxtaposes several representations and expressions of autistic experience manifested over time and across across perspectives: (1) in the early years of phenomenological psychiatry, usually within the context of schizophrenia; (2) as conceptualised within early developmental science; (3) as theorised in terms of recent clinical and scientific reviews; (4) as experienced in a current clinical context; and (5) as described by the voices of autistic people writing about themselves. These juxtapositions set up a series of resonating comparisons across historical time and between perspectives. The contemporary autistic voice, for instance, resonates in some striking ways with the early clinical phenomenology in descriptions of highly intense, elaborated and absorbing <italic>internal worlds</italic>. The quality of these descriptions, however, varies. They are framed pathologically in the early clinical accounts, linked, for instance, to analytic drive theory or a loss of being. They are framed more creatively and positively in contemporary accounts of autistic sensitivity, intuition, and hyper-awareness, although with concomitant vulnerability to being overwhelmed by environments and experiences. The contrasting idea from some early developmental science literature that the slower language and symbolic play development emphasised at that time<xref rid=\"fn0004\" ref-type=\"fn\">⁴</xref> was linked to an absence of internal-world fantasy is balanced by suggestions from some contemporary autistic accounts of very rich non-verbal sensory internal environments (such as auditory or visual); difficult to access but part of “spiky profiles” and enhanced sensory abilities in many autistic children (Ockelford, ##UREF##24##2015##). The re-introduction of interest in the phenomenology of these internal world experiences and potential abilities has substantial potential implications for developmental science concepts (see Evans, ##REF##24014081##2013##).</p>", "<p>Similarly common across history and theoretical positions, there is a described quality of <italic>dislocation</italic> between the autistic self and what is felt as a non-autistic reality. This is described from both clinical observation and subjective accounts of a sense of living in a parallel world—understandable within itself but subtly disconnected from a less understandable neuro-typically shared experience (Humphrey and Lewis, ##REF##18178595##2008##). What varies radically across historical time is the causal attributions made to this disconnect; early on, it is understood as driven by absorption in primary pathological fantasy and with a sense of shared sociability needing to be an achieved skill; later, as driven by a “poorness of fit” with or lack of adaptation <italic>from</italic> as well as <italic>to</italic> the neurotypical world. The profusion of acute sensitivity and openness manifest in many of the autistic-voice accounts in this study is a corrective to the intrinsic pathologising of earlier accounts. Yet, there is also a recognition of vulnerabilities and the perplexity associated with encountering a neurotypical assumptive world, which Murray et al. frame as a difficulty with “social joining” (Murray et al., ##REF##36183692##2023##). This (mutual) perplexity resonates with some of the qualitative literature and comes across in contemporary clinical phenomenology.</p>", "<p>One can perhaps generalise, in other words to an autistic experience characterised by a sense of difference or dislocation from the shared experience and communication between others, plus experiences of internal world difference and sensitivities, that under some circumstances, can confer strength and benefit and in others can contain a vulnerability to overwhelm and meltdown. This resonates with more general notions from the developmental science literature, for instance, of “differential susceptibility” and the idea of a continuum of underlying difference coupled with emergent difficulty, as articulated by Bervoets and Hens (##UREF##2##2020##), Bervoets (##UREF##1##2022##), and Green (##REF##36304560##2022##). For instance, Bervoets (##UREF##1##2022##) gives an account of “neuro gradualism” from which the (probabilistic rather than determined) emergence of intersubjective dislocation opens up both potential difficulties but also a space for moral imagination within and between both self and other. Naming autism here is a means toward human understanding rather than stigma and echoes back to the empathic interpersonal stance of Binswanger's phenomenology. The latter's sense of autism as a radical loss of being-in-the-world is answered by contemporary advocacy and action toward greater awareness and accommodation of neurodiverse difference within a shared community.</p>", "<p>The other thing that is juxtaposed in the accounts in this study are different positioned perspectives and associated partialities (or biases) in perception and evaluation. Such partiality is naturally most easily seen from a historical distance and is striking when one looks at the analytically informed accounts of the early twentieth century. Awareness of our equivalent partiality in the present takes more reflection. We have tried to emphasise the positionality of these different accounts in their context as they have been described. Sometimes, this partiality matters less. It takes only a few accounts of intense autistic worlds and elaborated sensitivity in autism to counter some of the more extreme deficit-based models which came from the behavioural turn and the downplaying of subjective experience, particularly within observation of children and a theoretical notion that autism was intrinsically related to language delay (something subsequently shown to be incorrect). But if we are to apply a more systematic and applicable phenomenology across the range of autism as currently apparent, then we are proposing the need for a more inclusive, systematic, and empirical approach—something that is outlined in the final section of the paper. This includes aspect of participatory science, new approaches to data collection and analysis (informed by citizen science models), and the development of new tools and technologies for non-speaking individuals to ensure that the range of autistic voices contributes to the development of the phenomenological research. This will not solve all the issues arising in this area, but it will be an approach that, if applied systematically over time, could overcome partiality in narratives and distil common representative elements of autistic phenomenology, foundational for a re-casting of the autism phenotype as it is currently framed.</p>" ]

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[ "<p>Edited by: Valeria Bizzari, Husserl-Archives: Centre for Phenomenology and Continental Philosophy, Belgium</p>", "<p>Reviewed by: Nicolas Tajan, Kyoto University, Japan</p>", "<p>Janko Nešić, Institute of Social Sciences (IDN), Serbia</p>", "<p>We are now at a transition point in autism conceptualisation, science, and clinical practise, where phenomenology could play a key role. This paper takes a broad view of the history of phenomenological perspectives on the autism concept and how this has evolved over time, including contemporaneous theory and methods. Early inquiry from a clinical perspective within the tradition of classical continental phenomenology, linked closely to the consideration of schizophrenia, is contrasted with emerging observations of child development and a period in the second half of the twentieth century of scientific inquiry into a behavioural autistic phenotype where there was little or no phenomenological aspect; a phenotype that has determined the recent scientific and clinical conceptualisation of autism within current nosology. We then mark a more recent reawakening of interdisciplinary interest in subjective experience and phenomenological inquiry, which itself coincides with the increasing prominence and salience of the neurodiversity movement, autistic advocacy, and critical autism studies. We review this emerging phenomenological work alongside a contemporaneous clinical phenomenology perspective and representations of autistic experience from within the extensive literature (including life writing) from autistic people themselves; all perspectives that we argue need now be brought into juxtaposition and dialogue as the field moves forward. We argue from this for a future which could build on such accounts at a greater scale, working toward a more co-constructed, systematic, representative, and empirical autistic phenomenology, which would include citizen and participatory science approaches. Success in this would not only mean that autistic experience and subjectivity would be re-integrated back into a shared understanding of the autism concept, but we also argue that there could be the eventual goal of an enhanced descriptive nosology, in which key subjective and phenomenological experiences, discriminating for autism, could be identified alongside current behavioural and developmental descriptors. Such progress could have major benefits, including increased mutual empathy and common language between professionals and the autistic community, the provision of crucial new foci for research through aspects of autistic experience previously neglected, and potential new supportive innovations for healthcare and education. We outline a programme and methodological considerations to this end.</p>" ]

[ "<title>The autistic concept in classical phenomenology</title>", "<p>Clinical descriptions are interested in both the <italic>content</italic> and <italic>form</italic> of another's experience to reach toward the empathic understanding of an individual, but also toward more general theories of mental function. The first conceptualisation of autism as a clinical concept is attributed to Bleuler from 1911 as part of early efforts to understand and theorise psychotic states in schizophrenia. He took the name from its Greek root to refer to an “elaborated self-ness” marked by an intense fantasy life linked with withdrawal or disconnection from a shared social world with others. The term was taken up by later psychoanalytic and developmental writers as they refracted evolving contemporary theories of mental life, a history that continues (Evans, ##REF##24014081##2013##). It became associated with psychoanalytic concepts of “primary narcissism”, and, as later amplified by Klein, an “autistic stage” became something conceptualised as part of all infants' early development before their emerging relational engagement with the world. Psychiatry linked to an evolving existential tradition focused more on the state of autistic being. Minkowski, for instance, emphasised the (autistic) “loss of vital contact with reality” and lack of interpersonal awareness or attunement to the environment (Parnas and Bovet, ##REF##2001623##1991##). For the existential psychiatrist Ludwig Binswanger, autism was understood as a radical loss of the central Heideggerian concept of <italic>Dasein</italic>—an authentic and freely embedded “being-in-the-world”. It was this “loss of self and the world” (Needleman, ##UREF##23##1968##, p. 388) that characterised the autistic state rather than an overly internally directed drive as understood from analytic theory. Binswanger's essentially relational understanding of being focused on the quality of interpersonal experience with autistic people—as a lack of their shared “common sense” or intersubjective horizon with others—which later came to be used as an intuitive criterion for clinicians in what was called atmospheric diagnosis (i.e., based on an experience of being with the other). Autistic experience is here characterised as something fundamentally differing and in parallel to a normatively shared world; for Jaspers, a differing world tending to be essentially “not knowable” (Erklären), while for Binswanger, fundamentally different but approachable with an effort of relational imagination (Hoffmann and Knorr, ##UREF##15##2018##).</p>", "<p>What can we take from this early tradition? There remains a question as to whether these early accounts were really of individuals we would now understand as autistic, but the articulated autism concept, in its location at the junction of selfhood and social life, encapsulated and became a focus-case for many general issues which preoccupied, and still preoccupy, the humanities and the human sciences. Autism raises questions around interpersonal being-in-the-world and inter-subjectivity that remain relevant and central to our tradition. These early clinical formulations of Bleuler's elaborated <italic>self-ness</italic> and Binswanger's sense of an autistic loss of absorption (he uses the term <italic>gestimmtheit</italic> or attunement) within a shared world, along with an apparent “unknowability” from others, receive vivid commentary (as well as some key challenges) in contemporary experiential accounts from autistic people themselves, as we will see below. Some of this classic clinical phenomenological writing has also continued into the present, but in the context of schizophrenia rather than autism itself, as a primary disruption of <italic>self-ness</italic> or <italic>ipseity</italic> and existential presentness as what are still termed “autistic” states within psychosis (for instance, Sass and Parnas, ##REF##14609238##2003##).</p>", "<p>From early on, autism was also a focus within an emerging child developmental science (methodologies directly involving children rather than inference from adult recollection). Piaget (##UREF##26##1929##) used observation and phenomenological-type interviews with a small group of children, particularly his own daughters, in conceptualising a sequence of emerging developmental engagement with reality from what he considered (echoing analytic stage theory) normative autistic beginnings to fully-fledged adult cognitive and moral engagement with the world. Gesell and Armatruda (##UREF##11##1945##) used more systematic empirical observation (including creative early photography and video) on larger samples in early child development to describe an “embryology of behaviour”, understood as an unfolding process analogous with foetal embryology. Rather than a normative stage, they inferred a developmental immaturity account of early psychosis and autistic behaviours; indeed, this work was antecedent to later developmental science demonstrating, in contrast to a normative autistic phase, the exquisite environmental responsiveness and connectedness of all babies. Normative child development milestones in language, motor, cognitive or social skills came to be codified based on significant divergence from population means. Clinicians such as Sukhareva and later Asperger published descriptive accounts of recognisably autistic teenagers and adults but less focussed on phenomenology, against a background of described intellectual disability and personality development. Sukhareva is notable particularly for her early clinical accounts from 1926, including with girls (Simmonds and Sukhareva, ##REF##31367779##2020##; Sher and Gibson, ##REF##34562153##2021##). The classic early clinical accounts of “infantile autism” (Kanner, ##REF##4880460##1943##) also grew from this framework of observed early child development but mixed still with psychoanalytic stage theory.</p>", "<title>A “behavioural turn” and loss of subjectivity</title>", "<p>An emerging empirical psychiatry in the 1960s and 70s took strategic decisions to focus exclusively on precise and repeatable observations of autistic behaviour, particularly within early childhood (in the tradition initiated by Gesell), actively excluding the subjective or experienced self from consideration. In this way, it radically re-cast the autism concept as an intended atheoretical and empirical “behavioural phenotype” within epidemiological enquiry (Evans, ##REF##24014081##2013##). This was partly to make the science easier but also to make decisive distance from ongoing analytically informed debates on the subjective internal world nature and causes of autism. This “behavioural turn” did meet resistance. A prominent clinical child psychiatrist wrote that it was “…impossible to use purely behavioural criteria if we were to convey what we all felt to be the heart of the matter—namely the presence of an impaired capacity for human relationships” (Creak, ##UREF##4##1961##). Later, the eminent US academic Leon Eisenberg worried about a move toward “mindless psychiatry” (Eisenberg, ##REF##3535971##1986##). Nevertheless, this behavioural turn can be credited with great scientific success in its own terms. Efforts toward a shared behavioural phenotypic account of autism, plus identifying it as an early childhood condition (with the associated views that internal mental states would be impossible to access), enabled a reliable psychometrically valid and shared empirical description to be established that facilitated much international science for 50 years. In turn, this behavioural focus was followed, by nosological and clinical language within the formulations of the International Classification of Diseases and Diagnostic and Statistical Manual of Mental Disorders. The autism concept was reframed and distinguished from both schizophrenia and phenomenology (Barale et al., ##UREF##0##2019##); however, any newer autistic phenomenology reflecting this emerging developmental understanding was missing. Evans (##REF##24014081##2013##) describes a key conceptual transformation in this, from understanding autistic states as an elaboration of recursive fantasy substituting for lack of contact with reality to the idea of autism as a “lack of affective contact” or a “social impairment”. Linked to it was the rise of a cognitive tradition of research focused on deficits in the “theory of mind” and reflexivity. This focus, in the establishment of the idea that autistic people were unable to be reflective or accurate witnesses to their own experience, became another reason for a first-person autistic phenomenology being thought impossible and a growing literature of subjective accounts of autism largely sidelined from scientific consideration.</p>", "<title>A shifting paradigm and re-emergence of phenomenology</title>", "<p>Much has changed in recent years to challenge this view. Firstly, the cognitive focus in autism has rather run its course; the differences in the “theory of mind” ended up being non-specific to autism and confounded by expressive language difficulties and IQ variation. Secondly, there has been the rise into prominence of a neurodiversity movement in which verbal and cognitively able autistic adults advocate for the significance of their experience and its qualities. The previous exclusion of these accounts can now be seen as a form of epistemic injustice (Fricker, ##UREF##8##2007##), as well as arguably having had the effect of narrowing scientific and clinical notions of what autism might be, with consequences for clinical neuroscientific accounts and research agendas. A new empirical autistic phenomenology now seems both necessary and feasible within a new phase in the historical evolution of the autism concept, re-attending to the phenomenological but doing so against a background of greatly more information about the normal range of development in brain and mind from neuroscience and developmental psychology research. Moving on from early functional notions of autistic withdrawal and defence, we are now able to frame instead the phenomenological impact of developmental brain differences, ranging from the subtle to the severely disruptive. Additional theories to explain the differences between autistic and non-autistic experiences have come from autistic scholarship, particularly around the social transaction and aspects of attention (e.g., Murray et al., ##REF##15857859##2005##; Milton, ##UREF##21##2012##; Milton et al., ##REF##36263746##2022##). Such conceptualisations, informed by the personal life experiences of autistic writers, are also beginning to have an impact on autism science. Theoretical directions and collaborations between autistic and non-autistic researchers are starting to be evident in contemporary scholarship on social interaction (Morrison et al., ##REF##31823656##2020##), masking (Pearson and Rose, ##REF##36601266##2021##), and participatory community research (Pickard et al., ##REF##34088215##2022##).</p>", "<p>Alongside, there has been a recent acceleration in editorial and position statements advocating a return to a phenomenology approach and phenomenology studies in psychiatry and psychology generally (Stanghellini and Broome, ##REF##25179621##2014##; Kyzar and Denfield, ##REF##36460728##2023##; Ritunnano et al., ##REF##36645112##2023##) and in autism, specifically (Zahavi and Parnas, ##UREF##37##2003##; Nilsson et al., ##REF##31170725##2019##). Phenomena around a set of primary defining experiences foundational to being can be distinguished within more general descriptive accounts with a broader focus that includes many general aspects of a lived life in society, workplace, and family. Particularly relevant to this paper is a phenomenology that foregrounds these primary distinctive aspects of autistic experience in a way that would, for instance, provide specific differentiation between autistic and non-autistic states and be relevant for autistic delineation. Nevertheless, as Boldsen (##UREF##3##2021##) notes, even such foregrounded essential autistic experience will always be situated; it remains important to include the social-spatial-material fields of the context within the interpretation. Much current phenomenological work published in autism studies contributes important evidence on the lives of autistic people and their needs with a broad focus (e.g., on social policy, education, and care) (DePape and Lindsay, ##UREF##6##2016##; Howard et al., ##REF##30672307##2019##; MacLeod, ##UREF##16##2019##; Williams et al., ##REF##29139322##2019##; Pellicano et al., ##REF##32474432##2020##). There is, however, emerging work on what could be considered more primary essential experiences. It is immediately apparent that one of the features of this recent phenomenological work is how it pushes back against recently received notions within the autism behavioural phenotype and associated cognitive research. For instance, a substantive body of work (Shanker, ##UREF##30##2004##; De Jaegher, ##REF##23532205##2013##, ##REF##34591703##2021##; Fuchs, ##UREF##10##2015##) critiques the previous explanatory notion of inter-subjectivity in terms of individuals' theory of mind deficits, introducing alternative framing and evidence on embodied experience in cognition and intersubjectivity, and accounts of autistic selfhood and self-experience that echo some of the recent discussion in schizophrenia (Sass and Parnas, ##REF##14609238##2003##). New participatory methodologies for investigating such experienced inter-subjectivity are also described (De Jaegher et al., ##UREF##5##2017##). These accounts provide important theoretical discussion and illustrate how a phenomenological perspective can open new conceptualisations and scientific areas for study.</p>", "<p>Other recent phenomenological writing takes a turn convergent with empirical evidence from contemporary neuroscience, which has moved beyond a search for specific deficits in the “social brain” to understand a more system-wide set of dislocation differences in early development. One summary of contemporary theoretical and empirical work (Barale et al., ##UREF##0##2019##) highlights aspects of the “Life World” of autistic people, drawing out common but inconsistent differences in the fundamentals of the experience of “temporality, trans modal perception, organisation of predictability and ‘forward thinking', affectivity and reciprocity, intentionality and praxic coordination”—a complex of differences in experience supported by recent developmental science (Jones et al., ##REF##24361967##2014##), including what can be characterised as the “sensorium”. They make the point that a search for unitary core <italic>deficit</italic> within autism is now past the point; it is more that the integration of multiple and varying fundamental aspects of being is different and confers vulnerability within physical as well as inter-subjective contexts (Barale et al., ##UREF##0##2019##). In this vein, Narzisi and Muccio (##REF##34356148##2021##) highlight work on altered perceptual and sensory sensitivity and reduced “priors” in cognition, and Rizzo and Röck (##UREF##28##2021##) describe the “Intense World Theory” of autistic experience, emphasising perceptual and sensory differences, and the consequent delights, capacities, and vulnerable difficulties associated with it. More focused recent qualitative studies using empirical methodology have addressed autistic sensory experience (Sibeoni et al., ##REF##35362340##2022##; Taels et al., ##REF##36876409##2023##), anxiety (Magiati et al., ##UREF##17##2017##), gender dysphoria (Cooper et al., ##REF##36089890##2023##), and the potential roots of some high level autistic sensory abilities (Ockelford, ##UREF##24##2015##). Despite this, there remains a relative lack of autistic researcher-mediated work or the direct autistic voice in this literature—something that we now represent further below.</p>", "<p>As a recent step toward a participatory autistic phenomenology, Murray et al. (##REF##36183692##2023##) reported an in-depth comparative peer-mediated phenomenological enquiry across the autism/non-autism divide between three autistic and one non-autistic participants. Many aspects of lived experience were found in common (see ##FIG##0##Figure 1##). For instance, the need for intimacy, trust and acceptance, and the enjoyment of social contact on the right terms; in a way that radically challenges classic notions of a fundamental disconnection from the world or of necessary intrinsic social avoidance and impairment in the current definition, as well as giving insights into common core needs and values across the “human spectrum”. But the areas of difference were also striking, particularly the distinctive experience of the autistic sensorium from early life and the early experience of being overwhelmed by experiences or misunderstood by others. This study echoes Barale et al. in arguing that it is likely that the core sensorium is foundational in the experience and reality of neurodivergence and may provide a key guide as to where neuroscience might look for future insights. A great challenge, clinically and methodologically, will be how to approach equivalent insights for young children and for older people with complex communication needs and to test how much a common experience may or may not underlie the heterogeneity in autistic people and their expression, and we return to this below.</p>", "<title>A contemporary clinical phenomenology</title>", "<p>It could be said that the practise of phenomenology concerns particular forms of attentiveness. Classical philosophical phenomenology is concerned with a quality of open attentiveness to one's own response to the world, removing as far as possible prior pre-conceptions to investigate the structure of consciousness and perception. Clinical phenomenology, on the other hand, pertains to the quality of attentiveness to the experience of another in an effort to understand their mind; as this evolved, the idea of clinical phenomenology progressed to describing another's state of mind in illness (Zahavi, ##UREF##36##2018##). Binswanger's emphasis on the relational, described above, anticipated a kind of attentive interpersonal listening that has become central to psychotherapy and much clinical mental health practise (Frie, ##UREF##9##2000##). In ideal terms, clinical description reports an encounter between the subject and an experienced and attentive other in a context of confidentially and intimacy. Binswanger spoke of a kind of agape (ethical love) aspired to in such clinical encounters (Hoffmann and Knorr, ##UREF##15##2018##), exemplifying a general human effort toward imaginative empathy across divides of otherness. Extended case descriptions, such as in Binswanger's of Lola Voss (Needleman, ##UREF##23##1968##), give vivid evidence of the autism concept of the time, just as have others more recently (Sacks, ##UREF##29##1995##). Nevertheless, there are inevitable partialities of the clinical context, such as ethical issues in relaying specific experiences while ensuring anonymity, selecting particular clients, or in a particular context in their lives, which may be different to a non-clinical context or autistic self-report. The clinician's accounts are also situated in the context of providing care, which may influence what they attend to and report. They also need to be understood explicitly in the context of the conceptual thinking of the time since, despite all efforts toward phenomenological reduction in setting aside prior concepts, the core organisation of a clinician's perception will still be informed by current ideas. To illustrate this point, it is enough to look back on clinicians' accounts from previous generations. The clinical accounts that follow are subject to all these considerations. They are vignettes, fully anonymised with key identifying details changed, and within which the quotes are illustrative close transliterations rather than verbatim recordings. They are offered to show what a contemporary clinical phenomenology can be like from the positionality of a clinician (JG) who is himself non-autistic and working in a period of neurodevelopmental conceptualisation of autism. This section can then be balanced against the following one of autistic auto-ethnography.</p>", "<title>Inter-subjective dislocation and social difficulties</title>", "<p>Two cases illustrate the experience of the <italic>inter-subjective dislocation and social difficulties</italic> spoken of by Barale et al. above. Brian is an autistic young man of 17 whom I met in the context of acute crises of anxiety and suicidality. Brian's main preoccupation is his experienced difficulty in negotiating the regular social world—in peer contact, social graces, and the ability to engage a girl he is attracted to. The phenomenology of his social difficulty is most clearly expressed in a distinction between his perplexity and anxiety at what he calls the “analogue reality” of every day, compared to his mastery of the “digital reality” of online environments. “<italic>When I'm with another person, I just can't tell what is going to happen next, what they mean with this look in their eyes or what they say; I always feel I'm going to make a mistake. When I'm online, I can understand the rules there and know what they're going to do</italic>”. He describes a core difficulty then around the inference of inter-subjective intent (the inferences beyond observed behaviour which make human interactions predictable), in contrast to his ease with algorithm rule-based interactions in the computer game. To be, in Temple Grandin's words, a person who has “to learn by trial and error what certain gestures and facial expressions mean” (Grandin, ##UREF##13##1995##). Brian is very popular in his peer group; they love him for his attractiveness and the self-deprecating humour he has developed to get by, but he is agonised by the sense he will never be able to have a relationship or family. He became passionately attracted to a girl in his class and then devastated by being unable to comprehend her non-response. In such moments of deep perplexity, he often decompensates, melts down, explodes with violence or worry with impulsive and serious suicidality, and needs to be rescued repeatedly from dangerous situations. As a participant commented in Murray et al. (##REF##36183692##2023##), “<italic>Not being able to bring one's gift to others does deep harm to people's lives</italic>”; echoing Binswanger's formulation of an effective and productive “being-in-the-world” as a necessity condition for mental health. In contrast to what is often implied by the older phenomenological literature as a primary withdrawal from reality into fantasy, the phenomenology here suggests rather that there is a primarily felt dislocation between self and other, with the internal world elaboration secondary. Adapting the external environment to be more in keeping with neurodiverse abilities and perceptions may mitigate this gap, but the depth of difficulties and normative aspirations for relationships remains a vulnerability (Barale et al., ##UREF##0##2019##). Brian is able to establish a bond with me within the adapted environment of a therapeutic encounter, and this helps his emotional self-regulation and suicidality. The task is to find a way to help him locate himself in the world (and the world to him). Practical strategies and specific supports can be rehearsed and go so far; there is a certain kind of existential adaptation that is needed to differentiate personally felt and socially normed expectations. With many autistic young people emerging into adulthood, the fundamental task is this accommodation to identity, adaptation, and effectiveness, given developmental differences. Brian does well and grows in confidence to the extent that he is able to take up a long-desired place at university. Great efforts go into constructing an adapted support network for him there, but sadly, it breaks down in the first semester; he is unable to manage life away from home in halls of residence and has to break off his studies.</p>", "<p>William was an autistic teenager of around 16 years with normal range cognition and language/communication ability who presented in the late 1980s with an acute mental health crisis related to anxiety, depression, and suicidal intent. I began work with him at that time using an adapted form of insight-orientated psychotherapy that I was using with other non-autistic teens. His father owned a furniture workshop, and it was his father's but also William's fervent wish as the eldest child to follow him into the family business. William became able to talk about an extraordinarily intense fantasy life, filled with the construction of a detailed imaginary space of an idealised workshop that he owned. He imbued it with socio-political ideals as a perfect world where workers and managers existed in harmony and that he would personally own and develop as he took over from his father. The workshop was laid out in vivid detail in his mind in spatial organisation, structure, and working relationships. Huge amounts of his time were spent organising details down to written worksheets, protocols, contracts, and vignettes about the working life and the development of the product line.</p>", "<p>As William got older, his father, concerned, gave him a role in the workshop as a store handler and warehouse boy. For William, anything related to social contact in the workshop caused him a huge amount of difficulty, as well as humiliation, in terms of his idealised future. The gap between his social life in the workplace and the leadership role in his imagined workshop was exquisitely painful.</p>", "<p>I began with techniques of insight-orientated therapy, aiming to use reflexivity and interpretation to link affect and thought through insight and work through realities about his position in the family, his likely capacities in future, and his ambivalent relationships with his parents. But it became clear that a core experiential difficulty for him was <italic>just this linking</italic>. It provoked a frantic perplexity and anxiety in the realisation of knowing that he did not know something but not knowing why he did not know it. Exploration around this, within what was manageable for him, just served to show how central this was in his autism. It revealed how difficult it was to approach and how much it showed characteristics of neurological difference rather than psychological reaction or defence (a key phenomenological distinction for clinical work). On the other hand, his skillfully elaborated internal world defied accepted views of the time on a developmental lack of capacity for internal fantasy in autism, instead being what can now be understood as an intense monotropism. In contrast to early theorising that such fantasy represented a primary pathology, this internal world elaboration appeared more to serve the function of rehearsal, meaning-making and problem-solving—his alternative to conventional insight.</p>", "<p>I came over time to work more with the grain of this imaginative fantasy and his autistic thinking rather than with insight as the mechanism for change and problem-solving. Working together over several years, covering his late adolescence and painful adult emergence, reduced his suicidality and led to a partial accommodation on behalf of both him and his family. As his father became more empathic and realistic (he had not accepted the autism diagnosis), the whole family structure became more reality orientated, and this began to help his distress. His personal experience was thus situated within a pattern of family and social expectations that needed understanding in order to help his development and work within his internal imaginative world; increased empathy from his family was his means to this. Such phenomenology around adult emergence is shared by many autistic teenagers—difficulty in working out implicit social rules and the perplexity that this involves, particularly in early adolescence when the child becomes more aware of complex social interaction; a sense that everyone else is in on the joke, and a phenomenological experience of opacity, of being unable to grasp something that they feel is implicitly apparent to others, and a painful sense of being out of step.</p>", "<title>Sensory sensitivity</title>", "<p>Sally is a 13-year-old autistic girl who was brought by her parents with longstanding concerns about anxiety, which had led to her being often homebound, avoiding outside situations, and not being able to go to school, although she is a talented team athlete. Her parents were sure that coming to the clinic would be disastrous and that she would not leave the car, but she did agree to come in. We spent the day with her; a multidisciplinary professional team, including psychiatrists, psychologists, nurses, speech therapists, and occupational therapists, focused on understanding her needs. On technical testing, she had an average range cognitive ability but, similar to many autistic children, a spiky profile of abilities within which we were able to understand in detail high levels of skill in quick and efficient visual processing but a much slower ability to sequence and process auditory and verbal information in her short-term memory. Her capacity to understand emotion and “theory of mind” was average for her age. When I talked with her, Sally is completely preoccupied with managing her sensory life every day. She feels frequently overwhelmed with disorientation in balance, coordination, and a sense of herself in space, in working out sounds and language. She loves movement as a way of dealing with aspects of her life that are more difficult, but without clear visual input (for instance, when on an escalator), she feels vertiginous and panicky. She hums and sings to drown out sounds when she finds them uncomfortable and panics about following verbal instructions.</p>", "<p>She can talk about anxiety feelings throughout the day, particularly with change or uncertainty, and two fundamental experiences related to this. Firstly, her constant sense that—particularly in complex situations, both social and non-social—“<italic>I just can't work things out quickly enough, and then I can't manage</italic>.” It is the sense that the world is going too fast in the moment for her to process things in time, related to her slower processing and working memory; “<italic>I can't slow down or stop to work things out—everything's going on anyway and won't slow down for me—and then it all goes wrong</italic>.” Secondly, in those moments, she then gets a cascade of anxious thoughts about remembered and anticipated catastrophic outcomes, panic rises in her body, and she must avoid and withdraw. On the other hand, there are situations, particularly in athletic classes, in an environment with people she knows and a physical routine that she understands, where she can get into a familiar movement flow and feels comfortable, confident and even ecstatic; a point where “<italic>I can keep up with myself</italic> ” in movement in time and space, and everything flows together (“flow states” of this kind are also discussed in Murray et al., ##REF##36183692##2023##, p. 4).</p>", "<p>We understand that over time and experience, these fundamental experiences have got overlaid with so many anxious worries and resistant avoidant management behaviours that their source is difficult to discern. Such disruption of the sensorium is commonly described in many autistic experiences, in the difficulty to integrate sensory experience in space and time, resulting in discomfort, disorientation, then a breakdown of sense of being and huge anxiety. And yet, in an environment where the balance is found, there can be the ecstasy of a flow state and the relief of integration. When the experienced pace and demands can be slowed, and trust in the environment is found, Sally is able to relax. Within a therapeutic relationship, she begins to be able to understand her sensory experiences. It becomes possible for her to construct better ways of managing her own functioning, gradually increasing her confidence within the experience of being autistic.</p>", "<p>For another 13-year-old girl, Tina, the pattern of sensory differences is subtly different; here, there is a slowness in registration of sensation, needing more time to process, meaning that she has difficulty reacting to rapidly presented or low-intensity stimuli, and she often needs to seek sensory input to keep focused. At some times, she finds auditory inputs hard and uncomfortable; at others, she instead seeks auditory input and likes having music around. During the interview, she talks about her sensory experiences, particularly at school, where she finds it very difficult coming down corridors with other students, overwhelmed with sounds and bustle that she cannot cope with. She has learned at these times to “space out” instead of getting upset—“getting upset would help me let other people know what was wrong, but spacing out ends up bottling it all up. When I space out, my attention goes, and I don't feel myself. Usually, I feel myself strongly, but when I space out, the sense of who I am goes.” “I really like painting and drawing, and if I can just slow down and not be rushed, then I can make sense of things.” For this girl, when her capacity to process and be in the world in space and time falls apart, she either gets very anxious or dissociates (spaces out), which works in the short term but cuts her off from everyone. She has anticipatory anxiety, which leads her to avoid school. Working with our therapist on understanding and managing these situations goes on for several months and is helpful. She can manage her sensory world more confidently now, and although things were up and down, she feels better about dealing with everyday activities most of the time. She is now able to go out with her family to restaurants and make a beginning at college since her obsessive checking and concerns about cleanliness and food have disappeared.</p>", "<p>These case studies of autistic girls and sensory features are also suggestive of how sensory features are foregrounded as a core characteristic. These findings are consistent with recent phenomenological studies of sensory experience (Sibeoni et al., ##REF##35362340##2022##, and particularly Taels et al., ##REF##36876409##2023##) and work on autism and gender (Osório et al., ##REF##34288517##2021##).</p>", "<title>Children with complex communication needs</title>", "<p>In a specialist school, I spend time with David, a 10-year-old boy with a significant learning disability and high-level needs—preverbal and within the very early stages of the individual curriculum. He is a boy showing high frequent levels of intense distress behaviours, self-harm, lashing out, screaming, and crying that do not necessarily have any clear environmental trigger. The challenge is to work back from these behaviours to understand something of his underlying internal experience, whether sensory, affective, or cognitive. By interacting with him I want to find out what gives him pleasure, what gives him distress, and in what sense he may be able to connect and use me as a comfort. I spend repeated time just lying beside David or in rough-and-tumble with him, watching for and mirroring his behaviour and reactions, waiting to find moments of reciprocity when my response to him leads to his response to me, the smallest signals in other words of reciprocal connection. From this, I try to infer the nature and quality of his experience and capacity, and also give him a sense of reciprocal relatedness. I can describe fleeting moments of reciprocity and relaxation, synchronised to and fro with transient pleasure and recognition; before his behaviour again settles away into something that feels out of reach.</p>", "<title>Contemporary autistic accounts</title>", "<p>A core feature in autistic life writing is the concept of the <italic>sensorium</italic> (Green, ##REF##36304560##2022##; Murray et al., ##REF##36183692##2023##), which interacts and intersects with several other thematic characteristics, particularly <italic>attention differences</italic> (monotropism, Murray et al., ##REF##15857859##2005##; Murray, ##UREF##22##2019##), <italic>sociality</italic> (double empathy theory, Milton, ##UREF##21##2012##; Milton et al., ##REF##36263746##2022##) and <italic>ontology</italic>, an existential theme that can be positive or negative. Its positive form is associated with a vivid sense of reality and presence in time and space, while the negative aspect involves intense overwhelm and undermining a person's sense of being in the world, the reality of their sense of self, and their relations with the environment and others.</p>", "<p>These themes can be seen in a diversity of autobiographical texts by autistic writers whose lived experiences encompass gender, race, and cognitive differences. For the purposes of this study, examples are drawn from four authors representing different perspectives (in terms of gender, ethnicity, and disability) and whose writing provides detailed accounts of childhood experiences: <italic>Born on a Blue Day</italic> (Tammet, ##UREF##33##2006##), <italic>The Reason I Jump</italic> (Higashida, ##UREF##14##2013##)<italic>, The Secret Life of a Black Aspie</italic> (Prahlad, ##UREF##27##2017##), and <italic>The Electricity of Every Living Thing</italic> (May, ##UREF##18##2018##).<xref rid=\"fn0002\" ref-type=\"fn\">²</xref> The collation is done by an arts-based scholar (NS) who identifies as autistic and neurodivergent.</p>", "<title>Sensorium</title>", "<p>The intensity of sensory experience and awareness is a prevalent and often defining theme of autistic life writing, indicating its significance for an autistic consciousness of being in the world. In Tammet's memoir, <italic>Born on a Blue Day</italic>, synaesthesia is part of the sensory textures. Numbers are described as a “first language, one I often feel and think in” (Tammet, ##UREF##33##2006##). For Tammet, reported to be “absorbed in my own world”, other children are remembered as “the background to [his] visual and tactile experiences”. Many aspects of Tammet's description of early years are characteristic of archetypal features of early autism as he recollects his fascination with shape, colour, objects, and patterns in memories of coloured beads, the “shape and motion” of blowing bubbles, his “obsession with hourglasses” and their “trickling flow of sand” (Narzisi and Muccio, ##REF##34356148##2021##), and “taking a coin and spinning it on the floor and watching it go round and round” (Murray et al., ##REF##36183692##2023##). This, however, is a memoir of a savant diagnosed with Asperger syndrome as an adult, whose abilities and experiences are radically different to most autistic people. As Tammet acknowledges, “I didn't rock my body continuously; I could talk and showed at least some ability to interact with the environment around me” (Tammet, ##UREF##33##2006##).</p>", "<p>Intense sensory engagement is similarly fundamental to Higashida's articulation of autistic experience and his sense of being in the world differently. Like Tammet, he enjoys spinning objects. “Everyday scenery doesn't rotate, so things that do spin simply fascinate us. Just watching spinning things fills us with a sort of everlasting bliss-for the time we sit watching them, they rotate with perfect regularity” (Higashida, ##UREF##14##2013##). This is sensory stimming, a key facet of repetitive autistic behaviour, also associated with sensory regulation and flow. “By performing whatever action it is, we feel a bit soothed and calmed down” (Higashida, ##UREF##14##2013##, p. 139). Descriptions of the pleasure and need for stimming as a means of sensory integration reverberate throughout autistic writing, creating temporal and spatial order in environments that threaten chaos.</p>", "<p>For Prahlad, like Tammet, autistic sensory experience intersects with synaesthesia: “People talk to me, and they assume I'm hearing and understanding their words. But usually, I'm listening to their colours. I'm seeing them. I'm feeling their temperatures. I'm smelling their scents. And whatever I see or smell or touch, I taste” (Prahlad, ##UREF##27##2017##). One of the distinctive features of Prahlad's autistic experience is difficulty with memory due to not thinking in pictures: “I think more in feelings and senses. In colours and sounds […] Not remembering things might have something to do with not seeing images in my mind. If you said, ‘Imagine a cat,' I would imagine the way cats make me feel. But I wouldn't get a picture of a cat” (Prahlad, ##UREF##27##2017##).</p>", "<p>Imagination is a feature of the autism phenotype that needs more research and more nuanced understanding, as many autistic people also experience <italic>aphantasia</italic>. Nevertheless, Prahlad's earliest memories are sensory and vividly detailed (detailed perception being a creative strength)—“the blossoms of a redbud outside the trailer window. I was stunned by their bright pink, pulpy stillness. I thought I was one of them. I thought my body hung somehow in sea blue, a cluster of soft petals, suspended and still, floating in space” (Prahlad, ##UREF##27##2017##). This sense of oneness with nature is also a common feature of autistic life writing.</p>", "<p>Katherine May's <italic>The Electricity of Every Living Thing</italic> narrates her experience of late diagnosis and shifting understanding of her personal history alongside a year-long series of walks as a psychological and physical journey. The electricity motif is inextricably related to the sensorium—“My world is made up of tiny electric shots. Every living thing carries its own current, and this finds its earth through me. Every unexpected touch, every glance, has a charge. I am a lightning rod, …., eternally braced for the metal-on-metal jolt of contact” (64). Electricity is both a positive and negative force, “people carry electricity for me; they have a current that surges around my body until I'm exhausted…unpredictable demands make the air thick, like humanity has…not a scent, but a texture. It makes me feel like I can't breathe” (9). It is also a maternal force. Her son is “the only person in my life whose electricity exactly matches my own, whose touch is as native to my skin as air or water” (245). Electricity for May is often triggered by sensory encounters, but touch is particularly acute and often negative.</p>", "<p>In <italic>The Reason I Jump</italic>, Higashida explains touch sensitivity that is consistent with its representation in autistic texts and certainly provides a strong connection to <italic>Electricity</italic> (despite the radical differences between these authors in terms of gender, age, and autistic experience)—“being touched by someone else means that the toucher is exercising control over a body, which even its owner can't properly control. It's as if we lose who we are” (55-6). This, again, is an indication of how the sensorium intersects with ontology.</p>", "<title>Attention differences (monotropism)</title>", "<p>The sensorium is inextricably linked to attention differences in autism, and this is particularly evident in autistic writing and scholarship. Monotropism is a term used to conceptualise thinking as being shaped by focused interests, flow states, and attention to detail (Murray et al., ##REF##15857859##2005##; Murray, ##UREF##22##2019##). This theorisation of the mind as an <italic>attention tunnel</italic> (Murray, ##UREF##22##2019##) is modelled on the capacity for autistic thinkers to become hyper-focussed on a particular topic or preference and for this to direct their attention almost exclusively. Sensory differences are also linked to monotropism, as Fergus Murray articulates: “If we can't tune an input out, it is often experienced as horribly intrusive… neural pathways that receive a lot of stimulation grow stronger, so perhaps autistic people are prone to long- term hyper-sensitivity in senses receiving intense attention, and under-sensitivity in channels we regularly tune out” (Murray, ##UREF##22##2019##, p. 46).</p>", "<p>This is richly evoked in autistic writing, as in the immersive pleasure of Prahlad's redbud memory. In a particularly vivid memory of his grandmother's pantry, there is an evocative entanglement of sensory experience and detailed perception:</p>", "<p>Detailed perception is similarly celebrated in <italic>The Reason I Jump</italic>:</p>", "<p>Detailed attention can also be experienced negatively, particularly in relation to sound. Prahlad refers to his experience of restaurants: “I can't stop hearing all of the conversation around me, or the patterns of silverware striking glasses and plates. […] And so, I hear only bits and pieces of what the people I'm sitting with are saying” (Prahlad, ##UREF##27##2017##).</p>", "<p>Attention to detail is linked to intense interests. In <italic>Electricity</italic>, May refers explicitly to the need for a deeper understanding of this autistic trait: “<italic>Detail has nuance. Detail has application. Not all detail is iterative, blunt, and competitive</italic>. I don't deny that my brain holds detail. I don't deny that it sucks in more detail than other people's brains, making it difficult to navigate simple situations due to an excess of input” (emphasis added) (p. 248). May describes the detail of her maternal understanding of her new-born son's needs and cues: “he would click his little tongue kkkk—in the seconds before he would start to wail his hunger […].” In this “call and response” attention to detail, “the wonderful and terrible pull of motherhood”, she emphasises that she is “not claiming any special powers” [as] those details “make me more like an ordinary woman than I've ever felt in my life before” (p. 250). However, the converse is worth considering as it may well be the case that for the ordinary non-autistic woman, the intensely detailed focus of the mother and baby relationship (positive or negative) is, in fact, a coming closer to the intensity of autistic experience in terms of detailed attention (monotropism) and a focus on survival.</p>", "<title>Empathy and sociality</title>", "<p>It is very evident in autistic autobiographical texts, as in the human spectrum paper (Murray et al., ##REF##36183692##2023##), that characteristics long thought intrinsic to autism, such as social misperception and reduced empathy, may be alternatively understood as state-dependent outcomes contingent on specific contexts and interactions. May, in <italic>Electricity</italic>, enjoys loving relationships with her husband and son and has a network of friends who accompany her on different stages of the coastal walk. She has a very high level of empathic understanding, is attuned to the feelings of others, and connects this to detailed perception: “Detail doesn't only lie in systematic knowledge of football cars or aeroplane serial numbers. There are other kinds of detail too […] there is the detail—quite mysterious to me—of noticing someone else's mood and knowing what will draw them toward their equilibrium” (p. 248). Higashida is reported to reiterate repeatedly that “he values the company of other people very much. But because communication is so fraught with problems, a person with autism tends to end up alone in a corner, where people then see him or her and think, ‘Aha, a classic sign of autism, that”' (Higashida, ##UREF##14##2013##).</p>", "<p>Tammet acknowledges the respite in solitude, the sense of outsider status and the difficulties of engaging with his peers:</p>", "<p>May similarly describes her sense of being an outsider and “poor at taking the social temperature” (May, ##UREF##18##2018##), particularly in childhood. In a passage that strikingly resonates with Tammet, she writes:</p>", "<p>Autism being experienced as speaking a “different language” with a need for translation is also a recurrent motif that has been theorised by Damian Milton in terms of a double empathy problem (Milton, ##UREF##21##2012##), with a mutual failure to understand both ways of communication between autistic and non-autistic people, as different neurotypes.<xref rid=\"fn0003\" ref-type=\"fn\">³</xref> For May: “I did everything I could to speak the same language as them, but I could see that it landed differently. I felt like a wild-eyed beast who speaks beautiful words, only to find them received as grunts and snarls. There was a translation error somewhere down the line” (May, ##UREF##18##2018##).</p>", "<p>Autistic life writing offers evidence of how “a different embodied way of being” contributes to “effects on social interactions and understanding” (Milton et al., ##REF##36263746##2022##) so that autistic sociality and community are prevalent and positive themes in this corpus of writing. Given the emphasis on intersubjectivity and early relations with caregivers in discussions of autism, it is noteworthy that in autistic writing, we frequently find examples of attention, attunement, and joy. Prahlad expresses one of the earliest memories of his mother's face as: “A maple-brown expanse of garden glowing, with lips that moved on her whispers and breath like a butterfly's wings when it sits on a blossom” (Prahlad, ##UREF##27##2017##). The engagement may be different, as Higashida explains in relation to eye contact: “What we're actually looking at is the other person's voice. Voices may not be visible things, but we're trying to listen to the other person with all of our sense organs” (Higashida, ##UREF##14##2013##). For many writers, neurodivergent parents are alluded to. For instance, Tammet's mother “had always considered herself an outsider as a child”, and he describes the pleasure of sitting on his parents' laps (who were dedicated readers) and enjoying the sensory aspects and shared pleasures of books as “the room would fill with silence” (Tammet, ##UREF##33##2006##).</p>", "<p>The complexities and dynamics of empathic engagement and disengagement are fundamental to the tensions inherent in autistic texts. In May's memoir, this is connected to detailed attention:</p>", "<p>A corollary of the double empathy problem is the extreme state of physical and psychic distress so frequently induced by sustained or intensive exposure to social space. In <italic>Electricity</italic>, these episodes punctuate the text like missiles, reminding the reader of the painful and felt reality of being neurodivergent in a non-autistic world.</p>", "<title>Ontology</title>", "<p>Prahlad refers to “always feeling disconnected” (4). This goes to the heart of the ontology theme, the existential need to feel a connection (to the environment, nature, people, and reality) and an associated experience of alienation. This ontology theme is similarly pervasive, often entangled with the sensorium and is also frequently associated with sociality, particularly when overwhelm is triggered by a social environment. A questioning of the reality of existence, feelings of disembodiment and the pull toward the natural environment (often seeking immersion) are shared features. This can take the form of self-harm, and it is striking to see head banging referenced so frequently—“at age two, I began to walk up to a particular wall in the living room and band my head against it. I would rock my body backwards and forwards, striking my forehead hard, repeatedly, and rhythmically against the wall” (Blue Day, 23). Tammet's reference (vestibular stimulation) connects to Higashida's jumping, which, in turn, links to the pull toward nature that can be observed across autistic texts: “When I'm jumping, it's as if my feelings are going upwards to the sky. Really, my urge to be swallowed by the sky is enough to make my heart quiver” (Tammet, ##UREF##33##2006##). The power of auditory perception, positively and negatively, is a core element of the ontology theme. As Higashida explains, “There are certain noises you don't notice, but that really gets to us… It's not quite that the noises grate on our nerves. Its more to do with a fear that if we keep listening, we'll lose all sense of where we are” (Higashida, ##UREF##14##2013##).</p>", "<title>Reflection</title>", "<p>Whilst there is considerable value in this rapidly developing corpus of autistic life writing (contributing to what is sometimes referred to as “autie-ethnography”; Yergeau, ##UREF##35##2013##), there are limitations that need to be highlighted. There is a long tradition of literary criticism foregrounding narrative methods in autobiography and memoir, questioning the truth status of the genre and conceptualising critical autobiography (Di Summa-Knoop, ##UREF##7##2017##). Like all life writing, autiebiography (Van Goidsenhoven, ##UREF##34##2017##) offers subjective perspectives that are necessarily selective in several ways. We need to be aware of issues of voice and representation in terms of who can write these accounts. The vast majority represent the perspectives and experiences of educated, articulate, and white autistic people. In <italic>The Reason I Jump</italic>, the form and style (questions and responses) are different to most autistic life writing. This dialectic account, foregrounding editorial collaboration, can be considered in the context of current practises of co-production (and in relation to more established methods of assisted communication). It is also worth noting the prevalence of autobiographies by women writers, which does not reflect the history of gender bias in diagnosis. Other factors concern the contexts in which books are produced and the publishing process. There is the influence of the selection criteria for publishers, the potential distortion through editorial intervention, as well as the prism of ghost-writers. Moreover, cultural preferences are also at play in terms of the vogue for overcoming illness and success narratives. A further limitation is not having a readily identifiable comparative element. Although autistic life writing and clinical phenomenological accounts allow rich insights into subjective experience, they are part of a complex picture framed by different lenses. They provide depth and vividness but are limited in several ways in terms of breadth.</p>", "<title>Toward a systematic autistic phenomenology</title>", "<p>The great majority of formal autistic phenomenology literature to date has been clinical phenomenology, originating in professionals' experience within clinical care. Direct autistic voice has been notably absent, represented in this paper by the preceding section. Going forward, we propose the need for a more systematic, empirical, and representative approach to autistic phenomenology, echoing calls from others in the field (Nilsson et al., ##REF##31170725##2019##). This is based on the view that what has been the essentially clinical case-report idiographic methods previously used need to be supplemented with large-scale representative methods, without sacrificing the deep and intimate subjective qualities of traditional phenomenology. The methodological challenges of doing this will be substantial, but the goal is ambitious—a reframed autism concept which includes the subjective and phenomenological, and one that may re-vitalise diagnostic language and provide new impetus to scientific enquiry through more focus around autistic experience.</p>", "<p>Several principled stages in this approach can be envisaged:</p>", "<p><bold>Co-construction:</bold> Such a project must be co-constructed collaboratively between often non-autistic researchers and autistic people themselves because, theoretically and practically, this collaboration will be necessary and because the absence of the autistic voice in a true autistic phenomenology constitutes a form of epistemic injustice, which needs to be corrected. There is now an evolving methodology to draw on Pellicano and den Houting (##REF##34730840##2022##) and Shaughnessy (##UREF##31##2022##) in which autistic and non-autistic researchers can find ways to collaborate on equal terms in ways that address concerns about inequalities of power and representation in such activity, building on good practise in participatory research.</p>", "<p><bold>Thematic analysis of existing accounts:</bold> This would begin with a collation (as we have begun in this study) of descriptive observation, clinical case studies, and phenomenology studies, along with autie-ethnographic first-person accounts of life experience within the autistic literature. Such a collation and review could then allow a descriptive analysis of common elements and themes across such diverse literature. Such analysis could serve to initially orientate the work and highlight foci and sampling strategy for the next stage of a more purposive, in-depth study of autistic consciousness and life experience.</p>", "<p><bold>Idiographic study:</bold> In-depth idiographic exploration for key themes, necessarily on a relatively small cohort but with purposive sampling. Various applied phenomenological methods could be utilised. The most empirically inductive might be semantic analysis techniques, such as in Petitmengin et al. (##UREF##25##2018##), taking narrative data from transcripts and encoding them into empirical clusters using inductive techniques, rather in the same way as natural language processing in artificial intelligence machine learning. Other approaches, whether based on transcript or interview, could be based more on a degree of inter-subjective meaning-making by the interviewer and/or interviewee. These might include the “descriptive phenomenological psychological method” of Giorgi (##UREF##12##2012##). Alternatively, it could include an “interpretive phenomenological analysis” (IPA; Smith et al., ##UREF##32##2009##) where each person's account receives systematic interpretative understanding in its own terms, using what Smith describes as a double hermeneutic where “the interviewer is trying to make sense of the participant trying to make sense of their experience”. Thereafter, convergences and divergences between individuals are explored.</p>", "<p>Whatever the methodological technique is used, there will be common (and challenging) issues to consider. Firstly, building on the principle of co-construction, there will need to be an approach to autistic participants that is carefully consensual and sensitive to the potential power and experiential differentials in such engagement. The involvement of the autistic community and autistic researchers in this work will be central to its acceptability and value. Secondly, a sampling strategy for these intrinsically small sample studies will need careful consideration, including key intersectionalities of, for instance, age, gender, and ability, and building on the prior thematic review. This will be necessary to begin approaching representativeness across autistic heterogeneity. Thirdly, an equivalent sampling from non-autistic experience will be arguably crucial. Only with this will one be able to establish the specificity and discrimination for autism itself of the accounts made. On a small scale, the work of Murray et al. (##REF##36183692##2023##), illustrates the insights that can arise from doing this.</p>", "<p><bold>Survey breadth:</bold> To achieve the necessary breadth required to inform a representative autistic phenomenology, the next step would be to iteratively to combine the in-depth idiographic data above with a broader approach to data collection offered by online digital technology platforms, such as those employing a citizen science approach. Citizen science, as a highly diverse practise, is already being used to enable people to contribute to research, both through crowdsourcing the collection of data as well as the processing and analyses of larger data sets (Haklay et al., ##REF##34457323##2021##). While online citizen science has not yet been applied to autism research at scale, it would be highly applicable for an “in-breadth” phenomenology of this kind. Research has already found that the structured format of existing social media sites can give autistic users the chance to contribute without having to worry about how they are being perceived (van Schalkwyk et al., ##REF##28616856##2017##), enabling the large-scale collection of autistic people's experiences. Themes and sub-themes capturing significant topics identified from analysis of the in-depth accounts would be used to generate specific prompts inviting autistic and relevant comparison groups to contribute their own related experiences at large scale across intersectionalties such as age, gender, and ethnicity. Data gathered via an online platform, in turn, will provide a test of the validity of the original, in-depth accounts. Such an approach could be used in conjunction with automated methods of analysis using machine learning/AI methods, such as Natural Language Processing (NLP) models, as well as other forms of qualitative analysis, such as content and framework or template analysis. In this way, a more representative account will be built, reflecting the diversity as well as the specificity of autistic subjective experiences and their similarities and differences to the experiences of relevant comparison groups. The big data sets potentially generated would allow the power for a stratified analytic strategy to investigate formally the specificity and representativeness of phenomenology across many sub-groupings of autistic experience.</p>", "<p><bold>Representation across ability:</bold> A key methodological challenge will be that of representativeness across ability, particularly within intellectual disability and participants with complex communication needs. Accounts within clinical phenomenology and autistic auto-ethnography almost wholly represent the experience of verbally and intellectually able autistic people and constitute, in Jaarsma and Welin's terms, a “narrow neurodiversity” concept (Jaarsma and Welin, ##REF##21311979##2012##) in contrast to a “broad neurodiversity”. While this is an essential beginning point to establish the parameters of autistic experience, we then need to have a method of testing out such constructs within the range of intellectual and language ability associated with autism. A key question here is whether an autism concept or experience is relatively continuous across these variables (the epidemiological, behavioural phenotype evidence suggests that it is, hence, the disaggregation of autism from IQ in current clinical nosology), and systematic phenomenology should have at least a role in clarifying this issue. Methods to be developed and validated for this could include new neuroscience methods (e.g., eye-tracking to signal intent and communication) for augmented or alternative communication, the use of ethnographic and participant observation methods such as PRISMA (De Jaegher et al., ##UREF##5##2017##), and related creative practises. They will allow us to test out or explore, beyond verbal communication, core elements of autistic people's experience. Such work will be particularly important if the outcome of the work is to influence clinical nosological definitions and to guide clinical neuroscience investigation since such differences and disabilities are key within much healthcare intervention.</p>", "<p><bold>A developmental perspective:</bold> Another central issue in interpreting subjective statements about autistic phenomenology will be its developmental nature. Does what one is tapping into constitute a primary aspect of autistic difference or the developmental outcome of a cascade of self-environment interactions over time? This is going to be a complex issue to work out, but one that epidemiology in other areas has methodologically addressed. One beginning will be the careful reconstruction of developmental experience in a sequenced fashion, for instance, using large datasets classified by age, and this will begin to help the discrimination of early experiences from later secondary ones. Exemplifying this process is evidence from reported phenomenology in early development (Murray et al., ##REF##36183692##2023##), supported by results from early intervention science (Green, ##REF##36304560##2022##). This evidence is convervent with more theoretical literature in suggesting that at least some of the social differences and impacts central to the currently described autistic behavioural phenotype are, in fact, secondarily emergent rather than primary. In other words, these differences may arise from transactions between neurodiverse individuals and their environment in the early years.</p>", "<title>Data availability statement</title>", "<p>The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.</p>", "<title>Ethics statement</title>", "<p>Written informed consent was not obtained from the individual(s) for the publication of any potentially identifiable images or data included in this article because the case reports have been fully anonymised, including fundamental adaptation of names and case details so as to be non-identifiable, with quotes that are illustrative transliterations rather than verbatim records.</p>", "<title>Author contributions</title>", "<p>JG: Conceptualisation, Investigation, Writing—original draft, Writing—review &amp; editing. NS: Conceptualisation, Writing—original draft, Writing—review &amp; editing.</p>" ]

[ "<p>This is trans-disciplinary work, and the authors would like to acknowledge the generous input and advice in preparing our manuscript from many colleagues across philosophy, psychology, qualitative methodology, and data science, with particular thanks to Dr. Jo Bervoets, Dr. Bastian Greshake Tzovaras, Joan O'Rafferty, Prof. Liz Pellicano, Prof. Jonathan Smith, and Dr. Emma Williams. Their inputs have been invaluable, but naturally, only we are responsible for what we have finally written. JG is a NIHR Senior Investigator (NIHR NF-SI-0617-10168) and Fellow of the Academy of Medical Sciences.</p>", "<title>Conflict of interest</title>", "<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>", "<title>Publisher's note</title>", "<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>", "<title>Author disclaimer</title>", "<p>The views expressed are those of the authors and not necessarily those of the NHS, the NIHR or the Department of Health.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><p>Derived phenomenological themes across autistic and non-autistic experience (reproduced with permission from Murray et al., ##REF##36183692##2023##, available at <ext-link xlink:href=\"https://doi.org/10.1159/000526213\" ext-link-type=\"uri\">https://doi.org/10.1159/000526213</ext-link>). Themes showing commonalities (red); differences (green); much overlap (blue).</p></caption></fig>" ]

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[ "<disp-quote><p>“<italic>The jars of jams sang and talked to me… One</italic> of my favourite things to do was to watch flecks of dust floating in streaks of light or sunshine. … <italic>The door was made of vertical rows of boards, painted light yellow on the outside, like washed-out mustard, like saffron or daffodils mixed with cream</italic>” (emphasis added) (26-7).</p></disp-quote>", "<disp-quote><p>“<italic>When you see an object, it seems that you see it as an entire thing first, and only afterward do its details follow on. But for people with autism, the details jump straight out at us first, and then, only gradually, detail by detail, does the whole image sort of float up into focus</italic>” (91-2).</p></disp-quote>", "<disp-quote><p>“<italic>Slowly, I think the feeling was creeping over me that I was different from the other children, but for some reason, it didn't bother me. I didn't yet feel any desire for friends; I was happy enough playing by myself. When the time came to play social games, such as musical chairs, I refused to join in. I was frightened by the thought of the other children touching me as they shoved one another</italic>”.</p></disp-quote>", "<disp-quote><p>“<italic>I had few friends as a child, and although I wouldn't say I liked it that way, I can't say it bothered me much either. I liked my own company. The painful part was being conscious of my own difference. I was another species entirely from the little girls in my class, with my big, awkward body and complete inability to relate to anything they said or did</italic>” (p. 47).</p></disp-quote>", "<disp-quote><p>“<italic>I over read other people's feelings to the point that they choke me…As a child, other people's emotions seemed to come out of nowhere. There would suddenly be crying or hysterical laughter, and I would flounder, wondering how on earth we got here</italic>” (p. 208).</p></disp-quote>", "<disp-quote><p>“<italic>I realise the whiteness has started […] a multiroom cloud of searing blankness. It feels like someone has pushed a knitting needle through my skull […] maybe their whole hand is in there palpating the matter it finds inside, squeezing the sense out of whole regions of my mind</italic>” (92).</p></disp-quote>" ]

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[ "<fn-group><fn id=\"fn0001\"><p>¹We begin by establishing our own positionally and perspectives as authors. The first author (Green) is non-autistic. He has worked clinically as a child psychiatrist with autistic children, young people, and families for 35 years and as a research professor in autism developmental and intervention research. The second author (Shaughnessy) identifies as neurodivergent and has interdisciplinary expertise in creative research methods, autism, and mental health.</p></fn><fn id=\"fn0002\"><p>²Daniel Tammet (b. 1979) is the eldest of nine children from a working-class East London family. <italic>Born on a Blue Day:A Memoir of Asperger's and an Extraordinary Mind</italic> describes his atypical early development in ways that are characteristic of childhood autism, also documenting the onset of epilepsy (aged four) and the associated acquisition of synaesthesia. Educated in local mainstream schools and excelling academically (particularly in maths and languages), he started his career as a language teacher, launching an online language learning company in 2002. He was diagnosed in 2004 with high-functioning autistic savant syndrome and quickly came to public attention for his award-winning recitation of Pi, at the age of 25. <italic>Blue Day</italic> was published in 2005, followed by <italic>Embracing the Wide Sky</italic> (2009) (on neuroscience) and the essay collection, <italic>Thinking in Numbers</italic> (2014).</p><p>Naoki Higashida (b. 1992) was educated in Japanese specialist schools after receiving an autism diagnosis at age five. He was supported to communicate using a Japanese alphabet grid and finger pointing which a helper transcribed. He wrote <italic>The Reason I Jump</italic> at age 13 which was published in Japan in 2007. It was translated into English and published in 2013 after being discovered by KA Yoshida, wife of the author David Mitchell who found the account pertinent to their autistic son. A sequel, <italic>Fall Down 7 Times Get Up 8</italic> (2017), documented Naoki's adolescent life. Whilst there is considerable controversy surrounding the authorship of these books due to criticism of the communication system and speculation about the writer's agency, <italic>The Reason I Jump</italic> (and the subsequent film documentary, 2020) is one of the few creative representations of young autistic people with complex communication needs.</p><p>Anand Prahlad (b. 1954) was born on a plantation in Virginia, attending “the only black school in the area” (63). Although he did not speak until the age of four and was acutely aware of being different to his peers, his diagnosis of Asperger's was when he was 57. <italic>The Secret Life of a Black Aspie</italic> is a memoir of growing up undiagnosed and how the intersections of race and neurodivergence shaped his subjectivity. After an MA and PhD, he became an academic (teaching creative writing, folklore, and disability studies) alongside a professional career as a poet, non-fiction author, and musician.</p><p>Katherine May (b.1977) was brought up in Kent and attended the University of Cambridge. She wrote <italic>The Electricity of Every Living Thing</italic> (2018) whilst going through the journey of diagnosis in her late 30s. The memoir is an example of a rapidly developing corpus of contemporary autobiographies by autistic women, whose experiences have until recently been under-represented. Following <italic>Electricity</italic>, May has further authored two acclaimed memoirs—<italic>Wintering</italic> (May, ##UREF##19##2020##) and <italic>Enchantment</italic> (May, ##UREF##20##2023##).</p></fn><fn id=\"fn0003\"><p>³Neurotypes is a term used by neurodiversity scholars referring to classes of “differently wired brains” (e.g., autistic, dyslexic, etc).</p></fn><fn id=\"fn0004\"><p>⁴An emphasis reduced as developmental science and epidemiology has progressed; with the disaggregation of language delay from autism nosology in DSM5.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"fpsyg-14-1287209-g0001\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Metabolic syndrome (MetS) is a collection of multiple metabolic abnormalities, such as hypertension, hyperglycemia, abdominal obesity, and atherogenic dyslipidemia. It remains one of the leading health challenges worldwide, affecting 25% of all adults according to the International Diabetes Federation definition. The syndrome contributes to a 1.5-fold increase in mortality risk and imposes an enormous burden on healthcare systems [##UREF##0##1##].</p>", "<p>A combination of genetic and environmental factors has been implicated in the etiology of MetS. Diet is a changeable environmental risk factor, and dietary modifications could significantly reduce the incidence and mortality of numerous diseases, including MetS [##REF##28539378##2##]. Increasing evidence shows that mild chronic metabolic acidosis may cause high cortisol production and therefore may be related to the development of MetS [##REF##33171835##3##]. Certain dietary factors have been shown to affect the body’s acid-base balance [##REF##9734733##4##]. Although compensatory physiological responses in the body maintain acid-base balance closely, adherence to diets rich in acid-forming factors (such as animal products, rice, and cheese) can decrease blood pH toward the lower limit of the normal physiological range [##REF##22972436##5##]. Purportedly alkaline dietary factors (such as fruits and vegetables) or homeostatic mechanisms do not compensate for this low limit of the acid-base balance. In this situation, mild chronic metabolic acidosis can result [##UREF##1##6##].</p>", "<p>Although mild chronic metabolic acidosis might be related to the development of metabolic abnormalities, numerous acid-forming foods (such as fish, nuts, eggs, and whole grains) are rich sources of beneficial compounds, including protein, unsaturated fatty acids (omega-3, eicosapentaenoic acid, docosahexaenoic acid, and monounsaturated fatty acid), vitamins (vitamins E, B6, and D, folic acid, riboflavin, and niacin), minerals (iron, zinc, calcium, magnesium, potassium, and copper), and bioactive compounds [##REF##23044160##7##, ####REF##22716911##8##, ##REF##14623484##9####14623484##9##]. In addition, whole grains provide mechanical advantages (primarily for the gastrointestinal tract) arising from the fiber content [##REF##22716911##8##]. Moreover, some dietary factors (such as butter and sugar) with negative effects on metabolic health have an approximately neutral effect on acid-base balance [##REF##22013455##10##].</p>", "<p>Given the inconsistent results of previous studies on the association between dietary acid load (DAL) and MetS [##UREF##2##11##, ####REF##31568970##12##, ##REF##32486113##13##, ##REF##32019435##14##, ##UREF##3##15##, ##REF##35045755##16####35045755##16##] and the different assumptions on the direction of this association, the present study aimed to examine the association between DAL and MetS and its components in a large sample of Iranian adults.</p>" ]

[ "<title>Materials and methods</title>", "<p>Study population</p>", "<p>Baseline data from the Prospective Epidemiological Research Studies in IrAN (PERSIAN) cohort study branch in Fasa city were used for this cross-sectional study. As part of the PERSIAN multicenter cohort study, the Fasa cohort study follows 10,138 adults aged 35-70 years old living in the Sheshdeh area of Fasa (Sheshdeh town and 24 villages surrounding it) in Fars province, Iran. The detailed protocol for the Fasa PERSIAN cohort study is available elsewhere [##REF##27756262##17##].</p>", "<p>Potential participants were selected through multistage cluster random sampling and recruited after they had provided written informed consent. Blood samples and information on general characteristics, demographic status, anthropometric indices, dietary intakes, and lifestyle-related factors were collected from eligible participants. All data were collected during a face-to-face interview with a pretested questionnaire [##REF##27756262##17##].</p>", "<p>Data for 10,138 participants were included in the present study. Participants with missing data for dietary intake (n = 20) or outcomes of interest (n = 15), as well as those who under- or over-reported calorie intake (&lt;800 kcal/d or &gt;4200 kcal/d; n = 1278), were excluded. In addition, pregnant and lactating women, and participants with a history of diseases such as kidney failure, diabetes, or hypertension were excluded from the study because of possible dietary modifications (n = 2429). Moreover, participants with a body mass index (BMI) of 40 or higher were excluded because they might under-report their dietary intakes (n = 40). Thus, 6356 individuals were included in the final analysis.</p>", "<p>Dietary assessment</p>", "<p>In face-to-face interviews with trained dietitians, the usual food intakes during the previous year were obtained using a validated block-format 125-item semi-quantitative food frequency questionnaire (FFQ) [##REF##19807937##18##]. As a part of the interview, participants were asked to report their average food intake frequency per day, week, or month based on household measurements. Then portion sizes of food items were converted to grams. The energy and nutrient contents of foods were derived from the US Department of Agriculture nutrient database modified for Iranian foods [##UREF##4##19##].</p>", "<p>Net endogenous acid production (NEAP) and potential renal acid load (PRAL) calculation</p>", "<p>Two scores are used to estimate DAL: NEAP and PRAL. The NEAP score was developed by Frassetto et al. [##REF##9734733##4##] and is based on total dietary protein intake and potassium intake. Low NEAP scores mean that the diet has lower acid-forming potential, while high scores indicate higher acid-forming potential. The PRAL score is another validated tool introduced by Remer and Manz [##REF##7797810##20##]. Estimates are derived based on average intestinal absorption rates of ingested total protein, as well as potassium, calcium, phosphorus, and magnesium. Diets with negative or positive PRAL scores tend to have base-forming and acid-forming properties, respectively [##REF##9734733##4##,##REF##7797810##20##]. The following formulas were used to calculate these scores:</p>", "<p>NEAP (mEq/d) = 54.5 × protein g/d/potassium (mEq/d) − 10.2.</p>", "<p>PRAL (mEq/d) = (0.49 × protein (g/d)) + (0.037 × phosphorus (mg/d)) − (0.021 × potassium (mg/d)) − (0.026 × magnesium (mg/d)) − (0.013 × calcium (mg/d)).</p>", "<p>Anthropometric and blood pressure measurements</p>", "<p>Weight was measured to the nearest 0.1 kg with a digital scale, and height was measured to the nearest 0.5 cm with a measuring tape, while participants wore light indoor clothing. BMI was calculated as body weight (kg) divided by height squared (m2). Waist circumference (WC) was measured after participants removed their clothing and exhaled normally, with no pressure applied to the body, at a distance halfway between the iliac crest and the lowest rib [##REF##12374515##21##]. To measure blood pressure with a mercury sphygmomanometer, participants were asked to rest for 10 minutes. Two measurements were taken from the right arm at 15-minute intervals, and the average of both measures was recorded.</p>", "<p>Biochemical assays</p>", "<p>Blood samples were collected after participants refrained from eating for 12-14 hours, and were stored at −70°C. Triglycerides (TG), fasting blood sugar (FBS), and high-density lipoprotein cholesterol (HDL-C) were measured with an autoanalyzer system (Selectra E, Vitalab, Holliston, the Netherlands) and Pars Azmoon kits (Tehran, Iran).</p>", "<p>Socioeconomic status (SES)</p>", "<p>The wealth score index (WSI) represents SES, estimated by multiple correspondence analysis of the following variables: access to a freezer, washing machine, dishwasher, computer, internet, motorcycle, car (no access, access to a car costing &lt;11,000 US dollars, or access to a car costing &gt;11,000 US dollars), vacuum cleaner, and color television (no color television or regular color television vs. plasma color television). Other variables included in the WSI are possessing a mobile phone, a personal computer, or a laptop, and having traveled abroad (never, pilgrimage only, both pilgrimage and non-pilgrimage trips) [##REF##27756262##17##].</p>", "<p>Assessment of other variables</p>", "<p>A pretested questionnaire was used in face-to-face interviews to collect information on age (continuous), sex (male/female), education (graduated from university vs. no university education), marital status (married vs. single or divorced), home ownership (owner vs. non-owner), active smoking (currently smoking at least one cigarette a day), previous diagnoses of fatty liver disease (yes/no), depression (yes/no) or thyroid disorder (yes/no), and use of calcium supplements (with or without vitamin D) (yes/no). The participants were questioned about their regular physical activity during the previous year, and their physical activity level was expressed as metabolic equivalent hours per week (METs h/w) [##REF##18053188##22##].</p>", "<p>Definition of MetS</p>", "<p>MetS was defined according to Adult Treatment Panel III criteria. The syndrome is considered to exist when at least three of the following components are present: (1) abdominal obesity (WC &gt; 102 cm for men and &gt;88 cm for women); (2) high serum TG levels (&gt;150 mg/dL); (3) low serum HDL-C levels (&lt;50 mg/dL for women and &lt;40 mg/dL for men); (4) elevated blood pressure (systolic blood pressure (SBP) ≥ 130 mm Hg and/or diastolic blood pressure (DBP) ≥ 85 mm Hg); and (5) abnormal glucose homeostasis (FBS ≥ 110 mg/dL) [##REF##14744958##23##].</p>", "<p>Statistical analysis</p>", "<p>To assess differences in variables across quintiles of NEAP and PRAL scores, one-way ANOVA (quantitative variables) and chi-squared tests (qualitative variables) were used [##UREF##5##24##]. Dietary intakes across quintiles of NEAP and PRAL scores were compared with analysis of covariance (ANCOVA) to adjust for energy intake (kcal/d) and age.</p>", "<p>Binary logistic regression with adjusted models was used to calculate the odds ratios (OR) and 95% confidence intervals (CI) for MetS and its components across quintiles of NEAP and PRAL scores in men and women. The first quintile group was used as the reference for ORs and their 95% CI estimates [##UREF##6##25##]. Model I was adjusted for the effects of age, energy intake (kcal/d), physical activity (continuous), education, marital status, home ownership, SES, history of obesity-related disease (fatty liver disease, depression, and thyroid disease), and use of calcium supplements (with or without vitamin D). Further adjustment for BMI was used in model II. All statistical analyses were done with SPSS version 21 (IBM Corp., Armonk, NY). P-values were considered significant at &lt;0.05.</p>" ]

[ "<title>Results</title>", "<p>After eligibility was verified according to the inclusion criteria, data for 6356 participants (mean age: 46.58 ± 8.82 years, mean BMI: 25.02 ± 4.60 kg/m2) were included in the present analysis. MetS was prevalent in 777 (12.2%) participants.</p>", "<p>The general characteristics of men and women across quintiles of NEAP and PRAL scores are shown in Table ##TAB##0##1##. Men in the highest PRAL quintile were more likely to have thyroid disease (p = 0.02) compared to men in the lowest quintile. In women, those in the highest PRAL quintile, compared to those in the lowest quintile, were more likely to be homeowners (p = 0.04) and less likely to have fatty liver disease (p = 0.004) and to have central obesity (0.03). In addition, women in the lowest NEAP quintile were more likely to have higher SES (p = 0.01) and fatty liver disease (p = 0.001) and to have central obesity (p = 0.01) than those in the highest quintile.</p>", "<p>The age- and energy-adjusted intakes of selected foods and nutrients across NEAP and PRAL quintiles in men and women are shown in Tables ##TAB##1##2##, ##TAB##2##3##, respectively. Men and women in the highest NEAP quintile had higher intakes of grains, meats, carbohydrates, total energy, protein, fat, cholesterol, folate, phosphorus, calcium, sodium, a higher sodium-to-potassium ratio, and lower intakes of dairy products, fruits, vegetables, dietary fiber, vitamin B12, potassium, and magnesium compared to those in the lowest quintile. Men and women in the highest PRAL quintile had greater intakes of grain, meat, total energy, protein, fat, cholesterol, folate, phosphorus (in men), calcium, sodium, and a higher sodium-to-potassium ratio, as well as lower intakes of vegetables, fruits, dairy products, carbohydrate, vitamin B12, phosphorus (in women), magnesium, and potassium compared to the lowest quintile (p &lt; 0.001 for all).</p>", "<p>Multivariable-adjusted OR for MetS and its components across NEAP and PRAL quintiles are presented in Table ##TAB##3##4##. Adherence to a diet with a high DAL (PRAL and NEAP) was not associated with increased odds of MetS in the crude or adjusted models. Among the components of MetS, in women, higher NEAP scores were associated with an increased odds of hypertriglyceridemia after adjusting for age, energy intake, physical activity, education, marital status, SES, home ownership, fatty liver disease, depression, thyroid disease, calcium supplementation, and calcium plus vitamin D supplementation in model I (OR: 1.54, 95% CI: 1.007-2.36, p trend = 0.02), and after further adjustment for BMI in the fully adjusted model (OR: 1.55, 95% CI: 1.01-2.40, p trend = 0.01). In the crude model, women in the bottom NEAP quintile were more likely to have elevated HDL-C than those in the top quintile (OR: 1.26, 95% CI: 1.01-1.56, p trend = 0.06). This association was still significant and even stronger after taking potential confounders into account (OR: 1.42, 95% CI: 1.001-2.03, p trend = 0.01 for model I; OR: 1.44, 95% CI: 1.009-2.06, p trend = 0.01 for model II). In addition, women in the highest PRAL quintile had greater odds of having elevated HDL-C than those in the fourth quintile in model I and the fully adjusted model (OR: 1.54, 95% CI: 1.08-2.19, p trend = 0.06 for model I; OR: 1.56, 95% CI: 1.10-2.23, p trend = 0.06 for model II).</p>", "<p>Lastly, in model I, men in the fourth NEAP quintile had 5.68-fold greater odds of hyperglycemia (OR: 5.68, 95% CI: 1.18-27.25, p trend = 0.11). Similar results were found after further adjustment for BMI in the fully adjusted model (OR: 5.89, 95% CI: 1.19-28.99, p trend = 0.54).</p>" ]

[ "<title>Discussion</title>", "<p>In this cross-sectional study, even after adjusting for potential confounders, adherence to a diet with a high DAL was not associated with increased odds of MetS in Iranian adults. Among the components of MetS, a significant association was observed between higher DAL (PRAL and NEAP) scores and increased odds of low HDL-C in women. In addition, higher DAL (NEAP) scores were significantly associated with greater odds of having hypertriglyceridemia. In men, moderate DAL (NEAP) was significantly associated with an increased odds of hyperglycemia.</p>", "<p>The PRAL and NEAP scores are used to measure DAL from dietary intakes. Meat, poultry, fish, dairy, eggs, grains, and alcohol have more acid precursors and are related to a higher DAL. In contrast, most fruits, nuts, legumes, potatoes, and vegetables have more alkaline precursors and are related to a lower DAL [##REF##7797810##20##]. Protein, sulfur, and phosphate are the acidic precursors, whereas alkaline precursors include calcium, potassium, and magnesium [##REF##9734733##4##,##REF##7797810##20##]. Thus, DAL may vary among different populations with different dietary habits and cultures. For example, the median NEAP score of the western dietary pattern rich in acid-forming foods is 34 to 76 mEq/d, whereas, for a vegan diet, the NEAP score is 7.26 mEq/d [##REF##7797810##20##].</p>", "<p>In the present study, neither PRAL nor NEAP was associated with MetS. In line with these findings, in a cross-sectional survey of 1430 Iranian adults [##REF##32019435##14##], no significant association was found between DAL and MetS. Moreover, another cross-sectional study of 371 Iranian women (20-50 years old) revealed no significant association between DAL and MetS [##REF##31568970##12##]. Research by Tangestani et al. did not reveal a statistically significant association between PRAL score and MetS in 246 Iranian women with overweight or obesity [##REF##35045755##16##]. However, two Japanese cross-sectional studies contradict these findings [##REF##32486113##13##,##UREF##3##15##]. Iwase et al. reported that increased DAL (both PRAL and NEAP) was associated with the prevalence of MetS in 260 Japanese patients with type 2 diabetes [##UREF##3##15##]. In addition, another study of Japanese participants (35-69 years old) found that higher NEAP was positively associated with the prevalence of MetS [##REF##32486113##13##].</p>", "<p>One reason for this discrepancy might be the between-population differences in mean DAL. As in most studies [##REF##31568970##12##,##UREF##7##26##, ####REF##32981831##27##, ##REF##27390726##28####27390726##28##] of Iranians with transitional dietary patterns, PRAL (mean: −11.63 mEq/d; median: −9.78) and NEAP (mean: 42.30 mEq/d; median: 40.74) values in the present study are much lower than the mean values reported for the western dietary pattern. Furthermore, differences in the study population, sociodemographic characteristics, the method of dietary intake assessment, criteria to identify MetS, behavioral and lifestyle factors (such as dietary patterns and habits), as well as the number and type of confounding factors controlled for in the analysis, may also explain the inconsistent results [##UREF##2##11##,##REF##32981831##27##].</p>", "<p>Although we found no association between DAL and MetS, the potential mechanism that links higher DAL with an increased odds of MetS may be a decrease in insulin sensitivity due to chronic metabolic acidosis induced by long-term consumption of an acidogenic diet. Increased DAL causes increased cortisol production, decreased urinary citrate secretion, and increased magnesium excretion, resulting in reduced insulin sensitivity [##REF##33171835##3##].</p>", "<p>In addition, we found that a higher DAL was associated with greater odds of having hypertriglyceridemia and low HDL-C in women. In line with the present findings, Bahadoran et al. reported that a higher DAL (PRAL and protein/potassium ratio (Pro/K)) was associated with higher TG and lower HDL-C [##UREF##7##26##]. Moreover, in a cross-sectional study by Kucharska et al. of 6170 Polish participants aged &gt;20 years, NEAP was positively associated with TG and negatively associated with HDL-C [##REF##30085432##29##]. In addition, other studies [##UREF##2##11##,##REF##31568970##12##] showed that DAL was positively associated with TG. In contrast, some cross-sectional studies found no significant association between DAL and TG or HDL-C [##REF##31568970##12##,##UREF##3##15##,##REF##27390726##28##, ####REF##30085432##29##, ##REF##25837215##30####25837215##30##].</p>", "<p>Little knowledge is available on the mechanisms behind alterations in TG and HDL-C levels associated with higher DAL scores. However, systemic metabolic acidosis results in the relocation of free fatty acid from adipocytes to the blood, which may elevate blood cortisol levels and stimulate lipolytic activity [##REF##17315601##31##]. Increased TG content with a high acid load may increase very low-density lipoprotein and TG concentrations in the liver [##REF##24875749##32##].</p>", "<p>The present study found an association between moderate DAL (NEAP) and increased odds of hyperglycemia in men. In line with this finding, Haghighatdoost et al. demonstrated that higher DAL (PRAL and Pro/K) was significantly associated with higher HbA1C, but that PRAL score was inversely associated with FBS [##REF##25837215##30##]. Similarly, Kucharska et al. reported that the NEAP score was positively associated with the prevalence of T2DM and FBS [##REF##30085432##29##]. In contrast, Amodu et al. found negative associations between higher NEAP scores and the prevalence of T2DM [##REF##24052219##33##].</p>", "<p>The lack of a significant direct association between the last NEAP quintile and hyperglycemia may be because men in this quintile consumed high amounts of protein-rich foods, which, in addition to containing acidic precursors, can reduce the glycemic load and glycemic index of the diet [##REF##1959475##34##]. In addition, the number of patients with MetS was low in the last quintile.</p>", "<p>A major strength of the present study is the large sample size, which made it possible to adjust for multiple covariates. A second strong point is that we considered calcium supplements (a crucial, commonly used alkaline-forming supplement in Iran [##UREF##8##35##]), in the adjusted analysis. Third, the data on dietary intakes were obtained with a validated, reliable FFQ. Lastly, our analysis excluded participants with a history of disorders that can affect renal function, which plays a crucial role in acid-base balance [##REF##19948674##36##].</p>", "<p>The present findings should nonetheless be interpreted in the context of certain limitations. First, because of the cross-sectional study design, the associations reported here do not necessarily indicate causation. Second, the PRAL and NEAP score estimations were based on self-reported dietary intake rather than objective assessment. There is a strong correlation between DAL scores and measured acid load based on 24-hour urine collection [##REF##9734733##4##,##REF##7797810##20##]. However, it is unclear whether the source of the observed associations is a high intake of anti-MetS foods (fruits and vegetables) and nutrients (potassium, calcium, magnesium) reflected in the DAL scores or a high DAL per se. Third, some observations may be biased by residual confounding or unmeasured factors. Fourth, this study was conducted in a developing country where dietary habits are changing; thus, our results may not be generalizable to other regions.</p>", "<p>Although we considered calcium supplement intakes in adjusted models, we propose to include the amount of calcium in the supplements in the PRAL equation to achieve a more accurate estimation of DAL. In addition, dietary fat and the proportion of fatty acids might be related to the development of metabolic abnormalities [##REF##31377181##37##]; however, they are not addressed by DAL scores. Moreover, although an increase in dietary protein might enhance weight reduction [##REF##26354540##38##], an increase in total protein intake is accompanied by an increase in DAL [##REF##7797810##20##], and they might cover the effects of each other in assessing the association between DAL and some metabolic risk factors (particularly obesity-related factors). Thus, in future studies, it might be helpful to consider the defects of these scores and find a way to cope with them.</p>" ]

[ "<title>Conclusions</title>", "<p>Our findings suggest that a higher DAL was inversely associated with HDL-C and directly associated with serum TG levels in women. Moreover, there was a positive association between the fourth quintile of NEAP and FBS in men. However, even after adjusting for potential confounders, there was no significant association between DAL and the odds of MetS in the sample of Iranian adults studied here. Longitudinal studies are warranted to shed further light on the replicability of these findings.</p>" ]

[ "<p>Introduction</p>", "<p>Metabolic syndrome (MetS) remains one of the leading health challenges worldwide. A combination of genetic and environmental factors has been implicated in the etiology of MetS. Diet is a changeable environmental risk factor, and dietary modifications could significantly reduce the incidence and mortality of numerous diseases, including MetS. Certain dietary factors may contribute to MetS by affecting the acid-base balance within the body. This study examined the association of dietary acid load (DAL) with MetS and its components in Iranian adults.</p>", "<p>Materials and methods</p>", "<p>This cross-sectional study was conducted in 2022 on 6356 Iranian adults aged 35-70 years. Potential renal acid load (PRAL) and net endogenous acid production (NEAP) as two indicators of DAL were calculated based on nutrient intake data from validated food frequency questionnaires. MetS and its components were defined according to the Adult Treatment Panel III criteria. Logistic regression analysis was used to explore the associations between DAL and MetS and its components. Age, energy intake, physical activity, education, marital status, home ownership, socioeconomic status, history of obesity-related disease, and calcium supplements were included in model I. Further adjustment in model II was made for body mass index.</p>", "<p>Results</p>", "<p>Higher NEAP scores were associated with increased odds of low high-density lipoprotein cholesterol (HDL-C) in the crude model (OR: 1.26, 95% CI: 1.01-2.56, p trend = 0.06) in women, which was confirmed in the adjusted models.</p>", "<p>In model I, women in the last quintile of NEAP had 54% greater odds of having hypertriglyceridemia compared to the first quintile (OR: 1.54, 95% CI: 1.007-2.36, p trend = 0.02). This association was still significant and even stronger after further adjustment for BMI (OR: 1.55, 95% CI: 1.01-2.40, p trend = 0.01). In addition, in model I, men in the fourth quintile of NEAP had 5.68-fold greater odds of hyperglycemia compared to the first quintile (OR: 5.68, 95% CI: 1.18-27.25, p trend = 0.11). Similar results were found in the fully adjusted model (OR: 5.89, 95% CI: 1.19-28.99, p trend = 0.54).</p>", "<p>Conclusion</p>", "<p>There was no significant association between DAL and MetS. DAL was positively associated with the odds of low HDL-C and hypertriglyceridemia in women. Moreover, moderate DAL (NEAP) was associated with an increased odds of hyperglycemia in men.</p>" ]

[]

[ "<p>We would like to thank the Research Council of the Nutrition and Food Security Research Center at Shahid Sadoughi University of Medical Sciences and the Noncommunicable Diseases Research Center at Fasa University of Medical Sciences for their support and funding. The authors thank all Fasa PERSIAN Cohort Study participants and the PERSIAN cohort study for their close collaboration. We also thank K. Shashok (AuthorAID in the Eastern Mediterranean) for checking the use of English in the manuscript. We would like to acknowledge the contribution of Reza Homayounfar as the first author.</p>" ]

[]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Characteristics of the participants across quintiles of PRAL and NEAP</title><p>Note: Data are presented as mean ± standard error or absolute number (percentage).</p><p>Abbreviations: PRAL, potential renal acid load; NEAP, net endogenous acid production; METs h/w: metabolic equivalent hours per week.</p><p>^a Obtained from one-way ANOVA or chi-squared tests, as appropriate.</p><p>^b General obesity was defined according to cutoff values established by the WHO (obesity: BMI ≥ 30 kg/m2).</p><p>^c Central obesity was defined according to the Adult Treatment Panel III (ATP III) criteria (waist circumference &gt;102 cm for men and &gt;88 cm for women).</p><p>^d Socioeconomic status is presented as the wealth score index (WSI).</p><p>^e Self-reported.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td colspan=\"5\" rowspan=\"1\">Quintiles of PRAL</td><td rowspan=\"2\" colspan=\"1\">p-value^a\n</td><td colspan=\"5\" rowspan=\"1\">Quintiles of NEAP</td><td rowspan=\"2\" colspan=\"1\">p-value^a\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Variables</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">5</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">5</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Men</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Range</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">-23.81, -12.95</td><td rowspan=\"1\" colspan=\"1\">-12.95, -5.15</td><td rowspan=\"1\" colspan=\"1\">-5.13, 4.19</td><td rowspan=\"1\" colspan=\"1\">&gt;4.19</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">&lt;29.77</td><td rowspan=\"1\" colspan=\"1\">29.77, 37.26</td><td rowspan=\"1\" colspan=\"1\">37.30, 44.32</td><td rowspan=\"1\" colspan=\"1\">44.33, 53.52</td><td rowspan=\"1\" colspan=\"1\">&gt;53.52</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">n</td><td rowspan=\"1\" colspan=\"1\">637</td><td rowspan=\"1\" colspan=\"1\">638</td><td rowspan=\"1\" colspan=\"1\">638</td><td rowspan=\"1\" colspan=\"1\">638</td><td rowspan=\"1\" colspan=\"1\">638</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">569</td><td rowspan=\"1\" colspan=\"1\">643</td><td rowspan=\"1\" colspan=\"1\">644</td><td rowspan=\"1\" colspan=\"1\">658</td><td rowspan=\"1\" colspan=\"1\">675</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Age (years)</td><td rowspan=\"1\" colspan=\"1\">47.95 ± 0.36</td><td rowspan=\"1\" colspan=\"1\">47.31 ± 0.35</td><td rowspan=\"1\" colspan=\"1\">47.00 ± 0.35</td><td rowspan=\"1\" colspan=\"1\">47.16 ± 0.35</td><td rowspan=\"1\" colspan=\"1\">46.80 ± 0.35</td><td rowspan=\"1\" colspan=\"1\">0.19</td><td rowspan=\"1\" colspan=\"1\">47.72 ± 0.39</td><td rowspan=\"1\" colspan=\"1\">47.53 ± 0.35</td><td rowspan=\"1\" colspan=\"1\">47 ± 0.35</td><td rowspan=\"1\" colspan=\"1\">46.86 ± 0.35</td><td rowspan=\"1\" colspan=\"1\">47.18 ± 0.34</td><td rowspan=\"1\" colspan=\"1\">0.41</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">General obesity (%)^b\n</td><td rowspan=\"1\" colspan=\"1\">51 (8)</td><td rowspan=\"1\" colspan=\"1\">57 (8.9)</td><td rowspan=\"1\" colspan=\"1\">46 (7.2)</td><td rowspan=\"1\" colspan=\"1\">50 (7.8)</td><td rowspan=\"1\" colspan=\"1\">37 (5.8)</td><td rowspan=\"1\" colspan=\"1\">0.29</td><td rowspan=\"1\" colspan=\"1\">41 (7.2)</td><td rowspan=\"1\" colspan=\"1\">64 (10)</td><td rowspan=\"1\" colspan=\"1\">46 (7.1)</td><td rowspan=\"1\" colspan=\"1\">46 (7)</td><td rowspan=\"1\" colspan=\"1\">44 (6.5)</td><td rowspan=\"1\" colspan=\"1\">0.14</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Central obesity (%)^c\n</td><td rowspan=\"1\" colspan=\"1\">63 (9.9)</td><td rowspan=\"1\" colspan=\"1\">56 (8.8)</td><td rowspan=\"1\" colspan=\"1\">58 (9.1)</td><td rowspan=\"1\" colspan=\"1\">57 (8.9)</td><td rowspan=\"1\" colspan=\"1\">55 (8.6)</td><td rowspan=\"1\" colspan=\"1\">0.94</td><td rowspan=\"1\" colspan=\"1\">49 (8.6)</td><td rowspan=\"1\" colspan=\"1\">63 (9.8)</td><td rowspan=\"1\" colspan=\"1\">58 (9)</td><td rowspan=\"1\" colspan=\"1\">59 (9)</td><td rowspan=\"1\" colspan=\"1\">60 (8.9)</td><td rowspan=\"1\" colspan=\"1\">0.96</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Marital status (married) (%)</td><td rowspan=\"1\" colspan=\"1\">621 (97.5)</td><td rowspan=\"1\" colspan=\"1\">622 (97.5)</td><td rowspan=\"1\" colspan=\"1\">617 (96.7)</td><td rowspan=\"1\" colspan=\"1\">621 (97.3)</td><td rowspan=\"1\" colspan=\"1\">610 (95.6)</td><td rowspan=\"1\" colspan=\"1\">0.23</td><td rowspan=\"1\" colspan=\"1\">554 (97.4)</td><td rowspan=\"1\" colspan=\"1\">626 (97.4)</td><td rowspan=\"1\" colspan=\"1\">621 (96.4)</td><td rowspan=\"1\" colspan=\"1\">639 (97.1)</td><td rowspan=\"1\" colspan=\"1\">651 (96.4)</td><td rowspan=\"1\" colspan=\"1\">0.75</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Smoking status (active smoker) (%)</td><td rowspan=\"1\" colspan=\"1\">342 (53.7)</td><td rowspan=\"1\" colspan=\"1\">349 (54.7)</td><td rowspan=\"1\" colspan=\"1\">362 (56.7)</td><td rowspan=\"1\" colspan=\"1\">357 (56)</td><td rowspan=\"1\" colspan=\"1\">351 (55)</td><td rowspan=\"1\" colspan=\"1\">0.84</td><td rowspan=\"1\" colspan=\"1\">305 (53.6)</td><td rowspan=\"1\" colspan=\"1\">362 (56.3)</td><td rowspan=\"1\" colspan=\"1\">359 (55.7)</td><td rowspan=\"1\" colspan=\"1\">377 (57.3)</td><td rowspan=\"1\" colspan=\"1\">358 (53)</td><td rowspan=\"1\" colspan=\"1\">0.49</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Education (academic degree) (%)</td><td rowspan=\"1\" colspan=\"1\">21 (3.3)</td><td rowspan=\"1\" colspan=\"1\">27 (4.2)</td><td rowspan=\"1\" colspan=\"1\">24 (3.8)</td><td rowspan=\"1\" colspan=\"1\">25 (3.9)</td><td rowspan=\"1\" colspan=\"1\">30 (4.7)</td><td rowspan=\"1\" colspan=\"1\">0.76</td><td rowspan=\"1\" colspan=\"1\">23 (4)</td><td rowspan=\"1\" colspan=\"1\">29 (4.5)</td><td rowspan=\"1\" colspan=\"1\">26 (4)</td><td rowspan=\"1\" colspan=\"1\">21 (3.2)</td><td rowspan=\"1\" colspan=\"1\">28 (4.1)</td><td rowspan=\"1\" colspan=\"1\">0.80</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Socioeconomic status^d\n</td><td rowspan=\"1\" colspan=\"1\">0.76 ± 0.09</td><td rowspan=\"1\" colspan=\"1\">0.49 ± 0.09</td><td rowspan=\"1\" colspan=\"1\">0.50 ± 0.09</td><td rowspan=\"1\" colspan=\"1\">0.56 ± 0.09</td><td rowspan=\"1\" colspan=\"1\">0.48 ± 0.09</td><td rowspan=\"1\" colspan=\"1\">0.22</td><td rowspan=\"1\" colspan=\"1\">0.76 ± 0.10</td><td rowspan=\"1\" colspan=\"1\">0.58 ± 0.09</td><td rowspan=\"1\" colspan=\"1\">0.61 ± 0.09</td><td rowspan=\"1\" colspan=\"1\">0.40 ± 0.08</td><td rowspan=\"1\" colspan=\"1\">0.48 ± 0.09</td><td rowspan=\"1\" colspan=\"1\">0.10</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Physical activity (METs h/w)</td><td rowspan=\"1\" colspan=\"1\">46.37 ± 0.56</td><td rowspan=\"1\" colspan=\"1\">45.83 ± 0.58</td><td rowspan=\"1\" colspan=\"1\">46.37 ± 0.57</td><td rowspan=\"1\" colspan=\"1\">44.79 ± 0.55</td><td rowspan=\"1\" colspan=\"1\">45.70 ± 0.57</td><td rowspan=\"1\" colspan=\"1\">0.27</td><td rowspan=\"1\" colspan=\"1\">46.09 ± 0.59</td><td rowspan=\"1\" colspan=\"1\">46.31 ± 0.58</td><td rowspan=\"1\" colspan=\"1\">45.85 ± 0.59</td><td rowspan=\"1\" colspan=\"1\">45.85 ± 0.56</td><td rowspan=\"1\" colspan=\"1\">45.01 ± 0.53</td><td rowspan=\"1\" colspan=\"1\">0.54</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Homeownership (owner) (%)</td><td rowspan=\"1\" colspan=\"1\">575 (90.3)</td><td rowspan=\"1\" colspan=\"1\">586 (91.8)</td><td rowspan=\"1\" colspan=\"1\">587 (92)</td><td rowspan=\"1\" colspan=\"1\">583 (91.4)</td><td rowspan=\"1\" colspan=\"1\">585 (91.7)</td><td rowspan=\"1\" colspan=\"1\">0.81</td><td rowspan=\"1\" colspan=\"1\">517 (90.9)</td><td rowspan=\"1\" colspan=\"1\">585 (91)</td><td rowspan=\"1\" colspan=\"1\">590 (91.6)</td><td rowspan=\"1\" colspan=\"1\">609 (92.6)</td><td rowspan=\"1\" colspan=\"1\">615 (91.1)</td><td rowspan=\"1\" colspan=\"1\">0.81</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Calcium supplementation (with or without vitamin D) (yes) (%)</td><td rowspan=\"1\" colspan=\"1\">54 (18.4)</td><td rowspan=\"1\" colspan=\"1\">54 (19.6)</td><td rowspan=\"1\" colspan=\"1\">64 (21)</td><td rowspan=\"1\" colspan=\"1\">43 (14.8)</td><td rowspan=\"1\" colspan=\"1\">38 (14.1)</td><td rowspan=\"1\" colspan=\"1\">0.12</td><td rowspan=\"1\" colspan=\"1\">49 (18.1)</td><td rowspan=\"1\" colspan=\"1\">62 (21.1)</td><td rowspan=\"1\" colspan=\"1\">53 (17.3)</td><td rowspan=\"1\" colspan=\"1\">48 (16.9)</td><td rowspan=\"1\" colspan=\"1\">41 (14.7)</td><td rowspan=\"1\" colspan=\"1\">0.37</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Fatty liver disease (yes) (%)^e\n</td><td rowspan=\"1\" colspan=\"1\">17 (2.7)</td><td rowspan=\"1\" colspan=\"1\">24 (3.8)</td><td rowspan=\"1\" colspan=\"1\">16 (2.5)</td><td rowspan=\"1\" colspan=\"1\">18 (2.8)</td><td rowspan=\"1\" colspan=\"1\">9 (1.4)</td><td rowspan=\"1\" colspan=\"1\">0.13</td><td rowspan=\"1\" colspan=\"1\">13 (2.3)</td><td rowspan=\"1\" colspan=\"1\">26 (4)</td><td rowspan=\"1\" colspan=\"1\">15 (2.3)</td><td rowspan=\"1\" colspan=\"1\">19 (2.9)</td><td rowspan=\"1\" colspan=\"1\">11 (1.6)</td><td rowspan=\"1\" colspan=\"1\">0.08</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Depression (yes) (%)^e\n</td><td rowspan=\"1\" colspan=\"1\">20 (3.1)</td><td rowspan=\"1\" colspan=\"1\">22 (3.4)</td><td rowspan=\"1\" colspan=\"1\">17 (2.7)</td><td rowspan=\"1\" colspan=\"1\">16 (2.5)</td><td rowspan=\"1\" colspan=\"1\">16 (2.5)</td><td rowspan=\"1\" colspan=\"1\">0.80</td><td rowspan=\"1\" colspan=\"1\">20 (3.5)</td><td rowspan=\"1\" colspan=\"1\">21 (3.3)</td><td rowspan=\"1\" colspan=\"1\">17 (2.6)</td><td rowspan=\"1\" colspan=\"1\">15 (2.3)</td><td rowspan=\"1\" colspan=\"1\">18 (2.7)</td><td rowspan=\"1\" colspan=\"1\">0.68</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Thyroid disease (yes) (%)^e\n</td><td rowspan=\"1\" colspan=\"1\">13 (2)</td><td rowspan=\"1\" colspan=\"1\">22 (3.4)</td><td rowspan=\"1\" colspan=\"1\">11 (1.7)</td><td rowspan=\"1\" colspan=\"1\">6 (0.9)</td><td rowspan=\"1\" colspan=\"1\">19 (3)</td><td rowspan=\"1\" colspan=\"1\">0.02</td><td rowspan=\"1\" colspan=\"1\">14 (2.5)</td><td rowspan=\"1\" colspan=\"1\">18 (2.8)</td><td rowspan=\"1\" colspan=\"1\">14 (2.2)</td><td rowspan=\"1\" colspan=\"1\">8 (1.2)</td><td rowspan=\"1\" colspan=\"1\">17 (2.5)</td><td rowspan=\"1\" colspan=\"1\">0.34</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Women</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Range</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">-25.76, -14.92</td><td rowspan=\"1\" colspan=\"1\">-14.89, -6.63</td><td rowspan=\"1\" colspan=\"1\">-6.53, 1.90</td><td rowspan=\"1\" colspan=\"1\">&gt;1.90</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">&lt;29.77</td><td rowspan=\"1\" colspan=\"1\">29.80, 37.26</td><td rowspan=\"1\" colspan=\"1\">37.31, 44.32</td><td rowspan=\"1\" colspan=\"1\">44.33, 53.52</td><td rowspan=\"1\" colspan=\"1\">&gt;53.54</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">n</td><td rowspan=\"1\" colspan=\"1\">633</td><td rowspan=\"1\" colspan=\"1\">634</td><td rowspan=\"1\" colspan=\"1\">633</td><td rowspan=\"1\" colspan=\"1\">634</td><td rowspan=\"1\" colspan=\"1\">633</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">702</td><td rowspan=\"1\" colspan=\"1\">628</td><td rowspan=\"1\" colspan=\"1\">628</td><td rowspan=\"1\" colspan=\"1\">613</td><td rowspan=\"1\" colspan=\"1\">596</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Age (years)</td><td rowspan=\"1\" colspan=\"1\">45.74 ± 8.28</td><td rowspan=\"1\" colspan=\"1\">45.87 ± 8.64</td><td rowspan=\"1\" colspan=\"1\">46.26 ± 8.63</td><td rowspan=\"1\" colspan=\"1\">45.66 ± 8.50</td><td rowspan=\"1\" colspan=\"1\">46.01 ± 8.74</td><td rowspan=\"1\" colspan=\"1\">0.74</td><td rowspan=\"1\" colspan=\"1\">46.16 ± 0.32</td><td rowspan=\"1\" colspan=\"1\">45.59 ± 0.34</td><td rowspan=\"1\" colspan=\"1\">45.72 ± 0.32</td><td rowspan=\"1\" colspan=\"1\">46.25 ± 0.35</td><td rowspan=\"1\" colspan=\"1\">45.79 ± 0.35</td><td rowspan=\"1\" colspan=\"1\">0.58</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">General obesity (%)^b\n</td><td rowspan=\"1\" colspan=\"1\">136 (21.5)</td><td rowspan=\"1\" colspan=\"1\">144 (22.7)</td><td rowspan=\"1\" colspan=\"1\">140 (22.1)</td><td rowspan=\"1\" colspan=\"1\">136 (21.5)</td><td rowspan=\"1\" colspan=\"1\">119 (18.8)</td><td rowspan=\"1\" colspan=\"1\">0.49</td><td rowspan=\"1\" colspan=\"1\">156 (22.2)</td><td rowspan=\"1\" colspan=\"1\">138 (22)</td><td rowspan=\"1\" colspan=\"1\">144 (22.9)</td><td rowspan=\"1\" colspan=\"1\">126 (20.9)</td><td rowspan=\"1\" colspan=\"1\">111 (18.6)</td><td rowspan=\"1\" colspan=\"1\">0.37</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Central obesity (%)^c\n</td><td rowspan=\"1\" colspan=\"1\">451 (71.2)</td><td rowspan=\"1\" colspan=\"1\">460 (72.6)</td><td rowspan=\"1\" colspan=\"1\">439 (69.4)</td><td rowspan=\"1\" colspan=\"1\">412 (56)</td><td rowspan=\"1\" colspan=\"1\">449 (70.9)</td><td rowspan=\"1\" colspan=\"1\">0.03</td><td rowspan=\"1\" colspan=\"1\">508 (72.4)</td><td rowspan=\"1\" colspan=\"1\">447 (71.2)</td><td rowspan=\"1\" colspan=\"1\">444 (70.7)</td><td rowspan=\"1\" colspan=\"1\">394 (64.3)</td><td rowspan=\"1\" colspan=\"1\">418 (70.1)</td><td rowspan=\"1\" colspan=\"1\">0.01</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Marital status (married) (%)</td><td rowspan=\"1\" colspan=\"1\">525 (82.9)</td><td rowspan=\"1\" colspan=\"1\">536 (84.5)</td><td rowspan=\"1\" colspan=\"1\">514 (81.2)</td><td rowspan=\"1\" colspan=\"1\">528 (83.3)</td><td rowspan=\"1\" colspan=\"1\">522 (52.5)</td><td rowspan=\"1\" colspan=\"1\">0.61</td><td rowspan=\"1\" colspan=\"1\">584 (83.2)</td><td rowspan=\"1\" colspan=\"1\">523 (83.3)</td><td rowspan=\"1\" colspan=\"1\">521 (83)</td><td rowspan=\"1\" colspan=\"1\">498 (81.2)</td><td rowspan=\"1\" colspan=\"1\">499 (83.7)</td><td rowspan=\"1\" colspan=\"1\">0.81</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Smoking status (active smoker) (%)</td><td rowspan=\"1\" colspan=\"1\">18 (2.8)</td><td rowspan=\"1\" colspan=\"1\">22 (3.5)</td><td rowspan=\"1\" colspan=\"1\">24 (3.8)</td><td rowspan=\"1\" colspan=\"1\">22 (3.5)</td><td rowspan=\"1\" colspan=\"1\">20 (3.2)</td><td rowspan=\"1\" colspan=\"1\">0.90</td><td rowspan=\"1\" colspan=\"1\">24 (3.4)</td><td rowspan=\"1\" colspan=\"1\">14 (2.2)</td><td rowspan=\"1\" colspan=\"1\">23 (3.7)</td><td rowspan=\"1\" colspan=\"1\">25 (4.1)</td><td rowspan=\"1\" colspan=\"1\">20 (3.4)</td><td rowspan=\"1\" colspan=\"1\">0.45</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Education (academic degree) (%)</td><td rowspan=\"1\" colspan=\"1\">13 (2.1)</td><td rowspan=\"1\" colspan=\"1\">11 (1.7)</td><td rowspan=\"1\" colspan=\"1\">8 (1.3)</td><td rowspan=\"1\" colspan=\"1\">11 (1.7)</td><td rowspan=\"1\" colspan=\"1\">9 (1.4)</td><td rowspan=\"1\" colspan=\"1\">0.83</td><td rowspan=\"1\" colspan=\"1\">15 (2.1)</td><td rowspan=\"1\" colspan=\"1\">11 (1.8)</td><td rowspan=\"1\" colspan=\"1\">10 (1.6)</td><td rowspan=\"1\" colspan=\"1\">10 (1.6)</td><td rowspan=\"1\" colspan=\"1\">6 (1)</td><td rowspan=\"1\" colspan=\"1\">0.62</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Socioeconomic status^d\n</td><td rowspan=\"1\" colspan=\"1\">-0.36 ± 0.06</td><td rowspan=\"1\" colspan=\"1\">-0.45 ± 0.06</td><td rowspan=\"1\" colspan=\"1\">-0.40 ± 0.06</td><td rowspan=\"1\" colspan=\"1\">-0.50 ± 0.06</td><td rowspan=\"1\" colspan=\"1\">-0.44 ± 0.06</td><td rowspan=\"1\" colspan=\"1\">0.69</td><td rowspan=\"1\" colspan=\"1\">-0.40 ± 0.06</td><td rowspan=\"1\" colspan=\"1\">-0.27 ± 0.06</td><td rowspan=\"1\" colspan=\"1\">-0.42 ± 0.06</td><td rowspan=\"1\" colspan=\"1\">-0.61 ± 0.06</td><td rowspan=\"1\" colspan=\"1\">-0.45 ± 0.07</td><td rowspan=\"1\" colspan=\"1\">0.012</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Physical activity (METs h/w)</td><td rowspan=\"1\" colspan=\"1\">38.31 ± 0.27</td><td rowspan=\"1\" colspan=\"1\">38.93 ± 0.28</td><td rowspan=\"1\" colspan=\"1\">38.46 ± 0.26</td><td rowspan=\"1\" colspan=\"1\">38.76 ± 0.26</td><td rowspan=\"1\" colspan=\"1\">38.93 ± 0.26</td><td rowspan=\"1\" colspan=\"1\">0.36</td><td rowspan=\"1\" colspan=\"1\">38.53 ± 0.27</td><td rowspan=\"1\" colspan=\"1\">38.63 ± 0.26</td><td rowspan=\"1\" colspan=\"1\">38.58 ± 0.26</td><td rowspan=\"1\" colspan=\"1\">38.52 ± 0.26</td><td rowspan=\"1\" colspan=\"1\">39.17 ± 0.27</td><td rowspan=\"1\" colspan=\"1\">0.41</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Homeownership (owner) (%)</td><td rowspan=\"1\" colspan=\"1\">563 (88.9)</td><td rowspan=\"1\" colspan=\"1\">563 (88.8)</td><td rowspan=\"1\" colspan=\"1\">576 (91.0)</td><td rowspan=\"1\" colspan=\"1\">590 (93.1)</td><td rowspan=\"1\" colspan=\"1\">579 (91.5)</td><td rowspan=\"1\" colspan=\"1\">0.04</td><td rowspan=\"1\" colspan=\"1\">625 (89)</td><td rowspan=\"1\" colspan=\"1\">567 (90.3)</td><td rowspan=\"1\" colspan=\"1\">567 (90.3)</td><td rowspan=\"1\" colspan=\"1\">563 (91.8)</td><td rowspan=\"1\" colspan=\"1\">549 (92.1)</td><td rowspan=\"1\" colspan=\"1\">0.29</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Calcium supplementation (with or without vitamin D) (yes) (%)</td><td rowspan=\"1\" colspan=\"1\">136 (39.7)</td><td rowspan=\"1\" colspan=\"1\">136 (42.4)</td><td rowspan=\"1\" colspan=\"1\">134 (41.9)</td><td rowspan=\"1\" colspan=\"1\">121 (37.8)</td><td rowspan=\"1\" colspan=\"1\">124 (39.2)</td><td rowspan=\"1\" colspan=\"1\">0.75</td><td rowspan=\"1\" colspan=\"1\">150 (38.1)</td><td rowspan=\"1\" colspan=\"1\">132 (41.4)</td><td rowspan=\"1\" colspan=\"1\">134 (41.7)</td><td rowspan=\"1\" colspan=\"1\">112 (39.4)</td><td rowspan=\"1\" colspan=\"1\">123 (40.7)</td><td rowspan=\"1\" colspan=\"1\">0.85</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Fatty liver disease (yes) (%)^e\n</td><td rowspan=\"1\" colspan=\"1\">80 (12.6)</td><td rowspan=\"1\" colspan=\"1\">74 (11.7)</td><td rowspan=\"1\" colspan=\"1\">84 (13.3)</td><td rowspan=\"1\" colspan=\"1\">48 (7.6)</td><td rowspan=\"1\" colspan=\"1\">58 (9.2)</td><td rowspan=\"1\" colspan=\"1\">0.004</td><td rowspan=\"1\" colspan=\"1\">90 (12.8)</td><td rowspan=\"1\" colspan=\"1\">74 (11.8)</td><td rowspan=\"1\" colspan=\"1\">83 (13.2)</td><td rowspan=\"1\" colspan=\"1\">44 (7.2)</td><td rowspan=\"1\" colspan=\"1\">53 (8.9)</td><td rowspan=\"1\" colspan=\"1\">0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Depression (yes) (%)^e\n</td><td rowspan=\"1\" colspan=\"1\">57 (9)</td><td rowspan=\"1\" colspan=\"1\">60 (9.5)</td><td rowspan=\"1\" colspan=\"1\">51 (8.1)</td><td rowspan=\"1\" colspan=\"1\">57 (9)</td><td rowspan=\"1\" colspan=\"1\">46 (7.3)</td><td rowspan=\"1\" colspan=\"1\">0.63</td><td rowspan=\"1\" colspan=\"1\">67 (9.5)</td><td rowspan=\"1\" colspan=\"1\">59 (9.4)</td><td rowspan=\"1\" colspan=\"1\">49 (7.8)</td><td rowspan=\"1\" colspan=\"1\">53 (8.6)</td><td rowspan=\"1\" colspan=\"1\">43 (7.2)</td><td rowspan=\"1\" colspan=\"1\">0.51</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Thyroid disease (yes) (%)^e\n</td><td rowspan=\"1\" colspan=\"1\">97 (15.3)</td><td rowspan=\"1\" colspan=\"1\">74 (11.7)</td><td rowspan=\"1\" colspan=\"1\">83 (13.1)</td><td rowspan=\"1\" colspan=\"1\">81 (12.8)</td><td rowspan=\"1\" colspan=\"1\">64 (10.1)</td><td rowspan=\"1\" colspan=\"1\">0.07</td><td rowspan=\"1\" colspan=\"1\">100 (14.2)</td><td rowspan=\"1\" colspan=\"1\">84 (13.4)</td><td rowspan=\"1\" colspan=\"1\">75 (11.9)</td><td rowspan=\"1\" colspan=\"1\">76 (12.4)</td><td rowspan=\"1\" colspan=\"1\">64 (10.7)</td><td rowspan=\"1\" colspan=\"1\">0.37</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>Dietary intakes of the participants across quintiles of NEAP scores</title><p>Note: Data are presented as mean ± standard deviation.</p><p>^a NEAP = net endogenous acid production.</p><p>^b Calculated with multivariate analysis of covariance (ANCOVA). All variables except energy were adjusted for both energy intake and age. Energy was adjusted for age.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td colspan=\"5\" rowspan=\"1\">Men</td><td rowspan=\"3\" colspan=\"1\">p-value^b\n</td><td colspan=\"5\" rowspan=\"1\">Women</td><td rowspan=\"3\" colspan=\"1\">p-value^b\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td colspan=\"5\" rowspan=\"1\">Quintiles of NEAP^a\n</td><td colspan=\"5\" rowspan=\"1\">Quintiles of NEAP</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Models</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">5</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">5</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Range</td><td rowspan=\"1\" colspan=\"1\">&lt;29.77</td><td rowspan=\"1\" colspan=\"1\">29.77, 37.26</td><td rowspan=\"1\" colspan=\"1\">37.30, 44.32</td><td rowspan=\"1\" colspan=\"1\">44.33, 53.52</td><td rowspan=\"1\" colspan=\"1\">&gt;53.52</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">&lt;29.77</td><td rowspan=\"1\" colspan=\"1\">29.80, 37.26</td><td rowspan=\"1\" colspan=\"1\">37.31, 44.32</td><td rowspan=\"1\" colspan=\"1\">44.33, 53.52</td><td rowspan=\"1\" colspan=\"1\">&gt;53.54</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">n</td><td rowspan=\"1\" colspan=\"1\">569</td><td rowspan=\"1\" colspan=\"1\">643</td><td rowspan=\"1\" colspan=\"1\">644</td><td rowspan=\"1\" colspan=\"1\">658</td><td rowspan=\"1\" colspan=\"1\">675</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">702</td><td rowspan=\"1\" colspan=\"1\">628</td><td rowspan=\"1\" colspan=\"1\">628</td><td rowspan=\"1\" colspan=\"1\">613</td><td rowspan=\"1\" colspan=\"1\">596</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Food groups (g/d)</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Grains</td><td rowspan=\"1\" colspan=\"1\">435.99 ± 162.33</td><td rowspan=\"1\" colspan=\"1\">547.24 ± 180.07</td><td rowspan=\"1\" colspan=\"1\">608.88 ± 187.28</td><td rowspan=\"1\" colspan=\"1\">687.46 ± 200.85</td><td rowspan=\"1\" colspan=\"1\">788.91 ± 224.05</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">419.51 ± 166.56</td><td rowspan=\"1\" colspan=\"1\">521.08 ± 170.05</td><td rowspan=\"1\" colspan=\"1\">576.24 ± 176.67</td><td rowspan=\"1\" colspan=\"1\">665.13 ± 203.38</td><td rowspan=\"1\" colspan=\"1\">776.17 ± 239.49</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Meats</td><td rowspan=\"1\" colspan=\"1\">69.49 ± 41.39</td><td rowspan=\"1\" colspan=\"1\">81.03 ± 44.14</td><td rowspan=\"1\" colspan=\"1\">94.36 ± 52.92</td><td rowspan=\"1\" colspan=\"1\">94.37 ± 57.58</td><td rowspan=\"1\" colspan=\"1\">98.95 ± 61.53</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">58.44 ± 34.47</td><td rowspan=\"1\" colspan=\"1\">74.66 ± 43.47</td><td rowspan=\"1\" colspan=\"1\">77.62 ± 46.00</td><td rowspan=\"1\" colspan=\"1\">86.00 ± 52.73</td><td rowspan=\"1\" colspan=\"1\">84.86 ± 62.47</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Dairy products</td><td rowspan=\"1\" colspan=\"1\">207.33 ± 161.08</td><td rowspan=\"1\" colspan=\"1\">216.28 ± 155.65</td><td rowspan=\"1\" colspan=\"1\">221.46 ± 166.19</td><td rowspan=\"1\" colspan=\"1\">187.44 ± 142.49</td><td rowspan=\"1\" colspan=\"1\">154.19 ± 120.99</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">214.62 ± 152.66</td><td rowspan=\"1\" colspan=\"1\">229.03 ± 177.26</td><td rowspan=\"1\" colspan=\"1\">205.42 ± 165.69</td><td rowspan=\"1\" colspan=\"1\">199.97 ± 155.68</td><td rowspan=\"1\" colspan=\"1\">149.59 ± 119.54</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Fruit</td><td rowspan=\"1\" colspan=\"1\">526.54 ± 359.24</td><td rowspan=\"1\" colspan=\"1\">440.15 ± 282.06</td><td rowspan=\"1\" colspan=\"1\">366.70 ± 225.16</td><td rowspan=\"1\" colspan=\"1\">286.78 ± 180.62</td><td rowspan=\"1\" colspan=\"1\">198.92 ± 127.37</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">525.04 ± 357.10</td><td rowspan=\"1\" colspan=\"1\">414.19 ± 266.13</td><td rowspan=\"1\" colspan=\"1\">329.20 ± 205.73</td><td rowspan=\"1\" colspan=\"1\">279.20 ± 178.23</td><td rowspan=\"1\" colspan=\"1\">189.04 ± 131.07</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Vegetable</td><td rowspan=\"1\" colspan=\"1\">685.63 ± 337.12</td><td rowspan=\"1\" colspan=\"1\">555.35 ± 256.05</td><td rowspan=\"1\" colspan=\"1\">495.63 ± 209.66</td><td rowspan=\"1\" colspan=\"1\">426.75 ± 191.66</td><td rowspan=\"1\" colspan=\"1\">304.66 ± 145.34</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">701.27 ± 377.40</td><td rowspan=\"1\" colspan=\"1\">572.21 ± 250.79</td><td rowspan=\"1\" colspan=\"1\">474.89 ± 203.73</td><td rowspan=\"1\" colspan=\"1\">426.55 ± 195.61</td><td rowspan=\"1\" colspan=\"1\">295.96 ± 145.16</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Nutrients</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Energy (kcal/d)</td><td rowspan=\"1\" colspan=\"1\">2514.33 ± 728.30</td><td rowspan=\"1\" colspan=\"1\">2637.30 ± 715.42</td><td rowspan=\"1\" colspan=\"1\">2747.08 ± 703.44</td><td rowspan=\"1\" colspan=\"1\">2803.64 ± 696.74</td><td rowspan=\"1\" colspan=\"1\">2888.41 ± 694.30</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">2373.47 ± 761.37</td><td rowspan=\"1\" colspan=\"1\">2536.54 ± 711.88</td><td rowspan=\"1\" colspan=\"1\">2550.67 ± 688.43</td><td rowspan=\"1\" colspan=\"1\">2740.78 ± 710.77</td><td rowspan=\"1\" colspan=\"1\">2799.46 ± 751.98</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Carbohydrate (g/d)</td><td rowspan=\"1\" colspan=\"1\">437.90 ± 138.88</td><td rowspan=\"1\" colspan=\"1\">445.09 ± 126.21</td><td rowspan=\"1\" colspan=\"1\">451.56 ± 123.67</td><td rowspan=\"1\" colspan=\"1\">461.40 ± 122.60 </td><td rowspan=\"1\" colspan=\"1\">472.60 ± 127.09</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">410.98 ± 141.81</td><td rowspan=\"1\" colspan=\"1\">422.68 ± 124.84</td><td rowspan=\"1\" colspan=\"1\">418.92 ± 120.03</td><td rowspan=\"1\" colspan=\"1\">446.08 ± 125.58</td><td rowspan=\"1\" colspan=\"1\">461.47 ± 130.82</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Protein (g/d)</td><td rowspan=\"1\" colspan=\"1\">72.55 ± 23.36</td><td rowspan=\"1\" colspan=\"1\">82.11 ± 24.62</td><td rowspan=\"1\" colspan=\"1\">90.04 ± 25.10</td><td rowspan=\"1\" colspan=\"1\">93.89 ± 26.96</td><td rowspan=\"1\" colspan=\"1\">99.27 ± 25.44</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">68.90 ± 23.39</td><td rowspan=\"1\" colspan=\"1\">79.88 ± 23.99</td><td rowspan=\"1\" colspan=\"1\">82.98 ± 24.85</td><td rowspan=\"1\" colspan=\"1\">91.23 ± 25.81</td><td rowspan=\"1\" colspan=\"1\">95.20 ± 27.46</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Fat (g/d)</td><td rowspan=\"1\" colspan=\"1\">60.68 ± 22.05</td><td rowspan=\"1\" colspan=\"1\">64.71 ± 22.41</td><td rowspan=\"1\" colspan=\"1\">69.13 ± 23.17</td><td rowspan=\"1\" colspan=\"1\">67.86 ± 21.23</td><td rowspan=\"1\" colspan=\"1\">67.85 ± 22.35</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">58.47 ± 22.50</td><td rowspan=\"1\" colspan=\"1\">64.31 ± 24.04</td><td rowspan=\"1\" colspan=\"1\">64.58 ± 21.63</td><td rowspan=\"1\" colspan=\"1\">68.84 ± 23.28</td><td rowspan=\"1\" colspan=\"1\">64.68 ± 21.69</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Fiber (g/d)</td><td rowspan=\"1\" colspan=\"1\">31.79 ± 12.10</td><td rowspan=\"1\" colspan=\"1\">30.08 ± 10.46</td><td rowspan=\"1\" colspan=\"1\">29.19 ± 9.53</td><td rowspan=\"1\" colspan=\"1\">27.84 ± 8.99</td><td rowspan=\"1\" colspan=\"1\">25.63 ± 9.96</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">31.47 ± 12.44</td><td rowspan=\"1\" colspan=\"1\">29.67 ± 10.38</td><td rowspan=\"1\" colspan=\"1\">27.31 ± 9.32</td><td rowspan=\"1\" colspan=\"1\">27.26 ± 8.90</td><td rowspan=\"1\" colspan=\"1\">25.14 ± 7.96</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Cholesterol (mg/d)</td><td rowspan=\"1\" colspan=\"1\">225.59 ± 121.98</td><td rowspan=\"1\" colspan=\"1\">254.45 ± 131.72</td><td rowspan=\"1\" colspan=\"1\">286.60 ± 146.76</td><td rowspan=\"1\" colspan=\"1\">273.82 ± 140.97</td><td rowspan=\"1\" colspan=\"1\">275.10 ± 149.27</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">198.64 ± 106.28</td><td rowspan=\"1\" colspan=\"1\">237.67 ± 126.08</td><td rowspan=\"1\" colspan=\"1\">236.52 ± 118.42</td><td rowspan=\"1\" colspan=\"1\">264.21 ± 143.78</td><td rowspan=\"1\" colspan=\"1\">244.81 ± 151.43</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Vitamin B12 (µg/d)</td><td rowspan=\"1\" colspan=\"1\">7.34 ± 0.16</td><td rowspan=\"1\" colspan=\"1\">6.93 ± 0.15</td><td rowspan=\"1\" colspan=\"1\">6.82 ± 0.15</td><td rowspan=\"1\" colspan=\"1\">5.89 ± 0.15</td><td rowspan=\"1\" colspan=\"1\">4.98 ± 0.15</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">7.04 ± 0.15</td><td rowspan=\"1\" colspan=\"1\">6.37 ± 0.16</td><td rowspan=\"1\" colspan=\"1\">5.76 ± 0.16</td><td rowspan=\"1\" colspan=\"1\">5.67 ± 0.16</td><td rowspan=\"1\" colspan=\"1\">4.28 ± 0.16</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Folate (µg/d)</td><td rowspan=\"1\" colspan=\"1\">720.22 ± 5.55</td><td rowspan=\"1\" colspan=\"1\">719.67 ± 5.19</td><td rowspan=\"1\" colspan=\"1\">734.82 ± 5.18</td><td rowspan=\"1\" colspan=\"1\">756.18 ± 5.13</td><td rowspan=\"1\" colspan=\"1\">785.04 ± 5.08</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">693.19 ± 5.09</td><td rowspan=\"1\" colspan=\"1\">697.06 ± 5.34</td><td rowspan=\"1\" colspan=\"1\">710.67 ± 5.33</td><td rowspan=\"1\" colspan=\"1\">721.29 ± 5.42</td><td rowspan=\"1\" colspan=\"1\">764.04 ± 5.51</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Phosphorus (mg/d)</td><td rowspan=\"1\" colspan=\"1\">1188.9 ± 387.61</td><td rowspan=\"1\" colspan=\"1\">1267.85 ± 389.43</td><td rowspan=\"1\" colspan=\"1\">1337.80 ± 377.68</td><td rowspan=\"1\" colspan=\"1\">1335.15 ± 371.92</td><td rowspan=\"1\" colspan=\"1\">1333.62 ± 341.79</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">1148.69 ± 385.94</td><td rowspan=\"1\" colspan=\"1\">1236.58 ± 376.29</td><td rowspan=\"1\" colspan=\"1\">1227.61 ± 376.31</td><td rowspan=\"1\" colspan=\"1\">1304.16 ± 371.46</td><td rowspan=\"1\" colspan=\"1\">1284.48 ± 366.68</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Potassium (mg/d)</td><td rowspan=\"1\" colspan=\"1\">4507.48 ± 1438.82</td><td rowspan=\"1\" colspan=\"1\">3989.28 ± 1189.10</td><td rowspan=\"1\" colspan=\"1\">3763.11 ± 1056.49</td><td rowspan=\"1\" colspan=\"1\">3398.52 ± 972.71</td><td rowspan=\"1\" colspan=\"1\">2871.79 ± 803.01</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">4373.59 ± 1493.07</td><td rowspan=\"1\" colspan=\"1\">3891.64 ± 1164.39</td><td rowspan=\"1\" colspan=\"1\">3482.00 ± 1048.83</td><td rowspan=\"1\" colspan=\"1\">3313.62 ± 950.30</td><td rowspan=\"1\" colspan=\"1\">2753.81 ± 837.64</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Calcium (mg/d)</td><td rowspan=\"1\" colspan=\"1\">1042.06 ± 387.83</td><td rowspan=\"1\" colspan=\"1\">1176.12 ± 394.49</td><td rowspan=\"1\" colspan=\"1\">1285.50 ± 414.46</td><td rowspan=\"1\" colspan=\"1\">1364.32 ± 447.18</td><td rowspan=\"1\" colspan=\"1\">1488.98 ± 478.19</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">1019.46 ± 395.94</td><td rowspan=\"1\" colspan=\"1\">1175.99 ± 384.82</td><td rowspan=\"1\" colspan=\"1\">1227.35 ± 425.92</td><td rowspan=\"1\" colspan=\"1\">1362.05 ± 460.86</td><td rowspan=\"1\" colspan=\"1\">1485.36 ± 513.64</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Magnesium (mg/d)</td><td rowspan=\"1\" colspan=\"1\">386.48 ± 117.53</td><td rowspan=\"1\" colspan=\"1\">377.14 ± 111.32</td><td rowspan=\"1\" colspan=\"1\">378.56 ± 106.68</td><td rowspan=\"1\" colspan=\"1\">366.10 ± 99.55</td><td rowspan=\"1\" colspan=\"1\">346.89 ± 87.87</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">372.84 ± 128.24</td><td rowspan=\"1\" colspan=\"1\">365.08 ± 108.03</td><td rowspan=\"1\" colspan=\"1\">348.66 ± 100.75</td><td rowspan=\"1\" colspan=\"1\">355.06 ± 99.13</td><td rowspan=\"1\" colspan=\"1\">336.35 ± 93.65</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Sodium (mg/d)</td><td rowspan=\"1\" colspan=\"1\">3808.35 ± 1385.88</td><td rowspan=\"1\" colspan=\"1\">4151.80 ± 1378.33</td><td rowspan=\"1\" colspan=\"1\">4486.92 ± 1453.65</td><td rowspan=\"1\" colspan=\"1\">4562.95 ± 1341.75</td><td rowspan=\"1\" colspan=\"1\">4887.86 ± 1402.78</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">3819.26 ± 1661.86</td><td rowspan=\"1\" colspan=\"1\">4222.49 ± 1449.05</td><td rowspan=\"1\" colspan=\"1\">4259.33 ± 1461.06</td><td rowspan=\"1\" colspan=\"1\">4705.45 ± 1518.34</td><td rowspan=\"1\" colspan=\"1\">4827.30 ± 1430.30</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Sodium/potassium</td><td rowspan=\"1\" colspan=\"1\">0.88 ± 0.33</td><td rowspan=\"1\" colspan=\"1\">1.07 ± 0.32</td><td rowspan=\"1\" colspan=\"1\">1.22 ± 0.36</td><td rowspan=\"1\" colspan=\"1\">1.38 ± 0.33</td><td rowspan=\"1\" colspan=\"1\">1.76 ± 0.48</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">0.90 ± 0.32</td><td rowspan=\"1\" colspan=\"1\">1.12 ± 0.40</td><td rowspan=\"1\" colspan=\"1\">1.25 ± 0.34</td><td rowspan=\"1\" colspan=\"1\">1.45 ± 0.40</td><td rowspan=\"1\" colspan=\"1\">1.82 ± 0.52</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB3\"><label>Table 3</label><caption><title>Dietary intakes of the participants across quintiles of PRAL scores</title><p>Note: Data are presented as mean ± standard deviation.</p><p>^a PRAL = potential renal acid load.</p><p>^b Calculated with multivariate analysis of covariance (ANCOVA). All variables except energy were adjusted for both energy intake and age. Energy was adjusted for age.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td colspan=\"5\" rowspan=\"1\">Men</td><td rowspan=\"3\" colspan=\"1\">p-value^b\n</td><td colspan=\"5\" rowspan=\"1\">Women</td><td rowspan=\"3\" colspan=\"1\">p-value^b\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td colspan=\"5\" rowspan=\"1\">Quintiles of PRAL^a\n</td><td colspan=\"5\" rowspan=\"1\">Quintiles of PRAL</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Models</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">5</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">5</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Range</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">-23.81, -12.95</td><td rowspan=\"1\" colspan=\"1\">-12.95, -5.15</td><td rowspan=\"1\" colspan=\"1\">-5.13, 4.19</td><td rowspan=\"1\" colspan=\"1\">&gt;4.19</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">-25.76, -14.92</td><td rowspan=\"1\" colspan=\"1\">-14.89, -6.63</td><td rowspan=\"1\" colspan=\"1\">-6.53, 1.90</td><td rowspan=\"1\" colspan=\"1\">&gt;1.90</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">n</td><td rowspan=\"1\" colspan=\"1\">637</td><td rowspan=\"1\" colspan=\"1\">638</td><td rowspan=\"1\" colspan=\"1\">638</td><td rowspan=\"1\" colspan=\"1\">638</td><td rowspan=\"1\" colspan=\"1\">638</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">633</td><td rowspan=\"1\" colspan=\"1\">634</td><td rowspan=\"1\" colspan=\"1\">633</td><td rowspan=\"1\" colspan=\"1\">634</td><td rowspan=\"1\" colspan=\"1\">633</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Food groups (g/d)</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Grains</td><td rowspan=\"1\" colspan=\"1\">517.34 ± 186.04</td><td rowspan=\"1\" colspan=\"1\">557.47 ± 208.25</td><td rowspan=\"1\" colspan=\"1\">598.42 ± 212.29</td><td rowspan=\"1\" colspan=\"1\">642.04 ± 205.49</td><td rowspan=\"1\" colspan=\"1\">784.18 ± 222.09</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">490.86 ± 184.94</td><td rowspan=\"1\" colspan=\"1\">516.30 ± 197.52</td><td rowspan=\"1\" colspan=\"1\">553.24 ± 200.53</td><td rowspan=\"1\" colspan=\"1\">613.02 ± 209.69</td><td rowspan=\"1\" colspan=\"1\">753.59 ± 241.49</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Meats</td><td rowspan=\"1\" colspan=\"1\">80.14 ± 45.48</td><td rowspan=\"1\" colspan=\"1\">80.56 ± 46.80</td><td rowspan=\"1\" colspan=\"1\">78.83 ± 47.87</td><td rowspan=\"1\" colspan=\"1\">90.40 ± 54.28</td><td rowspan=\"1\" colspan=\"1\">111.10 ± 63.83</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">67.60 ± 40.16</td><td rowspan=\"1\" colspan=\"1\">69.95 ± 41.68</td><td rowspan=\"1\" colspan=\"1\">69.67 ± 44.23</td><td rowspan=\"1\" colspan=\"1\">73.90 ± 45.75</td><td rowspan=\"1\" colspan=\"1\">97.72 ± 64.54</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Dairy products</td><td rowspan=\"1\" colspan=\"1\">239.65 ± 177.29</td><td rowspan=\"1\" colspan=\"1\">207.53 ± 157.44</td><td rowspan=\"1\" colspan=\"1\">194.40 ± 149.77</td><td rowspan=\"1\" colspan=\"1\">173.69 ± 130.39</td><td rowspan=\"1\" colspan=\"1\">167.98 ± 126.68</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">254.25 ± 181.41</td><td rowspan=\"1\" colspan=\"1\">208.82 ± 151.84</td><td rowspan=\"1\" colspan=\"1\">197.67 ± 159.32</td><td rowspan=\"1\" colspan=\"1\">174.14 ± 146.60</td><td rowspan=\"1\" colspan=\"1\">168.06 ± 130.56</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Fruit</td><td rowspan=\"1\" colspan=\"1\">606.00 ± 358.98</td><td rowspan=\"1\" colspan=\"1\">404.01 ± 235.75</td><td rowspan=\"1\" colspan=\"1\">312.47 ± 182.10</td><td rowspan=\"1\" colspan=\"1\">257.79 ± 166.01</td><td rowspan=\"1\" colspan=\"1\">210.25 ± 137.77</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">611.66 ± 366.67</td><td rowspan=\"1\" colspan=\"1\">389.83 ± 221.11</td><td rowspan=\"1\" colspan=\"1\">312.64 ± 186.91</td><td rowspan=\"1\" colspan=\"1\">247.80 ± 163.15</td><td rowspan=\"1\" colspan=\"1\">205.21 ± 142.95</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Vegetable</td><td rowspan=\"1\" colspan=\"1\">741.32 ± 316.99</td><td rowspan=\"1\" colspan=\"1\">536.80 ± 229.71</td><td rowspan=\"1\" colspan=\"1\">457.03 ± 185.82</td><td rowspan=\"1\" colspan=\"1\">385.74 ± 174.26</td><td rowspan=\"1\" colspan=\"1\">314.21 ± 157.53</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">787.66 ± 373.80</td><td rowspan=\"1\" colspan=\"1\">564.13 ± 215.71</td><td rowspan=\"1\" colspan=\"1\">453.99 ± 184.15</td><td rowspan=\"1\" colspan=\"1\">388.97 ± 175.37</td><td rowspan=\"1\" colspan=\"1\">312.01 ± 160.69</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Nutrients</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Energy (kcal/d)</td><td rowspan=\"1\" colspan=\"1\">2853.97 ± 684.92</td><td rowspan=\"1\" colspan=\"1\">2631.29 ± 737.98</td><td rowspan=\"1\" colspan=\"1\">2589.82 ± 722.81</td><td rowspan=\"1\" colspan=\"1\">2619.17 ± 706.13</td><td rowspan=\"1\" colspan=\"1\">2930.94 ± 670.92</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">2738.77 ± 724.65</td><td rowspan=\"1\" colspan=\"1\">2462.08 ± 725.50</td><td rowspan=\"1\" colspan=\"1\">2448.14 ± 724.12</td><td rowspan=\"1\" colspan=\"1\">2493.68 ± 725.71</td><td rowspan=\"1\" colspan=\"1\">2818.73 ± 729.92</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Carbohydrate (g/d)</td><td rowspan=\"1\" colspan=\"1\">497.02 ± 126.26</td><td rowspan=\"1\" colspan=\"1\">443.03 ± 127.57</td><td rowspan=\"1\" colspan=\"1\">429.75 ± 126.66</td><td rowspan=\"1\" colspan=\"1\">427.79 ± 121.81</td><td rowspan=\"1\" colspan=\"1\">473.99 ± 118.79</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">475.01 ± 133.05</td><td rowspan=\"1\" colspan=\"1\">412.54 ± 125.09</td><td rowspan=\"1\" colspan=\"1\">404.50 ± 124.09</td><td rowspan=\"1\" colspan=\"1\">406.91 ± 125.77</td><td rowspan=\"1\" colspan=\"1\">456.97 ± 127.84</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Protein (g/d)</td><td rowspan=\"1\" colspan=\"1\">85.64 ± 24.83</td><td rowspan=\"1\" colspan=\"1\">82.28 ± 26.35</td><td rowspan=\"1\" colspan=\"1\">83.29 ± 25.97</td><td rowspan=\"1\" colspan=\"1\">86.86 ± 26.42</td><td rowspan=\"1\" colspan=\"1\">102.26 ± 25.35</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">81.57 ± 24.93</td><td rowspan=\"1\" colspan=\"1\">77.25 ± 25.34</td><td rowspan=\"1\" colspan=\"1\">78.25 ± 25.81</td><td rowspan=\"1\" colspan=\"1\">81.53 ± 25.35</td><td rowspan=\"1\" colspan=\"1\">97.11 ± 27.37</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Fat (g/d)</td><td rowspan=\"1\" colspan=\"1\">66.98 ± 22.13</td><td rowspan=\"1\" colspan=\"1\">64.45 ± 22.70</td><td rowspan=\"1\" colspan=\"1\">63.67 ± 21.92</td><td rowspan=\"1\" colspan=\"1\">65.09 ± 22.16</td><td rowspan=\"1\" colspan=\"1\">70.79 ± 22.29</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">66.03 ± 22.42</td><td rowspan=\"1\" colspan=\"1\">61.40 ± 23.20</td><td rowspan=\"1\" colspan=\"1\">61.59 ± 22.72</td><td rowspan=\"1\" colspan=\"1\">62.80 ± 22.94</td><td rowspan=\"1\" colspan=\"1\">68.27 ± 22.44</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Fiber (g/d)</td><td rowspan=\"1\" colspan=\"1\">36.33 ± 11.10</td><td rowspan=\"1\" colspan=\"1\">29.40 ± 9.26</td><td rowspan=\"1\" colspan=\"1\">26.88 ± 8.77</td><td rowspan=\"1\" colspan=\"1\">25.39 ± 8.36</td><td rowspan=\"1\" colspan=\"1\">26.02 ± 7.84</td><td rowspan=\"1\" colspan=\"1\">0.000</td><td rowspan=\"1\" colspan=\"1\">36.65 ± 11.73</td><td rowspan=\"1\" colspan=\"1\">28.64 ± 8.94</td><td rowspan=\"1\" colspan=\"1\">26.23 ± 8.76</td><td rowspan=\"1\" colspan=\"1\">24.63 ± 8.33</td><td rowspan=\"1\" colspan=\"1\">25.27 ± 8.05</td><td rowspan=\"1\" colspan=\"1\">0.000</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Cholesterol (mg/d)</td><td rowspan=\"1\" colspan=\"1\">253.09 ± 130.92</td><td rowspan=\"1\" colspan=\"1\">253.77 ± 139.37</td><td rowspan=\"1\" colspan=\"1\">244.65 ± 130.05</td><td rowspan=\"1\" colspan=\"1\">262.08 ± 136.34</td><td rowspan=\"1\" colspan=\"1\">307.21 ± 155.62</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">225.83 ± 121.53</td><td rowspan=\"1\" colspan=\"1\">222.29 ± 115.70</td><td rowspan=\"1\" colspan=\"1\">215.37 ± 113.31</td><td rowspan=\"1\" colspan=\"1\">233.67 ± 127.92</td><td rowspan=\"1\" colspan=\"1\">279.21 ± 161.91</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Vitamin B12 (µg/d)</td><td rowspan=\"1\" colspan=\"1\">7.46 ± 0.15</td><td rowspan=\"1\" colspan=\"1\">6.67 ± 0.15</td><td rowspan=\"1\" colspan=\"1\">6.25 ± 0.15</td><td rowspan=\"1\" colspan=\"1\">5.82 ± 0.15</td><td rowspan=\"1\" colspan=\"1\">5.58 ± 0.15</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">7.19 ± 0.16</td><td rowspan=\"1\" colspan=\"1\">6.19 ± 0.16</td><td rowspan=\"1\" colspan=\"1\">5.60 ± 0.16</td><td rowspan=\"1\" colspan=\"1\">5.42 ± 0.16</td><td rowspan=\"1\" colspan=\"1\">4.94 ± 0.16</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Folate (µg/d)</td><td rowspan=\"1\" colspan=\"1\">741.80 ± 5.30</td><td rowspan=\"1\" colspan=\"1\">730.09 ± 5.29</td><td rowspan=\"1\" colspan=\"1\">749.77 ± 5.30</td><td rowspan=\"1\" colspan=\"1\">741.24 ± 5.29</td><td rowspan=\"1\" colspan=\"1\">758.08 ± 5.33</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">711.54 ± 5.41</td><td rowspan=\"1\" colspan=\"1\">707.57 ± 5.40</td><td rowspan=\"1\" colspan=\"1\">713.07 ± 5.41</td><td rowspan=\"1\" colspan=\"1\">718.50 ± 5.40</td><td rowspan=\"1\" colspan=\"1\">730.31 ± 5.44</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Phosphorus (mg/d)</td><td rowspan=\"1\" colspan=\"1\">1370.59 ± 381.51</td><td rowspan=\"1\" colspan=\"1\">1256.11 ± 388.85</td><td rowspan=\"1\" colspan=\"1\">1229.30 ± 373.65</td><td rowspan=\"1\" colspan=\"1\">1237.71 ± 366.12</td><td rowspan=\"1\" colspan=\"1\">1384.94 ± 346.16</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">1335.94 ± 383.15</td><td rowspan=\"1\" colspan=\"1\">1190.35 ± 364.81</td><td rowspan=\"1\" colspan=\"1\">1163.85 ± 375.47</td><td rowspan=\"1\" colspan=\"1\">1171.86 ± 364.59</td><td rowspan=\"1\" colspan=\"1\">1325.26 ± 371.50</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Potassium (mg/d)</td><td rowspan=\"1\" colspan=\"1\">4986.37 ± 1198.06</td><td rowspan=\"1\" colspan=\"1\">3882.09 ± 980.42</td><td rowspan=\"1\" colspan=\"1\">3417.17 ± 941.57</td><td rowspan=\"1\" colspan=\"1\">3120.22 ± 945.85</td><td rowspan=\"1\" colspan=\"1\">2984.40 ± 868.56</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">4992.59± 1276.18</td><td rowspan=\"1\" colspan=\"1\">3775.89 ± 931.66</td><td rowspan=\"1\" colspan=\"1\">3314.57 ± 941.48</td><td rowspan=\"1\" colspan=\"1\">2993.56 ± 910.07</td><td rowspan=\"1\" colspan=\"1\">2880.18 ± 897.30</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Calcium (mg/d)</td><td rowspan=\"1\" colspan=\"1\">1259.74 ± 424.31</td><td rowspan=\"1\" colspan=\"1\">1190.94 ± 433.45</td><td rowspan=\"1\" colspan=\"1\">1235.44 ± 449.91</td><td rowspan=\"1\" colspan=\"1\">1253.31 ± 442.35</td><td rowspan=\"1\" colspan=\"1\">1457.27 ± 468.65</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">1229.67 ± 426.45</td><td rowspan=\"1\" colspan=\"1\">1146.76 ± 423.87</td><td rowspan=\"1\" colspan=\"1\">1184.55 ± 446.96</td><td rowspan=\"1\" colspan=\"1\">1231.08 ± 455.33</td><td rowspan=\"1\" colspan=\"1\">1436.68 ± 514.13</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Magnesium (mg/d)</td><td rowspan=\"1\" colspan=\"1\">436.26 ± 105.30</td><td rowspan=\"1\" colspan=\"1\">371.28 ± 101.50</td><td rowspan=\"1\" colspan=\"1\">349.65 ± 101.12</td><td rowspan=\"1\" colspan=\"1\">337.60 ± 97.09</td><td rowspan=\"1\" colspan=\"1\">357.36 ± 92.69</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">429.29 ± 117.18</td><td rowspan=\"1\" colspan=\"1\">354.66 ± 98.24</td><td rowspan=\"1\" colspan=\"1\">332.16 ± 97.70</td><td rowspan=\"1\" colspan=\"1\">321.35 ± 94.83</td><td rowspan=\"1\" colspan=\"1\">343.60 ± 96.02</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Sodium (mg/d)</td><td rowspan=\"1\" colspan=\"1\">4302.67 ± 1466.07</td><td rowspan=\"1\" colspan=\"1\">4218.18 ± 1409.18</td><td rowspan=\"1\" colspan=\"1\">4264.09 ± 1432.23</td><td rowspan=\"1\" colspan=\"1\">4341.54 ± 1385.93</td><td rowspan=\"1\" colspan=\"1\">4867.51 ± 1438.86</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">4365.69 ± 1720.51</td><td rowspan=\"1\" colspan=\"1\">4107.00 ± 1478.26</td><td rowspan=\"1\" colspan=\"1\">4150.44 ± 1522.62</td><td rowspan=\"1\" colspan=\"1\">4315.02 ± 1465.19</td><td rowspan=\"1\" colspan=\"1\">4800.86 ± 1475.00</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Sodium/potassium</td><td rowspan=\"1\" colspan=\"1\">0.88 ± 0.28</td><td rowspan=\"1\" colspan=\"1\">1.09 ± 0.31</td><td rowspan=\"1\" colspan=\"1\">1.27 ± 0.38</td><td rowspan=\"1\" colspan=\"1\">1.43 ± 0.38</td><td rowspan=\"1\" colspan=\"1\">1.70 ± 0.51</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">0.89 ± 0.30</td><td rowspan=\"1\" colspan=\"1\">1.09 ± 0.31</td><td rowspan=\"1\" colspan=\"1\">1.27 ± 0.41</td><td rowspan=\"1\" colspan=\"1\">1.48 ± 0.43</td><td rowspan=\"1\" colspan=\"1\">1.74 ± 0.54</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB4\"><label>Table 4</label><caption><title>Odds ratios and 95% confidence intervals for prevalence of metabolic syndrome and its components across quintiles of PRAL and NEAP scores</title><p>Note: Data are presented as odds ratios and 95% confidence intervals. MetS and its components are defined according to the Adult Treatment Panel III (ATP III) criteria.</p><p>Abbreviations: PRAL, potential renal acid load; NEAP, net endogenous acid production; LDL-C, low-density lipoprotein cholesterol; HDL- C, high-density lipoprotein cholesterol; FBS, fasting blood sugar.</p><p>^a Obtained with binary logistic regression.</p><p>^b Adjusted for age, energy intake, physical activity, education, marital status, socioeconomic status, home ownership, fatty liver disease, depression, thyroid disease, calcium supplementation, and calcium plus vitamin D supplementation.</p><p>^c Additionally adjusted for BMI.</p><p>^d Central obesity was defined as waist circumference &gt;102 cm in men and &gt;88 cm in women.</p><p>^e Hypertension was defined as systolic blood pressure ≥ 130 mm Hg and/or diastolic blood pressure ≥ 85 mm Hg.</p><p>^f Low HDL-C was defined as HDL-C &lt;50 mg/dL for women and &lt;40 mg/dL for men.</p><p>^g Hyperglycemia was defined as FBS ≥ 110 mg/dL.</p><p>^h Hypertriglyceridemia was defined as triglycerides &gt;150 mg/dL.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td colspan=\"5\" rowspan=\"1\">Quintiles of PRAL</td><td rowspan=\"2\" colspan=\"1\">p trend^a\n</td><td colspan=\"5\" rowspan=\"1\">Quintiles of NEAP</td><td rowspan=\"2\" colspan=\"1\">p trend^a\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Models</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">5</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">5</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Men</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Range</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">-23.81, -12.95</td><td rowspan=\"1\" colspan=\"1\">-12.95, -5.15</td><td rowspan=\"1\" colspan=\"1\">-5.13, 4.19</td><td rowspan=\"1\" colspan=\"1\">&gt;4.19</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">&lt;29.77</td><td rowspan=\"1\" colspan=\"1\">29.77, 37.26</td><td rowspan=\"1\" colspan=\"1\">37.30, 44.32</td><td rowspan=\"1\" colspan=\"1\">44.33, 53.52</td><td rowspan=\"1\" colspan=\"1\">&gt;53.52</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">n</td><td rowspan=\"1\" colspan=\"1\">637</td><td rowspan=\"1\" colspan=\"1\">638</td><td rowspan=\"1\" colspan=\"1\">638</td><td rowspan=\"1\" colspan=\"1\">638</td><td rowspan=\"1\" colspan=\"1\">638</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">569</td><td rowspan=\"1\" colspan=\"1\">643</td><td rowspan=\"1\" colspan=\"1\">644</td><td rowspan=\"1\" colspan=\"1\">658</td><td rowspan=\"1\" colspan=\"1\">675</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Metabolic syndrome</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Crude</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.82 (0.53-1.26)</td><td rowspan=\"1\" colspan=\"1\">0.71 (0.45-1.11)</td><td rowspan=\"1\" colspan=\"1\">1.04 (0.69-1.57)</td><td rowspan=\"1\" colspan=\"1\">0.77 (0.50-1.20)</td><td rowspan=\"1\" colspan=\"1\">0.61</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.86 (0.64-1.14)</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.78-1.36)</td><td rowspan=\"1\" colspan=\"1\">0.91 (0.69-1.22)</td><td rowspan=\"1\" colspan=\"1\">1.01 (0.76-1.34)</td><td rowspan=\"1\" colspan=\"1\">0.92</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Model ɪ^b\n</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.54 (0.25-1.15)</td><td rowspan=\"1\" colspan=\"1\">0.68 (0.34-1.38)</td><td rowspan=\"1\" colspan=\"1\">1.24 (0.64-2.38)</td><td rowspan=\"1\" colspan=\"1\">1.11 (0.57-2.14)</td><td rowspan=\"1\" colspan=\"1\">0.21</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\"> 0.68 (0.33-1.40)</td><td rowspan=\"1\" colspan=\"1\">0.77 (0.38-1.53)</td><td rowspan=\"1\" colspan=\"1\">1.004 (0.52-1.92)</td><td rowspan=\"1\" colspan=\"1\">1.07 (0.56-2.04)</td><td rowspan=\"1\" colspan=\"1\">0.55</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Model ɪɪ^c\n</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.52 (0.23-1.16)</td><td rowspan=\"1\" colspan=\"1\">0.67 (0.31-1.42)</td><td rowspan=\"1\" colspan=\"1\">0.88 (0.43-1.79)</td><td rowspan=\"1\" colspan=\"1\">0.93 (0.46-1.87)</td><td rowspan=\"1\" colspan=\"1\">0.71</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.30 (0.13-0.73)</td><td rowspan=\"1\" colspan=\"1\">0.57 (0.27-1.24)</td><td rowspan=\"1\" colspan=\"1\">1.04 (0.51-2.12)</td><td rowspan=\"1\" colspan=\"1\">0.81 (0.39-1.68)</td><td rowspan=\"1\" colspan=\"1\">0.4</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Central obesity^d\n</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Crude</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.88 (0.60-1.28)</td><td rowspan=\"1\" colspan=\"1\">0.91 (0.63-1.32)</td><td rowspan=\"1\" colspan=\"1\">0.89 (0.61-1.30)</td><td rowspan=\"1\" colspan=\"1\">0.86 (0.59-1.26)</td><td rowspan=\"1\" colspan=\"1\">0.51</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.15 (0.77-1.70)</td><td rowspan=\"1\" colspan=\"1\">1.05 (0.70-1.56)</td><td rowspan=\"1\" colspan=\"1\">1.04 (0.70-1.55)</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.69-1.53)</td><td rowspan=\"1\" colspan=\"1\">0.91</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Model ɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.89 (0.48-1.64)</td><td rowspan=\"1\" colspan=\"1\">0.91 (0.50-1.66)</td><td rowspan=\"1\" colspan=\"1\">0.86 (0.47-1.59)</td><td rowspan=\"1\" colspan=\"1\">1.19 (0.67-2.12)</td><td rowspan=\"1\" colspan=\"1\">0.62</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.96 (0.52-1.76)</td><td rowspan=\"1\" colspan=\"1\">0.90 (0.49-1.65)</td><td rowspan=\"1\" colspan=\"1\">0.89 (0.47-1.67)</td><td rowspan=\"1\" colspan=\"1\">1.17 (0.64-2.12)</td><td rowspan=\"1\" colspan=\"1\">0.68</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Model ɪɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.48 (0.17-1.33_</td><td rowspan=\"1\" colspan=\"1\">0.81 (0.32-2.06)</td><td rowspan=\"1\" colspan=\"1\">0.63 (0.24-1.63)</td><td rowspan=\"1\" colspan=\"1\">0.67 (0.27-1.67)</td><td rowspan=\"1\" colspan=\"1\">0.59</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.36 (0.13-0.99)</td><td rowspan=\"1\" colspan=\"1\">0.59 (0.23-1.53)</td><td rowspan=\"1\" colspan=\"1\">0.44 (0.17-1.15)</td><td rowspan=\"1\" colspan=\"1\">0.79 (0.33-1.19)</td><td rowspan=\"1\" colspan=\"1\">0.84</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Hypertension^e\n</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Crude</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.77-1.38)</td><td rowspan=\"1\" colspan=\"1\">0.80 (0.59-1.09)</td><td rowspan=\"1\" colspan=\"1\">0.90 (0.67-1.12)</td><td rowspan=\"1\" colspan=\"1\">0.91 (0.67-1.23)</td><td rowspan=\"1\" colspan=\"1\">0.33</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.77 (0.57-1.05)</td><td rowspan=\"1\" colspan=\"1\">0.85 (0.63-1.15)</td><td rowspan=\"1\" colspan=\"1\">0.80 (0.59-1.08)</td><td rowspan=\"1\" colspan=\"1\">0.88 (0.65-1.18)</td><td rowspan=\"1\" colspan=\"1\">0.56</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Model ɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.85 (0.53-1.36)</td><td rowspan=\"1\" colspan=\"1\">0.81 (0.51-1.29)</td><td rowspan=\"1\" colspan=\"1\">0.81 (0.50-1.30)</td><td rowspan=\"1\" colspan=\"1\">0.60 (0.60-1.53)</td><td rowspan=\"1\" colspan=\"1\">0.75</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.59 (0.36-0.97)</td><td rowspan=\"1\" colspan=\"1\">0.84 (0.53-1.34)</td><td rowspan=\"1\" colspan=\"1\">0.95 (0.59-1.52)</td><td rowspan=\"1\" colspan=\"1\">0.85 (0.52-1.37)</td><td rowspan=\"1\" colspan=\"1\">0.88</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Model ɪɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.79 (0.49-1.28)</td><td rowspan=\"1\" colspan=\"1\">0.78 (0.48-1.26)</td><td rowspan=\"1\" colspan=\"1\">0.76 (0.47-1.24)</td><td rowspan=\"1\" colspan=\"1\">0.88 (0.55-1.41)</td><td rowspan=\"1\" colspan=\"1\">0.55</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.55 (0.33-0.91)</td><td rowspan=\"1\" colspan=\"1\">0.82 (0.51-1.31)</td><td rowspan=\"1\" colspan=\"1\">0.90 (0.55-1.45)</td><td rowspan=\"1\" colspan=\"1\">0.76 (0.46-1.23)</td><td rowspan=\"1\" colspan=\"1\">0.82</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Low HDL-C^f\n</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Crude</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.97 (0.77-1.22)</td><td rowspan=\"1\" colspan=\"1\">0.95 (0.75-1.20)</td><td rowspan=\"1\" colspan=\"1\">0.98 (0.78-1.24)</td><td rowspan=\"1\" colspan=\"1\">0.95 (0.75-1.20)</td><td rowspan=\"1\" colspan=\"1\">0.74</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.90 (0.71-1.14)</td><td rowspan=\"1\" colspan=\"1\">0.92 (0.73-1.17)</td><td rowspan=\"1\" colspan=\"1\">0.83 (0.65-1.05)</td><td rowspan=\"1\" colspan=\"1\">0.95 (0.75-1.20)</td><td rowspan=\"1\" colspan=\"1\">0.51</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Model ɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.06 (0.74-1.53)</td><td rowspan=\"1\" colspan=\"1\">0.98 (0.69-1.40)</td><td rowspan=\"1\" colspan=\"1\">1.04 (0.73-1.48)</td><td rowspan=\"1\" colspan=\"1\">0.91 (0.63-1.30)</td><td rowspan=\"1\" colspan=\"1\">0.62</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.87 (0.60-1.25)</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.72-1.47)</td><td rowspan=\"1\" colspan=\"1\">0.83 (0.57-1.20)</td><td rowspan=\"1\" colspan=\"1\">0.99 (0.69-1.43)</td><td rowspan=\"1\" colspan=\"1\">0.91</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Model ɪɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.72-1.49)</td><td rowspan=\"1\" colspan=\"1\">0.98 (0.69-1.40)</td><td rowspan=\"1\" colspan=\"1\">1.02 (0.71-1.45)</td><td rowspan=\"1\" colspan=\"1\">0.88 (0.61-1.27)</td><td rowspan=\"1\" colspan=\"1\">0.53</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.86 (0.60-1.24)</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.72-1.47)</td><td rowspan=\"1\" colspan=\"1\">0.81 (0.56-1.18)</td><td rowspan=\"1\" colspan=\"1\">0.96 (0.66-1.39)</td><td rowspan=\"1\" colspan=\"1\">0.76</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Hyperglycemia^g\n</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Crude</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.73 (0.36-1.46)</td><td rowspan=\"1\" colspan=\"1\">0.89 (0.46-1.072)</td><td rowspan=\"1\" colspan=\"1\">0.78 (0.39-1.55)</td><td rowspan=\"1\" colspan=\"1\">0.83 (0.43-1.64)</td><td rowspan=\"1\" colspan=\"1\">0.69</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">2.06 (0.97-4.37)</td><td rowspan=\"1\" colspan=\"1\">1.14 (0.49-2.63)</td><td rowspan=\"1\" colspan=\"1\">1.56 (0.71-3.42)</td><td rowspan=\"1\" colspan=\"1\">1.44 (0.65-3.11)</td><td rowspan=\"1\" colspan=\"1\">0.79</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Model ɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.24 (0.36-4.29)</td><td rowspan=\"1\" colspan=\"1\">0.81 (0.21-3.19)</td><td rowspan=\"1\" colspan=\"1\">1.97 (0.63-6.12)</td><td rowspan=\"1\" colspan=\"1\">1.54 (0.47-5.05)</td><td rowspan=\"1\" colspan=\"1\">0.29</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">4.29 (0.87-20.97)</td><td rowspan=\"1\" colspan=\"1\">1.98 (0.35-11.16)</td><td rowspan=\"1\" colspan=\"1\">5.68 (1.18-27.25)</td><td rowspan=\"1\" colspan=\"1\">3.81 (0.75-19.29)</td><td rowspan=\"1\" colspan=\"1\">0.11</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Model ɪɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.06 (0.30-3.80)</td><td rowspan=\"1\" colspan=\"1\">0.83 (0.21-3.30)</td><td rowspan=\"1\" colspan=\"1\">2.02 (0.63-6.46)</td><td rowspan=\"1\" colspan=\"1\">1.36 (0.41-4.57)</td><td rowspan=\"1\" colspan=\"1\">0.33</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">3.68 (0.72-18.64)</td><td rowspan=\"1\" colspan=\"1\">2.007 (0.34-11.58)</td><td rowspan=\"1\" colspan=\"1\">5.89 (1.19-28.99)</td><td rowspan=\"1\" colspan=\"1\">3.26 (0.62-17.05)</td><td rowspan=\"1\" colspan=\"1\">0.54</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Hypertriglyceridemia^h\n</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Crude</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.82 (0.64-1.05)</td><td rowspan=\"1\" colspan=\"1\">0.96 (0.76-1.22)</td><td rowspan=\"1\" colspan=\"1\">0.95 (0.75-1.21)</td><td rowspan=\"1\" colspan=\"1\">0.82 (0.64-1.04)</td><td rowspan=\"1\" colspan=\"1\">0.35</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.80 (0.63-1.03)</td><td rowspan=\"1\" colspan=\"1\">0.88 (0.68-1.12)</td><td rowspan=\"1\" colspan=\"1\">1.01 (0.79-1.29)</td><td rowspan=\"1\" colspan=\"1\">0.80 (0.63-1.03)</td><td rowspan=\"1\" colspan=\"1\">0.49</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Model ɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.80 (0.53-1.20)</td><td rowspan=\"1\" colspan=\"1\">0.91 (0.62-1.34)</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.70-1.51)</td><td rowspan=\"1\" colspan=\"1\">1.06 (0.72-1.56)</td><td rowspan=\"1\" colspan=\"1\">0.45</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.84 (0.56-1.27)</td><td rowspan=\"1\" colspan=\"1\">0.96 (0.65-1.44)</td><td rowspan=\"1\" colspan=\"1\">1.05 (0.70-1.57)</td><td rowspan=\"1\" colspan=\"1\">1.25 (0.74-1.87)</td><td rowspan=\"1\" colspan=\"1\">0.13</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Model ɪɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.72 (0.47-1.10)</td><td rowspan=\"1\" colspan=\"1\">0.89 (0.59-1.34)</td><td rowspan=\"1\" colspan=\"1\">0.96 (0.64-1.43)</td><td rowspan=\"1\" colspan=\"1\">0.97 (0.65-1.45)</td><td rowspan=\"1\" colspan=\"1\">0.68</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.79 (0.51-1.21)</td><td rowspan=\"1\" colspan=\"1\">0.95 (0.63-1.44)</td><td rowspan=\"1\" colspan=\"1\">0.99 (0.65-1.50)</td><td rowspan=\"1\" colspan=\"1\">1.13 (0.75-1.70)</td><td rowspan=\"1\" colspan=\"1\">0.31</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Women</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Range</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">-25.76, -14.92</td><td rowspan=\"1\" colspan=\"1\">-14.89, -6.63</td><td rowspan=\"1\" colspan=\"1\">-6.53, 1.90</td><td rowspan=\"1\" colspan=\"1\">&gt;1.90</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">&lt;29.77</td><td rowspan=\"1\" colspan=\"1\">29.80, 37.26</td><td rowspan=\"1\" colspan=\"1\">37.31, 44.32</td><td rowspan=\"1\" colspan=\"1\">44.33, 53.52</td><td rowspan=\"1\" colspan=\"1\">&gt;53.54</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">n</td><td rowspan=\"1\" colspan=\"1\">633</td><td rowspan=\"1\" colspan=\"1\">634</td><td rowspan=\"1\" colspan=\"1\">633</td><td rowspan=\"1\" colspan=\"1\">634</td><td rowspan=\"1\" colspan=\"1\">633</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">702</td><td rowspan=\"1\" colspan=\"1\">628</td><td rowspan=\"1\" colspan=\"1\">628</td><td rowspan=\"1\" colspan=\"1\">613</td><td rowspan=\"1\" colspan=\"1\">596</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Metabolic syndrome</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Crude</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.79 (0.59-1.06)</td><td rowspan=\"1\" colspan=\"1\">0.82 (0.62-1.10)</td><td rowspan=\"1\" colspan=\"1\">0.88 (0.66-1.16)</td><td rowspan=\"1\" colspan=\"1\">0.92 (0.69-1.22)</td><td rowspan=\"1\" colspan=\"1\">0.81</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.86 (0.64-1.14)</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.78-1.36)</td><td rowspan=\"1\" colspan=\"1\">0.91 (0.69-1.22)</td><td rowspan=\"1\" colspan=\"1\">1.01 (0.76-1.34)</td><td rowspan=\"1\" colspan=\"1\">0.8</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Model ɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.88 (0.56-1.37)</td><td rowspan=\"1\" colspan=\"1\">1.17 (0.77-1.79)</td><td rowspan=\"1\" colspan=\"1\">1.29 (0.84-2.001)</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.66-1.62)</td><td rowspan=\"1\" colspan=\"1\">0.37</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.77 (0.49-1.20)</td><td rowspan=\"1\" colspan=\"1\">0.92 (0.59-1.44)</td><td rowspan=\"1\" colspan=\"1\">1.09 (0.71-1.67)</td><td rowspan=\"1\" colspan=\"1\">0.89 (0.58-1.38)</td><td rowspan=\"1\" colspan=\"1\">0.82</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Model ɪɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.76 (0.48-1.22)</td><td rowspan=\"1\" colspan=\"1\">0.91 (0.57-1.44)</td><td rowspan=\"1\" colspan=\"1\">1.07 (0.68-1.68)</td><td rowspan=\"1\" colspan=\"1\">0.88 (0.56-1.38)</td><td rowspan=\"1\" colspan=\"1\">0.89</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.87 (0.55-1.39)</td><td rowspan=\"1\" colspan=\"1\">1.18 (0.76-1.83)</td><td rowspan=\"1\" colspan=\"1\">1.34 (0.85-2.11)</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.65-1.64)</td><td rowspan=\"1\" colspan=\"1\">0.37</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Central obesity</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Crude</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.06 (0.83-1.36)</td><td rowspan=\"1\" colspan=\"1\">0.91 (0.71-1.16)</td><td rowspan=\"1\" colspan=\"1\">0.74 (0.59-0.94)</td><td rowspan=\"1\" colspan=\"1\">0.98 (0.77-1.25)</td><td rowspan=\"1\" colspan=\"1\">0.15</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.94 (0.74-1.19)</td><td rowspan=\"1\" colspan=\"1\">0.92 (0.72-1.17)</td><td rowspan=\"1\" colspan=\"1\">0.68 (0.54-0.86)</td><td rowspan=\"1\" colspan=\"1\">0.89 (0.70-1.14)</td><td rowspan=\"1\" colspan=\"1\">0.04</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Model ɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.14 (0.76-1.71)</td><td rowspan=\"1\" colspan=\"1\">0.78 (0.42-1.17)</td><td rowspan=\"1\" colspan=\"1\">0.87 (0.59-1.28)</td><td rowspan=\"1\" colspan=\"1\">1.29 (0.86-1.96)</td><td rowspan=\"1\" colspan=\"1\">0.69</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.74 (0.50-1.09)</td><td rowspan=\"1\" colspan=\"1\">0.81 (0.55-1.21)</td><td rowspan=\"1\" colspan=\"1\">0.78 (0.52-1.16)</td><td rowspan=\"1\" colspan=\"1\">0.97 (0.64-1.47)</td><td rowspan=\"1\" colspan=\"1\">0.87</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Model ɪɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.61 (0.34-1.09)</td><td rowspan=\"1\" colspan=\"1\">0.70 (0.38-1.28)</td><td rowspan=\"1\" colspan=\"1\">0.83 (0.45-1.52)</td><td rowspan=\"1\" colspan=\"1\">1.11 (0.60-2.06)</td><td rowspan=\"1\" colspan=\"1\">0.58</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.55-1.91)</td><td rowspan=\"1\" colspan=\"1\">0.59 (0.31-1.11)</td><td rowspan=\"1\" colspan=\"1\">0.88 (0.47-1.64)</td><td rowspan=\"1\" colspan=\"1\">1.48 (0.79-2.78)</td><td rowspan=\"1\" colspan=\"1\">0.4</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Hypertension</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Crude</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.74 (0.54-1.03)</td><td rowspan=\"1\" colspan=\"1\">0.80 (0.58-1.10)</td><td rowspan=\"1\" colspan=\"1\">0.95 (0.70-1.29)</td><td rowspan=\"1\" colspan=\"1\">0.87 (0.64-1.19)</td><td rowspan=\"1\" colspan=\"1\">0.88</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.66 (0.48-0.91)</td><td rowspan=\"1\" colspan=\"1\">0.76 (0.56-1.04)</td><td rowspan=\"1\" colspan=\"1\">0.95 (0.70-1.28)</td><td rowspan=\"1\" colspan=\"1\">0.81 (0.59-1.11)</td><td rowspan=\"1\" colspan=\"1\">0.75</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Model ɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.68 (0.40-1.17)</td><td rowspan=\"1\" colspan=\"1\">0.80 (0.47-1.35)</td><td rowspan=\"1\" colspan=\"1\">0.88 (0.53-1.47)</td><td rowspan=\"1\" colspan=\"1\">1.05 (0.64-1.72)</td><td rowspan=\"1\" colspan=\"1\">0.61</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.65 (0.38-1.11)</td><td rowspan=\"1\" colspan=\"1\">0.84 (0.50-1.40)</td><td rowspan=\"1\" colspan=\"1\">1.003 (0.60-1.65)</td><td rowspan=\"1\" colspan=\"1\">0.96 (0.58-1.59)</td><td rowspan=\"1\" colspan=\"1\">0.7</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Model ɪɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.69 (0.40-1.18)</td><td rowspan=\"1\" colspan=\"1\">0.79 (0.46-1.35)</td><td rowspan=\"1\" colspan=\"1\">0.85 (0.51-1.43)</td><td rowspan=\"1\" colspan=\"1\">1.06 (0.64-1.73)</td><td rowspan=\"1\" colspan=\"1\">0.65</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.64 (0.38-1.10)</td><td rowspan=\"1\" colspan=\"1\">0.83 (0.50-1.39)</td><td rowspan=\"1\" colspan=\"1\">0.97 (0.58-1.63)</td><td rowspan=\"1\" colspan=\"1\">0.96 (0.58-1.60)</td><td rowspan=\"1\" colspan=\"1\">0.72</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Low HDL-C</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Crude</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.12 (0.90-1.39)</td><td rowspan=\"1\" colspan=\"1\">1.12 (0.90-1.40)</td><td rowspan=\"1\" colspan=\"1\">1.13 (0.91-1.41)</td><td rowspan=\"1\" colspan=\"1\">1.19 (0.96-1.49)</td><td rowspan=\"1\" colspan=\"1\">0.14</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.07 (0.86-1.33)</td><td rowspan=\"1\" colspan=\"1\">1.24 (1.006-1.54)</td><td rowspan=\"1\" colspan=\"1\">1.06 (0.85-1.32)</td><td rowspan=\"1\" colspan=\"1\">1.26 (1.01-1.56)</td><td rowspan=\"1\" colspan=\"1\">0.06</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Model ɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.27 (0.89-1.80)</td><td rowspan=\"1\" colspan=\"1\">1.55 (1.08-2.23)</td><td rowspan=\"1\" colspan=\"1\">1.54 (1.08-2.19)</td><td rowspan=\"1\" colspan=\"1\">1.30 (0.91-1.84)</td><td rowspan=\"1\" colspan=\"1\">0.06</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.16 (0.82-1.62)</td><td rowspan=\"1\" colspan=\"1\">1.63 (1.15-2.31)</td><td rowspan=\"1\" colspan=\"1\">1.46 (1.02-2.08)</td><td rowspan=\"1\" colspan=\"1\">1.42 (1.001-2.03)</td><td rowspan=\"1\" colspan=\"1\">0.01</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Model ɪɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.27 (0.89-1.81)</td><td rowspan=\"1\" colspan=\"1\">1.58 (1.09-2.28)</td><td rowspan=\"1\" colspan=\"1\">1.56 (1.10-2.23)</td><td rowspan=\"1\" colspan=\"1\">1.30 (0.91-1.86)</td><td rowspan=\"1\" colspan=\"1\">0.06</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.18 (0.83-1.66)</td><td rowspan=\"1\" colspan=\"1\">1.66 (1.17-2.35)</td><td rowspan=\"1\" colspan=\"1\">1.50 (1.05-2.15)</td><td rowspan=\"1\" colspan=\"1\">1.44 (1.009-2.06)</td><td rowspan=\"1\" colspan=\"1\">0.01</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Hyperglycemia</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Crude</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.75 (0.39-1.46)</td><td rowspan=\"1\" colspan=\"1\">1.05 (0.57-1.93)</td><td rowspan=\"1\" colspan=\"1\">1.002 (0.54-1.85)</td><td rowspan=\"1\" colspan=\"1\">0.86 (0.45-1.62)</td><td rowspan=\"1\" colspan=\"1\">0.95</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.92 (0.49-1.70)</td><td rowspan=\"1\" colspan=\"1\">0.87 (0.46-1.63)</td><td rowspan=\"1\" colspan=\"1\">1.15 (0.64-2.08)</td><td rowspan=\"1\" colspan=\"1\">0.76 (0.39-1.48)</td><td rowspan=\"1\" colspan=\"1\">0.73</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Model ɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.05 (0.68-1.62)</td><td rowspan=\"1\" colspan=\"1\">0.99 (0.63-1.55)</td><td rowspan=\"1\" colspan=\"1\">1.23 (0.81-1.90)</td><td rowspan=\"1\" colspan=\"1\">1.37 (0.90-2.08)</td><td rowspan=\"1\" colspan=\"1\">0.7</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.93 (0.35-2.46)</td><td rowspan=\"1\" colspan=\"1\">0.79 (0.28-2.23)</td><td rowspan=\"1\" colspan=\"1\">1.64 (0.67-4.01)</td><td rowspan=\"1\" colspan=\"1\">0.44 (0.13-1.48)</td><td rowspan=\"1\" colspan=\"1\">0.6</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Model ɪɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.44 (0.14-1.37)</td><td rowspan=\"1\" colspan=\"1\">0.84 (0.32-2.24)</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.41-2.55)</td><td rowspan=\"1\" colspan=\"1\">0.48 (0.16-1.04)</td><td rowspan=\"1\" colspan=\"1\">0.6</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.95 (0.35-2.56)</td><td rowspan=\"1\" colspan=\"1\">0.80 (0.28-2.30)</td><td rowspan=\"1\" colspan=\"1\">1.60 (0.64-4.01)</td><td rowspan=\"1\" colspan=\"1\">0.42 (0.12-1.45)</td><td rowspan=\"1\" colspan=\"1\">0.14</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Hypertriglyceridemia</td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Crude</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.79-1.35)</td><td rowspan=\"1\" colspan=\"1\">0.87 (0.67-1.15)</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.79-1.35)</td><td rowspan=\"1\" colspan=\"1\">1.14 (0.88-1.48)</td><td rowspan=\"1\" colspan=\"1\">0.37</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.99 (0.76-1.29)</td><td rowspan=\"1\" colspan=\"1\">1.08 (0.83-1.40)</td><td rowspan=\"1\" colspan=\"1\">1.03 (0.79-1.35)</td><td rowspan=\"1\" colspan=\"1\">1.20 (0.93-1.56)</td><td rowspan=\"1\" colspan=\"1\">0.17</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Model ɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">0.44 (0.14-1.36)</td><td rowspan=\"1\" colspan=\"1\">0.86 (0.33-2.27)</td><td rowspan=\"1\" colspan=\"1\">1.10 (0.45-2.70)</td><td rowspan=\"1\" colspan=\"1\">0.50 (0.17-1.45)</td><td rowspan=\"1\" colspan=\"1\">0.1</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.07 (0.69-1.65)</td><td rowspan=\"1\" colspan=\"1\">1.34 (0.88-2.05)</td><td rowspan=\"1\" colspan=\"1\">1.41 (0.91-2.18)</td><td rowspan=\"1\" colspan=\"1\">1.54 (1.007-2.36)</td><td rowspan=\"1\" colspan=\"1\">0.02</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Model ɪɪ</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.04 (0.67-1.62)</td><td rowspan=\"1\" colspan=\"1\">0.98 (0.62-1.55)</td><td rowspan=\"1\" colspan=\"1\">1.24 (0.80-1.91)</td><td rowspan=\"1\" colspan=\"1\">1.37 (0.89-2.09)</td><td rowspan=\"1\" colspan=\"1\">0.1</td><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">1.07 (0.69-1.66)</td><td rowspan=\"1\" colspan=\"1\">1.35 (0.88-2.07)</td><td rowspan=\"1\" colspan=\"1\">1.45 (0.93-2.25)</td><td rowspan=\"1\" colspan=\"1\">1.55 (1.01-2.40)</td><td rowspan=\"1\" colspan=\"1\">0.01</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Seyede Hamide Rajaie, Sayyed Saeid Khayyatzadeh, Shiva Faghih, Yaser Mansoori, Mohammad Mehdi Naghizadeh, Mojtaba Farjam, Reza Homayounfar, Hassan Mozaffari-Khosravi</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Seyede Hamide Rajaie, Sayyed Saeid Khayyatzadeh, Shiva Faghih, Yaser Mansoori, Mohammad Mehdi Naghizadeh, Mojtaba Farjam, Reza Homayounfar, Hassan Mozaffari-Khosravi</p><p><bold>Drafting of the manuscript:</bold>  Seyede Hamide Rajaie, Sayyed Saeid Khayyatzadeh, Shiva Faghih, Yaser Mansoori, Mohammad Mehdi Naghizadeh, Mojtaba Farjam, Reza Homayounfar, Hassan Mozaffari-Khosravi</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Seyede Hamide Rajaie, Sayyed Saeid Khayyatzadeh, Shiva Faghih, Yaser Mansoori, Mohammad Mehdi Naghizadeh, Mojtaba Farjam, Reza Homayounfar, Hassan Mozaffari-Khosravi</p><p><bold>Supervision:</bold>  Seyede Hamide Rajaie, Sayyed Saeid Khayyatzadeh, Shiva Faghih, Yaser Mansoori, Mohammad Mehdi Naghizadeh, Mojtaba Farjam, Reza Homayounfar, Hassan Mozaffari-Khosravi</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study. Ethics Committee of Shahid Sadoughi University of Medical Sciences, Yazd, Iran and Ethics Committee of Fasa University of Medical Sciences, Fasa, Iran issued approval IR.SSU.SPH.REC.1399.168 and IR.FUMS.REC.1399.145. This study was approved by the Ethics Committee of Shahid Sadoughi University of Medical Sciences, Yazd, Iran (Ethics code: IR.SSU.SPH.REC.1399.168) and the Ethics Committee of Fasa University of Medical Sciences, Fasa, Iran (Ethics code: IR.FUMS.REC.1399.145). All methods in this study were used in accordance with the Declaration of Helsinki, and all participants provided their informed consent in writing.</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Animal Ethics</title><fn fn-type=\"other\"><p><bold>Animal subjects:</bold> All authors have confirmed that this study did not involve animal subjects or tissue.</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Mucopolysaccharidoses (MPSs) are a subgroup of inherited lysosomal storage disorders caused by the deficiency of specific lysosomal enzymes on the glycosaminoglycans (GAG) catabolism pathway. The consequent storage of GAG in tissues leads to progressive multisystemic damage [##REF##30057281##1##].</p>", "<p>Mucopolysaccharidosis type I (MPS I) is an autosomal recessive disorder caused by alpha-L-iduronidase deficiency. This enzyme catalyzes the degradation of the GAG dermatan and heparan sulphate. Therefore, pathological accumulation of both GAGs occurs in patients with MPS I, with manifestations in multiple organs. This disorder has traditionally been divided into three syndromes, namely, Hurler syndrome (severe form), Hurler-Scheie syndrome (moderate form), and Scheie syndrome (mild form). However, phenotypes are present on a spectrum of severity, no biochemical differences have been identified, and clinical findings overlap. Currently, affected individuals are better divided into severe (Hurler) and attenuated (Hurler-Scheie, Scheie) forms as this distinction influences therapeutic options [##REF##33572941##2##].</p>", "<p>MPS I is a life-threatening condition with severe disease burden and premature death. Without treatment, children presenting with the severe form usually die in the first decade of life with multisystemic disease, which also affects the brain [##REF##18796143##3##]. At birth, the neonate appears healthy. Early symptoms are non-specific, such as recurrent respiratory tract infections, umbilical and inguinal hernias, thoracolumbar kyphosis, and hepatosplenomegaly. Coarse facial features may not become distinguishable until after one year of age. More specific symptoms beginning in the first year include impaired hearing and vision, progressive skeletal dysplasia, followed by neurodevelopmental delay, and cardiorespiratory disease [##REF##32764324##4##]. In patients with the attenuated form, symptoms usually emerge after three years of age and range in severity and rate of progression.</p>", "<p>Early diagnosis allows timely treatment that aims to slow disease progression and improve quality of life. Currently approved treatments for the prevention of primary manifestations include enzyme replacement therapy (ERT) and hematopoietic stem cell transplantation (HSCT). The recommended treatment for patients with the severe form is HSCT, and patients with the attenuated form are often treated with ERT alone [##REF##19117856##5##].</p>", "<p>This work presents the case of a child with severe MPS I with classic clinical features with early diagnosis that allowed HSCT and slowed disease progression.</p>" ]

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[ "<title>Discussion</title>", "<p>MPS I has an estimated prevalence of 0.69 to 3.8 per 100,000 live births [##REF##16435194##7##]. At birth, the neonate can look healthy. Early symptoms include progressive coarse facial features, recurrent rhinitis, upper air obstruction, abdominal or inguinal hernias, thoracolumbar kyphosis, and hepatosplenomegaly. Upper air obstruction can result in sleep apnea and respiratory tract and ear infections [##REF##23151682##8##]. Cardiac involvement includes thickened cardiac valves, which can lead to poor mobility, regurgitation, or stenosis [##REF##21744090##9##]. Dysostosis multiplex is also one of the more common manifestations [##UREF##1##10##]. Spine involvement includes abnormalities affecting the intervertebral discs, vertebrae, odontoid process, and dura, causing spinal deformity and cord compression [##REF##27296532##11##]. Central nervous system involvement can include white matter injury, enlargement of perivascular spaces, hydrocephalus, brain atrophy, characteristic enlargement of the subarachnoid spaces, and compressive myelopathy, among other manifestations [##REF##29752520##12##]. Neurodevelopment can become impaired in the first year of life, with progressive decline after the second year [##REF##29751845##13##]. Ophthalmic manifestations include corneal clouding, ocular hypertension/glaucoma, retinal degeneration, optic nerve swelling/atrophy, refractive errors, and ocular motility abnormalities [##REF##16414358##14##,##REF##31540112##15##].</p>", "<p>The child described in this report presented with typical clinical manifestations, namely, multiple hernias, recurrent otitis media, rhinitis, wheezing, coarse facial features, craniosynostosis, thoracolumbar kyphosis, hepatomegaly, dysplastic mitral valve with insufficiency, aortic insufficiency, and mild corneal opacity. These features led to MPS suspicion before two years of age.</p>", "<p>Urinary GAG analysis should be conducted upon suspicion as a fast and efficient method to screen for the disease. Enzyme testing (measurement of alfa-L-iduronidase activity) is the gold standard for diagnosis and, in HSCT-treated patients, is paired with urinary GAG for treatment efficacy monitoring [##REF##30057281##1##]. Molecular diagnosis is the confirmatory test. The alpha-I-iduronidase gene has been mapped to chromosome 4p16.3. The c.1205G&gt;A (p.Trp402Ter), c.208C&gt;T (p.Gln70Ter), and c.1598C&gt;G (p.Pro533Arg) mutations are mainly responsible for the disease in the European population [##REF##2220820##16##, ####REF##1301196##17##, ##REF##1505961##18####1505961##18##]. Our patient’s genetic study identified two of the most common mutations in Europe.</p>", "<p>According to the European consensus (2011) [##REF##21831279##19##], the preferred treatment strategy is HSCT for patients with MPS diagnosed before 2.5 years, with a development quotient over 70.</p>", "<p>ERT, which must be administrated intravenously every week, is ineffective in preventing neurocognitive and behavioral impairment as the recombinant enzyme does not cross the blood-brain barrier. Nevertheless, it remains the best option for late-presenting or late-diagnosed patients as it improves some of the life-threatening manifestations, such as respiratory impairment. Although it is a high-risk procedure, with non-negligible mortality and morbidity, early HSCT may prevent cognitive decline and improve other clinical manifestations of the disease with a better quality of life in the long term.</p>", "<p>However, how early is early enough? On retrospective analysis of our patient’s history, we can conclude that it took too long before the MPS I diagnosis and treatment were assumed: from 15 months of age to 29 months (ERT) and 33 months (HSCT). She was transplanted over the recommended age limit, according to European Guidelines [##REF##21831279##19##]. Although neurologic disease progression is assumedly better after HSCT than on ERT alone, she presented some intellectual and adaptive impairment, which might have been prevented with an earlier diagnosis. Training of healthcare professionals and dissemination among the public are needed to valorize the first signs of the disease as when the complete picture is present, diagnosis is easy but late.</p>", "<p>Neonatal screening of MPS I is being advocated based on the knowledge that GAG accumulation is prenatal and progressive and on the improved prognosis of MPS I younger siblings submitted to earlier treatment [##REF##36412587##20##]. Presently, it is not included in the Portuguese Newborn Screening Program.</p>", "<p>Currently, the only way to treat MPS I patients adequately is through improving MPS awareness among healthcare professionals.</p>" ]

[ "<title>Conclusions</title>", "<p>MPS I is an inherited autosomal recessive disorder with a multisystemic (including neurological) progressive course. Clinicians may suspect this diagnosis in children with coarse features, abdominal/inguinal hernias, recurrent upper respiratory infections, or dorsolumbar kyphosis. The Portuguese FIND Project provides physicians easy and quick access to MPS I enzyme diagnosis in a dried blood spot sample. Early diagnosis and treatment with HSCT can slow disease progression and improve quality of life. In this context, neonatal screening for MPS I may be helpful.</p>" ]

[ "<p>Mucopolysaccharidoses are rare lysosomal storage disorders in which glycosaminoglycans accumulate in tissues, causing multiorgan dysfunction. Mucopolysaccharidosis type I is an autosomal recessive disease caused by a deficiency of the enzyme alpha-L-iduronidase, resulting in the accumulation of dermatan and heparan sulfate. Early diagnosis is crucial for early treatment and improved outcomes.</p>", "<p>We report the case of a female child with classic clinical features who was diagnosed early which allowed hematopoietic stem cell transplantation and slowed disease progression. She presented at birth with linea alba and umbilical and inguinal hernias. Since the first months of life, she had recurrent respiratory infections. At nine months, a motor delay was noticed, and at 20 months, craniosynostosis was corrected with surgery. Coarse facial features, thoracolumbar kyphosis, and hepatomegaly prompted a urinary glycosaminoglycan study at 22 months, which showed elevated levels. Alfa-L-iduronidase activity in dried blood spot testing was low, compatible with mucopolysaccharidosis type I. Molecular testing of gene <italic>IDUA, </italic>performed for genetic counseling, revealed the pathogenic variants c.1205G&gt;A (p.Trp402Ter) and c.1598C&gt;G (p.Pro533Arg) in compound heterozygosity. At 26 months, her development quotient was average for her age. She started enzyme replacement therapy at 29 months and underwent hematopoietic stem cell transplantation at 33 months, which softened the coarse features, reduced respiratory infections, and improved hepatomegaly. However, at age five, her development quotient was 76 (mean = 100, standard deviation = 15). This intellectual impairment might have been prevented with an earlier diagnosis and treatment.</p>" ]

[ "<title>Case presentation</title>", "<p>In 2017, a 15-month-old girl was referred to a neuropediatric consultation because of motor delay. She was the first child of healthy, non-consanguineous parents. The gestation was uneventful. Linea alba and umbilical and bilateral inguinal hernias were noticed at birth. Surgical correction of the right inguinal hernia was performed at two months old (the left one had spontaneous resolution). By nine months old, she was unable to sit without support, raising concerns about neurodevelopment. Since the first months of life, she had recurrent otitis media, rhinitis, and wheezing.</p>", "<p>Clinical findings</p>", "<p>Physical examination showed coarse facial features (prominent supraorbital ridges, hypertelorism, broad nose, and thick lips) (Figure ##FIG##0##1##), dolichocephaly (craniosynostosis suspicion), mild thoracolumbar kyphosis, hepatomegaly, and umbilical hernia. The remaining examination was otherwise normal.</p>", "<p>The evolution of weight and length was in the 15th-50th percentile, and the cephalic perimeter was in the 50th-85th percentile.</p>", "<p>At 18 months, a computed cranial tomography showed a sagittal and lambdoid craniosynostosis, with increased anterior-posterior diameter (Figure ##FIG##1##2##). At 20 months, she underwent a cranial vault remodeling.</p>", "<p>Diagnostic assessment</p>", "<p>MPS was suspected at 22 months old, and the urinary GAG study showed elevated levels at 308.2 mg/L (normal range = 39.5-51.5 mg/L).</p>", "<p>The child was then referred to Genetic and Inherited Metabolic Disorders consultations. Alfa-L-iduronidase activity in dried blood spot testing (FIND Project [##UREF##0##6##]) was low: 0.01 nmol/h/spot (normal range = 0.1-0.9 nmol/h/spot), suggestive of MPS I.</p>", "<p>Molecular testing of the gene <italic>IDUA </italic>revealed the nonsense pathogenic variant c.1205G&gt;A (p.Trp402Ter) in exon 9 and the missense pathogenic variant c.1598C&gt;G (p.Pro533Arg) in exon 11, in compound heterozygosity.</p>", "<p>Further evaluations at diagnosis and during evolution were performed according to MPS guidelines [##REF##19117856##5##] and are presented in Table ##TAB##0##1##. Figure ##FIG##2##3## shows spine radiography performed at 27 months.</p>", "<p>Neurodevelopment evaluation with Griffiths Mental Development Scale and adaptive behavior with Vineland Adaptive Behaviour Scale at 26 months old was average for age (Figure ##FIG##3##4##).</p>", "<p>Therapeutic intervention</p>", "<p>The child started intravenous ERT with laronidase (Aldurazyme®) at 100 U/kg/dose weekly at 29 months old. Additionally, she was proposed for HSCT, which occurred at 33 months old. Full-donor chimerism was achieved four months after HSCT. She maintained ERT weekly until six months after the HSCT. The evolution of alfa-L-iduronidase activity levels and urinary GAG is shown in Figure ##FIG##4##5##.</p>", "<p>The main complications after HSCT were grade II graft-versus-host disease (cutaneous involvement, Figure ##FIG##5##6##), autoimmune hemolytic anemia, reinfections of cytomegalovirus and Epstein-Barr virus, hypogammaglobinaemia secondary to immunosuppression, and iron deficiency. All resolved with standard therapy (corticoids, immunoglobulin, antivirals, monoclonal antibody, and iron supplementation).</p>", "<p>Currently, she is seven years old, maintains follow-up by a multidisciplinary team, and is clinically stable. The coarse features softened (Figure ##FIG##6##7##), and hepatomegaly improved (Figure ##FIG##7##8##). Although upper airway infections reduced in number, she was submitted to adenoidectomy and myringotomy at the age of seven due to chronic suppurative otitis (Table ##TAB##0##1##).</p>", "<p>The last neurodevelopment evaluation at five years of age showed a Griffiths general quotient of 76 (average = 100, standard deviation = 15) and a Vineland’s adaptative behavior composite score of 61 (average = 100, standard deviation = 15) (Figure ##FIG##8##9##). She is in regular school with education support.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Coarse facial features and thoracolumbar kyphosis at 15 months.</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>Cranial computed tomography performed at 18 months.</title><p>(A) Increased anteriorposterior diameter (dolichocephaly). Also, left temporoparietal suture has decreased patency. (B) Sagittal and lambdoid sutures are closed bilaterally.</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG3\"><label>Figure 3</label><caption><title>Spine radiography at 27 months (lateral view).</title><p>Absence of natural spine curves, with a mid-thoracolumbar kyphosis (mostly positional).</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG4\"><label>Figure 4</label><caption><title>Griffiths Scale of Child Development and Vineland Adaptive Behaviour Scale at 26 months old (medium 100 = standard deviation = 15).</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG5\"><label>Figure 5</label><caption><title>Evolution of alfa-L-iduronidase (IDUA) activity in leucocytes and urinary glycosaminoglycans (GAG).</title><p>IDUA activity normal range levels are 53-105 nmol/h/mg protein (y axis). Urinary GAG normal range levels are 4-11 mg/mmol creatinine (y axis).</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG6\"><label>Figure 6</label><caption><title>Cutaneous graft-versus-host disease. An umbilical hernia can also be noticed.</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG7\"><label>Figure 7</label><caption><title>Photograph at age five showing softening of coarse features.</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG8\"><label>Figure 8</label><caption><title>Liver ultrasound at 28 months (A) and 44 months (B).</title><p>(A) Hepatomegaly at 28 months, with right lobe sizing 118.8 mm. (B) Right lobe with 101.7 mm, showing liver size reduction at 44 months.</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG9\"><label>Figure 9</label><caption><title>Griffiths Scale of Child Development and Vineland Adaptive Behaviour Scale at 60 months old (medium = 100, standard deviation = 15).</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Clinical manifestations, complementary evaluations, and treatment during follow-up.</title><p>According to mucopolysaccharidosis type I guidelines for management and treatment [##REF##19117856##5##], patients should receive a comprehensive baseline evaluation, including neurologic, ophthalmologic, auditory, cardiac, respiratory, gastrointestinal, and musculoskeletal assessments, with monitoring every 6-12 months, by individualized specialty assessments.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Consultation</td><td rowspan=\"1\" colspan=\"1\">Clinical manifestations and treatment</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Pneumology</td><td rowspan=\"1\" colspan=\"1\">Recurrent rhinitis and wheezing, snoring – inhaled and nasal corticosteroids</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Cardiology</td><td rowspan=\"1\" colspan=\"1\">No symptoms – normal electrocardiography; echocardiogram: dysplastic mitral valve with moderate insufficiency and moderate aortic insufficiency – medicated with lisinopril</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Surgery</td><td rowspan=\"1\" colspan=\"1\">Mild umbilical hernia – corrective surgery at 40 months</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Neurosurgery</td><td rowspan=\"1\" colspan=\"1\">No need for further neurosurgeries</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Orthopedics</td><td rowspan=\"1\" colspan=\"1\">Bilateral carpal tunnel syndrome – corrective surgery at the age of six</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ophthalmology</td><td rowspan=\"1\" colspan=\"1\">No symptoms – mild right corneal opacity, normal fundus evaluation, and normal intraocular pressure</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Otorhinolaryngology</td><td rowspan=\"1\" colspan=\"1\">Recurrent rhinitis, otitis media (normal audiometry and tympanogram) – myringotomy at 28 months and adenotonsillectomy at 30 months old, repeated adenoidectomy and myringotomy at the age of seven due to chronic suppurative otitis</td></tr><tr style=\"background-color:#ccc\"><td colspan=\"2\" rowspan=\"1\">Exam evaluations</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Abdominal echography (28 months, 32 months and 44 months)</td><td rowspan=\"1\" colspan=\"1\">At 28 months, mild hepatomegaly, right lobe with 120 mm, inferior vena cava plane 120 mm, and mild splenomegaly with 88 mm. At 44 months, the right lobe was 101 mm, and the left lobe was 81 mm, with the spleen measuring 71 mm</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Musculoskeletal radiography (multiple from 15 months)</td><td rowspan=\"1\" colspan=\"1\">Incomplete fusion of C4 vertebra, beaking of thoracolumbar vertebra bodies, dorsolumbar kyphoscoliosis (Figure ##FIG##2##3##), right acetabular dysplasia, distal femur diaphysis widening</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Cranial magnetic resonance imaging (27 months)</td><td rowspan=\"1\" colspan=\"1\">Enlarged perivascular spaces in the peri trigonal regions, with hyperintense signal in DP, T2, and fluid-attenuated inversion recovery, of the white substance, involving periventricular regions, corona radiata, external and subcortical capsules, that can be secondary to gliosis, demyelination or dysmyelination. J-shaped sella turcica</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Spine magnetic resonance imaging (27 months and seven years old)</td><td rowspan=\"1\" colspan=\"1\">Odontoid hypoplasia, mild periodontal ligament thickening, reduction of the subarachnoid space in this location</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Polysomnography (32 months)</td><td rowspan=\"1\" colspan=\"1\">Moderate obstructive sleep apnea syndrome</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Thoracic computed tomography (seven years old)</td><td rowspan=\"1\" colspan=\"1\">Bronchi walls parietal thickening with inferior lobe predominance. Bilateral densification of medium lobe and lingula, compatible with atelectases</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Rui Diogo, Luísa Diogo, Alexandra Oliveira</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Rui Diogo, Luísa Diogo, Rute Serra, Joana Almeida, Alexandra Oliveira</p><p><bold>Drafting of the manuscript:</bold>  Rui Diogo, Luísa Diogo, Alexandra Oliveira</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Rui Diogo, Luísa Diogo, Rute Serra, Joana Almeida, Alexandra Oliveira</p><p><bold>Supervision:</bold>  Rui Diogo, Luísa Diogo, Alexandra Oliveira</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050595-i01\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050595-i02\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050595-i03\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050595-i04\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050595-i05\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050595-i06\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050595-i07\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050595-i08\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050595-i09\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Tuberculosis is a major health problem, with 10.6 million new cases in 2021 according to the Global Tuberculosis Report 2022. Paradoxical deterioration upon tuberculosis chemotherapy, known as tuberculosis-immune reconstitution inflammatory syndrome (TB-IRIS), is a well-described entity in human immunodeficiency virus (HIV) patients after receiving antiretroviral therapy (ART) [##REF##18652998##1##]. The pathology of TB-IRIS is attributed to cytokine release, specifically that of interleukin 6 and tumor necrosis factor (TNF) [##REF##20817712##2##]. A similar reaction exists in non-HIV patients, first described in 1954 [##REF##19535147##3##]. Paradoxical reaction (PR) is frequent in non-HIV extrapulmonary tuberculosis, occurring in 20%-25% of patients, and has an excellent prognosis except in central nervous system (CNS) involvement [##REF##19535147##3##,##REF##23203899##4##]. Pathogenesis is possibly secondary hypersensitivity to tuberculoprotein and a decrease in immune suppression [##REF##12461590##5##,##REF##601870##6##].</p>", "<p>PR/IRIS is a diagnosis of exclusion, requiring the elimination of drug toxicity or reaction, tuberculosis treatment failure caused by resistance, another infection, and poor adherence to treatment [##REF##12461590##5##]. Indication for therapy depends on both severity and presentation. Corticosteroid therapy has proven effective in the context of constrictive pericarditis and meningitis and for the prevention of TB-IRIS in high-risk patients upon ART treatment [##REF##15496623##7##, ####REF##2891992##8##, ##REF##31042844##9####31042844##9##]. In the absence of life-threatening conditions, corticosteroids have been anecdotally used to relieve symptoms and nodes or abscess compression, but with a lack of controlled trials [##REF##22752438##10##, ####REF##17488505##11##, ##REF##23107635##12####23107635##12##].</p>", "<p>Anti-TNF is a potential treatment in cases of PR/IRIS resistant to corticosteroids, especially in CNS involvement. In a recent review of the usage of anti-TNF in PR/IRIS, 24 cases were identified. Despite being used as a second-line treatment and as a salvage therapy in severe cases, the rate of improvement was 100% [##REF##36795280##13##]. Here, we present a case report emphasizing the potential benefit of TNF antagonism in PR.</p>" ]

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[ "<title>Discussion</title>", "<p>The patient was initially treated for acute miliary tuberculosis complicated by multiorgan system failure (i.e., shock, respiratory, and renal failure). The initial sites of involvement were the lungs, lymphatic system, liver, and urinary system. Brain extension was not explored at that time. At presentation, the patient had no neurological manifestation, which was first noted after one month of treatment. Although the paradoxical progression of the intracranial tuberculoma on antituberculosis treatment could not be well evaluated due to the absence of an anterior exam, we think that the patient could fall into this category because of the absence of neurological manifestation at presentation and its appearance on antituberculosis treatment.</p>", "<p>Despite intensified tuberculosis therapy and the reintroduction of corticosteroids, there was no improvement in neurological symptoms. Anti-TNF therapy (infliximab) was initiated on day 10, leading to positive progression, extubation, and transfer to the regular floor after two months.</p>", "<p>Whether it was a worsening of a preexisting lesion or an appearance of a new one, this progression could not be explained by a subtherapeutic treatment due to close surveillance of the pharmacodynamics of antituberculosis treatment, nor by resistance, as the organism was sensitive. In several studies on this form of PR, surgery had no role, and prolonged antituberculosis therapy along with corticosteroids was the standard of care [##REF##7616286##14##,##REF##9143584##15##]. Unfortunately, our patient did not improve on the same standard of care. Here, the significance of anti-TNF therapy comes into play as salvage therapy. In a literature review conducted by Armange et al., all 24 patients on TNF-α antagonists showed improvement [##REF##36795280##13##]. Anti-TNF therapy is considered a safe and effective salvage therapy in severe PR.</p>" ]

[ "<title>Conclusions</title>", "<p>PRs warrant close attention in patients presenting with neurological manifestations during tuberculosis treatment. Anti-TNF therapy has emerged as a promising salvage option in severe cases of PR. The potential benefits of utilizing anti-TNF therapy as a first-line treatment or as a corticoid-sparing modality should be carefully evaluated to tailor therapies effectively for these patients.</p>" ]

[ "<p>We report the case of a 42-year-old immunocompetent Indian patient presenting with miliary tuberculosis complicated by respiratory failure requiring intubation. Conventional quadritherapy was initiated for wild-type <italic>Mycobacterium tuberculosis</italic>. On day 29 of antibiotic treatment, persistent fever and neurological deterioration prompted the diagnosis of multiple brain and medullary tuberculomas, some surrounded by edema. Laboratory investigations ruled out meningitis and subtherapeutic drug concentrations. To enhance cerebrospinal fluid penetration, ethambutol was replaced with levofloxacin on day 30, and rifampicin doses were increased to 30 mg/kg. Dexamethasone was introduced on day 30 to address the paradoxical response to antituberculosis therapy, but neurological deterioration persisted, leading to hemiparesis and coma, with concurrent development of acute respiratory distress syndrome. As salvage therapy, an anti-tumor necrosis factor agent, infliximab (IFX), was administered on day 40. Rapid clinical improvement was observed, marked by awakening and subsequent weaning from respiratory ventilation just eight days after the first IFX infusion. The patient was discharged from the intensive care unit 10 days post-IFX initiation, with steroids discontinued one month after IFX introduction. Both antituberculosis treatment and IFX infusions (seven in total) were maintained for one year. Clinical and radiological evaluation at one year demonstrated complete clinical and radiological recovery.</p>" ]

[ "<title>Case presentation</title>", "<p>A 42-year-old male of Indian origin, living in France since 2002, presented to our hospital with a one-month history of general status alteration. He reported asthenia and 19 kg of unintentional weight loss during an interval of less than six months. The patient sought medical care due to the appearance of shortness of breath graded on the modified Medical Research Council Dyspnea Scale. He denied experiencing night sweats, fever, cough, gastrointestinal, or genitourinary symptoms. The patient was not known to be immunocompromised, with negative HIV results, no hypogammaglobulin levels, and no history of immunosuppressive therapy or recurrent infections. No known past medical history was reported for tuberculosis or screening upon his arrival to France for latent tuberculosis.</p>", "<p>Upon arrival at the emergency department, the patient exhibited a heart rate of 143 beats per minute, an oxygen saturation of 88% on room air, a respiratory rate of 40 breaths per minute, and a temperature of 37.8°C. Physical examination revealed a normal neurological status, no palpable adenomegaly, no signs of hypoperfusion, regular heart sounds with no added sounds, bilateral basal crepitation on lung auscultation, a soft non-tender abdomen, and no hepatomegaly or splenomegaly. An electrocardiogram showed sinus tachycardia. Laboratory findings ruled out severe acute respiratory syndrome coronavirus 2, influenzae, respiratory syncytial virus, and urinary tract infection. However, elevated C-reactive protein levels were noted (50 mg/L), along with the absence of leukocytosis and an increased D-dimer concentration (6,535 ng/mL).</p>", "<p>A chest X-ray revealed diffuse bilateral opacity (Figure ##FIG##0##1##, Panel a). A computed tomography pulmonary angiogram confirmed the presence of diffuse micronodules of hematogenous spread, compatible with military tuberculosis, along with bilateral small pleural effusion. The patient was administered oxygen (6 L/minute) and transferred to the pulmonary and infectious disease floor.</p>", "<p>On day two, the patient underwent a bronchoscopy for microbiological documentation. The procedure was performed in the intensive care unit (ICU) for monitoring due to his critical condition. Unfortunately, respiratory failure progressed during the intervention, followed by shock. As a result, the patient was rapidly sedated and intubated. The relevant arterial blood gas parameters were a lactate level of 7.8 mmol/L, a pH of 7.24, a PaO₂/FiO₂ ratio of 111, and a positive end-expiratory pressure of 6 cmH₂O.</p>", "<p>The patient was placed on noradrenaline and received antibiotherapy with spiramycin, cefotaxime, linezolid, and amikacin. To enhance diagnostic yield, a lumbar puncture and a transjugular hepatic biopsy were conducted. The lumbar puncture ruled out meningitis, revealing cytology with a red blood cell count of 39/mm³ and leukocytes of 4/mm³ (no differential was performed). Glucose levels were 4.7 mmol/L, protein levels were elevated at 0.7 mmol/L, and both polymerase chain reaction (PCR) and culture results were negative. The patient’s urine direct exam was negative, but cultures returned positive after 30 hours. The hepatic biopsy revealed a non-caseating granuloma on histopathology. A workup for acquired immunodepression showed negative serology for HIV.</p>", "<p>On day three, the PCR results from bronchoalveolar lavage returned positive for tuberculosis with an absence of mutation associated with resistance to rifampicin of the <italic>rpoB</italic> gene and isoniazid of the <italic>KatG</italic> and <italic>InhA </italic>genes, whereas the lumbar puncture remained negative for tuberculosis. A quadritherapy regimen with intravenous (IV) rifampicin 600 mg (10 mg/kg), isoniazid 200 mg (3.5 mg/kg), ethambutol 1,200 mg (20 mg/kg), and per os pyrazinamide 1,750 mg (30 mg/kg) was initiated. The antibiotherapy was de-escalated to cefazoline due to the identification of methicillin-resistant <italic>Staphylococcus aureus</italic> as a probable bacterial superinfection. Adjunct corticosteroid therapy with prednisolone 1 mg/kg was considered necessary due to significant inflammation.</p>", "<p>The patient’s ICU stay was further complicated by ventilator-associated pneumonia due to <italic>Pseudomonas</italic>, drug reaction to ceftazidime, volume overload, and pneumothorax. These complications necessitated adjustments in antibiotherapy, continuous renal replacement therapy, and chest tube insertion with drainage. The patient showed improvement in his ventilator parameters, with an FiO₂% of 25% and a PaO₂/FiO₂ ratio of 305, and weaning from vasopressors.</p>", "<p>On day 29, following sedation withdrawal, the patient exhibited left hemiparesis and hyperactive deep tendon reflexes. Neuroimaging via brain and medullary magnetic resonance imaging (MRI) revealed multiple lesions compatible with tuberculoma, along with a left parietal hematoma surrounded by edema, moderate hydrocephalus with signs of resorption, arachnoiditis, and lumbar epiduritis (Figure ##FIG##0##1##, Panels b and c). A lumbar puncture indicated moderate intracranial hypertension with a pressure of 24 cmH₂O. PCR testing of his tuberculosis cerebrospinal fluid (CSF) remained negative, and a fundoscopic exam showed no papilledema. Urgent CSF shunt surgery was not indicated by neurosurgeons. The rifampicin dose was increased to 20 mg/kg/day to optimize pharmacodynamic targeting in the CSF. Corticosteroids were reintroduced, this time with dexamethasone 25 mg.</p>", "<p>An expert opinion was obtained from the Centre National de Référence des Mycobactéries, suggesting further optimization of the tuberculosis therapy. Rifampicin doses were increased to 1,800 mg (30 mg/kg), isoniazid to 300 mg (5 mg/kg), and ethambutol was switched to levofloxacin 500 mg due to better CSF penetration, with a CSF-to-serum ratio of the area under the curve of approximately 1. Additionally, infliximab 300 mg was initiated initially every 15 days as a treatment for a probable PR. Prophylaxis for opportunistic infection with atovaquone was also added.</p>", "<p>Anti-TNF therapy (infliximab) was administered after 10 days of intensifying antibiotherapy due to persistent neurological symptoms. The patient’s subsequent evolution was favorable, with successful weaning from mechanical ventilation on day 48, leading to a transfer to the regular floor after approximately two months.</p>", "<p>On the regular floor, the rifampicin dose was reduced to 20 mg/kg on day 27 of therapy, and the route of administration was switched to oral. Weaning of steroids was initiated after one month of therapy, while infliximab was continued as maintenance immunosuppressive therapy. A brain MRI performed on the ninth week of antituberculosis treatment showed regression of the hematoma and decreased contrast enhancement.</p>", "<p>After three months of hospitalization, the patient was discharged to a rehabilitation center where he underwent physiotherapy while maintaining antituberculosis therapy and infliximab perfusions. Regular follow-up consultations were conducted to monitor for PR relapse. The intensive phase of antituberculosis therapy was extended to six months (isoniazid/rifampicin/pyrazinamide/levofloxacin), followed by six additional months of isoniazid/rifampicin during the continuation phase. The total duration of antituberculosis therapy was one year, given in association with infliximab. The last consultation, conducted after a year of therapy, revealed that the patient had no neurological or respiratory symptoms. The follow-up imaging revealed the resolution of the miliary pattern on the chest X-ray (Figure ##FIG##0##1d##), the complete disappearance of multiple intracranial tuberculomas, and the chronic hematoma on the MRI (Figure ##FIG##0##1##, Panels e and f).</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Evolution of intracranial and thoracic findings: a year-long imaging assessment.</title><p>Imaging at baseline:</p><p>a. Chest X-ray reveals numerous bilateral small opacities resembling millet seeds.</p><p>b. Cerebral magnetic resonance imaging displays left median and paramedian pontine lesions.</p><p>c. Left parietal hematoma observed with perilesional edema.</p><p>Imaging after a year:</p><p>d. Resolution of miliary opacity.</p><p>e. Disappearance of pontine lesions observed.</p><p>f. Residual sequelae of the parietal hematoma and resolution of its perilesional edema.</p></caption></fig>" ]

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[ "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050596-i01\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Hidradenitis suppurativa (HS) is a chronic inflammatory skin disorder characterized by recurrent, painful, and deep-seated abscesses, primarily in areas with apocrine sweat glands, such as the axillae, groin, buttocks, and infra-mammary regions [##REF##22236226##1##, ####REF##32165620##2##, ##REF##32460374##3####32460374##3##]. This condition typically presents with painful nodules, sinus tracts, and scarring, often leading to considerable morbidity and reduced quality of life [##REF##22236226##1##,##REF##32165620##2##]. The global prevalence of HS is estimated to range from 0.00033% to 4.1% [##REF##32460374##3##]. Women are disproportionately affected, with a two to three times higher incidence compared to men [##REF##22236226##1##]. The highest occurrence of HS is observed in patients aged between 18 and 44 years [##REF##37801538##4##]. Management options include a combination of medical therapies, such as antibiotics, immunosuppressants, and biologics [##REF##35409118##5##,##UREF##0##6##], associated with lifestyle modifications, as well as surgical interventions, including incision and drainage, wide local excision, or laser therapy [##REF##35409118##5##]. The prognosis of HS can vary widely among individuals, with some experiencing chronic, relapsing disease, while others achieve prolonged periods of remission [##REF##22236226##1##,##REF##32165620##2##]. Early diagnosis and comprehensive management are crucial for optimizing outcomes in affected patients.</p>", "<p>Anti-tumor necrosis factor-alpha (TNF-α) is a pro-inflammatory cytokine with a significant role in the pathogenesis of chronic inflammatory diseases and other immune-mediated conditions [##REF##35409118##5##]. TNF-α increases endothelial permeability, enhances leukocyte migration, and activates neutrophils and eosinophils functionally. Consequently, it is involved in a range of adaptive immune responses, including granuloma formation, phagosome development, macrophage activation and differentiation, and the immune response against viral pathogens [##REF##32098513##7##].</p>", "<p>Adalimumab, a TNF-α inhibitor, has shown promise in the treatment of HS, a debilitating chronic skin condition [##REF##35409118##5##]. By targeting the pro-inflammatory cytokine TNF-α, adalimumab can help mitigate the inflammatory processes associated with HS, reducing the frequency and severity of painful abscesses, sinus tracts, and scarring [##REF##35409118##5##,##UREF##0##6##]. It is used for moderate to severe HS cases that are refractory to conventional therapies [##REF##35409118##5##], demonstrating the potential to improve quality of life and induce long-term disease remission in some individuals [##REF##35409118##5##, ####UREF##0##6##, ##REF##32098513##7##, ##UREF##1##8##, ##REF##11971010##9####11971010##9##]. Adalimumab is an important therapeutic tool in the management of this challenging condition; however, like other biologics, its use comes with considerations for potential side effects and increases susceptibility to severe and opportunistic infections, including tuberculosis (TB) [##UREF##0##6##,##REF##32098513##7##].</p>", "<p>TB remains a leading global cause of mortality [##UREF##2##10##]. Despite a decline in new cases in Portugal, the country still maintains the highest TB incidence rate in Western Europe. In 2021, Portugal reported a total of 1,513 TB cases, equating to a notification rate of 14.6 cases per 100,000 residents. Notably, the Northern region and the Lisbon and Tagus Valley region continue to exhibit the highest incidence rates: 17 cases per 100,000 inhabitants [##UREF##3##11##]. Severe forms of TB, i.e., disseminated, meningeal, or involving the central nervous system (CNS), account for 5.3% of the total number of cases [##UREF##3##11##].</p>", "<p>This report delineates a case of a female patient afflicted with HS, who, two years after commencing adalimumab therapy, manifested CNS TB. Although CNS TB is a known consequence of TNF-α therapy, it is crucial to maintain awareness of latent tuberculosis infection (LTBI), particularly in regions with high prevalence. This vigilance is essential to reduce adverse outcomes. This report offers a comprehensive analysis encompassing the clinical facets, diagnostic considerations, therapeutic approaches, and prognostic outlook for this particular presentation.</p>" ]

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[ "<title>Discussion</title>", "<p>Adalimumab is a fully human IgG1 monoclonal antibody that specifically targets TNF-α. It is currently approved for the treatment of various inflammatory diseases, including moderate to severe HS [##UREF##4##12##,##REF##27518661##13##]. While its ability to neutralize TNF-α contributes to disease control through immunomodulation and immunosuppression, it can also render individuals more susceptible to infections [##REF##32098513##7##,##UREF##4##12##, ####REF##27518661##13##, ##REF##34009114##14####34009114##14##].</p>", "<p>In the context of LTBI, mycobacteria can persist within macrophages [##REF##11971010##9##]. Anti-TNF-α agents have the potential to trigger the reactivation or dissemination of <italic>Mycobacterium tuberculosis</italic>, leading to active TB [##REF##32098513##7##, ####UREF##1##8##, ##REF##11971010##9####11971010##9##]. Conversely, TNF-α has not been implicated in the innate immune response against extracellular bacterial pathogens [##REF##11971010##9##,##UREF##4##12##]. Furthermore, this therapy theoretically increases susceptibility to other pathogens such as bacteria (<italic>Listeria monocytogenes</italic> or <italic>Salmonella</italic> spp.) and viruses (hepatitis B virus, VZV, and human polyomavirus John Cunningham) [##REF##32098513##7##,##UREF##4##12##].</p>", "<p>In the case of this patient, she initially presented with a VZV infection. Despite being on immunosuppressive therapy, vaccination is not recommended due to her age [##UREF##4##12##]. This is because even under these conditions, the infection is rare within her age group. A higher frequency is observed in patients over 60 years on anti-TNF-α therapy, although the risk in this age range, compared to the general population, is not universally agreed to be increased [##REF##32098513##7##,##UREF##4##12##,##REF##34009114##14##].</p>", "<p>Among patients with TB, approximately 1% to 5% are complicated by CNS TB. In regions with low TB prevalence, such as North America and Western Europe, extrapulmonary manifestations of TB are primarily observed in adults with reactivated disease, with CNS TB, particularly in the form of meningitis, being the predominant presentation [##UREF##5##15##]. Following the inhalation of aerosol droplets containing <italic>M. tuberculosis</italic>, the host may either immediately eliminate the organism, develop primary active disease, remain with LTBI, or progress to active disease after a period of latent infection [##UREF##4##12##,##REF##34009114##14##]. Among individuals with LTBI and no underlying medical conditions, reactivation disease occurs in approximately 5% to 10% of cases [##UREF##5##15##]. The risk of reactivation significantly increases in immunosuppressed patients, particularly those on anti-TNF-alpha therapy, estimated to be two to four times higher. While TB can result from new exposure, in the context of immunosuppressive therapy, it typically stems from the reactivation of latent infection [##REF##23348114##16##].</p>", "<p>To mitigate this risk, screening is recommended for patients who are candidates to initiate biological immunosuppression, both before its commencement and subsequently on an annual basis or sooner if there is exposure to a patient with active TB [##REF##23348114##16##]. The screening process involves, first and foremost, ruling out active disease through symptom inquiry, assessment of signs consistent with TB disease, and chest radiography. Subsequently, it involves the exclusion of LTBI by inquiring about past TB and/or potential high-risk contacts [##UREF##4##12##,##REF##34009114##14##,##REF##23348114##16##]. There is no gold standard analytical method for diagnosing LTBI [##REF##34009114##14##,##REF##23348114##16##]. These tests, in addition to relying on an intact immune system, which may compromise their interpretation in patients under immunosuppressive therapy, are imperfect in assessing the risk of progression to TB disease. Among the available tests, IGRA exhibits higher sensitivity compared to the tuberculin skin test (TST) [##UREF##4##12##,##REF##34009114##14##,##REF##23348114##16##]. In the absence of past TB treatment, a previously positive IGRA before the initiation of therapy is an indication for LTBI treatment, typically involving monotherapy for four to nine months [##REF##34009114##14##]. In the case of our patient, we did not have access to the results of pre-treatment tests and we know that the annual control was not carried out.</p>", "<p>Diagnosing active TB in these patients can be challenging since symptoms related to cellular immune response (e.g., fever) and corresponding signs of inflammation (pulmonary infiltrates) may be absent or reduced [##REF##29223319##17##]. Diagnostic investigation in these patients should follow the traditional approach, involving the collection of microbiological specimens for direct examination (which has lower sensitivity in immunosuppressed patients), culture, and nucleic acid amplification tests for TB and other potential opportunistic diseases [##REF##34009114##14##]. In patients on anti-tumor necrosis factor (TNF) medications, the disease presents as extrapulmonary or disseminated in 25-48% of cases [##REF##29223319##17##], with central nervous system involvement still being rare, primarily manifesting as meningitis [##REF##29223319##17##, ####REF##31697616##18##, ##REF##28283299##19####28283299##19##].</p>", "<p>Neurotuberculosis can manifest in various ways, categorized as extra-axial and intra-axial presentations. Extra-axial manifestations include the most common tuberculous meningitis (leptomeningitis) and the rare tuberculous pachymeningitis. Intra-axial manifestations encompass intracranial tuberculous granuloma (tuberculoma), focal tuberculous cerebritis, intracranial tuberculous abscess, tuberculous rhombencephalitis, and tuberculous encephalopathy [##REF##31697616##18##]. The cavernous sinus, middle cranial fossa floor, tentorium, and cerebral convexity are commonly involved sites [##REF##28283299##19##]. Differential diagnoses for neurotuberculosis include other infectious causes of meningitis/encephalitis, neurosarcoidosis, meningioma, as well as neoplastic conditions such as metastasis and lymphoma. Differential diagnosis is made through lesion biopsy [##REF##31697616##18##].</p>", "<p>Considering the patient's clinical features, which include immunosuppression due to anti-TNF therapy, the absence of a personal history of TB, a positive IGRA result (despite prior BCG vaccination, which may influence the outcome), and the occupational exposure risk linked to her profession, a decision was made to commence empirical treatment and monitor the response to therapy. Chemoprophylaxis has been a topic of discussion for at-risk groups; however, there is no consensus recommending its implementation in patients with negative IGRA [##REF##23976779##20##].</p>", "<p>Timely identification with an accurate diagnosis, discontinuation of anti-TNF medications, and prompt initiation of treatment are crucial measures for achieving successful therapy [##UREF##4##12##,##REF##34009114##14##]. These steps can help prevent disease exacerbation, minimize the risk of neurological complications, and reduce the potential for long-term health effects and mortality [##UREF##4##12##,##REF##34009114##14##,##REF##23348114##16##].</p>" ]

[ "<title>Conclusions</title>", "<p>This case of neurotuberculosis in a young individual following immunosuppressive treatment for HS underscores the critical importance of early screening for opportunistic diseases in patients undergoing immunosuppressive therapies. Notably, the initial unknown serological status before treatment initiation and the preceding episode of trigeminal herpes zoster, which initially obscured the patient's complaints, add complexity to the diagnostic challenge. The manifestation of cerebral TB in this context highlights the potential reactivation of latent infections that can occur when the immune system is compromised, emphasizing the need for vigilant monitoring before, during, and after immunosuppressive treatment.</p>", "<p>Furthermore, this case reinforces the significance of thoroughly assessing and investigating patients’ complaints, particularly those related to new neurological symptoms, irrespective of the initial pathology. The successful management of this case underlines the importance of adopting a holistic approach in medical practice. Careful assessment of underlying conditions, diligent monitoring during immunosuppressive treatment, and comprehensive investigation of patient complaints are pivotal in preventing and managing complications stemming from opportunistic diseases.</p>" ]

[ "<p>Hidradenitis suppurativa (HS) is a chronic inflammatory skin disease with limited therapeutic options. Adalimumab, an anti-tumor necrosis factor-alpha (TNF-α) monoclonal antibody, was the first biological agent approved for the treatment of moderate to severe HS. Tuberculosis (TB) is a highly prevalent global public health problem, affecting individuals worldwide. Continuous immunosuppression from TNF-α treatment increases the risk of TB development. Isolated neurotuberculosis, in the absence of other symptoms, emerges as a rarely observed infection pattern in such patients.</p>", "<p>We present a case of a 23-year-old woman with severe HS undergoing treatment with adalimumab. After two years, she developed a pronounced occipital tension headache, constant nausea, and persistent fever. The patient's latent TB status was unknown without annual screening. Subsequent magnetic resonance imaging revealed a lesion in the cerebellar vermis. Immunosuppressive therapy was suspended and an etiological study was conducted; the only positive result was the interferon-gamma release assay. Empirically, antituberculosis treatment and prednisolone were initiated, leading to clinical and neurological improvement. After one year of treatment, symptoms resolved without neurological sequelae.</p>", "<p>This case highlights the importance of vigilant monitoring before, during, and after immunosuppressive treatment. Early recognition, discontinuation of anti-tumor necrosis factor medications, and appropriate management of TB are crucial to prevent complications.</p>" ]

[ "<title>Case presentation</title>", "<p>This case described a 23-year-old woman medical student in Portugal. She had a medical history dating back to the age of 12 years, marked by the presence of HS. In early 2019, she started adalimumab due to extensive suppurative lesions with fistulous tracts in various areas of her body (medial thighs, inguinal region, pubic mound, gluteal region, axillary, and inframammary region). In October 2020, she developed varicella-zoster virus (VZV) affecting the trigeminal nerve and was treated with valacyclovir and corticosteroids. Over the next four months, she experienced neuropathic facial pain, which improved and eventually resolved.</p>", "<p>In September 2021, the patient presented with a recurrence of neuropathic pain in the ophthalmic and maxillary branches of the trigeminal nerve, accompanied by low-grade fever, nausea, malaise, and tensional occipital pain. Her primary care physician initially diagnosed the condition as post-herpetic neuritis. However, the patient felt that her symptoms were atypical and requested a cerebral magnetic resonance imaging (MRI) that revealed a lesion in the superior vermis of the cerebellum, involving the cuneus, hyperintense on T2-weighted images (WI) and hypointense on T1-WI, with minimal mass effect and no contrast enhancement (Figure ##FIG##0##1##).</p>", "<p>Following the referral to the internal medicine department for further evaluation, the patient presented with an intense occipital tension headache (7 out of 10 on the numerical pain scale), constant nausea, and daily persistent fever. There were no other constitutional symptoms or meningeal signs. Laboratory tests revealed mild normocytic, normochromic anemia, leukocytosis with neutrophilia, elevated lactate dehydrogenase, C-reactive protein of 81.4 mg/L, and an erythrocyte sedimentation rate of 89 mm/h. A lumbar puncture showed clear cerebrospinal fluid (CSF) with mononuclear cells, normal glucose, elevated protein, negative viral polymerase chain reaction (PCR), and negative <italic>Mycobacterium tuberculosis</italic> deoxyribonucleic acid (DNA). CSF samples were examined using Ziehl-Neelsen (ZN) stain for acid-fast bacilli, Gram stain for bacteria, India ink preparations for fungi, and an antigen test for <italic>Cryptococcus neoformans</italic>. All results were negative. The patient was unaware of the result of the interferon-gamma release assay (IGRA) test for TB before starting treatment with adalimumab, and she did not undergo an annual check-up. As a child, she had been vaccinated with Bacillus Calmette-Guérin (BCG).</p>", "<p>In the following days, as cultural results were pending, the patient’s headaches and nausea worsened despite receiving antiviral empiric therapy with intravenous acyclovir. A cerebral MRI, conducted two weeks later, indicated worsening infiltrative changes in the dura mater, primarily in the torcular region, vermis, and right occipito-polar region. It suggested an inflammatory process, possibly granulomatous, with the involvement of the cerebral venous sinuses (Figure ##FIG##1##2##). The IGRA test returned positive.</p>", "<p>Given the deteriorating condition, empirical anti-tuberculosis treatment was initiated with isoniazid, rifaximin, ethambutol, pyrazinamide, and pyridoxine, alongside prednisolone. Within a week, her symptoms improved significantly. Five weeks after starting treatment, the lesion had regressed in MRI (Figure ##FIG##2##3##).</p>", "<p>After 42 days, mycobacterial cultures turned out negative. She completed 12 months of anti-tuberculosis treatment, and a follow-up MRI one year after the initiation of treatment revealed a significant improvement in the dural thickening and the disappearance of the previously observed imaging abnormalities (Figure ##FIG##3##4##). The patient showed a favorable response to anti-tuberculosis therapy, with significant clinical and radiological improvement, without neurological sequelae.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>(a) T2WI-FLAIR shows a subtle area of hyperintensity in the posterior region of the cerebellar vermis (arrow). There is no evidence of restricted diffusion, suggesting probable vasogenic edema. (b) Contrast-enhanced T1WI reveals pachymeningeal diffuse thickening and enhancement in the falcotentorial posterior region (arrow).</title><p>T2WI: T2-weighted imaging; FLAIR: fluid-attenuated inversion recovery; T1WI: T1-weighted imaging.</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>Two weeks later, (a) T2WI shows an increase in hyperintensity in two-thirds of the superior vermis and adjacent areas, indicating an increase in vasogenic edema (arrow). (b) Contrast-enhanced T1WI shows an increase in irregular pachymeningeal enhancement adjacent to the cerebellar vermis, without restricted diffusion (arrow).</title><p>T2WI: T2-weighted imaging; T1WI: T1-weighted imaging.</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG3\"><label>Figure 3</label><caption><title>Five weeks after the initial cerebral MRI, (a) T2WI-FLAIR shows a reduction in edema in the vermis (arrow). (b) Contrast-enhanced T1WI reveals a decrease in pachymeningeal enhancement (arrow).</title><p>T2WI: T2-weighted imaging; FLAIR: fluid-attenuated inversion recovery; T1WI: T1-weighted imaging.</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG4\"><label>Figure 4</label><caption><title>One year after the initial cerebral MRI, (a) T2WI shows complete resolution of the signal alteration in the vermis (arrow), medial cerebellar hemispheres, and occipito-polar region (edema). (b) Contrast-enhanced T1WI reveals reduced thickening and pachymeningeal enhancement previously observed in the region of the torcula (arrow).</title><p>T2WI: T2-weighted imaging; T1WI: T1-weighted imaging.</p></caption></fig>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Francisca Martins, Alexandra Rodrigues, João Fonseca Oliveira, Rui Malheiro, Luís Cerqueira</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Francisca Martins, Alexandra Rodrigues</p><p><bold>Drafting of the manuscript:</bold>  Francisca Martins, Alexandra Rodrigues</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Francisca Martins, João Fonseca Oliveira, Rui Malheiro, Luís Cerqueira</p><p><bold>Supervision:</bold>  João Fonseca Oliveira, Rui Malheiro, Luís Cerqueira</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Social determinants of health (SDoH) are non-medical factors that influence health outcomes, including the conditions in which people are born, grow, work, live, and age, as well as the wider set of forces and systems shaping daily life, such as economic policies, development agendas, social norms, and political systems.##REF##15781105##1–4## These factors contribute significantly to health disparities due to systemic disadvantages and biases.##UREF##3##5##^,##REF##32349541##6## Systemic disadvantages refer to unequal distribution of resources and opportunities, while bias refers to unfair treatment based on social, economic, or demographic characteristics. Health inequities, which are unfair and avoidable differences in health among population groups, can arise from these determinants and warrant ethical consideration.##REF##18994664##7## For example, mental health during pregnancy plays a pivotal role in both the mother’s and the unborn child’s well-being.##REF##14999160##8## In a similar vein, lifestyle choices and living environments are intricately linked to the health outcomes of diabetes patients, with significant correlations observed.##UREF##3##5## These examples illustrate how systemic disadvantages and biases contribute to health inequities, underlining the importance of addressing SDoH in medical treatments for these conditions.##UREF##3##5##^,##REF##29897915##9–11##</p>", "<p>Social work notes written by social workers contain comprehensive information on SDoH compared to other common clinical note types documented by clinicians or medical professionals. Examples of social aspects covered in social work notes include living conditions, family support, access to transportation, employment status, and education level. While other types of notes such as nursing notes, discharge summaries, and hospital progress notes may include some SDoH-related information such as insurance status, and health-related aspect such as food and physical environment, they typically focus on specific aspects of patient care and may not provide as extensive information on SDoH as social work notes, which are written to provide a more complete view of these factors.##REF##35202844##12–14## However, our capacity to research sociodemographic and socioeconomic health outcomes is still quite constrained. Most assessments of SDoH are not present in structured data.##REF##34613399##15## Instead, much of this information is collected in unstructured notes, making the information largely inaccessible without advanced technical processing. The inability to easily extract this information limits research into the effects of SDoH on care delivery and success.</p>", "<p>To understand the information embedded in the social work notes and to characterize specific SDoH factors covered across nearly one million notes, we explored the use of unsupervised methods for topic modeling. Topic modeling methods based on Latent Dirichlet Allocation (LDA) have been previously successful in finding hidden structures (topics) from large corpora,##REF##25291807##16##^,##REF##32663156##17## the utility of which we further explored in this study. The large collection of social work notes analyzed in this study spanned a diverse cohort of patient demographics and disease groups. This allowed us to develop a comprehensive understanding of the underlying SDoH topics from different note types for a variety of disease chapters. We explored several methods to circumvent the inherent limitations of topic modeling approaches, such as pre-determining a fixed number of clusters, intrinsic randomness, and need for human-based interpretation.</p>" ]

[ "<title>Materials and methods</title>", "<title>Data sources and patient demographics</title>", "<p>This study uses the deidentified clinical notes at UCSF recorded between 2012 and 2021.##UREF##12##31## The study was approved by the Institutional Review Board (IRB) of the University of California, San Francisco (UCSF; IRB #18-25163). Our cohort consists of the following demographic distribution: Gender—Male: 95 387 (52.5%), Female: 85 635 (47.1%); Race—White: 22 839 (12.6%), Black: 21 120 (11.6%), Asian: 47 723 (26.3%), Native American: 14 813 (8.2%), Other: 75 149 (41.4%); Age—Median: 33 years (Range: 12-58); Ethnicity—Hispanic: 41 386 (22.8%), Non-Hispanic: 128 018 (70.5%).</p>", "<title>Data preprocessing</title>", "<p>We initiated our research by collecting clinical notes from a de-identified dataset, specifically selecting those entries where the metadata contained the term “social”—case-insensitive—within the encounter type, department name, specialty, or provider type, thus designating these as “social work notes.” From the extensive corpus of 106 million notes representing 1.2 million patients, this focused query yielded 2.5 million social work notes attributed to 181 644 unique patients. To ensure the quality and relevance of our data, we excluded notes under 30 characters, anticipating they would not provide substantial content. Duplicate notes were also removed to eliminate redundancy and decrease computational demands. Following this stringent quality control process, we distilled the dataset down to 1 million notes corresponding to the same 181 644 patients, which formed the basis for our downstream topic modeling analysis, as depicted in ##FIG##0##Figure 1##.</p>", "<title>Topic modeling with LDA analysis</title>", "<p>While word frequency calculations can provide preliminary insights about term relevance, this view is too limited to understand what broader topics may be contained within social work notes. In contrast, topic modeling is a field of unsupervised learning that learns statistical associations between words or groups of words to identify “topics”: clusters of words that tend to co-occur within the same document.</p>", "<p>LDA is a generative probabilistic model, which assumes that each document is a combination of a few different topics, and that each word’s presence can be attributed to particular topics in the document. The result is a list of clusters, each of which contains a collection of distinct words. The combination of words in a cluster can be used for topic model interpretation. Python package <italic toggle=\"yes\">gensim</italic> was used for the implementation.##UREF##13##32## We used <italic toggle=\"yes\">gensim.models.ldamodel.LdaModel</italic> for the actual analysis. The core estimation code is based on Hoffman et al.##UREF##14##33##</p>", "<p>Python package <italic toggle=\"yes\">nltk</italic> was used. As a preprocessing step, English language stop words and special characters including “\\t,” “\\n,” “\\s” were removed from note text. The resulting text from all social work notes were vectorized and topics were inferred with the LDA algorithm. In addition to the analysis on the complete cohort of social work notes, in order to investigate the topic distribution across specific social work note categories, we additionally analyzed the 4 largest categories of social work notes: <italic toggle=\"yes\">Progress Notes</italic>, <italic toggle=\"yes\">Interdisciplinary</italic>, <italic toggle=\"yes\">Telephone encounters</italic>, and <italic toggle=\"yes\">Group Notes</italic>. We also extended the investigation to social work note subsets across 10 ICD-10 disease chapters. These subsets were determined by investigating encounter-specific ICD-10 diagnostic codes. The common stop words were also excluded, using <italic toggle=\"yes\">stopwords.words(“english”)</italic> from <italic toggle=\"yes\">nltk</italic> package.##UREF##15##34## To overcome the inherent stochasticity of topic modeling approaches and ensure the reliability of our findings, we ran 5 independent modeling analyses for each category of notes. This allowed us to capture consistent patterns and topics across different iterations, increasing our confidence in the identified topics and their relevance to the respective disease groups. Another critical step in LDA topic modeling was determining the optimal cluster number, which is further discussed in the next subsection. Furthermore, when extending the analysis to different note types, we labeled the inferred topics using heuristics described further.</p>", "<title>Determining the optimal number of topics for notes</title>", "<p>One of the most important hyper-parameters for LDA analysis is the number of topics K. Generally, if K is chosen to be too small, the model will lack the capacity to provide a holistic summary of complex document collections; and returned topical vectors may combine semantically unrelated words/tokens.##UREF##16##35## Conversely, if K is chosen to be too large, the returned topical vectors may be redundant, and a parsimonious explanation of a complex phenomenon may not be achieved. We used 2 evaluation metrics, topic coherence##UREF##17##36##^,##UREF##18##37## and topic similarity,##UREF##19##38## to systematically determine the optimal number of clusters. Topic Coherence (C) quantifies the score of a single topic by measuring the degree of semantic similarity between high-scoring words in the topic.##UREF##20##39## The measure helps distinguish between topics that are semantically interpretable and those that are artifacts of statistical inference. The coherence metric we compute is based on a sliding window, one-set segmentation of the top words and an indirect confirmation measure that uses normalized pointwise mutual information and the cosine similarity.##UREF##18##37## Similarly, topic similarity (S) measures how similar 2 clusters are considering the words contained in the topics. The lower the values are, the less redundant the topic distribution is. For quantifying topic similarity, we use Jaccard similarity.##UREF##19##38## Furthermore, there are alternative ways to evaluate the quality of topic discovery, such as assessing “topic diversity.”##UREF##21##40## Considering these evaluation metrics in future work may provide further insights into the performance of our methods.</p>", "<p>An ideal solution would have a high topic coherence and low similarity metric. To decide the optimal number of clusters, for each analysis, we ran the LDA analysis with the number of clusters K ranging from 10 to 50, simultaneously computing C and S scores. The number having the <italic toggle=\"yes\">i_th</italic> highest C value, <italic toggle=\"yes\">j_th</italic> smallest S value, and the minimum <italic toggle=\"yes\">i</italic> + <italic toggle=\"yes\">j</italic> among all runs was selected as the final number of topics (##SUPPL##0##Figure S1A##). We found that the best cluster number for analyzing the entire notes repository was 17.</p>", "<title>Topic modeling per notes with certain type</title>", "<p>In order to investigate the topic distribution across specific note categories, we applied topic modeling on the 4 largest categories of social work notes: <italic toggle=\"yes\">Progress Notes</italic>, <italic toggle=\"yes\">Interdisciplinary</italic>, <italic toggle=\"yes\">Telephone encounters</italic>, and <italic toggle=\"yes\">Group Notes</italic>. This approach allowed us to gain insights into the prevalence of certain topics within these major categories and assess their potential impact on the overall topic modeling results. We used the same pipeline for identifying the optimal number of clusters as described earlier in the Materials and methods section. To ensure robustness in our results, given the inherent randomness of the LDA method, we conducted each analysis across 5 different iterations for every category. This approach allowed us to capture a broader range of variability, thereby increasing the reliability of our findings. The results from these 5 iterations were then pooled together. This pooling strategy was instrumental in developing a well-grounded heuristic for labeling the topic clusters, ensuring our results were reflective of consistent patterns observed across all iterations, rather than being influenced by any single run’s anomalies.</p>", "<p>In our analysis, we determined that the optimal number of clusters for most of the analyses we conducted is ∼20. This balances the trade-off between coherence and similarity metrics, ensuring that we obtain semantically interpretable and non-redundant topic clusters, which provide meaningful insights into the underlying document collection. Consequently, we used 20 clusters for the majority of our note analyses, including those focused on note subtypes or disease chapters. However, we found that the best cluster number for analyzing the entire notes repository was 17, so we utilized 17 clusters for the topic modeling of all notes combined (see previous session).</p>", "<title>Topic labeling heuristics</title>", "<p>Apart from labeling topics determined from the entire cohort of social work notes, our analysis screened 20 topic clusters (determined experimentally; see Results) for all 14 categories of notes (10 disease chapters and 4 social work note types) for 5 independent runs (to reduce stochasticity), thereby resulting in 1400 topic clusters that required further labeling. To assign labels to all 1400 topics, we developed a heuristic to automatically assign topic labels for subsequent analyses, the details of which are discussed next.</p>", "<p>We first constructed the dictionary of topic names and the corresponding words by manually analyzing the topic modeling results for one run on the complete corpus of 0.95 million social work notes at UCSF. Then we expanded the individual topic clusters by first retrieving 20 most similar words to the words comprising topic clusters based on the cosine similarity of their word embeddings.##UREF##22##41## Any words that were not relevant to the topic label, as determined through manual review, were not considered further. The final dictionary of topic labels and the set of words used to label the topics is shown in ##TAB##1##Table 2##. In our approach, we automatically assigned topic labels to individual word clusters by calculating the intersection over union (IOU) ratio for the words in a cluster. This enabled us to assign labels to all 1400 topic clusters from our analysis. The details can be found in the pseudo-code below. To address your professor’s concerns, we used the IOU of word frequencies within each cluster. We assigned the label with the maximum IOU, but only if there was an overlap of at least 2 words. If none of the topics met this criterion, we did not assign a topic to the word cluster.</p>", "<p>\n<italic toggle=\"yes\">Heuristics of automatic assigning topic names for the individual topic cluster</italic>\n</p>", "<p>\n<bold>\n<italic toggle=\"yes\">Begin:</italic>\n</bold>\n</p>", "<p>\n<italic toggle=\"yes\">Construct the dictionary of topic names and the words comprising this topic;</italic>\n</p>", "<p>\n<italic toggle=\"yes\">Expand the individual topic cluster space;</italic>\n</p>", "<p>\n<italic toggle=\"yes\">&gt; Enrichment[k|h] means the frequency of words belong to topic h for word cluster k</italic>\n</p>", "<p>&gt; means intersect, means union</p>", "<p>\n<bold>\n<italic toggle=\"yes\">For</italic>\n</bold>\n<italic toggle=\"yes\">each iteration of the topic modeling results</italic> <bold><italic toggle=\"yes\">do:</italic></bold></p>", "<p> <bold><italic toggle=\"yes\">For</italic></bold><italic toggle=\"yes\">every word cluster k</italic> <bold><italic toggle=\"yes\">do:</italic></bold></p>", "<p>  <bold><italic toggle=\"yes\">For</italic></bold><italic toggle=\"yes\">topic name h, corresponding word set</italic> <bold><italic toggle=\"yes\">in topic</italic></bold><italic toggle=\"yes\">dictionary</italic> <bold><italic toggle=\"yes\">do:</italic></bold></p>", "<p>   overlap = word cluster k topic h</p>", "<p>  total = word cluster k topic h</p>", "<p>   <italic toggle=\"yes\">Enrichment[k|h]</italic> <bold><italic toggle=\"yes\">=</italic></bold><italic toggle=\"yes\">overlap/total</italic></p>", "<p>  <italic toggle=\"yes\">The topic assigned to word cluster k = max([Enrichment[k|h] for h in H])</italic></p>", "<p>\n<bold>\n<italic toggle=\"yes\">End</italic>\n</bold>\n</p>", "<p>Code for the paper is available on <ext-link xlink:href=\"https://github.com/ShenghuanSun/LDA_TM\" ext-link-type=\"uri\">https://github.com/ShenghuanSun/LDA_TM</ext-link></p>", "<title>Word frequency calculation</title>", "<p>To perform a preliminary investigation of disease-specific features in the social work notes, 10 disease chapters were identified with ICD-10 codes: (1) Diseases of the nervous system (G00-G99), (2) Diseases of the circulatory system (I00-I99), (3) Diseases of the respiratory system (J00-J99), (4) Diseases of the digestive system (K00-K95), (5) Diseases of the musculoskeletal system and connective tissue (M00-M99), (6) Diseases of the genitourinary system (N00-N99), (7) Pregnancy, childbirth and the puerperium (O00-O9A), (8) Congenital malformations, deformations and chromosomal abnormalities (Q00-Q99) (9) Neoplasms (C00-D49), and (10) Diseases of the blood and blood-forming organs and certain disorders involving the immune mechanism (D50-D89).</p>", "<p>Chi-squared statistics was used to compare the frequency of words across different note categories (χ² function from <italic toggle=\"yes\">sklearn.feature</italic> selection was used to this end). After ranking the <italic toggle=\"yes\">P</italic> values and removing stop words, the top 5 potential meaningful words were visualized by the word frequency calculation. Python package <italic toggle=\"yes\">scikit-learn</italic> was used to conduct the analysis.##UREF##23##42## To embed and tokenize the unstructured notes, <italic toggle=\"yes\">text</italic>. <italic toggle=\"yes\">CountVectorizer</italic> function from <italic toggle=\"yes\">sklearn.feature</italic> extraction package was used.</p>" ]

[ "<title>Results</title>", "<p>We retrieved a total of 0.95 million de-identified clinical social work notes generated between 2012 and 2021 (see Materials and methods) from our UCSF Information Commons##UREF##12##31## (##FIG##0##Figure 1##). The majority of notes were classified as Progress Notes, Interdisciplinary Notes, or Telephone Encounter Notes; other note categories included Patient Instructions, Group Note, Letter, which comprised fewer than 5% each. These notes covered 181 644 patients of which 95387 (52.5%) were female. The median age of these patients was 33 years. Among them, 69 211 patients had only one note; 65 100 patients had between 2 and 5 notes, and 47 333 patients had more than 5 notes (##SUPPL##0##Table S1##, ##SUPPL##0##Figure S2B##). The demographics distribution is presented in ##TAB##0##Table 1##. No demographic feature was statistically associated with the number of notes for each patient (##SUPPL##0##Table S1##).</p>", "<p>In addition to analyzing the number of notes, we were also interested in exploring the medical conditions associated with patients who received social work notes. This aspect can provide valuable insights into the factors contributing to the need for social work intervention. To investigate this, we collected the ICD-10 codes for the encounters during which social work notes were recorded for the patients. These ICD-10 codes were then mapped at the chapter level.##UREF##14##33## The 3 most frequent ICD-10 chapters found to be associated with a social work note were “Mental, Behavioral and Neurodevelopmental disorders,” “Factors influencing health status and contact with health services,” and “Symptoms, signs and abnormal clinical and laboratory findings, not elsewhere classified” (##SUPPL##0##Table S2##).</p>", "<title>Using LDA to extract topics in social work notes</title>", "<p>Looking at the word components of each topic (##TAB##0##Table 1##), we discovered a few diverse clusters that cover many different social aspects of patients including social service (Topic 11), abuse history (Topic 14), phone call/online communications (Topic 12), living condition/lifestyle (Topic 16), risk of death (Topic 8), group session (Topic 7), consultation/appointment (Topic 5), family (Topic 4, 6), and mental health (Topic 1). Many of these topics are consistent with topics covering SDoH; most importantly, most of the information potentially conveyed through these topics are absent in the structured data. Of note, in our parameter exploration, we found that increasing the number of clusters can lead to additional recognizable topics, such as food availability (data not shown), although we also obtain redundant topics.</p>", "<title>Topic modeling on specific note categories</title>", "<p>Analyzing the topics appearance in each note subtype, we found that social work notes in the Progress Notes category contained a higher percentage of clinically related topics, such as Mental Health (4.32%) and Clinician/Hospital/Medication-related information (8.40%), along with a smaller proportion of SDoH-related topics like Insurance/Income, Abuse history, Social support (10.46%), and Family (6.29%). Compared to Progress Notes, Telephone Encounter notes contained a larger proportion of topics related to Insurance/Income (3.93%), Phone call/Online (7.47%), Social support (11.56%), and Family (8.08%). Interestingly, telephone encounter notes lacked information about the Risk of death (0%), which may be because the discussions on this topic are not appropriate for telephone encounters. Furthermore, Group Notes, which are the notes taken during group therapy, describe the group’s progress and dynamics. As expected, Group Notes have a more uneven topic category distribution, with a higher percentage of Group session (24.69%) and Phone call/Online (12.71%) related topics (##FIG##1##Figure 2A##).</p>", "<p>We also applied LDA analysis to the social work notes associated with 10 ICD-10 chapters described earlier (##FIG##1##Figure 2B##). We observed that most diseases have a similar topic proportion distribution, for example, most of them are enriched for Social support and Family topics. In particular, Social support is highly represented in notes related to Neoplasms (21.51%) and Diseases of the digestive system (22.47%). Family topics are also frequently mentioned in notes associated with Diseases of the nervous system (23.31%), Pregnancy, childbirth, and the puerperium (20.1%), and Congenital malformations, deformations, and chromosomal abnormalities (21.43%). However, some differences were identified between the ICD-10 chapters. Notes associated with disorders of mental health and pregnancy contain a higher percentage of SDoH topics on mental health, as would be expected. Mental health topics are more frequently mentioned in clinical notes around pregnancy than even in nervous system disorders. Interestingly, the Family topic area was often mentioned in notes associated with congenital malformation abnormalities. In summary, the analysis demonstrated both the commonness and uniqueness of topics around SDoH covered across the various diseases and conditions which afflict patients.</p>", "<title>Word frequency on individual disease</title>", "<p>In addition to performing topic modeling on social work notes associated with 10 ICD-10 chapters, we also conducted a word frequency analysis. This analysis highlighted that note from each ICD-10 chapter contained both disease-specific terms and a limited number of disease-specific SDoH topics. For instance, notes from patients with neoplasms frequently mentioned terms like “oncology,” “chemotherapy,” and “tumor,” while those associated with musculoskeletal disorders often included words such as “arthritis” and “rheumatology.” In addition to these disease-specific words, there were observable patterns in the prevalence of certain SDoH-related terms. Words like “mindfulness” appeared predominantly in chapters on Pregnancy and the Nervous System, and “wheelchair” was a recurrent term in Musculoskeletal disorders. Notably, conditions related to pregnancy showed a significant presence of mental health topics, indicating a frequent assessment of this aspect in social work notes for pregnancy care (##FIG##2##Figure 3##).</p>", "<p>Overall, the word frequency analysis serves as a complementary tool to topic modeling. While topic modeling is adept at uncovering general patterns, predominantly SDoH topics, in social work notes, word frequency analysis, with its focused approach, tends to reveal features specific to particular diseases, especially when comparing different ICD-10 chapters.</p>" ]

[ "<title>Discussion</title>", "<p>We used an unsupervised topic modeling method called LDA modeling on our corpus of 0.95 million de-identified clinical social work notes. We showed that topic modeling can be used to (1) extract the hidden themes from this huge corpus of clinical notes and identify the critical information embedded in the notes, namely SDoH factors; and (2) calculate the proportion of each theme across different subsets of the note corpus and systemically characterize notes of different types. Using simple term frequency methods on this large corpus, we found that specific SDoH terms tend to be enriched in notes from patients within different disease categories, including wheelchair for patients with musculoskeletal disorders and depression for patients with pregnancy diagnoses, suggesting that these populations may be more at risk for these SDoH features.</p>", "<p>We extracted several concrete SDoH-related topics, thus providing insight into the information that may be extracted from these corpora for facilitating future work around understanding how these topics correlate with health outcomes. During our comparison of notes of different subtypes, we found that the topic distribution of notes for specific types of diseases contains similar information but showed different levels of enrichment, representing the unique features of each disease set. As one of many examples, our analysis shows how mental health issues are frequently documented around pregnancy (##FIG##1##Figure 2B##). This type of information can help us better understand the social determinants of most concern to patients when interacting with the health system.</p>", "<p>The specific topics identified in our study were in line with findings from a previous publication.##REF##35202844##12## This recent research extracted information on physical, mental, and social health by applying the non-negative matrix factorization (NMF) topic modeling method to 382 666 primary care clinical notes. However, that study exclusively examined physician-generated notes, whereas our focus was on social work notes, enabling us to uncover a broader range of SDoH topics. In our paper, we identified several additional topics, including but not limited to Living Condition/Lifestyle, Family, Risk of Death, and Abuse History.</p>", "<p>Our research has several potential use cases. First, it aids computational sociology and epidemiology studies by identifying key factors that influence health outcomes. This extraction process lays the groundwork for in-depth analysis within these fields. Second, the findings from computational analyses can substantiate policy decisions. By providing empirical evidence, these findings can guide regulations and interventions aimed at health equity. Lastly, for participating healthcare providers, these extracted SDoH factors offer insights for effective resource allocation, particularly in supporting vulnerable groups. Overall, understanding the distribution of SDoH topics in patient records is crucial for developing targeted interventions and preventive strategies, aimed at addressing the root causes of health disparities.</p>", "<p>Our study has several strengths. We performed analysis on a large corpus of notes, which to our knowledge, is the largest social work notes dataset to be used in a similar study. Instead of focusing on a single disease category or specific medical topic, we aimed at comprehensively finding the potential SDoH topics in all types of clinical social notes for a variety of diseases. Furthermore, to obtain a thorough understanding of the information embedded in social worker notes and capture the richness and complexity of the rhetoric in these notes, we conducted complementary analyses: a word frequency enrichment analysis allowed us to identify specific terms more frequently associated with particular ICD-10 chapters, which demonstrated the prevalence of disease-related terms in social work notes, providing a more granular view of the data. Second, the use of LDA allowed us to identify broader topics of increased relevance in these disease groups. It helped us uncover patterns related to SDoH, offering a higher-level perspective on the data.</p>", "<p>Recognizing the intrinsic instability of LDA topic modeling methods, we enhanced the robustness of our results by independently searching for optimal hyperparameters to predefine topic numbers. Additionally, we ensured reliability by conducting each analysis across 5 iterations for every category (see Materials and methods). However, it is possible to still obtain different topic clusters with a different set of hyperparameters. Moreover, other topic modeling algorithms, such as NMF##REF##35202844##12##^,##REF##10548103##43## and BERTopic,##UREF##24##44## could be explored to compare their performance and suitability for our specific task. In addition, we developed topic labeling heuristics that allow us to assign topics to the individual clusters. However, the heuristics may not cover all topic-related keywords, and in the future, it may be interesting to revisit our heuristic to expand upon the topic clusters further to make them more generalizable. State-of-the-art large language models like ChatGPT offer significant potential for improving our pipeline, particularly in the nuanced task of assigning topic labels.##UREF##25##45–47## With effective prompt engineering, these models could systematically extract patterns from social work notes, enhancing the depth and accuracy of our statistical analyses, and potentially uncovering new insights in SDoH. We also exclusively utilized ICD-10 codes, acknowledging the prospective merit of incorporating ICD-9 in future research. Another limitation of our study is the lack of structured Electronic Health Record (EHR) data for recording comorbidities, insurance, and living status. These factors are relevant to SDoH and could provide valuable insights into the relationships between health outcomes and social determinants. The absence of such data may limit our ability to fully capture the complex interplay of these factors and their effects on health. Finally, we did not explicitly exclude negations or the lengthy expression, as they still contribute to the overall discussion of certain topics. However, we acknowledge that the consideration of negation is crucial for a more nuanced understanding of the information contained in clinical notes, and for more accurate analysis of the semantic meaning of the identified topics.</p>", "<p>Our study opens pathways for several key areas of future research. For data scientists and computational researchers, future research should focus on combining these identified themes with predictive modeling techniques to assess their correlation with future health outcomes. This integration would not only validate the relevance of the identified SDoH themes but also provide a more holistic understanding of patient care dynamics and health outcomes. For healthcare practitioners, the challenge lies in integrating SDoH insights into patient care and public health policies. This demands not only an understanding of clinical informatics but also an insight into health policy and administration. Collaborating with experts in these fields could lead to developing actionable strategies that utilize our findings to improve healthcare delivery and policy decisions.</p>" ]

[ "<title>Conclusion</title>", "<p>Social work notes contain rich and unique information about SDoH factors, frequently only recorded in text notes. SDoH factors are critical for analyzing health outcomes, and this study identified detailed categories of SDoH information covered by social work notes. Furthermore, the study demonstrated that different categories of notes emphasize different aspects of SDoH, despite belonging to social work consultations. The findings from this study would form a basis of potential future research questions around this utilizing SDoH to uncover health disparities and SDoH-associated disease trajectories, as well as methods to extract comprehensive SDoH-related information from clinical notes.</p>" ]

[ "<p>Drs Madhumita Sushil and Atul J. Butte contributed equally to this work.</p>", "<title>Abstract</title>", "<title>Objective</title>", "<p>Existing research on social determinants of health (SDoH) predominantly focuses on physician notes and structured data within electronic medical records. This study posits that social work notes are an untapped, potentially rich source for SDoH information. We hypothesize that clinical notes recorded by social workers, whose role is to ameliorate social and economic factors, might provide a complementary information source of data on SDoH compared to physician notes, which primarily concentrate on medical diagnoses and treatments. We aimed to use word frequency analysis and topic modeling to identify prevalent terms and robust topics of discussion within a large cohort of social work notes including both outpatient and in-patient consultations.</p>", "<title>Materials and methods</title>", "<p>We retrieved a diverse, deidentified corpus of 0.95 million clinical social work notes from 181 644 patients at the University of California, San Francisco. We conducted word frequency analysis related to ICD-10 chapters to identify prevalent terms within the notes. We then applied Latent Dirichlet Allocation (LDA) topic modeling analysis to characterize this corpus and identify potential topics of discussion, which was further stratified by note types and disease groups.</p>", "<title>Results</title>", "<p>Word frequency analysis primarily identified medical-related terms associated with specific ICD10 chapters, though it also detected some subtle SDoH terms. In contrast, the LDA topic modeling analysis extracted 11 topics explicitly related to social determinants of health risk factors, such as financial status, abuse history, social support, risk of death, and mental health. The topic modeling approach effectively demonstrated variations between different types of social work notes and across patients with different types of diseases or conditions.</p>", "<title>Discussion</title>", "<p>Our findings highlight LDA topic modeling’s effectiveness in extracting SDoH-related themes and capturing variations in social work notes, demonstrating its potential for informing targeted interventions for at-risk populations.</p>", "<title>Conclusion</title>", "<p>Social work notes offer a wealth of unique and valuable information on an individual’s SDoH. These notes present consistent and meaningful topics of discussion that can be effectively analyzed and utilized to improve patient care and inform targeted interventions for at-risk populations.</p>" ]

[ "<title>Background and significance</title>", "<p>Computational understanding of the free text in clinical notes is well known to be an open challenge, including the extraction of structured information from these documents.##REF##21233086##18## Some progress has been made in extracting SDoH factors from clinical text using named entity recognition (NER), a Natural Language Processing (NLP) method of extracting pre-defined concepts from text.##REF##30975223##19##^,##REF##37080559##20## Both machine learning-based and traditional rule-based NER have been developed and tested.##REF##37080559##20–22## While NER approaches have been shown to be effective, they can be time-consuming.##UREF##5##23##</p>", "<p>Topic modeling methods have been widely applied for unbiased topic discovery from large collections of documents##UREF##6##24–26## and have been used in the fields of social science,##UREF##9##27## environmental science,##UREF##10##28## political science,##UREF##11##29## and in biological and medical contexts.##REF##35202844##12## Recent studies, such as work by Meaney et al,##REF##35202844##12## have begun to explore latent topics in clinical notes. However, to our knowledge, topic modeling has not been heavily used to assess corpora of social work notes for SDoH factors, likely due to the general availability of sufficiently large corpora.</p>", "<p>Clinical social workers are licensed professionals who specialize in identifying and addressing social and environmental barriers experienced by patients. In particular, text notes documented by clinical social workers are an invaluable data resource for understanding SDoH information in patients. As such, the clinical notes written by social workers often include specific text capturing an individual’s SDoH. Yet, to date, social work notes have been a relatively under-utilized data source and have not been extensively investigated for understanding SDoH.##REF##34549294##30##</p>", "<p>This study aims to explore the potential of social work notes as a rich source of data on SDoH by analyzing the most meaningful social work terminology across different disease chapters and applying LDA topic modeling to identify robust topics of discussion within a large cohort of social work notes. By doing so, we seek to uncover clinically relevant SDoH information contained in these notes and their potential impact on patient and public health, demonstrating the value of social work notes in understanding SDoH factors.</p>", "<title>Supplementary Material</title>" ]

[ "<title>Acknowledgments</title>", "<p>We thank all researchers, clinicians, and social workers who helped create and collect the deidentified clinical notes in our UCSF Information Commons. We thank everyone in Dr Atul J. Butte’s lab for the helpful discussion and feedback. We thank staff members in the Bakar Computational Health Sciences Institute and UCSF IT Services who build and maintain the UCSF Information Commons. We thank the Wynton High-Performance Computing (HPC) cluster for making available the needed computation capacity.</p>", "<title>Author contributions</title>", "<p>A.J.B. put forward the research idea. S.S., M.S., and A.J.B. designed the study. S.S. developed the methods, analyzed the data, and drafted the article. A.J.B. and M.S. supervised the study. All authors contributed to manuscript review and editing.</p>", "<title>Funding</title>", "<p>This publication was supported by the National Center for Advancing Translational Sciences, National Institutes of Health, through UCSF-CTSI Grant Number UL1 TR001872. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the NIH.</p>", "<title>Conflict of interest</title>", "<p>A.J.B. is a co-founder and consultant to Personalis and NuMedii; consultant to Mango Tree Corporation, and in the recent past, Samsung, 10x Genomics, Helix, Pathway Genomics, and Verinata (Illumina); has served on paid advisory panels or boards for Geisinger Health, Regenstrief Institute, Gerson Lehman Group, AlphaSights, Covance, Novartis, Genentech, and Merck, and Roche; is a shareholder in Personalis and NuMedii; is a minor shareholder in Apple, Meta (Facebook), Alphabet (Google), Microsoft, Amazon, Snap, 10x Genomics, Illumina, Regeneron, Sanofi, Pfizer, Royalty Pharma, Moderna, Sutro, Doximity, BioNtech, Invitae, Pacific Biosciences, Editas Medicine, Nuna Health, Assay Depot, and Vet24seven, and several other non-health related companies and mutual funds; and has received honoraria and travel reimbursement for invited talks from Johnson and Johnson, Roche, Genentech, Pfizer, Merck, Lilly, Takeda, Varian, Mars, Siemens, Optum, Abbott, Celgene, AstraZeneca, AbbVie, Westat, and many academic institutions, medical or disease specific foundations and associations, and health systems. AJB receives royalty payments through Stanford University, for several patents and other disclosures licensed to NuMedii and Personalis. A.J.B.’s research has been funded by NIH, Peraton (as the prime on an NIH contract), Genentech, Johnson and Johnson, FDA, Robert Wood Johnson Foundation, Leon Lowenstein Foundation, Intervalien Foundation, Priscilla Chan and Mark Zuckerberg, the Barbara and Gerson Bakar Foundation, and in the recent past, the March of Dimes, Juvenile Diabetes Research Foundation, California Governor’s Office of Planning and Research, California Institute for Regenerative Medicine, L’Oreal, and Progenity. The authors have declared that none of these entities affected the research or its results.</p>", "<title>Data availability</title>", "<p>The data that support the findings of this study are available from the Information Commons platform at UCSF, but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the authors upon reasonable request and with permission of UCSF.</p>" ]

[ "<fig position=\"float\" id=\"ooad112-F1\"><label>Figure 1.</label><caption><p>Retrieval of clinical social work notes for the study. The social work notes from the UCSF Information Commons between 2012 and 2021 were initially retrieved. Notes that were duplicated or extremely short were excluded, which resulted in a corpus of 0.95 million notes. Later, the notes were analyzed using 2 methods: word frequency calculation (Bottom Left) and topic modeling (Bottom Right). Later, the word frequency was compared between different disease chapters. For topic modeling, Latent Dirichlet Allocation was used to identify the topics in individual social work notes. Topic coherence metric and Jaccard distance were implemented to decide the optimal clustering results.</p></caption></fig>", "<fig position=\"float\" id=\"ooad112-F2\"><label>Figure 2.</label><caption><p>Topic proportion comparison for different categories. (A) Topic proportion comparison for different note types. (B) Topic proportion comparison for different disease chapters. Size and color of the circle represent proportion of each topic.</p></caption></fig>", "<fig position=\"float\" id=\"ooad112-F3\"><label>Figure 3.</label><caption><p>Word frequency calculation for social work notes associated with each ICD-10 chapter. The proportion of the words in social work notes associated with each ICD-10 chapter is shown by the heatmap.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"ooad112-T1\"><label>Table 1.</label><caption><p>Topic modeling results for all social work notes.</p></caption><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col valign=\"top\" align=\"left\" span=\"1\"/><col valign=\"top\" align=\"left\" span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\">Clusters</th><th rowspan=\"1\" colspan=\"1\">Key words</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">goal, anxiety, problem, term, depression, mood, therapy, symptom, long, treatment</td></tr><tr><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">recommendation, wife, education, treatment, patient, form, appearance, ongoing, advocate, trauma</td></tr><tr><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">hospital, self, day, pain, other, connection, recent, feeling, side, number</td></tr><tr><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">mother, father, family, room, information, nurse, source, concrete, control, instruction</td></tr><tr><td rowspan=\"1\" colspan=\"1\">5</td><td rowspan=\"1\" colspan=\"1\">session, consultation, telehealth, location, time, tool, objective, parking, other, treatment</td></tr><tr><td rowspan=\"1\" colspan=\"1\">6</td><td rowspan=\"1\" colspan=\"1\">parent, family, school, child, sister, support, place, year, well, initial</td></tr><tr><td rowspan=\"1\" colspan=\"1\">7</td><td rowspan=\"1\" colspan=\"1\">group, intervention, patient, discussion, response, time, summary, progress, participant, skill</td></tr><tr><td rowspan=\"1\" colspan=\"1\">8</td><td rowspan=\"1\" colspan=\"1\">risk, chronic, thought, normal, imminent, status, testing, intervention, speech, suicide</td></tr><tr><td rowspan=\"1\" colspan=\"1\">9</td><td rowspan=\"1\" colspan=\"1\">client, health, service, caregiver, mental, therapist, therapy, behavioral, individual, group</td></tr><tr><td rowspan=\"1\" colspan=\"1\">10</td><td rowspan=\"1\" colspan=\"1\">well, when, time, week, also, able, state, more, friend, very</td></tr><tr><td rowspan=\"1\" colspan=\"1\">11</td><td rowspan=\"1\" colspan=\"1\">social, service, support, family, assessment, medical, time, note, concern, ongoing,</td></tr><tr><td rowspan=\"1\" colspan=\"1\">12</td><td rowspan=\"1\" colspan=\"1\">care, home, plan, phone, contact, work, information, resource, call, support</td></tr><tr><td rowspan=\"1\" colspan=\"1\">13</td><td rowspan=\"1\" colspan=\"1\">time, clinician, name, date, code, behavior, risk, number, plan, provider</td></tr><tr><td rowspan=\"1\" colspan=\"1\">14</td><td rowspan=\"1\" colspan=\"1\">history, child, other, factor, current, none, substance, abuse, psychiatric, year</td></tr><tr><td rowspan=\"1\" colspan=\"1\">15</td><td rowspan=\"1\" colspan=\"1\">donor, donation, potential, employment, understanding, risk, decision, independent, process, care</td></tr><tr><td rowspan=\"1\" colspan=\"1\">16</td><td rowspan=\"1\" colspan=\"1\">night, morning, hour, sleep, house, already, less, past, aggressive, evening</td></tr><tr><td rowspan=\"1\" colspan=\"1\">17</td><td rowspan=\"1\" colspan=\"1\">transplant, medication, post, support, health, insurance, husband, psychosocial, message, history</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"ooad112-T2\"><label>Table 2.</label><caption><p>Topic assignment heuristic.</p></caption><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col valign=\"top\" align=\"left\" span=\"1\"/><col valign=\"top\" align=\"left\" span=\"1\"/></colgroup><thead><tr><th rowspan=\"1\" colspan=\"1\">Topics</th><th rowspan=\"1\" colspan=\"1\">Keywords</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">Mental health</td><td rowspan=\"1\" colspan=\"1\">mental, depression, anxiety, mood, psychological, physical, cognitive, emotional, mind, psychiatric</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Family</td><td rowspan=\"1\" colspan=\"1\">family, parent, father, mother, child, children, sister, parents, relatives, clan, childhood, friends</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Consultation/appointment</td><td rowspan=\"1\" colspan=\"1\">appointment, consultation, consult, questionnaire, question, advice, biographical, wikipedia, relevant, questions, know, documentation</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Group session</td><td rowspan=\"1\" colspan=\"1\">group, intervention, session, interpers, community, class, organization, together, part, organization</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Risk of death</td><td rowspan=\"1\" colspan=\"1\">suicide, suicidal, risk, crisis, homicide, murder, commit, bombing, murdered, murders, bomber, killing, convicted, victims</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Clinician/hospital/medication</td><td rowspan=\"1\" colspan=\"1\">patient, medication, hospital, medical, clinic, clinician, treatment, therapy, surgery, symptoms, patients, drugs, diagnosis, treatments, prescribed</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Living condition/lifestyle</td><td rowspan=\"1\" colspan=\"1\">shelter, housing, house, living, sleep, bedtime, building, buildings, urban, employment, suburban, campus, acres</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Social support</td><td rowspan=\"1\" colspan=\"1\">social, service, support, referral, recommendation, recommend, worker, resource, supports, provide, supporting, supported, allow, providing, assistance, benefit, help</td></tr><tr><td rowspan=\"1\" colspan=\"1\">TelephoneEcounter/online communication</td><td rowspan=\"1\" colspan=\"1\">telehealth, phone, call, video, telephone, mobile, wireless, gsm, cellular, dial, email, calling, networks, calls, messages, telephones, internet</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Abuse history</td><td rowspan=\"1\" colspan=\"1\">abuse, history, addiction, alcohol, drugs, allegations, victim, violence, sexual, rape, dependence</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Insurance/income</td><td rowspan=\"1\" colspan=\"1\">insurance, income, coverage, financial, contracts, banking, finance, liability, private, pay</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><fn id=\"tblfn2\"><p>Each row is an inferred topic, which is composed of 10 words.</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"tblfn1\"><p>The words in the <italic toggle=\"yes\">Keywords</italic> column are the representative words used to define the topics.</p></fn></table-wrap-foot>" ]

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[ "<title>Introduction</title>", "<p>To achieve a precise and prompt diagnosis, physicians should meticulously consider the interplay between patient history, thorough physical examination, and diagnostic imaging. This comprehensive approach frequently unveils distinctive and intriguing cases. The presence of odynophagia and posterior neck pain can occasionally signal serious infectious conditions, such as deep neck infection and meningitis. Symptoms associated with soft tissue calcification in the neck may also manifest similarly. While the differential diagnosis can be challenging, the synergistic application of a characteristic patient history, thorough physical examination, and diagnostic imaging can significantly enhance the quality of diagnostic assessments. </p>" ]

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[ "<title>Discussion</title>", "<p>Acute calcific tendinitis of the longus colli muscle (ACTLC) is characterized by the deposition of calcium hydroxyapatite and subsequent inflammation in the longus colli muscle, primarily affecting its superior oblique portion [##REF##14239862##1##,##REF##32720813##2##]. Common symptoms include neck pain (experienced by over 90% of patients), limited range of motion (in about half of patients), and neck stiffness (about a half) [##REF##19540570##3##]. ACTLC can impact the retropharyngeal space, leading to symptoms like odynophagia, dysphagia, sore throat, and difficulty in mouth opening [##REF##32720813##2##,##REF##19540570##3##]. Some cases manifest with torticollis [##UREF##0##4##]. Risk factors include repetitive trauma and recent injuries [##REF##23525848##5##]. The mechanism of developing ACTLC is not fully understood. One hypothesis suggests that trauma, degeneration, or ischemia of the tendon may lead to the deposition of crystals as a compensatory mechanism for reduced tendon quality [##UREF##1##6##]. ACTLC may go underdiagnosed due to its nonspecific symptoms, self-limiting nature, and lack of familiarity among physicians [##REF##23525848##5##].</p>", "<p>Diagnosis is supported by identifying calcification of the tendon anterior to the atlas on CT scans, found in about nine out of 8416 consecutive neck CT scans with no other apparent cause for the patient's symptoms [##REF##28744692##7##]. However, not all cases display calcification [##REF##32720813##2##]. Contrast-enhanced CT showing uniform fluid retention without rim-enhancing effects in the anterior space of the first to sixth cervical vertebrae and an absence of suppurative retropharyngeal lymphadenopathy or other structural abnormalities can substantiate the diagnosis [##REF##32720813##2##,##REF##23525848##5##]. Magnetic resonance imaging (MRI) can identify prevertebral edema and fluid effusion but lacks the capability to detect calcium deposits. Therefore, CT may work better than MRI in diagnosing prevertebral calcification [##REF##29145108##8##]. Differential diagnosis is crucial, especially to distinguish ACTLC from conditions like retropharyngeal abscess and meningitis [##REF##23525848##5##]. In some cases, the diagnosis becomes complex when bacterial infection occurs alongside asymptomatic calcification [##REF##33717788##9##,##REF##33795240##10##]. Given that ACTLC pain swiftly resolves with non-steroidal anti-inflammatory drug administration, close monitoring is essential [##REF##19540570##3##,##UREF##0##4##]. Physicians should be aware that ACTLC rarely occurs in individuals younger than 20 years old [##REF##28744692##7##].</p>" ]

[ "<title>Conclusions</title>", "<p>A typical history of preceding neck strain, symptoms including odynophagia and limited motion of the neck, and a CT finding of retropharyngeal calcification may indicate ACTLC. It is imperative for physicians to recognize the clinical characteristics of ACTLC and conduct thorough follow-ups, confirming that the pain completely dissipates within a few days, to exclude infection. This approach is essential for ensuring an efficient and accurate diagnosis.</p>" ]

[ "<p>A man in his 40s complained of posterior neck pain and headache after a local festival. The patient also developed mild fever, odynophagia, and difficulty opening his mouth widely. Physical examination revealed mild rightward torticollis and limited ranges of neck motion. A neck computed tomography (CT) revealed calcification on the tendon of the lingus colli muscle. The pain decreased rapidly after acetaminophen and loxoprofen administration. Physicians should recognize the clinical characteristics of acute calcific tendinitis of the longus colli muscle (ACTLC) and conduct thorough follow-ups to exclude infection.</p>" ]

[ "<title>Case presentation</title>", "<p>A previously healthy man at age 42 presented to our emergency department with complaints of posterior neck pain and headache. He reported participating in a local festival two days prior, during which he carried a Mikoshi, or portable shrine, by placing the carrying pole between his right shoulder and the right side of his neck. The neck pain and headache began one day before his presentation. On the morning of the presentation day, he experienced mild fever, pain on the left side of his throat when swallowing, and difficulty opening his mouth widely. Physical examination revealed mild rightward torticollis. While he was able to move his head back and forth with mild to moderate pain, he could not rotate it due to severe pain. There were no signs of redness in the pharynx and tonsils, and no tenderness was noted over the thyroid or jugular veins. There were no swollen or tender lymph nodes.</p>", "<p>Considering his history of carrying the Mikoshi, the limited head rotation, and the posterior neck pain accompanied by pain during swallowing, a diagnosis of acute tendinitis of the left longus colli muscle was suspected. A neck computed tomography (CT) revealed calcification on the tendon (Figure ##FIG##0##1##). He received 1000 mg of intravenous acetaminophen, leading to a rapid decrease in pain. He was prescribed 60 mg tablets of loxoprofen, and the pain subsided within a few days.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>(A) A horizontal view of the neck CT. An arrow indicates the calcification of the longus colli muscle. (B) A sagittal view. An arrow indicates the calcification</title><p>CT, computed tomography</p></caption></fig>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Junki Mizumoto</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Junki Mizumoto</p><p><bold>Drafting of the manuscript:</bold>  Junki Mizumoto</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Junki Mizumoto</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050599-i01\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Situs inversus (SI) is an autosomal recessive congenital abnormality characterized by a complete mirror reversal of both the abdominal and the thoracic organs [##REF##31181389##1##]. The normal human body’s composition is termed situs solitus with the classic external human body demonstrating bilateral symmetry with internal asymmetry [##REF##31181389##1##]. Even paired organs that we commonly think of, such as the lungs and kidneys, show some degree of asymmetry. This degree of asymmetry is especially important for surgeons as procedures can differ due to varying degrees of vascularity and the composition of the anatomy. In this article, we present a case of a 26-year-old male with a past medical history of suicidal ideations and gallstones complaining of left upper quadrant pain for two weeks.</p>" ]

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[ "<title>Discussion</title>", "<p>SI is a rare autosomal recessive congenital abnormality characterized by a complete mirror reversal arrangement of the internal organs along the left-right axis [##REF##32111882##2##]. While it is difficult to estimate the real frequency of this abnormality, a systematic review from Eitler et al. found that there was a 1:10,000 prevalence rate, with similar rates found in Adams et al. in 1937 and Lin et al. in 2000 [##REF##35264880##3##].</p>", "<p>One of the first reported cases of SI was by Aristotle who noticed this reversal in animals. In the 17th century, Fabricius and Blalock described the first case of a person with liver and spleen reversal, and Kuchenmeister in the 19th century initially recognized the condition in a living person through a thorough physical exam [##REF##35264880##3##]. In the modern era, initial imaging with ultrasound or plain films is the method of choice to view and clarify this abnormality [##REF##35264880##3##].</p>", "<p>Upon literature review, the first successful case of a laparoscopic cholecystectomy was performed by Mouret in 1987, whose technique has now become the gold standard [##UREF##0##4##]. In 1991, the first successful case of laparoscopic cholecystectomy in a patient with SI was performed by Campos and Sipes [##UREF##0##4##]. Since then, 91 successful cases have been reported in the literature without significant postoperative complications [##UREF##0##4##]. Thus, SI is not a contraindication to laparoscopic cholecystectomy. However, careful consideration must be met when deciding to operate on a patient with SI due to the left-right mirror reversal of the visceral organs that can make it difficult for an accurate diagnosis and surgical approach [##REF##23810920##5##]. While SI does not predispose one to increased gallbladder disease, it can lead to diagnostic confusion as most patients present with left upper quadrant abdominal pain [##REF##23810920##5##]. Ultrasonography or plain film X-rays are usually the initial choice for imaging, and CT or MRI can be used to view more detailed anatomy and possible pathological findings [##REF##23810920##5##].</p>", "<p>While there is no specific standard of operating procedure for laparoscopic cholecystectomy in patients with SI, the position of the trocars and consideration of the handedness of the lead surgeon are of utmost importance. In addition, rearrangement of surgical equipment and allowing extra time to recognize the mirrored visceral organs to prevent iatrogenic injury are crucial to success in the operating room [##REF##23810920##5##]. The advantage of doing laparoscopic procedures is that they can be modified and customized to a patient’s varying anatomy without significant deviation from standard procedures [##UREF##0##4##]. The most widely used technique for patients with SI is having the surgeon and camera assistant stand on the right side of the patient, while the first assistant and monitor are situated on the left [##REF##31181389##1##]. Salama et al. utilized two ports placed in the epigastric and subumbilical areas and two ports placed in the left midclavicular and left anterior axillary lines [##REF##23810920##5##]. Eitler et al. utilized the technique of placing an umbilical trocar for the camera and three trocars in the standard subcostal positions but on the left rather than the standard right [##REF##35264880##3##]. The surgeon can then hold the infundibulum (Hartmann’s pouch) with the left hand through the subxiphoid port and use the right hand to perform the dissection through the left midclavicular port [##REF##35264880##3##].</p>", "<p>While there is a preferential bias for left-handed surgeons operating on patients with SI, utilization of good ergonomics, solid knowledge of anatomy, and rearrangement of surgical instruments are good markers of a successful operation.</p>" ]

[ "<title>Conclusions</title>", "<p>This case represents a rare example of surgical management of cholecystitis in a patient with SI. Accurate diagnosis of cholecystitis can be made with a thorough understanding of anatomy as well as the utilization of imaging such as ultrasound and plain films. Once a diagnosis is made and the patient elects to undergo a laparoscopic cholecystectomy, extra time must be utilized to prepare an operating room that is ergonomically efficient for the surgeon and the team. A significant deviation from the standard procedure is not necessary except for the rearrangement of tools, the placement of trocars, and the position of the surgeon and assistants, which must be carefully considered. Despite the challenging nature of such a case, SI is not a contraindication to laparoscopic cholecystectomy.</p>" ]

[ "<p>Situs inversus (SI) is an autosomal recessive congenital abnormality in which there is a complete mirror reversal of visceral organs. In this article, we present the case of a 26-year-old male with a past medical history of suicidal ideations, gallstones, and SI who complained of left upper quadrant pain for two weeks. After admission for acute cholecystitis, he underwent a successful laparoscopic cholecystectomy without postoperative complications. Due to the anatomical deviation characteristic of SI, it can be challenging for surgeons to accurately diagnose and perform laparoscopic cholecystectomies. Careful consideration must be given when deciding to do a laparoscopic cholecystectomy, as the placement of not only the trocars and surgical instruments but also the position of the surgeon and assistants needs to be deliberated.</p>" ]

[ "<title>Case presentation</title>", "<p>A 26-year-old male presented to the emergency department with left upper quadrant and epigastric abdominal pain for two weeks. He had been diagnosed with gallstones two years prior but never pursued surgery. His medical history was significant for SI and gallstones with a previous history of a laparoscopic appendectomy. He was admitted for management of acute cholecystitis.</p>", "<p>Pertinent admitting labs included a complete blood count within normal limits, electrolytes within normal limits, elevated glucose, alkaline phosphatase within normal limits, alanine transaminase in the upper limit of normal, aspartate transaminase within normal limits, lipase within normal limits, and total bilirubin in the upper limit of normal (Table ##TAB##0##1##). A CT scan and abdominal ultrasound were performed, which showed gallbladder wall thickening and gallstones suggesting cholecystitis (Figure ##FIG##0##1##).</p>", "<p>While admitted, he was scheduled for a laparoscopic cholecystectomy. He was taken to the operating room on the same day as his presentation. He was placed on the operating room table, intubated, and started on general anesthesia. His abdomen was prepped and draped in a sterile fashion. A small incision was made just above the umbilicus, and a Veress needle was used to insufflate the abdomen to 15 mmHg. An additional 12-mm port was placed in the epigastric region, and two 5-mm ports were placed in the left upper quadrant. The abdomen was briefly explored. There were no signs of an iatrogenic injury. The gallbladder was grasped and retracted. It had omental adhesions that were taken down. The cystic duct and cystic artery were dissected circumferentially, clipped, and ligated. The gallbladder was removed from the gallbladder fossa using an electrocautery. The gallbladder was then removed from the abdomen in an EndoCatch bag. The epigastric port site was closed with a 0-Vicryl suture using a Carter-Thomason device, and the skin was closed with a 4-0 Monocryl. The patient was extubated, transferred to a gurney, and taken to the postanesthesia care unit for recovery. Figure ##FIG##1##2## demonstrates the port site placement. He had an uneventful postoperative recovery and was discharged home the following day.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>CT scan of the patient with SI. Stones in the gallbladder can be seen</title><p>The surgeon was on the patient's right side and the assistant on the patient's left side</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>Port sites for laparoscopic cholecystectomy in the patient with SI</title><p>Image credits: Andrew McCague</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Labs at the time of presentation</title><p>WBC: white blood cells, ALT: alanine transaminase, AST: aspartate transaminase</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Parameters</td><td rowspan=\"1\" colspan=\"1\">Results</td><td rowspan=\"1\" colspan=\"1\">Reference range</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Hemoglobin (g/dL)</td><td rowspan=\"1\" colspan=\"1\">15.8</td><td rowspan=\"1\" colspan=\"1\">Male 13.5-17.5 g/dL</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">WBC (cells/mcL)</td><td rowspan=\"1\" colspan=\"1\">10.7</td><td rowspan=\"1\" colspan=\"1\">4.5-11</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Platelet (×10⁹)</td><td rowspan=\"1\" colspan=\"1\">278</td><td rowspan=\"1\" colspan=\"1\">150-400</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Glucose (mg/dL)</td><td rowspan=\"1\" colspan=\"1\">116</td><td rowspan=\"1\" colspan=\"1\">70-110</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Hematocrit</td><td rowspan=\"1\" colspan=\"1\">45.4%</td><td rowspan=\"1\" colspan=\"1\">Male 41-53%</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Sodium (mEq/L)</td><td rowspan=\"1\" colspan=\"1\">136</td><td rowspan=\"1\" colspan=\"1\">136-146</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Potassium (mEq/L)</td><td rowspan=\"1\" colspan=\"1\">3.7</td><td rowspan=\"1\" colspan=\"1\">3.5-5.0</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Chloride (mEq/L)</td><td rowspan=\"1\" colspan=\"1\">98</td><td rowspan=\"1\" colspan=\"1\">95-105</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Bicarbonate (mEq/L)</td><td rowspan=\"1\" colspan=\"1\">23</td><td rowspan=\"1\" colspan=\"1\">22-28</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Total Bilirubin (mg/dL)</td><td rowspan=\"1\" colspan=\"1\">0.9</td><td rowspan=\"1\" colspan=\"1\">0.1-1.0</td></tr><tr><td rowspan=\"1\" colspan=\"1\">ALT (U/L)</td><td rowspan=\"1\" colspan=\"1\">40</td><td rowspan=\"1\" colspan=\"1\">10-40</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">AST (U/L)</td><td rowspan=\"1\" colspan=\"1\">27</td><td rowspan=\"1\" colspan=\"1\">12-38</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Alkaline phosphatase (U/L)</td><td rowspan=\"1\" colspan=\"1\">81</td><td rowspan=\"1\" colspan=\"1\">25-100</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Lipase (U/L)</td><td rowspan=\"1\" colspan=\"1\">39</td><td rowspan=\"1\" colspan=\"1\">0-160</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050598-i01\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050598-i02\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction and background</title>", "<p>Fatty liver disease, characterized by fat accumulation in the liver, has two primary variants: alcoholic fatty liver disease, or steatohepatitis, and non-alcoholic fatty liver disease (NAFLD). The latter, which is unrelated to excessive alcohol consumption, can be divided into two categories. First, there's a simple fatty liver, characterized by fat accumulation without significant hepatocyte damage or inflammation. The second category is non-alcoholic steatohepatitis (NASH), which involves fat buildup, hepatocellular injury, and inflammation [##REF##31428186##1##].</p>", "<p>Conversely, alcoholic steatohepatitis is a consequence of heavy alcohol consumption. It results from an immune system response to toxic metabolites generated during hepatic alcohol metabolism, leading to inflammatory harm to hepatocytes [##REF##30179104##2##]. This spectrum ranges from alcoholic fatty liver disease to alcoholic hepatitis and cirrhosis [##REF##31428186##1##,##REF##30179104##2##].</p>", "<p>The global landscape bears witness to an alarming escalation in fatty liver disease prevalence. It is important to note that the prevalence varies widely depending on the population studied and the definition used. However, it is estimated that NAFLD affects a quarter of the world's population [##REF##22488764##3##]. Additionally, nearly half of the afflicted individuals in the United States are middle-aged [##REF##22488764##3##]. Furthermore, estimates indicate that 20% to 30% of newly diagnosed NAFLD cases may have progressed to NASH, with a subsequent 10% to 20% progressing to cirrhosis or hepatocellular carcinoma [##REF##25461851##4##]. NASH remains the leading cause of liver disease among individuals awaiting liver transplants in the United States. The underpinning premise linking NAFLD to metabolic syndromes, incorporating insulin resistance, dyslipidemia, and type 2 diabetes mellitus, has garnered scholarly attention [##REF##22488764##3##].</p>", "<p>Conventionally, percutaneous liver biopsy serves as the gold standard for liver disease diagnosis [##REF##22707395##5##]. It affords confirmation of steatosis and quantification of fibrosis, ballooning, and lobular inflammation. Noteworthy scoring systems, such as the SAF score and NAFLD activity score, evaluate disease severity post-biopsy by scrutinizing fibrosis, activity, and steatosis parameters, yielding objective and comprehensive insights [##REF##22707395##5##]. However, inherent limitations encompass tissue sampling variability, inter-observer discrepancies, invasiveness-associated risks, and discomfort [##REF##24574716##6##]. Additional limitations of such scoring systems include sampling error, resource-intensive nature, and invasiveness [##REF##24574716##6##]. Serial biopsies, though informative, present several substantial challenges. These include their invasiveness, the associated risks, and the patient's reluctance to undergo multiple invasive procedures over time. Moreover, serial biopsies are resource-intensive and may not accurately represent the dynamic nature of the disease, potentially missing temporal changes. In turn, this limitation has a significant impact on patient care. The inability to perform serial biopsies hampers our ability to continuously monitor disease progression, assess the efficacy of treatments, and adapt patient care plans accordingly. It underscores the urgency of developing non-invasive methodologies for diagnosing, screening, and monitoring fatty liver disease [##REF##31391806##7##]. These non-invasive approaches are essential not only for reducing the burden on patients but also for ensuring that healthcare providers have access to real-time, comprehensive data for effective disease management and personalized treatment strategies.</p>", "<p>Artificial intelligence (AI), a field of computer science that focuses on the development of algorithms and models enabling machines to perform tasks that typically require human intelligence, has become a transformative force increasingly integrated into imaging and clinical screening systems to bolster diagnostic accuracy [##REF##22488764##3##]. This is especially true for machine learning and deep learning, which are subsets of AI. The past decade has witnessed AI's prowess in discerning patterns and correlations within vast datasets across medical disciplines, thus making it aptly poised for advancing disease diagnostics [##REF##35589253##8##].</p>", "<p>The existing literature on AI applications in diagnosing fatty liver disease highlights several important aspects. In a meta-analysis to evaluate the diagnostic accuracy and reliability of ultrasonography for detecting fatty liver [##REF##29427488##9##], ultrasonography was found to be a reliable method for diagnosing fatty liver, but there may be limitations in its accuracy. Barre et al. reviewed the application of AI in gastroenterology and hepatology and discussed its potential in aiding diagnosis and prognosis [##REF##31593701##10##]. They emphasized the need for further randomized, controlled studies to validate AI techniques. Furthermore, Lin et al. compared the diagnostic criteria for NAFLD and metabolic-associated fatty liver disease (MAFLD) in the real world [##REF##32478487##11##]. They highlighted the novelty of the MAFLD concept and the need for validation in real-world settings, which extends to diagnosis using AI.</p>", "<p>Moreover, Wai et al. reviewed the confounding factors of non-invasive tests for NAFLD [##REF##32451628##12##]. They emphasized the importance of considering these factors when interpreting test results. Finally, Decharatanachart et al. conducted a systematic review and meta-analysis on the application of AI in chronic liver diseases, reporting that AI techniques have the potential to diagnose liver fibrosis [##REF##33407169##13##]. Overall, the existing literature highlights the potential of AI in diagnosing fatty liver disease, but further research is needed to validate its effectiveness and address the limitations of current diagnostic methods. Our study attempts to fill in this gap in the literature.</p>", "<p>Notably, AI applications have played a pivotal role in liver disease management, encompassing the prognostication of liver decompensation, facilitating transplant recipient selection, and predicting transplant complications and survival [##REF##35589253##8##]. The integration of AI extends across various domains in healthcare, including electronic health records, digital pathology, and medical imaging. Within the field of medical imaging, AI augments diagnostic precision, particularly in the context of fatty liver diseases such as NAFLD. AI-driven diagnostic systems have exhibited remarkable accuracy, aided by a spectrum of commonly employed algorithms, including support vector machines (SVMs), convolutional neural networks (CNNs), multilayer perceptron, fuzzy Sugeno, and probabilistic neural networks (PNNs), among others. Despite these advancements, the existing literature lacks comprehensive systematic reviews or meta-analyses that synthesize the efficacy of AI-assisted diagnostic systems, especially concerning fatty liver diseases using imaging data. In line with this perspective, our systematic review aims to elucidate the performance and viability of AI-assisted systems in diagnosing fatty liver disease via imaging data.</p>" ]

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[ "<title>Conclusions</title>", "<p>In synthesis, this systematic review underscores the promise of AI-assisted diagnosis for various types of fatty liver disease, including NAFLD, through the integration of imaging data and diverse diagnostic methods. Across a diverse range of studies with varying sample sizes, AI classifiers, and geographical locations, AI consistently demonstrated impressive diagnostic accuracy, specificity, and sensitivity. Notably, neural network-based AI systems exhibited superior performance, emphasizing the role of advanced machine learning techniques in achieving remarkable diagnostic precision.</p>", "<p>The encouraging outcomes highlight AI's potential in identifying NASH, fibrosis, and steatosis, though the limited number of studies in certain areas necessitates cautious interpretation. Future large-scale studies are vital to substantiate the positive impact of AI-assisted diagnosis in fatty liver disease, paving the way for its transformative integration into clinical practice. Challenges and considerations in conducting these large-scale studies include the need for standardized AI algorithms, diverse patient populations, and real-world clinical settings for validation. Overcoming these challenges will be essential in ensuring the seamless integration of AI into the broader landscape of clinical practice and healthcare delivery, ultimately enhancing the precision and accessibility of fatty liver disease diagnosis.</p>" ]

[ "<p>Fatty liver disease, also known as hepatic steatosis, poses a significant global health concern due to the excessive accumulation of fat within the liver. If left untreated, this condition can give rise to severe complications. Recent advances in artificial intelligence (AI, specifically AI-based ultrasound imaging) offer promising tools for diagnosing this condition. This review endeavors to explore the current state of research concerning AI's role in diagnosing fatty liver disease, with a particular emphasis on imaging methods. To this end, a comprehensive search was conducted across electronic databases, including Google Scholar and Embase, to identify relevant studies published between January 2010 and May 2023, with keywords such as \"fatty liver disease\" and \"artificial intelligence (AI).\" The article selection process adhered to the PRISMA framework, ultimately resulting in the inclusion of 13 studies. These studies leveraged AI-assisted ultrasound due to its accessibility and cost-effectiveness, and they hailed from diverse countries, including India, China, Singapore, the United States, Egypt, Iran, Poland, Malaysia, and Korea. These studies employed a variety of AI classifiers, such as support vector machines, convolutional neural networks, multilayer perceptron, fuzzy Sugeno, and probabilistic neural networks, all of which demonstrated a remarkable level of precision. Notably, one study even achieved a diagnostic accuracy rate of 100%, underscoring AI's potential in diagnosing fatty liver disease. Nevertheless, the review acknowledged certain limitations within the included studies, with the majority featuring relatively small sample sizes, often encompassing fewer than 100 patients. Additionally, the variability in AI algorithms and imaging techniques added complexity to the comparative analysis. In conclusion, this review emphasizes the potential of AI in enhancing the diagnosis and management of fatty liver disease through advanced imaging techniques. Future research endeavors should prioritize the execution of large-scale studies that employ standardized AI algorithms and imaging techniques to validate AI's utility in diagnosing this prevalent health condition.</p>" ]

[ "<title>Review</title>", "<p>Methodology</p>", "<p>Search Strategy</p>", "<p>We followed the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) framework for our study's methodology. Our goal was to find research studies that used AI to diagnose and categorize fatty liver disease using imaging data. To do this, we looked for relevant articles in major databases like Google Scholar and Embase. We focused on articles published between January 2010 and May 2023, avoiding older ones to stay up to date with current AI techniques.</p>", "<p>We used a targeted search strategy with specific keywords and Boolean operators to ensure comprehensive results. Our search terms included \"fatty liver disease\" AND (\"AI\" OR \"artificial intelligence\") AND (\"diagnos*\" OR \"categoriz*\") AND (\"imaging data\" OR \"hepatic imaging\") AND (\"cirrhosis\" OR \"fibrosis\" OR \"steatohepatitis\" OR \"NASH\" OR \"NAFLD\" OR \"MAFLD\") AND (\"deep learning\" OR \"machine learning\" OR \"neural network\"). The protocol for this review was not registered. This decision was made to allow flexibility in responding to the rapid emergence of relevant research in the field and undergoing this research effort in a very quick manner, ensuring our systematic review remains up-to-date and comprehensive. Table ##TAB##0##1## shows the full search strategy utilized in the search.</p>", "<p>Inclusion and Exclusion Criteria</p>", "<p>To provide a clear and coherent focus within our study, we implemented specific inclusion and exclusion criteria. In selecting studies for inclusion, we prioritized those employing AI techniques in the context of fatty liver disease diagnosis. These selected studies were required to present sufficient data, specify the AI classification used, and provide comprehensive details about their diagnostic methodology. This emphasis on clear descriptions aimed to ensure the transparency and reliability of the studies included in our review.</p>", "<p>Conversely, our exclusion criteria were designed to maintain the rigor and quality of the review. We excluded studies that did not report the desired outcomes or those with insufficient data. Additionally, studies lacking vivid descriptions of validation cohorts and the characteristics of training or validation methods were omitted. The decision to exclude non-English studies was made to ensure the accessibility of our review to a wider audience and to minimize potential language-related interpretation issues.</p>", "<p>We also excluded abstracts, conference proceedings, editorials, and reviews that lacked sufficient data regarding source image features or the study population. Furthermore, studies without the requisite data for constructing the 2x2 table were excluded to maintain the analytical robustness of our review. These criteria were carefully applied to ensure the integrity and quality of our study, with the goal of providing a comprehensive and reliable analysis of AI-based diagnostic methods for fatty liver disease. For a comprehensive outline of the detailed inclusion and exclusion criteria, please consult Table ##TAB##1##2##.</p>", "<p>Data Extraction</p>", "<p>The screening of titles and abstracts was conducted by the main author to identify studies for further assessment in the full-text review phase. Following the completion of the screening process, the subsequent step involved data extraction, which was undertaken independently by the author and cross-verified as necessary. In cases where uncertainty arose regarding the eligibility of specific data for inclusion in the study, a comprehensive cross-checking procedure was revisited to ensure a thorough examination. The extracted data encompassed various specific details, including validation cohorts, training and validation method particulars, study design (whether retrospective or prospective cohort), study location, publication year, and authorship. Additionally, the data extraction process captured specificity and sensitivity values. In instances where investigations encompassed multiple AI classifiers, the focus was directed toward the AI classifier demonstrating optimal performance, as assessed by the highest area under the curve or superior accuracy.</p>", "<p>Results</p>", "<p>Literature Search</p>", "<p>Retrieved articles were entered into a reference software manager, and duplicates (n = 129) were removed. As illustrated in Figure ##FIG##0##1##, the titles and abstracts of all search results (n = 267) were independently screened by the investigators for relevance. Following duplication removal, of the remaining 138 articles, some were not original research (such as editorials and reviews) (n = 65), with other studies examining diseases other than any of the types of fatty liver disease (n = 14), studies that had focused on animal research (n = 12), and those that had not been documented in the English language (n =9). Some studies also failed to report validation population characteristics or the desired outcomes (n = 3), while others did not apply AI to imaging data to diagnose fatty liver disease (n = 5). Some studies focused on other parameters like histology, risk, and outcome prediction (n = 17). Eventually, 13 articles were found that employed AI-assisted ultrasonography. Ultrasound was predominantly employed in almost all the selected studies as the primary imaging modality. This preference for ultrasound was primarily due to its affordability and accessibility. Given its cost-effectiveness and widespread availability, ultrasound served as the imaging method of choice in the majority of these studies. Additionally, several of the selected studies harnessed the power of artificial neural networks (ANNs) to enhance the diagnostic capabilities of ultrasound. Some of the types of AI that the selected studies embraced included ANNs, CNNs, SVMs, and multiple AI models. ANNs and CNNs are specialized neural networks designed for image analysis, while SVM is a machine learning algorithm used for classification and regression tasks. The PRISMA protocol depicting this literature search process is highlighted in summary form, as shown in Figure ##FIG##0##1##.</p>", "<p>Table ##TAB##2##3## offers a summary of the findings regarding the characteristics of the studies and their associated findings.</p>", "<p>The studies included in this comprehensive review collectively underscore the transformative potential of AI integration in enhancing the accuracy and reliability of ultrasound analysis for fatty liver disease diagnosis. These selected studies reveal that the incorporation of AI into ultrasound analysis not only improves overall performance but also serves as an effective means to mitigate potential human errors, thereby enhancing diagnostic precision. Notably, the remarkable capacity of AI-assisted ultrasound for the early detection of steatosis stands out as a critical advancement in the field, offering the potential for timely intervention and improved patient outcomes.</p>", "<p>In our comprehensive review of various studies, we observed a wide array of AI classifiers being employed, predominantly focused on ultrasound diagnostics. The classifiers ranged from CNN, with specific instances like Inception v3, to more complex models such as fuzzy Sugeno CNN, PNN, decision tree (DT), SVM, Bayesian network models, the J48 algorithm, and voting-based classifiers. This diversity reflects the dynamic and evolving nature of AI applications in medical imaging.</p>", "<p>Delving into the methodologies of specific studies, Rhyou et al. employed a meticulous approach where medical experts annotated ultrasound images, categorizing them into four levels of steatosis: normal, mild, moderate, or severe. Subsequently, the dataset was divided into a 6:2:2 ratio for training, validation, and testing, ensuring a balanced approach for model training and performance evaluation. This methodological rigor aids in mitigating the risk of overfitting, a critical aspect of machine learning models [##REF##34450746##14##].</p>", "<p>Contrasting this, Zamanian et al. adopted a more comprehensive approach for network training by utilizing multiple image collections rather than the single image classification method prevalent in earlier studies. This broader approach likely enhances the model's ability to generalize better across various cases, an important consideration in the practical application of AI in medical diagnostics [##REF##33564642##20##].</p>", "<p>Furthermore, it is crucial to acknowledge the dependency of AI systems on their training and data quality. The effectiveness of these systems in diagnosing conditions like steatosis is heavily influenced by the data they are trained on, which was a key factor across all studies. Moreover, the role of the examiner is paramount in ensuring the accuracy of AI classifiers. The skill with which examiners provide relevant and quality material for AI training cannot be overstated and is a consistent theme across the reviewed literature, highlighting the synergy between human expertise and AI in advancing medical diagnostics.</p>", "<p>Sample sizes in the studies ranged from 100 in a Singapore-based study to an impressive 10,508 in a China-based study, reflecting a diverse spectrum of research populations and allowing for a comprehensive assessment of AI's effectiveness across different sample sizes. Furthermore, the AI classifiers employed in these studies exhibited remarkable diversity, including DTs, PNNs, and CNNs, underscoring the versatility of AI techniques in fatty liver disease diagnosis. These studies were conducted over a span of several years, from 2016 to 2022, demonstrating the continued evolution of AI methodologies in this field.</p>", "<p>In addition to the diversity in sample sizes and AI classifiers, the studies in this review were conducted across diverse populations and geographical regions, encompassing Korea, the United States, Malaysia, Poland, Singapore, Iran, and India. This diversity in study locations and patient demographics enriches the generalizability of the findings and highlights the global relevance of AI-assisted fatty liver disease diagnosis.</p>", "<p>Despite these promising advancements, there remain several avenues for further refinement that warrant scholarly exploration. To address the prevailing challenges faced by current AI classifiers, a concerted effort is needed to ameliorate specific aspects of the analysis process.</p>", "<p>Firstly, enhancing image accuracy upon extraction from expansive datasets is crucial. This challenge involves ensuring the quality and fidelity of input data to maintain the integrity of AI-assisted diagnostics. Secondly, the reduction of dimensionality for improved computational efficiency is paramount. Simplifying complex datasets and algorithms can significantly impact the speed and scalability of AI models, making them more practical and accessible for clinical use. Thirdly, mitigating issues related to computational time is essential. Improving the efficiency and speed of AI algorithms is a key consideration for real-time or near-real-time diagnosis. Lastly, bridging the semantic gap and addressing speckle noise in ultrasound images are critical steps toward harnessing the full potential of AI-based analysis, ultimately contributing to more reliable and precise diagnoses.</p>", "<p>The semantic gap refers to the disparity between the raw data in medical images and the clinical understanding required for an accurate diagnosis. This gap can hinder the performance of AI-based systems as they struggle to comprehend the nuanced features of pathology. For example, when interpreting ultrasound images, AI may encounter difficulties in distinguishing between benign and malignant lesions due to the subtleties in tissue appearance. Researchers are actively working on semantic segmentation models and natural language processing techniques to bridge this gap, enabling AI to provide more clinically relevant insights. Speckle noise, an inherent interference pattern in ultrasound images, can introduce artifacts and affect the accuracy of AI analysis. For instance, it may lead to false positives or negatives in the diagnosis of fatty liver disease. Solutions to mitigate speckle noise include advanced image processing techniques, such as despeckling filters and deep learning-based noise reduction algorithms. These strategies are pivotal in enhancing image quality and, subsequently, the reliability of AI-assisted diagnosis. Addressing these challenges in bridging the semantic gap and mitigating speckle noise is paramount to realizing the full potential of AI-based analysis. By developing and implementing effective solutions, researchers can contribute to more reliable and precise diagnoses, ultimately improving patient care and clinical decision-making.</p>", "<p>Furthermore, the exploration of parameter effects and optimization strategies is vital to driving continuous improvements in model performance. As the field advances toward enhanced accuracy and clinical applicability, the acquisition of more extensive and diverse datasets, encompassing inputs from various operators and patients across different healthcare centers, becomes pivotal. Additionally, expanding the consideration of disease stages and employing advanced image improvement techniques prior to analysis could yield enriched insights, further enhancing the diagnostic capabilities of AI in the context of fatty liver disease.</p>", "<p>In summary, the findings presented in this review not only highlight the significant achievements in AI-assisted fatty liver disease diagnosis but also underscore the potential areas for refinement within this rapidly evolving field. The integration of AI holds great promise for revolutionizing clinical practice by improving diagnostic accuracy, reducing errors, and facilitating early disease detection. Continued research efforts aimed at addressing existing challenges and optimizing AI-based diagnostic models will play a crucial role in maximizing the impact of AI in the diagnosis and management of fatty liver disease.</p>", "<p>Discussion</p>", "<p>This systematic review delved into diverse AI-assisted techniques for the diagnosis of fatty liver disease. The results unveiled a compelling narrative, showcasing the remarkable performance of AI-enhanced ultrasonography in diagnosing various types of fatty liver disease, particularly NAFLD. Notably, the majority of the studies reviewed reported very high diagnostic accuracy, with figures ranging from 83% to 100%. This range not only underscores the efficiency of AI-enhanced ultrasonography but also highlights its reliability. Moreover, these studies consistently demonstrated high specificity and sensitivity, reinforcing the potential of AI as a powerful tool in the early and accurate diagnosis of fatty liver diseases. Furthermore, this diagnostic prowess was complemented by the identification of relatively low heterogeneity across a substantial portion of the studies.</p>", "<p>However, it's important to recognize that variations in clinical input did contribute to instances of elevated heterogeneity. These variations may stem from differences in patient populations, clinical settings, or diagnostic methodologies across the included studies. Such heterogeneity is significant for the interpretation of study results, as it highlights the need for a nuanced understanding of AI's performance in diverse clinical scenarios. These variations underscore the importance of considering the context in which AI-assisted diagnostics are applied and the potential impact of these factors on diagnostic outcomes. Amidst these nuanced findings, a consistent theme emerged: when AI is seamlessly integrated into ultrasonography, the diagnostic landscape of fatty liver disease experiences a significant uplift. A crucial aspect to bear in mind is the widespread availability of ultrasonography across healthcare facilities and hospitals, characterized by its non-invasiveness and cost-effectiveness [##REF##22488764##3##]. However, the dependence on user interpretation in ultrasonography analysis could potentially introduce intra- and inter-observer variations, ultimately challenging the reliability of conventional ultrasonography in diagnosing fatty liver disease, including NAFLD. This systematic review substantiates the notion that the fusion of AI with ultrasonography effectively minimizes human-related errors, culminating in a promising enhancement of diagnostic performance. Consequently, these findings substantiate the robustness and advantages inherent in the utilization of AI-assisted ultrasonography. Nevertheless, the ongoing pursuit of clinical validation necessitates randomized controlled trials pitting conventional imaging modalities against AI-assisted systems to rigorously validate performance disparities.</p>", "<p>NAFLD diagnosis, particularly in the identification of fibrosis and NASH, holds significant clinical importance. Identifying the degree of fibrosis and the presence of NASH is pivotal for tailoring patient management strategies and determining the appropriate interventions. NASH, characterized by inflammation and hepatocellular injury, signifies a higher risk of disease progression and complications, making its accurate diagnosis crucial for timely therapeutic measures. AI's diagnostic acumen in identifying NASH appears encouraging, accompanied by an acceptable level of sensitivity. However, it's pertinent to acknowledge that the observed heterogeneity, albeit influenced in part by varying populations and diagnostic methods, prompts reflection on the scarcity of studies available. The limited study pool underscores the imperative for increased research efforts, fostering an expanded repertoire of investigations to enable more comprehensive and reliable analyses.</p>", "<p>In the pursuit of advancing fatty liver disease diagnostics, AI stands as a catalyst, offering insights into the potential of combining cutting-edge technology with clinical practice. The avenues for AI's integration are abundant, promising substantial improvements in accuracy and clinical decision-making across critical areas such as early disease detection, treatment selection, and patient management. The outcomes of this systematic review lay a foundation, both affirming current advancements and underscoring the critical necessity for continued research and robust validation efforts in the dynamic landscape of AI-enhanced diagnostics.</p>", "<p>Strengths and Limitations</p>", "<p>This systematic review leverages a rigorous methodology based on the PRISMA framework, ensuring comprehensive coverage of the relevant literature on AI-assisted diagnosis of fatty liver disease. The inclusion of major databases, such as Google Scholar and Embase, enhances the breadth of the literature search. The detailed search strategy, as presented in Table ##TAB##0##1##, facilitated the identification of a diverse range of studies addressing the intersection of AI and fatty liver disease diagnosis. Notably, this diversity includes studies from various geographical regions, adding an additional layer of depth and context to our analysis as well as improving the generalizability of the findings. The systematic approach employed for data extraction and analysis reinforces the reliability and validity of the findings.</p>", "<p>The review's focus on AI integration with ultrasonography contributes to its clinical relevance, given the widespread use of ultrasonography in healthcare settings. By highlighting the potential of AI in enhancing the accuracy and reliability of ultrasonography-based fatty liver disease diagnosis, this review underscores the imminent translational impact of AI technologies.</p>", "<p>Despite the comprehensive methodology employed, this review is not immune to certain limitations. Firstly, the limited number of studies retrieved from the literature review is a limitation to the level of evidence synthesized. The identified studies predominantly concentrated on AI applications in ultrasonography, potentially limiting the generalizability of the findings to other imaging modalities. The lack of studies directly comparing AI-assisted systems with conventional imaging modalities restricts the ability to draw conclusive performance comparisons. Furthermore, heterogeneity observed across studies, attributed to varying clinical inputs and methodologies, introduces a degree of complexity in the interpretation of the collective findings.</p>", "<p>Additionally, it's crucial to acknowledge that the exceptionally high accuracy rates reported in some studies may raise concerns about overfitting, wherein AI models perform exceedingly well on the training data but may not generalize optimally to new, unseen data. This aspect warrants careful consideration when interpreting the results. Moreover, the relatively limited number of studies addressing specific aspects, such as the identification of NASH, warrants cautious interpretation and calls for additional research efforts in these domains.</p>", "<p>The review's scope extends only to studies published in English, which may introduce potential language bias and exclude relevant non-English publications. Finally, while efforts were made to encompass a range of studies, it is plausible that some studies within the rapidly evolving field of AI-assisted diagnosis were not captured in the search.</p>", "<p>Future Research</p>", "<p>Moving forward, this systematic review spotlights crucial avenues for future research in the domain of AI-assisted diagnosis of fatty liver disease. While the integration of AI with ultrasonography has demonstrated remarkable potential, the need for head-to-head comparisons with conventional imaging modalities through randomized controlled trials remains paramount. Rigorous validation efforts are essential to not only corroborate the superiority of AI-assisted systems but also to ascertain the specific contexts in which these systems excel. However, it's essential to acknowledge the potential challenges and ethical considerations inherent in such trials, including issues related to patient consent, data security, and ensuring equitable access to AI-assisted diagnostic technologies.</p>", "<p>Moreover, as AI's diagnostic prowess expands beyond NAFLD diagnosis, the intricate landscape of fibrosis and NASH assessment warrants further exploration. The heterogeneous findings observed in this review underscore the necessity for a larger and more diverse pool of studies to provide a robust foundation for comprehensive analyses. As the field advances, it is imperative to consider not only diagnostic accuracy but also the integration of AI into clinical workflows and its potential impact on patient outcomes.</p>", "<p>Additionally, the refinement of AI algorithms to address the challenges of image accuracy, dimensionality reduction, computational efficiency, and noise reduction remains a crucial research frontier. Future studies should explore the optimization of AI models through the examination of parameter effects, leveraging expansive datasets from diverse patient populations and healthcare centers.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>PRISMA study selection flow diagram</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Comprehensive search strategy for identifying studies on AI-assisted diagnosis and categorization of fatty liver disease using imaging data</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Database</td><td rowspan=\"1\" colspan=\"1\">Search terms</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Google Scholar</td><td rowspan=\"1\" colspan=\"1\">(\"fatty liver disease\" AND (\"AI\" OR \"artificial intelligence\") AND (\"diagnos*\" OR \"categoriz*\") AND (\"imaging data\" OR \"hepatic imaging\") AND (\"cirrhosis\" OR \"fibrosis\" OR \"steatohepatitis\" OR \"NASH\" OR \"NAFLD\" OR \"MAFLD\") AND (\"deep learning\" OR \"machine learning\" OR \"neural network\")) AND (\"2010-01-01\" [Date - Publication]: \"2023-05-31\" [Date - Publication])</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Embase</td><td rowspan=\"1\" colspan=\"1\">(\"fatty liver disease\"/exp OR \"fatty liver disease\" OR \"hepatic steatosis\" OR \"non-alcoholic fatty liver disease\" OR \"NAFLD\" OR \"NASH\" OR \"non-alcoholic steatohepatitis\" OR \"MAFLD\" OR \"AI\" OR \"artificial intelligence\") AND (\"diagnos*\"/exp OR \"categoriz*\"/exp OR \"diagnos*\" OR \"categoriz*\") AND (\"imaging data\"/exp OR \"imaging data\" OR \"hepatic imaging\") AND (\"cirrhosis\" OR \"fibrosis\" OR \"steatohepatitis\" OR \"deep learning\" OR \"machine learning\" OR \"neural network\") AND (\"2010-01-01\" : \"2023-05-31\")</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>Detailed inclusion and exclusion criteria for the selection of studies on AI-assisted diagnosis and categorization of fatty liver disease using imaging data</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Inclusion criteria</td><td rowspan=\"1\" colspan=\"1\">Exclusion criteria</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Studies employing AI for fatty liver disease diagnosis</td><td rowspan=\"1\" colspan=\"1\">Studies not reporting desired outcomes or having insufficient data</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Studies providing sufficient data, specifying AI class, and detailing diagnostic method</td><td rowspan=\"1\" colspan=\"1\">Studies lacking clear descriptions of validation cohorts and training/validation methods</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Studies utilizing either definition for fatty liver disease (biopsy-based definition or imaging characteristics)</td><td rowspan=\"1\" colspan=\"1\">Studies reported in languages other than English</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Abstracts, conference proceedings, editorials, and reviews lacking adequate data on source image features or study population</td></tr><tr><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Studies without adequate data for constructing the 2x2 table</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB3\"><label>Table 3</label><caption><title>Summary of results from included studies on AI-assisted diagnosis and categorization of fatty liver disease using imaging data</title><p>CNN: convolutional neural network, PNN: probabilistic neural network, SVM: support vector machine, DT: decision tree</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Author</td><td rowspan=\"1\" colspan=\"1\">Year of publication</td><td rowspan=\"1\" colspan=\"1\">Country/research setting</td><td rowspan=\"1\" colspan=\"1\">Sample size </td><td rowspan=\"1\" colspan=\"1\">AI classifier</td><td rowspan=\"1\" colspan=\"1\">Diagnostic method</td><td rowspan=\"1\" colspan=\"1\">Sensitivity as extracted from the article</td><td rowspan=\"1\" colspan=\"1\">Specificity as extracted from the article </td><td rowspan=\"1\" colspan=\"1\">Diagnostic accuracy as extracted from the article</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Rhyou and Yoo [##REF##34450746##14##]</td><td rowspan=\"1\" colspan=\"1\">2021</td><td rowspan=\"1\" colspan=\"1\">Korea</td><td rowspan=\"1\" colspan=\"1\">3200</td><td rowspan=\"1\" colspan=\"1\">CNN (Inception v3)</td><td rowspan=\"1\" colspan=\"1\">Ultrasound</td><td rowspan=\"1\" colspan=\"1\">99.78%</td><td rowspan=\"1\" colspan=\"1\">100%</td><td rowspan=\"1\" colspan=\"1\">99.91%</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Acharya et al. [##UREF##0##15##]</td><td rowspan=\"1\" colspan=\"1\">2016</td><td rowspan=\"1\" colspan=\"1\">Singapore</td><td rowspan=\"1\" colspan=\"1\">100</td><td rowspan=\"1\" colspan=\"1\">Fuzzy Sugeno</td><td rowspan=\"1\" colspan=\"1\">Ultrasound</td><td rowspan=\"1\" colspan=\"1\">100%</td><td rowspan=\"1\" colspan=\"1\">100%</td><td rowspan=\"1\" colspan=\"1\">100%</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Gummadi et al. [##UREF##1##16##]</td><td rowspan=\"1\" colspan=\"1\">2020</td><td rowspan=\"1\" colspan=\"1\">U.S</td><td rowspan=\"1\" colspan=\"1\">905</td><td rowspan=\"1\" colspan=\"1\">CNN</td><td rowspan=\"1\" colspan=\"1\">Ultrasound</td><td rowspan=\"1\" colspan=\"1\">88.6%</td><td rowspan=\"1\" colspan=\"1\">95.3%</td><td rowspan=\"1\" colspan=\"1\">92.3%</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Acharya et al. [##REF##27825038##17##]</td><td rowspan=\"1\" colspan=\"1\">2016</td><td rowspan=\"1\" colspan=\"1\">Malaysia</td><td rowspan=\"1\" colspan=\"1\">150</td><td rowspan=\"1\" colspan=\"1\">PNN</td><td rowspan=\"1\" colspan=\"1\">Ultrasound</td><td rowspan=\"1\" colspan=\"1\">96%</td><td rowspan=\"1\" colspan=\"1\">100%</td><td rowspan=\"1\" colspan=\"1\">97.33%</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Byra et al. [##REF##30094778##18##]</td><td rowspan=\"1\" colspan=\"1\">2018</td><td rowspan=\"1\" colspan=\"1\">Poland</td><td rowspan=\"1\" colspan=\"1\">550</td><td rowspan=\"1\" colspan=\"1\">CNN</td><td rowspan=\"1\" colspan=\"1\">Ultrasound</td><td rowspan=\"1\" colspan=\"1\">100%</td><td rowspan=\"1\" colspan=\"1\">88.2%</td><td rowspan=\"1\" colspan=\"1\">96.3%</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Acharya et al. [##REF##22830759##19##]</td><td rowspan=\"1\" colspan=\"1\">2012</td><td rowspan=\"1\" colspan=\"1\">Singapore</td><td rowspan=\"1\" colspan=\"1\">100</td><td rowspan=\"1\" colspan=\"1\">DT</td><td rowspan=\"1\" colspan=\"1\">Ultrasound </td><td rowspan=\"1\" colspan=\"1\">88.9%</td><td rowspan=\"1\" colspan=\"1\">100%</td><td rowspan=\"1\" colspan=\"1\">93.3%</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Zamanian et al. [##REF##33564642##20##]</td><td rowspan=\"1\" colspan=\"1\">2021</td><td rowspan=\"1\" colspan=\"1\">Iran</td><td rowspan=\"1\" colspan=\"1\">550</td><td rowspan=\"1\" colspan=\"1\">SVM</td><td rowspan=\"1\" colspan=\"1\">Ultrasound</td><td rowspan=\"1\" colspan=\"1\">97.2%</td><td rowspan=\"1\" colspan=\"1\">100%</td><td rowspan=\"1\" colspan=\"1\">98.64%</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Neogi et al. [##UREF##2##21##]</td><td rowspan=\"1\" colspan=\"1\">2018</td><td rowspan=\"1\" colspan=\"1\">India</td><td rowspan=\"1\" colspan=\"1\">340</td><td rowspan=\"1\" colspan=\"1\">PNN</td><td rowspan=\"1\" colspan=\"1\">Ultrasound</td><td rowspan=\"1\" colspan=\"1\">100%</td><td rowspan=\"1\" colspan=\"1\">87%</td><td rowspan=\"1\" colspan=\"1\">99%</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Zhang et al. [##UREF##3##22##]</td><td rowspan=\"1\" colspan=\"1\">2019</td><td rowspan=\"1\" colspan=\"1\">China</td><td rowspan=\"1\" colspan=\"1\">500</td><td rowspan=\"1\" colspan=\"1\">CNN</td><td rowspan=\"1\" colspan=\"1\">Ultrasound</td><td rowspan=\"1\" colspan=\"1\">83%</td><td rowspan=\"1\" colspan=\"1\">95%</td><td rowspan=\"1\" colspan=\"1\">90%</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Han et al. [##REF##32096706##23##]</td><td rowspan=\"1\" colspan=\"1\">2020 </td><td rowspan=\"1\" colspan=\"1\">U.S</td><td rowspan=\"1\" colspan=\"1\">204 </td><td rowspan=\"1\" colspan=\"1\">CNN</td><td rowspan=\"1\" colspan=\"1\">Ultrasound</td><td rowspan=\"1\" colspan=\"1\">97%</td><td rowspan=\"1\" colspan=\"1\">94%</td><td rowspan=\"1\" colspan=\"1\">96%</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Reddy et al. [##UREF##4##24##]</td><td rowspan=\"1\" colspan=\"1\">2018</td><td rowspan=\"1\" colspan=\"1\">India</td><td rowspan=\"1\" colspan=\"1\">157</td><td rowspan=\"1\" colspan=\"1\">CNN </td><td rowspan=\"1\" colspan=\"1\">Ultrasound</td><td rowspan=\"1\" colspan=\"1\">95%</td><td rowspan=\"1\" colspan=\"1\">85%</td><td rowspan=\"1\" colspan=\"1\">90.6%</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Ma et al. [##REF##30402478##25##]</td><td rowspan=\"1\" colspan=\"1\">2018</td><td rowspan=\"1\" colspan=\"1\">China</td><td rowspan=\"1\" colspan=\"1\">10508</td><td rowspan=\"1\" colspan=\"1\">Bayesian network model</td><td rowspan=\"1\" colspan=\"1\">Ultrasound</td><td rowspan=\"1\" colspan=\"1\">67.5%</td><td rowspan=\"1\" colspan=\"1\">87.8%</td><td rowspan=\"1\" colspan=\"1\">83%</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Gaber et al. [##UREF##5##26##]</td><td rowspan=\"1\" colspan=\"1\">2022</td><td rowspan=\"1\" colspan=\"1\">Egypt</td><td rowspan=\"1\" colspan=\"1\">300</td><td rowspan=\"1\" colspan=\"1\">J48 algorithm, voting-based classifier</td><td rowspan=\"1\" colspan=\"1\">Ultrasound</td><td rowspan=\"1\" colspan=\"1\">97.05%</td><td rowspan=\"1\" colspan=\"1\">94.44%</td><td rowspan=\"1\" colspan=\"1\">95.71% (voting-based classifier), 93.12% (J48 algorithm)</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Yazan A. Al-Ajlouni, Basil N. Nduma</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Yazan A. Al-Ajlouni, Basile Njei</p><p><bold>Concept and design:</bold>  Basile Njei, Basil N. Nduma</p><p><bold>Supervision:</bold>  Basile Njei</p><p><bold>Drafting of the manuscript:</bold>  Basil N. Nduma</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050601-i01\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Ulnar collateral ligament (UCL) tear is a common injury sustained by overhead athletes that results from repetitive valgus stresses on the elbow joint [##REF##32112818##1##]. It’s becoming a more prevalent injury as baseball pitching velocity has increased, especially among youth baseball players [##REF##32112818##1##]. Recent statistics indicate an annual incidence of elbow injury ranging from 2.3% among adolescent pitchers to as high as 40.6% in youth pitchers, with a peak age range of 15 to 24 years [##REF##29335854##2##]. The UCL complex is comprised of three components: the anterior oblique, posterior oblique, and transverse bands. The anterior band is most commonly injured [##REF##33487638##3##]. Currently, treatment ranges from conservative to surgical intervention pending degree of injury with a growing trend towards conservative approaches.</p>", "<p>UCL tears typically present with symptoms such as pain on the medial side of elbow, reduced throwing velocity and accuracy, and sometimes a “pop” or tearing sensation during an athlete's throwing motion. Athletes may also experience elbow instability, leading to difficulty performing their sport-related activities [##REF##31028010##4##].</p>", "<p>UCL tears can be classified as grade 1, grade 2, and grade 3 injuries. Grade 1 tears are low-grade injuries that consist of edema around the ligament [##REF##31028010##4##]. Grade 2 tears are medium-grade injuries, with disruption to some of the ligament, and grade 3 tears are high-grade injuries that consist of full ligament disruption [##REF##31028010##4##,##REF##33330225##5##]. Low- and medium-grade tears can typically be managed non-operatively with relative rest and then a progressive throwing program over the course of a few months [##REF##29085844##6##].</p>", "<p>Typical conservative, non-operative, management consists of pain control with non-steroidal anti-inflammatory medications, relative rest and rehabilitation [##REF##32112818##1##]. Return to play for non-operative cases varies anywhere from 42-84% for those following a rehabilitation protocol [##REF##29085844##6##]. Platelet-rich plasma (PRP) has gained traction as a treatment option in the management of musculoskeletal conditions, due to its demonstrated ability in in vitro studies to increase proliferation of ligament cells and collagen creation [##REF##37588639##7##]. PRP uses an individual’s blood to harvest a concentrated injectate of platelets and growth factors. Thus far, there have been some case reports documenting the utility of PRP being used to successfully treat low- and medium-grade UCL tears, however, not high-grade tears [##UREF##0##8##]. In prior studies, use of PRP to treat partial tears resulted in an 88% success rate in returning athletes to play after an average of 12 weeks [##REF##29085844##6##].</p>", "<p>Those who sustain a high-grade tear upon injury as well as those who fail conservative management of low- and medium-grade tears are candidates for operative repair. The current standard surgical treatment for athletes with a ruptured, or high-grade, UCL tear looking to return to play is UCL reconstruction also known as “Tommy John” surgery. This procedure was initially developed by Dr. Frank Jobe, MD in 1974 and since its advent has significantly improved the prognosis of many overhead throwers returning to competition following UCL injury [##REF##29335854##2##]. Athletes who undergo surgical correction are typically unable to return to competitive play until 12-15 months post-op [##REF##32112818##1##].</p>", "<p>This case explores the utility of a minimally invasive approach including PRP injection for a high-grade UCL tear.</p>" ]

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[ "<title>Discussion</title>", "<p>The presented case illustrates an intriguing instance of a high-grade partial UCL tear in a young baseball pitcher managed successfully through non-operative means. UCL tears are a considerable concern among throwing athletes, primarily due to the high valgus stresses imposed on the elbow during overhead activities. While surgical intervention has traditionally been favored for high-grade tears, there is an emerging interest in conservative approaches to manage UCL injuries, especially for partial tears [##REF##29085844##6##]. The patient's responsiveness to non-operative management was notable in regard to decreased pain, increased function, improvement in the appearance of the ligament and return to activity. In comparison to traditional surgical intervention, his recovery was much faster requiring only 12 weeks of therapy before returning to pitching at full speed. In comparison, following surgical repair, a thrower is unable to begin a throwing protocol until four to five months post-op.</p>", "<p>Ultrasound-guided UCL PRP injection, a minimally invasive technique, has gained traction due to its potential to facilitate tissue healing and repair. PRP has demonstrated success in returning pitchers to throwing quickly and successfully 73-96% of the time following medium-grade UCL tears or UCL insufficiency [##REF##37588639##7##]. Typically in high-grade UCL tears, PRP has shown little benefit, with a 12.5% success rate while low-grade I and II tears are more frequently successfully treated with PRP and physical therapy [##UREF##0##8##]. The combination of PRP with a structured physical therapy program tailored to the individual's needs likely played a pivotal role in the patient's recovery.</p>", "<p>Dynamic ultrasound imaging provided a real-time assessment of the ligament's healing process and joint widening. This underscores the value of such imaging modalities in monitoring the response to treatment, potentially reducing unnecessary invasive interventions [##REF##32112818##1##,##REF##29335854##2##]. This also allows for earlier referral for surgical intervention in the event healing is not demonstrated during ultrasound evaluation. Dynamic testing can be performed by applying a valgus stress to the elbow while maintaining visibility of the UCL on ultrasound. Here, the UCL can be assessed for any changes within the ligament itself (hypoechoic areas), as well as increased laxity in the UCL. The patient's ability to resume throwing activities without pain and with a healthy-appearing UCL on ultrasound indicates that conservative management can yield satisfactory outcomes.</p>", "<p>While this case is encouraging, individualized treatment strategies remain crucial. Factors such as tear severity, patient compliance, and the specific demands of the sport must be considered [##REF##30963080##9##]. Additionally, it is important to consider the potential complications that may be associated with use of PRP injections. This includes but may not be limited to onset of ulnar nerve irritation [##REF##33796592##10##]. Further research is warranted to determine the optimal protocols for conservative management, including the role of PRP injections, the length of time PRP resolves symptoms, the rate of re-tearing of the UCL after PRP injection, and tailored rehabilitation programs in various grades of UCL tears.</p>" ]

[ "<title>Conclusions</title>", "<p>This case report sheds light on the potentially successful opportunity for non-operative management of a high-grade partial UCL tear in a competitive baseball pitcher. The patient's significant improvement in pain, coupled with objective findings of structural healing and joint widening on dynamic ultrasound, demonstrates the potential effectiveness of guided PRP injection and intensive physical therapy in promoting tissue repair and functional recovery. While surgical intervention remains a common approach, this case underscores the importance of considering conservative strategies, especially for partial UCL tears. Tailored rehabilitation programs and close monitoring through dynamic imaging modalities can contribute to positive outcomes. Nonetheless, a personalized approach, accounting for tear severity, patient compliance, and athletic demands, remains paramount in determining the most appropriate management pathway. Further research is necessary to refine non-operative protocols for UCL injuries and to establish their broader applicability within the spectrum of throwing-related sports injuries.</p>" ]

[ "<p>Ulnar collateral ligament (UCL) tears of the elbow are prevalent injuries among throwing athletes and are associated with excessive or repeated valgus forces at the elbow. We present the case of an 18-year-old male baseball pitcher with an 18-month history of progressive right elbow pain, notably worsened during his fastball pitching. Clinical assessment revealed tenderness with dynamic stressing of the right UCL. Imaging analyses, including magnetic resonance imaging (MRI) and dynamic ultrasound, confirmed a high-grade partial tear of the UCL at its origin. Non-operative management was pursued, which included an ultrasound-guided platelet-rich plasma (PRP) injection and intensive physical therapy. Follow-up evaluations at six and 12 weeks demonstrated a noteworthy improvement in subjective pain descriptions and structural healing of the UCL. After the patient completed a therapy and rehabilitation program, throwing activities at full strength were able to be resumed. This case underscores the potential efficacy of conservative approaches in handling UCL tears with the inclusion of PRP as a viable treatment option.</p>" ]

[ "<title>Case presentation</title>", "<p>Case history</p>", "<p>An 18-year-old right-hand-dominant competitive high school baseball pitcher presented for evaluation of right elbow pain. Over the course of the past 18 months, the pain had progressively worsened. Notably, the pain's intensity varied based on the type of pitch thrown and was most pronounced during fastball deliveries, reaching speeds of up to 90 miles per hour. As a starting pitcher with a rotation every five days, he experienced pain after completing three innings. Importantly, while the pain did not impact his pitching speed, it did influence his choice of pitches. The discomfort was localized to the inner aspect of the right elbow. Notably, the patient maintained consistent pitching mechanics and had not pursued any form of physical therapy before seeking medical attention. He denied experiencing any radiation of pain, numbness, or weakness in the right arm. Furthermore, his medical history was unremarkable, with no previous injuries, surgeries, or instances of trauma.</p>", "<p>Physical exam</p>", "<p>The patient's posture was consistent with their age and gender, showing no significant deformities or abnormalities in alignment. There were no signs of swelling, redness, or muscle wasting. Upon palpation, tenderness was evident at the points corresponding to the origin and insertion of the ulnar collateral ligament in the right elbow. Dynamic stressing of the right UCL triggered pain. However, no outright instability of either the UCL or lateral collateral ligament (LCL) was observable. The patient's radial and ulnar pulses were graded as 2+, and sensation from C5 to T1 dermatomes was intact and symmetrical. Furthermore, strength testing of the right elbow yielded normal results. MRI demonstrated a near full-thickness, at least high-grade partial, tearing of the anterior bundle proximal portion of the UCL attachment on the distal aspect of the medial epicondyle (Figure ##FIG##0##1##). The zoomed image is from the associated diagnostic ultrasound which demonstrates hypoechoic thickening and loss of normal ligamentous echotexture involving the proximal portion of the anterior band of the UCL (Figure ##FIG##0##1##).</p>", "<p>Treatment and patient course</p>", "<p>The MRI of the right elbow unveiled a high-grade partial tear originating from the UCL. A dynamic ultrasound assessment of the right elbow revealed UCL stability. Opting for a non-operative approach, the patient underwent an ultrasound-guided PRP injection into the affected UCL. Subsequently, the patient engaged in an intensive three-month physical therapy regimen, encompassing graduated exercises and progressive simulations of throwing.</p>", "<p>At the initial six-week follow-up, the patient reported a notable reduction in subjective pain. A limited ultrasound at this point indicated an enhancement in ligamentous echotexture and an expansion in joint space. Accordingly, an additional six weeks of therapy were recommended. Upon evaluation at the twelve-week mark, the patient's pain had fully subsided. The limited ultrasound imagery at this juncture displayed a healthy appearance of the UCL, with sustained joint expansion (Figure ##FIG##1##2##). Following the completion of the 12-week therapy protocol, the patient embarked on throwing exercises from the pitcher's mound, commencing at 50% effort and progressively advancing to full-strength throwing.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Magnetic Resonance Imaging and Ultrasound of the right elbow. </title><p>Magnetic resonance imaging (MRI) demonstrates a near full-thickness, at least high-grade partial, tearing of the anterior bundle proximal portion of the ulnar collateral ligament (UCL) attachment on the distal aspect of the medial epicondyle. The zoomed image is from the associated diagnostic ultrasound which demonstrates hypoechoic thickening and loss of normal ligamentous echotexture involving the proximal portion of the anterior band of the ulnar collateral ligament.</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>Post-treatment 12-week ultrasound. </title><p>Demonstrates improved ligamentous echotexture and a healthy-appearing ulnar collateral ligament (arrow). </p></caption></fig>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Rock P. Vomer II, Emma York, George G. A. Pujalte, Sara Memon, Daniel P. Montero, Shane Shapiro, Chris Fungwe</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Rock P. Vomer II, Emma York, George G. A. Pujalte, Sara Memon, Daniel P. Montero, Shane Shapiro, Chris Fungwe</p><p><bold>Drafting of the manuscript:</bold>  Rock P. Vomer II, Emma York, George G. A. Pujalte, Sara Memon, Daniel P. Montero, Shane Shapiro, Chris Fungwe</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Rock P. Vomer II, Emma York, George G. A. Pujalte, Sara Memon, Daniel P. Montero, Shane Shapiro, Chris Fungwe</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050600-i01\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050600-i02\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"P3\">Photoacoustic (PA) imaging combines optical contrast with ultrasound (US) image formation. It exploits the PA effect in which time-modulated light is absorbed in chromophores within biological tissue, inducing differential thermoelastic expansion forming acoustic waves (or PA signals) [##UREF##0##1##], [##UREF##1##2##]. The common imaging framework is to emit a sequence of nanosecond laser pulses into the tissue region of interest (ROI) and detect induced PA signals using an array of acoustic sensors [##REF##22866233##3##], [##REF##32708869##4##], [##REF##19175133##5##]. The primary advantage of this approach compared to other pure optical methods is that light diffusion within tissue does not affect PA image resolution at any depth because it is entirely defined by US (acoustic resolution). This enables optical detection of vascular diseases and cancers or monitoring physiological changes in relatively deep tissue areas [##UREF##2##6##], [##UREF##3##7##], [##UREF##4##8##] through PA imaging of light absorption. Combined with medical US, PA imaging brings a molecular dimension using targeted molecular contrast agents such as dyes and various nanoparticles [##REF##27467727##9##], [##REF##36004990##10##].</p>", "<p id=\"P4\">Many PA systems have been developed that optimize the laser source, irradiation geometry, and sound detectors for a specific application [##REF##31871888##11##], [##REF##31463194##12##], [##REF##30923674##13##], [##REF##30603201##14##], [##UREF##5##15##], [##REF##30603199##16##], [##UREF##6##17##], [##UREF##7##18##], [##REF##36925681##19##], [##REF##35186358##20##], [##REF##31293884##21##], [##REF##35268261##22##]. Among them, PA-computed tomography (PACT) reconstructs a macroscopic image with relatively deep penetration (a few centimeters) by strategically combining PA signals emitted from the entire ROI and detected by multiple sensors [##REF##27086868##23##], [##UREF##8##24##]. Specifically, a broadened laser beam in tissue can access a large ROI for a short time, and a sensor array can acquire PA signals over time at different positions for each laser firing. The universal back-projection (UBP) algorithm derived from the spherical Radon transform reconstructs a quantitative map of optical absorption from PA measurements provided that the detection array is full view and full bandwidth [##UREF##9##25##], the laser fluence distribution in the medium is known, and the medium is acoustically and thermally homogeneous. PACT generally uses a hemisphere or cylindrical sensor array to surround a target, so the main applications have been small animal imaging or monitoring human breast disease [##REF##36925681##19##].</p>", "<p id=\"P5\">Although nearly ideal for small animal studies, the PACT model is limited for general clinical applications by the array geometry. Due to limited acoustic access, as well as cost and physical complexity, large-scale hemispherical and cylindrical arrays are impractical for most clinical applications. An alternative approach is to integrate an optical delivery system within a standard US scanner for interleaved, real-time PAUS imaging [##REF##30603199##16##], [##REF##33397941##26##], [##REF##25754364##28##], [##REF##28442746##29##], [##UREF##11##30##], [##REF##23221479##31##]. A handheld US transducer contains a piezoelectric sensor array. For conventional 1-D arrays providing real-time 2-D images, optical fibers or light-emitting diode (LED) sources [##REF##32349414##32##], [##REF##29234601##33##] are located near/on the transducer surface to deliver laser light into the tissue volume (see ##FIG##0##Fig. 1##). Since the transducer is planar or almost planar (typically convex), unlike the circular geometry, users can flexibly position it on any surface of the body. Laser and fiber delivery systems and scanning protocols have been developed to generate simultaneous US and PA images at real-time imaging rates (&gt;20 Hz) for clinical use [##REF##33397941##26##], [##REF##28442746##29##].</p>", "<p id=\"P6\">The primary advantage of the PAUS approach is that it leverages not only PA imaging but also the current state of the art in real-time US. Thus, it is appropriate for a number of clinical applications and is particularly well suited for image-guided interventional procedures, such as drug delivery and surgeries, where PA imaging provides a molecular dimension missing from current US guidance systems [##REF##25643081##34##], [##REF##32817994##35##]. Some companies have already released commercial PAUS scanners for animal studies, which display real-time PA and conventional US images. Many academic research groups have also implemented PAUS systems by simply modifying a commercial US platform. Schellenberg and Hunt [##REF##30073147##36##] specifically reviewed such systems and associated clinical trials.</p>", "<p id=\"P7\">However, PAUS imaging has not yet been routinely adopted for human applications in the clinic. The critical hurdles are still low image quality and inaccurate quantitative measures. Specifically, PA image reconstruction from raw sensor data represents a severely ill-posed problem due to the limited view and narrow bandwidth of clinical US transducers [##UREF##12##37##], [##REF##34585543##38##]. As a result, the PA image is far from a one-to-one map of optical absorbers, and a range of diverse artifacts complicates image interpretation. These images are further degraded by common acoustic issues such as reverberations or clutter, reflection artifacts, and speed of sound (SOS) aberrations [##REF##3514956##39##], [##UREF##13##40##], [##UREF##14##41##].</p>", "<p id=\"P8\">Designing a transducer array simultaneously optimizing PA and US imaging poses significant challenges given the practical constraints of handheld operation. In medical US, to effectively convert electrical power to acoustic waves and form transmit acoustic beams, US transducers must be relatively narrowband. As a consequence, US B-mode images exhibit speckle due to local heterogeneities in US scattering. A large transducer aperture and a broad view are not usually required, and for most applications, the transducer should be quite compact to enable access to different organs within the human body.</p>", "<p id=\"P9\">In contrast, a PA image is formed by the distribution of heat release in the medium induced by pulsed laser irradiation. Reconstructing the spatial distribution of heat release is mathematically very different from reconstructing a local scattering function. It requires a detection system with ultrabroadband detection and a geometry that captures all potential propagation paths from sources (i.e., full view). Consequently, the optimal detection configurations are very different for US and PA modalities.</p>", "<p id=\"P10\">Recent studies have explored a hemispherical handheld array [##UREF##15##42##] to enhance PA image quality using tomographic reconstruction, but this probe is also suboptimal for conventional US imaging due to its limited effective field of view [##REF##30073147##36##], [##UREF##15##42##]. Standard US probes (transducers) are typically designed as linear or convex arrays, taking various factors into account, including not only the scan view but also cost, image quality, scanning convenience, and clinical applications. We believe that for widespread adoption of PAUS systems, the probe and system must be optimized for high-quality, real-time US imaging. This means that PA image quality will be sacrificed. If the transducer characteristics and geometry cannot be optimized in the PAUS configuration, advanced reconstruction methods that can compensate for the transducer’s limited view and bandwidth are in high demand. Consequently, PA advanced reconstruction through data processing could potentially improve overall PAUS image quality to a level enabling largescale clinical applications such as procedure guidance. For instance, it can assist in guiding drug release to an optimal target position, ablation procedures, and biopsy needles [##UREF##16##43##], [##REF##37636547##44##].</p>", "<p id=\"P11\">Deep learning (DL) techniques [##REF##26017442##45##], [##UREF##17##46##], [##UREF##18##47##] have significantly impacted biomedical imaging in areas such as microscopy [##REF##31463194##12##], histology [##REF##33990804##48##], MRI [##UREF##19##49##], [##UREF##20##50##], and CT [##UREF##21##51##], [##REF##33637310##52##]. They also have the potential to address the primary limitations of PAUS imaging because of their strong generalizability and efficiency. To handle ill-conditioned problems, standard mathematical or handcrafted models require additional human knowledge, specific hypotheses, and/or physical phenomena that are often difficult to generalize for all data acquisition environments. In contrast, DL is a data-driven approach without priors that can be trained with many plausible data samples to capture the essential features of real cases.</p>", "<p id=\"P12\">Currently, many neural network types have demonstrated their superior ability to adapt to new data [##UREF##22##53##]. In addition, computational time is much lower compared to standard model-based techniques depending on iterative schemes. Thus, the core procedure is: 1) develop a concise DL network to automatically extract features that reduce data redundancy and narrow possible solutions against ill-posed conditions and 2) create optimal training samples to guide the network to adapt to a wide distribution of real samples.</p>", "<p id=\"P13\">Overall, DL studies for PA imaging were well summarized in [##REF##33837678##54##], [##REF##33717977##55##], [##REF##35529338##56##], [##REF##33425679##57##], and [##UREF##23##58##]. They encompassed various DL applications, including image reconstruction, image understanding (classification and segmentation), and quantitative imaging. In this article, we focus on imaging (specifically the reconstruction of initial pressure images) in the geometry provided by clinical US that enables PAUS imaging using handheld probes and conceptually review current work exploiting DL frameworks to overcome fundamental PAUS limitations. Specifically, in <xref rid=\"S2\" ref-type=\"sec\">Section II</xref>, we outline the PA signal acquisition geometry and standard image reconstruction procedure. <xref rid=\"S6\" ref-type=\"sec\">Section III</xref> describes DL work, including in-silico, in vitro, or in vivo data generation and neural network construction to process these data. <xref rid=\"S10\" ref-type=\"sec\">Section IV</xref> summarizes the findings and discusses remaining challenges and opportunities.</p>" ]

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[ "<title>Conclusion and Discussion</title>", "<title>PAUS Imaging</title>", "<p id=\"P50\">PAUS imaging systems integrate a fiber-optic delivery system within a conventional clinical US transducer, enabling simultaneous PA and US imaging with flexible physical manipulation for human subject scanning. They can potentially image microvascular structures and blood perfusion in localized areas, leveraging the strong optical absorption of blood for contrast and high US frequencies for fine spatial resolution. Consequently, integrated PAUS imaging has the potential to detect and quantify vascular diseases such as atherosclerosis or stroke, as well as monitor angiogenesis. With a few cm light penetration depth (approximately 3–5 cm, depending on the optical scattering and absorption of background tissue for a specific application), PAUS imaging may also be well suited to image different forms of cancer, such as melanoma, ovarian, thyroid, muscle carcinoma, and breast cancers [##REF##31293884##21##], [##REF##27669961##108##].</p>", "<p id=\"P51\">Real-time PAUS imaging may also bring molecular sensitivity to conventional US. Spectroscopic PA imaging leverages the optical absorption spectrum of molecules to identify specific species in the body. Endogenous molecular imaging primarily exploits the molecular characteristics of hemoglobin and, under controlled conditions, can measure the local oxygenation state of blood [##REF##22734732##61##]. Exogenous molecular imaging exploits the specific absorption spectrum of contrast agents for a range of applications in molecular diagnostics and therapy. Of particular interest are applications where molecular labeling is combined with real-time spectroscopic PAUS to guide interventions such as drug delivery, surgeries, and therapies [##REF##26148989##109##].</p>", "<p id=\"P52\">Although the promise of PAUS imaging is substantial, the poor image quality of PA images reconstructed using the limited view and bandwidth of handheld US arrays has severely limited clinical adoption. As discussed in <xref rid=\"S2\" ref-type=\"sec\">Section II</xref>, PA imaging to determine initial PA pressure (or heat function) serves as a crucial preliminary step before subsequent quantification of target absorbers. However, as evidenced by simulation results (see ##FIG##3##Fig. 4##) for this acquisition geometry, scanning areas encompassing numerous absorbers produce PA images with significant artifacts and shape-distortions using conventional approaches. These images are not accurate and do not faithfully depict the distribution of initial PA pressure. Here, we have presented a review of diverse DL frameworks focused on overcoming these limitations of clinical PAUS imaging with a handheld probe.</p>", "<title>DL Reconstructions in PAUS Imaging</title>", "<p id=\"P53\">In particular, DL techniques can mitigate the physical limitations imposed by the PAUS platform, potentially translating this important tool into clinical applications. As demonstrated in <xref rid=\"S2\" ref-type=\"sec\">Section II</xref>, PA image reconstruction (i.e., the acoustic inverse problem) is ill-posed for limited-view and limited-bandwidth data. To address this issue, target absorbers were inevitably constrained to point-like, small circular, or vascular objects. However, conventional methods like DAS or TR still produced low-contrast and low-resolution images with artifacts. The typical DL framework serves as a postprocessing tool, taking a conventionally reconstructed image (DAS or TR) as input and producing a higher quality output image. In particular, the UNET model is commonly chosen for this problem because both the input and output belong to the image domain and share the same size. In the model, encoder and decoder components are explicitly designed to extract and combine multiscale features, respectively.</p>", "<p id=\"P54\">To further enhance image quality, numerous studies utilizing the UNET framework extended the model’s access to additional information and modified the network structure to accommodate these changes. Additional information includes channel data, tensor data derived from channel data, combinations of channel data and a DAS image, or DAS images obtained using different sound speeds. In some cases, the model incorporates additional skip connections between the encoder and decoder or replaces existing skip connections with convolutional layers. These modifications access features at various levels of abstraction, enabling the network to leverage both low-level and high-level features for a more comprehensive representation.</p>", "<p id=\"P55\">In addition, alternative frameworks such as RNN or GAN have also been explored. For instance, one study combined RNN with CNN to address multiframe images and strategically average them. Another study adopted a GAN architecture, where the UNET-based generator aimed to generate images that closely resembled reference images by attempting to deceive the discriminator beyond standard loss metrics like mse. These novel frameworks offer additional avenues to improve PAUS image quality.</p>", "<p id=\"P56\">As the size of input data increases, there is a growing need for enhanced DL network efficiency. In line with this, as discussed in <xref rid=\"S6\" ref-type=\"sec\">Section III</xref>, several research groups have incorporated a simple attention mechanism into the UNET architecture to capture spatial dependencies beyond the limitations of a convolutional filter [##UREF##42##84##]. Currently, the vision transformer (ViT) [##REF##35180075##110##], which leverages the concept of self-attention, has performed at a high level for various medical imaging tasks by effectively capturing long-range dependencies. Unlike convolutional architectures, self-attention can capture relationships between different subsections in the full image domain regardless of their positions [##UREF##64##111##]. The ViT model is clearly a viable option for complex and voluminous PA data.</p>", "<p id=\"P57\">The majority of papers reviewed here focused on supervised approaches where in-silico data served as a reference for model training. However, the inherent discrepancy between in-silico training data and real test data is a major issue for clinical translation. Acquiring a large volume of ground-truth data (gold standards) is challenging, leading research groups to rely on synthetic data to train their models. Consequently, performance in a clinical setting heavily depends on the similarity between synthetic and real tissue models. To mitigate this issue, plausible ranges or distributions of optical parameter values for each tissue have been used, with the aim of facilitating successful transfer learning.</p>", "<p id=\"P58\">Despite these efforts, unexpected artifacts have been reported in many papers due to the underlying disparities between synthetic and real data. This is exemplified in ##FIG##8##Figs. 9## and ##FIG##9##10##, where the yellow arrows indicate the likelihood of these artifacts occurring. One presumable cause for these artifacts could be that the synthetic model is based on a 2-D geometry, while real data represent a 3-D environment. As discussed in <xref rid=\"S2\" ref-type=\"sec\">Section II</xref>, the influence of absorbers extends beyond the plane of interest (in-plane) to those located out-of-plane in PA data acquisition. Given that each transducer element has elevational directivity during acquisition, misinterpreting PA signals from out-of-plane sources as originating from in-plane sources could be a plausible explanation for such artifacts. Thus, the development of more realistic 3-D tissue and transducer models holds the potential to bridge the gap between training and test data, leading to artifact reduction.</p>", "<p id=\"P59\">An alternative that can address this challenge is unsupervised learning using approaches such as CycleGAN [##UREF##65##112##]. CycleGAN uses GANs for image-to-image translation or raw data-to-image translation without requiring paired training data. The fundamental idea behind CycleGAN is cycle consistency loss, which forces the translated image to be accurately reconstructed back to its original form. For instance, in the case of translating a horse image to a zebra image to deceive a discriminator, the translated zebra image is not random but rather constrained to closely resemble the original image due to the loss. In the field of medical imaging, CycleGAN learned mappings between different domains, such as CT and MRI [##UREF##66##113##], enabling image transfer while preserving content. In the context of PA imaging, where obtaining paired data is challenging, exploiting CycleGAN can provide a more convenient approach for image reconstruction tasks.</p>", "<p id=\"P60\">PAUS systems offer a significant advantage by providing both a standard US image and a PA image [##REF##34584840##114##]. This implies that DL networks can leverage US data to enhance PA imaging using acoustic features. For instance, US B-mode images can be particularly valuable in obtaining accurate SOS measurements. In a tissue domain with large vessels, US data can provide essential information about their position and shape, which may not be readily observed in PA images. In cases where involuntary movement occurs during scanning or data acquisition, US speckle can be used for motion tracking and compensation [##REF##33397941##26##]. Furthermore, two different domains, US and PA, present a promising opportunity for unsupervised learning such as CycleGAN, offering considerable enhancements in real-time PAUS images. They have the potential to strengthen PA/US dual-mode imaging [##UREF##67##115##], [##REF##30306043##116##], [##UREF##68##117##], offering complementary information that can enhance its translation into practical clinical applications.</p>", "<title>Extension of DL Frameworks</title>", "<p id=\"P61\">DL frameworks can be extended to further reinforce PAUS imaging. First, they can potentially mitigate clutter signals caused by sound reverberation between reflectors in tissue. For instance, Allman et al. [##UREF##69##118##] presented a CNN-based model designed to identify reflection artifacts and source signals, focusing on scenarios where the object was limited to point-like targets. In the domain of standard US imaging, numerous DL techniques have been introduced to suppress reverberation clutter [##UREF##70##119##], [##UREF##71##120##], [##UREF##72##121##].</p>", "<p id=\"P62\">Second, as 3-D US imaging evolves, free-hand PA imaging with DL can be extended to include 3-D imaging capabilities. In traditional US imaging, a 2-D matrix probe is employed to simultaneously scan multiple image planes and acquire volumetric data in real-time. However, this approach requires a higher cost to achieve high spatial resolution, and the computational demands are intense. One approach to tackle these challenges is the use of a sparse 2-D matrix probe, in which elements are intentionally skipped or spaced further apart to reduce the number of physical elements in the probe. In this context, DL plays a crucial role in enhancing image reconstruction from undersampled data, thus compensating for element reductions compared to dense arrays [##UREF##73##122##]. Additionally, panoramic imaging techniques can help create extended 3-D field-of-view images by stitching together multiple 2-D images obtained by sweeping a standard probe. Presently, DL methods were employed to estimate the probe’s position and movement without requiring any additional positional sensors [##UREF##74##123##], [##REF##29936399##124##].</p>", "<p id=\"P63\">Finally, DL frameworks hold the potential to enhance spectroscopic PA imaging, mapping chromophore concentration using PA spectra acquired at different wavelengths. Accurately estimating chromophore concentrations poses a significant challenge due to spectral distortion [##REF##22734732##61##], [##REF##32670789##125##]. The PA spectrum at a specific spatial position is influenced not only by the linear combination of intrinsic absorption spectra of chromophores at that position but also by wavelength-varying optical fluence. DL methods [##REF##29905680##126##], [##UREF##75##127##], [##UREF##76##128##], [##REF##32840068##129##], [##UREF##77##130##] offered a solution to this challenge by mapping from PA images acquired at multiple wavelengths, eliminating the need for prior knowledge such as fluence maps and intrinsic absorption spectra of chromophores. Additionally, spectroscopic PA imaging holds promise for automatic segmentation and isolation of target objects. Currently, multispectral imaging combined with DL techniques improved task performance [##REF##32840068##129##], [##UREF##78##131##], [##REF##35371919##132##].</p>", "<p id=\"P64\">Although these methods have been validated in simulation settings, comprehensive validation in vivo remains largely unexplored. Presently, tissue phantom models used to generate training data are often considered overly simplistic to adequately simulate real-world scenarios. To tackle this limitation, some researchers have embarked on alternate approaches, such as creating phantoms that integrate information from other imaging modalities. For instance, Yang and Gao [##UREF##76##128##] designed a 3-D heterogeneous tissue structure by leveraging an MR breast database, subsequently assigning optical and acoustic parameters to replicate a more realistic environment. In existing literature, PA images reconstructed using DAS or TR methods are commonly used as input to DL models [##REF##29905680##126##], [##UREF##75##127##], [##UREF##76##128##], [##REF##32840068##129##], [##UREF##77##130##]. Expanding upon this framework, two approaches will be pursued: 1) using two separate DL networks, where one network focuses on generating clear PA images from PA signals, while the other network is dedicated to generating the map of chromophore concentration from PA images and 2) alternatively, a single DL network will be trained using an end-to-end learning approach, directly mapping PA signals to the predicted concentration map.</p>", "<title>Challenges</title>", "<p id=\"P65\">A current challenge to DL approaches for PA reconstruction is the lack of performance comparisons between DL methods in the current literature, mainly due to individual model optimization using separate datasets. To address this issue, the availability of a publicly accessible framework to evaluate novel DL methods using identical reference data, including both phantom and real data, would prove invaluable. This framework would facilitate the comparison of different methods, thereby expediting the advancement of PA reconstruction techniques. Open frameworks have already been established in some medical imaging fields to address similar challenges. For instance, the Challenge on Ultrasound Beamforming with DL (CUBDL) was offered during the 2020 IEEE International Ultrasonics Symposium [##UREF##79##133##].</p>", "<p id=\"P66\">Additionally, a significant challenge is in vivo validation of all DL techniques. This presents significant hurdles in translating these techniques to clinical practice and, to date, none have achieved clinical translation. While many DL studies have been validated in simulation by quantifying mse-based metrics between estimates and the ground-truth of initial PA pressure, their validation has often been limited to in vitro phantom experiments. The focus of these studies to date has been to assess the clarity of reconstructed absorber shapes and the reduction of artifacts arising from ill-posed conditions. Although some studies have extended validation to in vivo scenarios, these evaluations mainly focus on the plausibility of reconstructed absorber shapes, which relies heavily on prior anatomical knowledge. Given the scarcity of quantitative ground-truth maps in vivo, most papers have omitted quantitative evaluation of their DL results. For instance, in situations where the target is blood vessels, the initial PA pressure can exhibit positional variations within vessels, a factor often overlooked in evaluations. Therefore, a compelling future challenge involves addressing this limitation and developing methodologies to quantify the performance of DL techniques in vivo.</p>", "<title>Conclusion</title>", "<p id=\"P67\">In conclusion, reconstructing PAUS images with DL is very new, but there is no doubt that this framework holds significant potential to improve the modality. With the advent of next-generation computing systems, more complex and realistic tissue models can be created through simulations in a shorter time. The development of efficient networks trained on large volumes of data will greatly facilitate successful transfer learning from virtual environments to real-world applications. Additionally, unsupervised techniques, especially those incorporating US data, can potentially improve performance and make DL systems more robust for real clinical applications. Similar to recent advances in CT and MR imaging resulting from DL tools, as PAUS systems and imaging techniques become more standardized, a wealth of patient data will become available. Abundant human subject data will provide ample opportunities to thoroughly evaluate DL methods, leading to increased trust and confidence in their clinical utility.</p>" ]

[ "<title>Conclusion and Discussion</title>", "<title>PAUS Imaging</title>", "<p id=\"P50\">PAUS imaging systems integrate a fiber-optic delivery system within a conventional clinical US transducer, enabling simultaneous PA and US imaging with flexible physical manipulation for human subject scanning. They can potentially image microvascular structures and blood perfusion in localized areas, leveraging the strong optical absorption of blood for contrast and high US frequencies for fine spatial resolution. Consequently, integrated PAUS imaging has the potential to detect and quantify vascular diseases such as atherosclerosis or stroke, as well as monitor angiogenesis. With a few cm light penetration depth (approximately 3–5 cm, depending on the optical scattering and absorption of background tissue for a specific application), PAUS imaging may also be well suited to image different forms of cancer, such as melanoma, ovarian, thyroid, muscle carcinoma, and breast cancers [##REF##31293884##21##], [##REF##27669961##108##].</p>", "<p id=\"P51\">Real-time PAUS imaging may also bring molecular sensitivity to conventional US. Spectroscopic PA imaging leverages the optical absorption spectrum of molecules to identify specific species in the body. Endogenous molecular imaging primarily exploits the molecular characteristics of hemoglobin and, under controlled conditions, can measure the local oxygenation state of blood [##REF##22734732##61##]. Exogenous molecular imaging exploits the specific absorption spectrum of contrast agents for a range of applications in molecular diagnostics and therapy. Of particular interest are applications where molecular labeling is combined with real-time spectroscopic PAUS to guide interventions such as drug delivery, surgeries, and therapies [##REF##26148989##109##].</p>", "<p id=\"P52\">Although the promise of PAUS imaging is substantial, the poor image quality of PA images reconstructed using the limited view and bandwidth of handheld US arrays has severely limited clinical adoption. As discussed in <xref rid=\"S2\" ref-type=\"sec\">Section II</xref>, PA imaging to determine initial PA pressure (or heat function) serves as a crucial preliminary step before subsequent quantification of target absorbers. However, as evidenced by simulation results (see ##FIG##3##Fig. 4##) for this acquisition geometry, scanning areas encompassing numerous absorbers produce PA images with significant artifacts and shape-distortions using conventional approaches. These images are not accurate and do not faithfully depict the distribution of initial PA pressure. Here, we have presented a review of diverse DL frameworks focused on overcoming these limitations of clinical PAUS imaging with a handheld probe.</p>", "<title>DL Reconstructions in PAUS Imaging</title>", "<p id=\"P53\">In particular, DL techniques can mitigate the physical limitations imposed by the PAUS platform, potentially translating this important tool into clinical applications. As demonstrated in <xref rid=\"S2\" ref-type=\"sec\">Section II</xref>, PA image reconstruction (i.e., the acoustic inverse problem) is ill-posed for limited-view and limited-bandwidth data. To address this issue, target absorbers were inevitably constrained to point-like, small circular, or vascular objects. However, conventional methods like DAS or TR still produced low-contrast and low-resolution images with artifacts. The typical DL framework serves as a postprocessing tool, taking a conventionally reconstructed image (DAS or TR) as input and producing a higher quality output image. In particular, the UNET model is commonly chosen for this problem because both the input and output belong to the image domain and share the same size. In the model, encoder and decoder components are explicitly designed to extract and combine multiscale features, respectively.</p>", "<p id=\"P54\">To further enhance image quality, numerous studies utilizing the UNET framework extended the model’s access to additional information and modified the network structure to accommodate these changes. Additional information includes channel data, tensor data derived from channel data, combinations of channel data and a DAS image, or DAS images obtained using different sound speeds. In some cases, the model incorporates additional skip connections between the encoder and decoder or replaces existing skip connections with convolutional layers. These modifications access features at various levels of abstraction, enabling the network to leverage both low-level and high-level features for a more comprehensive representation.</p>", "<p id=\"P55\">In addition, alternative frameworks such as RNN or GAN have also been explored. For instance, one study combined RNN with CNN to address multiframe images and strategically average them. Another study adopted a GAN architecture, where the UNET-based generator aimed to generate images that closely resembled reference images by attempting to deceive the discriminator beyond standard loss metrics like mse. These novel frameworks offer additional avenues to improve PAUS image quality.</p>", "<p id=\"P56\">As the size of input data increases, there is a growing need for enhanced DL network efficiency. In line with this, as discussed in <xref rid=\"S6\" ref-type=\"sec\">Section III</xref>, several research groups have incorporated a simple attention mechanism into the UNET architecture to capture spatial dependencies beyond the limitations of a convolutional filter [##UREF##42##84##]. Currently, the vision transformer (ViT) [##REF##35180075##110##], which leverages the concept of self-attention, has performed at a high level for various medical imaging tasks by effectively capturing long-range dependencies. Unlike convolutional architectures, self-attention can capture relationships between different subsections in the full image domain regardless of their positions [##UREF##64##111##]. The ViT model is clearly a viable option for complex and voluminous PA data.</p>", "<p id=\"P57\">The majority of papers reviewed here focused on supervised approaches where in-silico data served as a reference for model training. However, the inherent discrepancy between in-silico training data and real test data is a major issue for clinical translation. Acquiring a large volume of ground-truth data (gold standards) is challenging, leading research groups to rely on synthetic data to train their models. Consequently, performance in a clinical setting heavily depends on the similarity between synthetic and real tissue models. To mitigate this issue, plausible ranges or distributions of optical parameter values for each tissue have been used, with the aim of facilitating successful transfer learning.</p>", "<p id=\"P58\">Despite these efforts, unexpected artifacts have been reported in many papers due to the underlying disparities between synthetic and real data. This is exemplified in ##FIG##8##Figs. 9## and ##FIG##9##10##, where the yellow arrows indicate the likelihood of these artifacts occurring. One presumable cause for these artifacts could be that the synthetic model is based on a 2-D geometry, while real data represent a 3-D environment. As discussed in <xref rid=\"S2\" ref-type=\"sec\">Section II</xref>, the influence of absorbers extends beyond the plane of interest (in-plane) to those located out-of-plane in PA data acquisition. Given that each transducer element has elevational directivity during acquisition, misinterpreting PA signals from out-of-plane sources as originating from in-plane sources could be a plausible explanation for such artifacts. Thus, the development of more realistic 3-D tissue and transducer models holds the potential to bridge the gap between training and test data, leading to artifact reduction.</p>", "<p id=\"P59\">An alternative that can address this challenge is unsupervised learning using approaches such as CycleGAN [##UREF##65##112##]. CycleGAN uses GANs for image-to-image translation or raw data-to-image translation without requiring paired training data. The fundamental idea behind CycleGAN is cycle consistency loss, which forces the translated image to be accurately reconstructed back to its original form. For instance, in the case of translating a horse image to a zebra image to deceive a discriminator, the translated zebra image is not random but rather constrained to closely resemble the original image due to the loss. In the field of medical imaging, CycleGAN learned mappings between different domains, such as CT and MRI [##UREF##66##113##], enabling image transfer while preserving content. In the context of PA imaging, where obtaining paired data is challenging, exploiting CycleGAN can provide a more convenient approach for image reconstruction tasks.</p>", "<p id=\"P60\">PAUS systems offer a significant advantage by providing both a standard US image and a PA image [##REF##34584840##114##]. This implies that DL networks can leverage US data to enhance PA imaging using acoustic features. For instance, US B-mode images can be particularly valuable in obtaining accurate SOS measurements. In a tissue domain with large vessels, US data can provide essential information about their position and shape, which may not be readily observed in PA images. In cases where involuntary movement occurs during scanning or data acquisition, US speckle can be used for motion tracking and compensation [##REF##33397941##26##]. Furthermore, two different domains, US and PA, present a promising opportunity for unsupervised learning such as CycleGAN, offering considerable enhancements in real-time PAUS images. They have the potential to strengthen PA/US dual-mode imaging [##UREF##67##115##], [##REF##30306043##116##], [##UREF##68##117##], offering complementary information that can enhance its translation into practical clinical applications.</p>", "<title>Extension of DL Frameworks</title>", "<p id=\"P61\">DL frameworks can be extended to further reinforce PAUS imaging. First, they can potentially mitigate clutter signals caused by sound reverberation between reflectors in tissue. For instance, Allman et al. [##UREF##69##118##] presented a CNN-based model designed to identify reflection artifacts and source signals, focusing on scenarios where the object was limited to point-like targets. In the domain of standard US imaging, numerous DL techniques have been introduced to suppress reverberation clutter [##UREF##70##119##], [##UREF##71##120##], [##UREF##72##121##].</p>", "<p id=\"P62\">Second, as 3-D US imaging evolves, free-hand PA imaging with DL can be extended to include 3-D imaging capabilities. In traditional US imaging, a 2-D matrix probe is employed to simultaneously scan multiple image planes and acquire volumetric data in real-time. However, this approach requires a higher cost to achieve high spatial resolution, and the computational demands are intense. One approach to tackle these challenges is the use of a sparse 2-D matrix probe, in which elements are intentionally skipped or spaced further apart to reduce the number of physical elements in the probe. In this context, DL plays a crucial role in enhancing image reconstruction from undersampled data, thus compensating for element reductions compared to dense arrays [##UREF##73##122##]. Additionally, panoramic imaging techniques can help create extended 3-D field-of-view images by stitching together multiple 2-D images obtained by sweeping a standard probe. Presently, DL methods were employed to estimate the probe’s position and movement without requiring any additional positional sensors [##UREF##74##123##], [##REF##29936399##124##].</p>", "<p id=\"P63\">Finally, DL frameworks hold the potential to enhance spectroscopic PA imaging, mapping chromophore concentration using PA spectra acquired at different wavelengths. Accurately estimating chromophore concentrations poses a significant challenge due to spectral distortion [##REF##22734732##61##], [##REF##32670789##125##]. The PA spectrum at a specific spatial position is influenced not only by the linear combination of intrinsic absorption spectra of chromophores at that position but also by wavelength-varying optical fluence. DL methods [##REF##29905680##126##], [##UREF##75##127##], [##UREF##76##128##], [##REF##32840068##129##], [##UREF##77##130##] offered a solution to this challenge by mapping from PA images acquired at multiple wavelengths, eliminating the need for prior knowledge such as fluence maps and intrinsic absorption spectra of chromophores. Additionally, spectroscopic PA imaging holds promise for automatic segmentation and isolation of target objects. Currently, multispectral imaging combined with DL techniques improved task performance [##REF##32840068##129##], [##UREF##78##131##], [##REF##35371919##132##].</p>", "<p id=\"P64\">Although these methods have been validated in simulation settings, comprehensive validation in vivo remains largely unexplored. Presently, tissue phantom models used to generate training data are often considered overly simplistic to adequately simulate real-world scenarios. To tackle this limitation, some researchers have embarked on alternate approaches, such as creating phantoms that integrate information from other imaging modalities. For instance, Yang and Gao [##UREF##76##128##] designed a 3-D heterogeneous tissue structure by leveraging an MR breast database, subsequently assigning optical and acoustic parameters to replicate a more realistic environment. In existing literature, PA images reconstructed using DAS or TR methods are commonly used as input to DL models [##REF##29905680##126##], [##UREF##75##127##], [##UREF##76##128##], [##REF##32840068##129##], [##UREF##77##130##]. Expanding upon this framework, two approaches will be pursued: 1) using two separate DL networks, where one network focuses on generating clear PA images from PA signals, while the other network is dedicated to generating the map of chromophore concentration from PA images and 2) alternatively, a single DL network will be trained using an end-to-end learning approach, directly mapping PA signals to the predicted concentration map.</p>", "<title>Challenges</title>", "<p id=\"P65\">A current challenge to DL approaches for PA reconstruction is the lack of performance comparisons between DL methods in the current literature, mainly due to individual model optimization using separate datasets. To address this issue, the availability of a publicly accessible framework to evaluate novel DL methods using identical reference data, including both phantom and real data, would prove invaluable. This framework would facilitate the comparison of different methods, thereby expediting the advancement of PA reconstruction techniques. Open frameworks have already been established in some medical imaging fields to address similar challenges. For instance, the Challenge on Ultrasound Beamforming with DL (CUBDL) was offered during the 2020 IEEE International Ultrasonics Symposium [##UREF##79##133##].</p>", "<p id=\"P66\">Additionally, a significant challenge is in vivo validation of all DL techniques. This presents significant hurdles in translating these techniques to clinical practice and, to date, none have achieved clinical translation. While many DL studies have been validated in simulation by quantifying mse-based metrics between estimates and the ground-truth of initial PA pressure, their validation has often been limited to in vitro phantom experiments. The focus of these studies to date has been to assess the clarity of reconstructed absorber shapes and the reduction of artifacts arising from ill-posed conditions. Although some studies have extended validation to in vivo scenarios, these evaluations mainly focus on the plausibility of reconstructed absorber shapes, which relies heavily on prior anatomical knowledge. Given the scarcity of quantitative ground-truth maps in vivo, most papers have omitted quantitative evaluation of their DL results. For instance, in situations where the target is blood vessels, the initial PA pressure can exhibit positional variations within vessels, a factor often overlooked in evaluations. Therefore, a compelling future challenge involves addressing this limitation and developing methodologies to quantify the performance of DL techniques in vivo.</p>", "<title>Conclusion</title>", "<p id=\"P67\">In conclusion, reconstructing PAUS images with DL is very new, but there is no doubt that this framework holds significant potential to improve the modality. With the advent of next-generation computing systems, more complex and realistic tissue models can be created through simulations in a shorter time. The development of efficient networks trained on large volumes of data will greatly facilitate successful transfer learning from virtual environments to real-world applications. Additionally, unsupervised techniques, especially those incorporating US data, can potentially improve performance and make DL systems more robust for real clinical applications. Similar to recent advances in CT and MR imaging resulting from DL tools, as PAUS systems and imaging techniques become more standardized, a wealth of patient data will become available. Abundant human subject data will provide ample opportunities to thoroughly evaluate DL methods, leading to increased trust and confidence in their clinical utility.</p>" ]

[ "<p id=\"P1\">Photoacoustic (PA) imaging provides optical contrast at relatively large depths within the human body, compared to other optical methods, at ultrasound (US) spatial resolution. By integrating real-time PA and US (PAUS) modalities, PAUS imaging has the potential to become a routine clinical modality bringing the molecular sensitivity of optics to medical US imaging. For applications where the full capabilities of clinical US scanners must be maintained in PAUS, conventional limited view and bandwidth transducers must be used. This approach, however, cannot provide high-quality maps of PA sources, especially vascular structures. Deep learning (DL) using data-driven modeling with minimal human design has been very effective in medical imaging, medical data analysis, and disease diagnosis, and has the potential to overcome many of the technical limitations of current PAUS imaging systems. The primary purpose of this article is to summarize the background and current status of DL applications in PAUS imaging. It also looks beyond current approaches to identify remaining challenges and opportunities for robust translation of PAUS technologies to the clinic.</p>", "<title>Graphical Abstract</title>", "<p id=\"P2\">\n\n</p>" ]

[ "<title>Photoacoustic Imaging</title>", "<p id=\"P14\">This section briefly describes the fundamentals of PA imaging. Further details can be found in [##UREF##24##59##], [##UREF##25##60##], and [##REF##22734732##61##]. ##FIG##1##Fig. 2## illustrates PA signal generation and acquisition through two distinct processes. The first, known as the optical forward problem, determines the initial pressure generated by chromophores within the medium. Each endogenous or exogenous chromophore possesses a unique absorption coefficient at a specific light wavelength [##REF##23666068##62##]\n\nwhere denotes the number of chromophore types and and denote the concentration and unit optical absorption spectrum of the chromophore, respectively.</p>", "<p id=\"P15\">The ultimate goal of PA imaging is to reconstruct the concentration of chromophores at each position using the known spectrum . For example, a primary target for many PAUS applications is the local blood concentration and its oxygenation level, which can be reconstructed from the PA-reconstructed optical absorption estimated at a number of wavelengths. The concentration of each chromophore contributes to the medium’s optical absorption coefficient at any specific wavelength. At the same time, most biological tissues are highly light scattering (or turbid), and light scattering also a function of both location and wavelength) is many times larger than optical absorption. The combination of optical absorption and scattering within the medium defines the optical fluence distribution , thereby determining the distribution of absorbed energy\n\nand, subsequently, the pressure excitation through thermalization\n\nwhere is the Gruneisen coefficient, is the sound speed, is the coefficient of volumetric thermal expansion, and is the specific heat at constant pressure, which, in general, are all functions of . The second process, referred to as the acoustic forward model, determines the US signals acquired by the imaging system arising from the initial pressure. PA data are influenced by both the acoustic properties of the medium and the characteristics of the detector(s).</p>", "<p id=\"P16\">To quantify the volumetric distribution of chromophore concentration, the overall inverse problem must be solved. First, to determine the initial pressure distribution from recorded data, the acoustic inverse problem must be addressed. This process, and the resultant map of initial pressure, are commonly referred to as “PA reconstruction” and the “PA image,” respectively.</p>", "<p id=\"P17\">The subsequent step estimates chromophore concentrations using PA images and volumetric maps of and . Multiple optical wavelengths are often used to improve these estimates since each chromophore has a unique optical absorption spectrum. This approach is commonly called “PA spectroscopic imaging” or “PA quantitative imaging.” It is not trivial and requires separate analysis. Details on optical fluence reconstruction methods are summarized in [##REF##22734732##61##], [##REF##32405456##63##], and [##REF##32517204##64##].</p>", "<p id=\"P18\">The simple sensor geometry used in many PAUS systems is determined by the physical access available to US probes for a specific medical application. The limited size and bandwidth of these probes affects the quality of reconstructed PA images, often greatly misrepresenting the shape of volumetric chromophore distributions (endogenous or exogeneous). For example, large blood vessels and microvessel networks containing strongly absorbing blood can be greatly distorted. In <xref rid=\"S3\" ref-type=\"sec\">Sections II-A</xref>–<xref rid=\"S5\" ref-type=\"sec\">II-C</xref>, these technical difficulties will be described in detail. Thus, this article primarily focuses on reconstructing the volumetric shape of absorbers, an essential component of complete PA inversion.</p>", "<title>Photoacoustic Signal</title>", "<p id=\"P19\">The spatio-temporal pressure at time after initial pressure generation is given by the PA equation [##UREF##24##59##]\n\nIf the excitation laser pulse is short enough to satisfy stress and thermal confinement conditions, it can be approximated as an infinitesimally short pulse, and hence the can be represented as . Then, the temporal profile of pressure at the position of an acoustic detector, , can be expressed as a Rayleigh integral over the distribution of heat release [##UREF##8##24##]\n\nAssume a transducer contains detection elements. Then, the signal recorded by the element can be represented as\n\nwhere and denote the system function and acquisition noise, respectively. The goal of PA image reconstruction is to map initial PA pressure [or the function ] from the measurements .</p>", "<title>Detection Element Geometry</title>", "<p id=\"P20\">An ideal PACT system must have a cylindrical or spherical geometry for the transducer sensor array surrounding the measurement volume to detect all PA signals originating from every chromophore in the volume [##UREF##9##25##], [##UREF##26##65##]. For 2-D sectional imaging, therefore, the object must be enclosed by a circular array as shown in ##FIG##2##Fig. 3(a)##. The spatial image resolution is determined by the frequency response of a single sensor in the array assuming that the detectors are point-like. If the detectors are not point-like, then their specific geometry must be taken into account. For instance, a focused array is usually employed to realize high elevational resolution.</p>", "<p id=\"P21\">Artifacts are likely present in 2-D imaging because each sensor can inevitably receive signals from outside the imaging plane since light is diffused over three dimensions. To illustrate, consider a scenario where a strong point absorber lies outside the plane of the detector but is close to the origin of the circular array. Even if the array has a tight elevational focus, acoustic waves generated by this absorber will be detected by all array elements. Since the arrival time of these signals does not coincide with the in-plane propagation time from the center of the ring to a given detector, any 2-D reconstruction cannot eliminate this signal, resulting in a “blob” artifact rather than a well-defined point at the image center. Although these artifacts can be significant, we will limit the scope of this article to 2-D reconstructions neglecting out-of-plane artifacts. Future studies must address full 3-D reconstructions to ensure robust PA imaging under all conditions.</p>", "<p id=\"P22\">The typical PAUS platform does not even approach an ideal 2-D geometry because a standard clinical transducer, typically a linear sensor array as shown in ##FIG##2##Fig. 3(b)##, has a greatly limited view, i.e., PA signals are recorded within an aperture much less than 180°. This geometry can be easily manipulated and brings PAUS imaging to a wide range of medical applications where US is currently used. However, it creates an ill-posed condition that degrades absorber shapes in the reconstruction process. Thus, PA signals recorded under limited view conditions impose severe shape artifacts even for simple objects. The condition is exacerbated if the target is both discrete and not small compared to an acoustic wavelength at the central operating frequency of the array.</p>", "<p id=\"P23\">These points were thoroughly examined in simulation. ##FIG##3##Fig. 4## displays reconstructions using a standard method (introduced in <xref rid=\"S5\" ref-type=\"sec\">Section II-C</xref>) under various acquisition conditions. To simplify these simulations, light and US attenuations were omitted, and postimage processing steps were skipped to focus solely on visualizing the pattern changes caused by ill-posed conditions. The details are summarized in the <xref rid=\"APP1\" ref-type=\"app\">Appendix</xref>. When the geometry used a circular array with full bandwidth, accurate reconstruction was achieved, as shown in ##FIG##3##Fig. 4(b)##. However, narrowing the bandwidth during acquisition preserved the object shapes but introduced ripple artifacts, as demonstrated in ##FIG##3##Fig. 4(c)##. On the other hand, when the acquisition view was limited, the shapes became distorted, as illustrated in ##FIG##3##Fig. 4(d)##–##FIG##3##(e)##.</p>", "<p id=\"P24\">Image artifacts are exacerbated if the target was not small compared to an acoustic wavelength at the central operating frequency of the array. If the absorption field at a specific optical wavelength slowly varies around position , the emitted signal from is extremely weak because of the derivative term with respect to time in ##FORMU##28##(5)##. This signal was even weaker if the sensor has limited bandwidth. For example, as shown in ##FIG##2##Fig. 3##, a circular absorber generates a bandwidth-limited N-shaped signal. If the diameter of the absorber is large, it causes relatively strong signals at boundaries but weak signals around the center. Only a full aperture and wide signal bandwidth can recover the low frequencies required for a faithful reconstruction of a circle. Thus, only targets limited to strong, sparse absorbers whose shape is point-like or finely vascular, protruding from other weak absorbers regarded as background in a medium, can lessen the ill-posedness of this geometry. In the frequency domain, the signal components for this class of absorber are distributed evenly across the total domain, or dominantly in the high-frequency domain. As a result, even though the derivative term and limited bandwidth may significantly weaken low-frequency components, PAUS image reconstruction is still tractable. However, one exception is the vertical vascular shape, as shown in ##FIG##3##Fig. 4##, because the array sensors cannot receive plane waves propagating horizontally. This effect is explained in the frequency domain in [##UREF##12##37##]. Simulations in [##UREF##27##66##], [##UREF##28##67##], and [##UREF##29##68##] have also revealed similar artifacts for this detection geometry.</p>", "<title>Conventional Image Reconstructions Schemes</title>", "<p id=\"P25\">Many papers proposed analytical approaches to map the initial pressure [or heating function ] from PA measurements given a well-posed condition. When the acquisition view and detector bandwidth are full, the detector function is linear and the noise is zero, the simplified UBP method [##UREF##9##25##], [##UREF##30##69##] can be expressed as\n\nwhere the constant depends on the transducer geometry. If the density of detector elements is above the spatial Nyquist sampling rate, the discrete version of UBP can reconstruct PA sources perfectly from measurements as\n\nThe UBP method can also be used when the view and bandwidth are limited. As shown in <xref rid=\"S4\" ref-type=\"sec\">Section II-B</xref>, the main target should be small or vessel-like. Since the strong signals from compact absorption sites against a uniform background are short pulses, the derivative term can be ignored in ##FORMU##48##(8)##. Instead, postprocessing to smooth the wave oscillation can be used as\n\nwhere denotes a processing operator, such as the Hilbert transform, and . This approach is very similar to delay and sum (DAS) beamforming used in radar applications or clinical US imaging [##UREF##31##70##]. As shown in ##FORMU##49##(9)##, before summing, a delay is applied to account for the variable propagation distance/time from the source at position to the sensor at position .</p>", "<p id=\"P26\">Another reconstruction approach uses time reversal (TR) to solve the wave inversion equation by simulating a wave back-propagating to the image field from each sensor [##UREF##13##40##]. TR utilizes time-reversed reemission of received signals to focus the energy at the desired imaging location. By iteratively computing the wave field, TR can account for aberrations if medium heterogeneities are not high, but the computational burden associated with the iteration process is a practical limitation. If tissue is not fully enclosed by detectors, the resulting image quality is compromised because TR uses the DAS framework.</p>", "<p id=\"P27\">##FORMU##13##Equations (3)##, ##FORMU##28##(5)##, and ##FORMU##31##(6)## can be simply expressed as where denotes the forward operator generating measurement from source . Both DAS and TR methods cannot invert this operation uniquely due to limited view and bandwidth conditions. Some groups [##UREF##32##71##], [##REF##24410918##72##] adopted penalties (regularizers) based on prior knowledge to obtain a more plausible solution, where optimization takes the form\n\nand denotes the penalty term. However, it is challenging to identify a penalty function that is general enough for all samples. In most cases, there is no closed-form solution available. Iterative algorithms approaching real-time rates are possible for simple objects and PA data from high SNR, broad bandwidth, and near-full view tomographic detection [##UREF##33##73##], [##REF##28248347##74##], or for aberration correction induced by variance in US speed [##UREF##34##75##]. However, iterative inversion methods have not yet been proven or experimentally demonstrated to converge to the actual volumetric distribution of heat release for very sparse PA data (very limited bandwidth and view, and typically low SNR) acquired from a conventional PAUS geometry [##UREF##35##76##], [##UREF##36##77##].</p>", "<title>Deep Learning for Imaging</title>", "<title>Supervised Learning</title>", "<p id=\"P28\">DL is part of a broader family of machine learning methods based on artificial neural networks. The fundamental learning technique fits large sets of training data using the model to find features (patterns) to adapt properly to new data [##UREF##37##78##]. Given each data sample as an input, the model outputs the scalar, vector, matrix, or higher dimensional tensor type, depending on the imaging task. Supervised learning [##UREF##38##79##] takes advantage of an instructor concept to optimize model parameters that minimize the cost (loss) function measuring the discrepancy between ground truth (answer) and model output. During training (learning), the model decomposes data into shape, texture, or abstract features to facilitate the recovery of intended visual data.</p>", "<p id=\"P29\">The learning process can be expressed as\n\nwhere denotes the number of training samples, denotes the loss function, denotes the ground-truth, and denotes the predicted output of the DL model with the parameter set when the input is the data sample . To find the optimal set , each parameter is gradually updated by gradient descent approaches. The stochastic gradient descent framework using batches can leverage GPU parallel computing to train large-scale neural networks efficiently.</p>", "<p id=\"P30\">As described in [##UREF##37##78##], this approach can be viewed as minimizing the Kullback–Leibler (KL) divergence where denotes the probability distribution over data space given by the input and parameter set , and denotes the empirical distribution defined by the training data. In this context, the optimization process aims to align the model distribution with the empirical distribution, ideally representing the true data-generating distribution . Using Gaussians for the distributions minimizes mean squared error (mse) as the loss function .</p>", "<title>Deep Learning Model</title>", "<p id=\"P31\">Here, we briefly introduce DL models that are best suited to medical imaging. A fully connected network (FCN) is the basic DL model, as shown in ##FIG##4##Fig. 5(a)##, where it contains an input layer, hidden layers, and an output layer. Every neuron (perceptron) in a hidden layer is connected to the neurons in the previous and next layers and sums all inputs, applying a nonlinear operation (activation) to the resultant as\n\nwhere denotes the output of neurons in the layer or the input of neurons in the layer, denotes the weights, denotes biases, denotes the activation function, such as a rectified linear unit (Relu), and denotes the output of neurons in the layer. For a regression task, the output layer has no special activation function. It has been shown that the hierarchical model excels at capturing complex nonlinear relationships in data and extracting abstract features relevant across different instances of a problem [##UREF##37##78##].</p>", "<p id=\"P32\">Convolutional neural networks (CNNs) have performed well for various imaging tasks because they leverage common statistical properties of images such as local invariance [##UREF##39##80##], [##REF##33425651##81##]. The basic network block is the convolutional layer illustrated in ##FIG##4##Fig. 5(b)##. Assume one image is input to the layer; as each small filter travels over the entire image, it can highlight the specific pattern in a local area and store the degree in the feature map. In the next layer, the feature maps are convolved with new filter banks to extract deeper features and store the results in the new feature maps [see ##FIG##4##Fig. 5(b)##]. This process is repeated for the next layers as\n\nwhere the operation denotes convolution, denotes the number of feature maps (channels) in the layer, denotes the feature map (channel) in the layer, denotes the filter (patch or kernel) in the layer generating the feature map in the next layer, denotes the bias, and denotes the activation function. Convolutional layers can reduce computational complexity due to parameter sharing and spatial localization properties. They have a significantly lower number of connections (trainable parameters) compared to fully connected layers, and are suitable for large-scale datasets or resource-limited scenarios.</p>", "<p id=\"P33\">UNET [##UREF##40##82##] is one of the CNN networks well-suited to image-to-image mapping. As shown in ##FIG##4##Fig. 5(b)##, the structure consists of: 1) the encoder conducting multiscale image decomposition using convolutional layers and downsampling operators and 2) the decoder recovering an image from multiscale feature maps using convolutional layers and upsampling operators. The concept is similar to discrete wavelet decomposition and reconstruction using filter banks to identify multiresolution features [##UREF##41##83##]. The “skip connection” concatenates feature maps in the decoder with those in the encoder, so that the decoder can access not only deep features but also low-level features. Currently, UNET has been modified by adding attention modules in the convolutional layers or replacing skip connections into them [##UREF##42##84##]. Attention focuses the model on key feature maps and suppresses redundant features [##UREF##43##85##], [##UREF##44##86##]. For example, channel attention and spatial attention assign weights to different channels and spatial locations based on their importance for the task, respectively.</p>", "<p id=\"P34\">The hybrid architecture combining a CNN with a recurrent neural network (RNN) has been developed for multiframe images [##UREF##45##87##], [##UREF##46##88##]. As shown in ##FIG##5##Fig. 6(a)##, the RNN structure is specialized to sequential data by inputting data at every time-step. The network has recurrent connections between hidden units, simply expressed as\n\nwhere denotes the time-step, denotes the hidden unit, denotes the input unit, and denotes the sharing neural network with trainable parameters over . The network can produce an output at every time-step or at specific time-step (in general, the last step). Like CNN, RNN can reduce complexity due to parameter sharing and localization across time-steps.</p>", "<p id=\"P35\">Generative adversarial networks (GANs) [##UREF##47##89##], [##UREF##48##90##] can output more plausible images using ingenious cost (loss) functions for optimization beyond standard metrics such as mse or mean absolute error (MAE). The GAN contains two neural networks: a generator and a discriminator. The generator captures the real image distribution and creates a realistic fake image while the discriminator discriminates fake from real samples. As shown in ##FIG##5##Fig. 6(b)##, the generator maps from random (noise) space to image space , and the discriminator takes either the generated image or real image to output the probability that the image came from real samples rather than fake samples. GAN updates the parameters in the two networks using the minmax optimization problem as\n\nwhere denotes the distribution of real image samples. For instance, if the input is real, the discriminator attempts to output the number closest to 1 by maximizing the cost. This architecture can be used for data augmentation in medical imaging [##UREF##49##91##], [##UREF##50##92##].</p>", "<p id=\"P36\">When the task is image enhancement or reconstruction from low-quality image or raw data, the conditional GAN (CGAN) [##UREF##51##93##] has been adopted, as shown in ##FIG##5##Fig. 6(c)##. Pix2pix [##UREF##52##94##] is one of the best-known CGANs for image-to-image translations. In this architecture, the noise vector is replaced by the image or data as a condition, and the generator is trained to create the image close to a reference by minimizing the combination of the KL-based GAN cost and mse or MAE. Using only mse/MAE, a blurry image is often produced [##UREF##53##95##]. The addition of GAN cost, however, helps extract details in the reference and create a more sophisticated image by attempting to deceive the discriminator.</p>", "<title>Deep Learning Frameworks for PA Image Reconstruction</title>", "<p id=\"P37\">Several studies have explored DL frameworks for the PAUS geometry. The selection of papers for this review aimed to highlight recent discoveries concerning learning structure and/or experimental in vivo results in PA reconstruction within the context of the PAUS geometry. All PA imaging work presented here adopted supervised learning to overcome limited view and bandwidth problems. In every reconstruction task, the output was commonly a PA image (2-D matrix) mapping initial pressure . However, the input to the DL model varied considerably. As shown in ##FIG##6##Fig. 7##, input data can be categorized into three types: 1) sensor, or channel, data; 2) preprocessed (transformed) channel data; and 3) reconstructed images using a conventional method such as DAS or TR. ##TAB##0##Table I## summarizes data acquisition conditions and proposed DL models for the work reviewed here.</p>", "<p id=\"P38\">Waibel et al. [##UREF##10##27##] proposed two distinct DL architectures. The first utilized a standard UNET framework with a rough DAS image as the input. The second model, derived from the UNET backbone, replaced each skip connection with a convolutional layer featuring a large kernel size and a large step size (called stride) at which the kernel moves across the data. This modification converted the high-sampled temporal domain in the encoder to the low-sampled spatial domain in the decoder. They used simulated (in silico) data from a linear transducer array obtained solely from circular targets that mimicked vessel cross sections.</p>", "<p id=\"P39\">##FIG##7##Fig. 8(a)##–##FIG##7##(d)## demonstrated that both models reconstructed target shapes more accurately compared to standard methods. For targets located in the far-field [see ##FIG##7##Fig. 8(e)##], the models predominantly restored the objects [see ##FIG##7##Fig. 8(g)## and ##FIG##7##(h)##], unlike DAS [see ##FIG##7##Fig. 8(f)##]. Interestingly, despite extremely faint object traces in the DAS image, the first DL model, which was fed with the DAS image, restored the objects. The second model, which used channel data as input, had a larger number of trainable parameters than the first. However, it produced more distorted results compared to the first model. This suggests that the translation from data to image is considerably more challenging than image-to-image translation, necessitating more sophisticated structures specifically designed for this mapping. Although the parameters of the tissue-mimicking phantom used in the simulations may not perfectly represent real-world situations, this study is valuable as it represents one of the initial attempts to apply a DL approach to PA reconstruction and highlights the potential of DL in the field.</p>", "<p id=\"P40\">As observed in the literature, mapping channel data directly to an image is challenging, even though channel data contain more physical information about the target and acquisition conditions. Although the drawbacks of mapping image-to-image are rarely discussed in the literature, it can lead to artifacts and low generalization, especially when dealing with complex targets due to limited information. Lan et al. [##REF##32612929##96##] developed a UNET-based model called YNET, which addresses these challenges by simultaneously feeding both channel data and a DAS image into the network. The model consisted of two encoder modules inputting both channel data and the image, and one decoder module that produced the final image. The key concept behind this approach was that the two encoders shared their feature maps with the decoder using skip connections at every scale.</p>", "<p id=\"P41\">Compared to a method employing two independent networks, the shared decoder in YNET could leverage features from both channel data and image domains, while also reducing the number of trainable parameters. Channel data were acquired using a 7-MHz linear transducer with 80% bandwidth. For target objects in simulations, vascular structures were extracted from fundus oculi images [##UREF##57##101##], and training data were synthesized under limited view and bandwidth conditions. The DL model was trained with synthetic data and the model was tested using simulation data, in vitro chicken breast data, and in vivo human palm data, demonstrating that the proposed model significantly improved imaging performance, as shown in ##FIG##8##Fig. 9##. The method presented targets with higher contrast compared to standard methods (DAS and TR) and fewer artifacts than a UNET model fed by only a DAS image.</p>", "<p id=\"P42\">Kim et al. [##UREF##12##37##] proposed a new form of input data for a UNET model. As shown in ##FORMU##49##(9)##, channel data can be transformed to based on the time of flight of an US pulse from a potential PA source. ##FIG##6##Fig. 7(c)## illustrates this conversion when the imaging plane is . Discretization of creates multichannel 2-D matrices (3-D tensor). Specifically, is sampled using image pixel positions and assigned channel data samples based on the time-of-flight from each pixel position to each sensor (channel). Using this multichannel data as input, the DL model can more effectively access the primary data samples contributing to each pixel position. If the pixel resolution is sufficiently high, the model can handle raw data with minimal information loss in both the encoder and decoder.</p>", "<p id=\"P43\">The target of this study was vascular structures, and thus a fundus oculi database [##UREF##57##101##] was also employed as a reference. During simulation, real acquisition conditions were mimicked (linear probe, center frequency: 15 MHz, 3-dB bandwidth: 8 MHz), and synthetic raw data were generated to train the model. Results showed the effectiveness of feeding preprocessed data using synthetic vascular data, in vitro data (w-shape wire), and in vivo data (human finger). This approach restored more detailed structures with fewer artifacts compared to inputting the DAS image, as illustrated in ##FIG##9##Fig. 10##.</p>", "<p id=\"P44\">Vu et al. [##UREF##54##97##] introduced a GAN-based model to enhance images acquired with the PAUS geometry. While the traditional GAN loss function [see (##FORMU##113##15##)] is typically based on KL or Jensen–Shannon (JS) divergence [##UREF##47##89##], these approaches often fail because of gradient vanishing and mode collapse [##UREF##58##102##], [##UREF##59##103##]. The metric turns infinite when the generated distribution does not overlap with the real distribution. Instead, these researchers adopted Wasserstein GAN (WGAN) [##UREF##58##102##], which utilizes the continuous loss called the Wasserstein distance (also known as Earth’s Mover distance) to enhance stability (convergence) during min-max optimization. They employed a CGAN framework and constructed a loss function that combines GAN loss with mse.</p>", "<p id=\"P45\">As described in <xref rid=\"S8\" ref-type=\"sec\">Section III-B</xref>, GAN loss guides the generator to produce image samples aligned well with the distribution of real image samples, thus deceiving the discriminator. The generator in their model used an initial TR image. For reference images, they generated simulated images with randomly distributed circular disks, and they also employed a brain vascular database obtained through two-photon microscopy [##UREF##60##104##]. Training and testing data were generated in a simulation environment assuming a linear transducer with a center frequency of 5 MHz and a 3-dB bandwidth of 60%. Additionally, for in vivo testing, they imaged skin vasculature in the trunk of a mouse. As demonstrated in ##FIG##10##Fig. 11##, the proposed model improved the visibility of target structures, including vertical vessels. Compared to a standard UNET model, the proposed model preserved fine structural details with higher contrast.</p>", "<p id=\"P46\">An LED has some advantages as a light source in PAUS systems. It is cost effective, can operate at very high repetition rates, and can switch between different optical wavelengths quickly. However, its low fluence produces very weak PA signals. In their study, Hariri et al. [##UREF##55##98##] developed a DL framework specifically designed to enhance image contrast in LED-based PAUS systems.</p>", "<p id=\"P47\">To simulate complex vascular networks, they constructed an in vitro phantom using 3-D printing with a light-absorbing material. Additionally, TiO2-based optical scatters were introduced into the phantom to acquire low-fluence data typical of in vivo conditions. PA images from scattering and nonscattering media using a 15-MHz linear transducer served as training input and reference data, respectively. The authors employed a multilevel wavelet-CNN architecture, which was also based on the U-Net backbone. Common pooling operations in the U-Net architecture, typically employed to enlarge the receptive field, often result in irreversible information loss [##UREF##61##105##], [##UREF##62##106##]. Therefore, in this study, these pooling operations were replaced with discrete wavelet and inverse-wavelet transforms, gradually restoring image resolution to access multiscale features. As illustrated in ##FIG##11##Fig. 12##, the DL model provided higher contrast-to-noise ratio (CNR) images compared to input images in in vivo experiments involving mice injected with contrast agents.</p>", "<p id=\"P48\">A common approach to improve image quality in LED systems is to average multiple image frames, taking advantage of fast data acquisition. Anas et al. [##UREF##56##99##] proposed a DL model that effectively combines low-quality images to generate an enhanced image. The DL model integrated CNNs and RNNs, where the CNN extracted spatial features from each image frame and the RNN combined these features by considering temporal dependencies. To train and test the model, images were acquired from an in vitro phantom containing gold magnetic nanoparticles or wires. A reference image was generated by averaging multiple frames and filtering noise. The results demonstrated that the DL model produced clearer images compared to standard averaging techniques, as depicted in ##FIG##12##Fig. 13##.</p>", "<p id=\"P49\">The studies reviewed above assumed a static and known SOS in tissue. However, discrepancies between the assumed value and the actual value can lead to noticeable phase aberrations. While TR is a well-known technique for aberration correction, its main practical challenge lies in its limitation to coherent reflectors, such as a point target at the focal point [##UREF##63##107##]. In their research, Jeon et al. [##REF##34665732##100##] proposed a UNET-based model named SegU-net to accurately determine the true static SOS and minimize aberration artifacts. The model was trained using multichannel images, each reconstructed using raw data and a different SOS within the range of 1460–1600 m/s. A PA image obtained using the true SOS served as the ground truth. These authors modified the UNET architecture by incorporating additional links between the encoder and decoder to facilitate detailed feature extraction. Both in-silico and in vivo experiments demonstrated that the network effectively enhanced the main lobe while suppressing sidelobes, reducing aberration artifacts even in heterogeneous media as illustrated in ##FIG##13##Fig. 14##.</p>" ]

[ "<p id=\"P68\">This work was supported in part by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIT) under Grant 2021R1A2C2094778; in part by the Institute of Information and Communications Technology Planning and Evaluation (IITP) under the Artificial Intelligence Convergence Innovation Human Resources Development Grant funded by the Korean Government (MSIT) under Grant IITP-2023-RS-2023–00254177; and in part by the NIH under Grant HL-125339, Grant EY-026532, and Grant EB-030484.</p>", "<p id=\"P70\">\n\n</p>", "<p id=\"P71\"><bold>MinWoo Kim</bold> received the M.S. degree from the Department of Bio and Brain Engineering, Korea Advanced Institute of Science and Technology, Daejeon, South Korea, in 2009, and the Ph.D. degree from the Department of Electrical and Computer Engineering, University of Illinois at Urbana–Champaign, Champaign, IL, USA, in 2018.</p>", "<p id=\"P72\">He was a Research Engineer with Samsung Medison Company, Ltd., Seoul, South Korea, from 2009 to 2012, where he specialized in ultrasonic imaging equipment. He was a Senior Fellow with the Department of Bioengineering, University of Washington, Seattle, WA, USA, from 2018 to 2020. He is currently an Assistant Professor of biomedical convergence engineering with the Center for Artificial Intelligence Research, Pusan National University, Busan, South Korea. His current research interests include ultrasound and photoacoustic imaging, biomedical signal processing, and machine learning.</p>", "<p id=\"P73\">\n\n</p>", "<p id=\"P74\"><bold>Ivan Pelivanov</bold> (Member, IEEE) has been working in the areas of photoacoustics (PA) and laser-ultrasonics (LU) since 1994 when the fields were still very new, well before their recent boom in both the optics and ultrasound communities. He addressed a very diverse spectrum of problems, both fundamental and applied, from NDE to biomedical diagnostics and imaging with a combination of light and ultrasound. A large part of his research has focused on new imaging approaches, techniques, and devices among which: ultrawideband PVDF transducers of different shapes and sizes for PA signal reception, fully noncontact laser-ultrasound systems for industrial NDE applications, noncontact optical coherence elastography for in vivo volumetric mapping of soft tissue elastic moduli, and different PA systems, including the most recent fast-swept PAUS.</p>", "<p id=\"P75\">\n\n</p>", "<p id=\"P76\"><bold>Matthew O’Donnell</bold> (Life Fellow, IEEE) was with General Electric CRD, Niskayuna, NY, USA; the University of Michigan, Ann Arbor, MI, USA, where he was the Chair of the BME Department from 1999 to 2006; and the University of Washington (UW), Seattle, WA, USA, where he was the Frank and Julie Jungers Dean of engineering from 2006 to 2012. He is currently a Professor of bioengineering at UW. His most recent research has focused on elasticity imaging, optoacoustic devices, photoacoustic contrast agents, laser ultrasound systems, and catheter-based devices.</p>", "<p id=\"P77\">Dr. O’Donnell is a fellow of the American Institute of Medical and Biological Engineering (AIMBE) and is an Elected Member of the Washington State Academy of Sciences and the National Academy of Engineering.</p>", "<title>Reconstruction Simulation</title>", "<p id=\"P69\">We employed a custom simulation method outlined in ##FORMU##28##(5)## and ##FORMU##31##(6)## for data generation and ##FORMU##48##(8)## for image reconstruction. To simplify the simulation, in ##FORMU##31##(6)##, the system function was set as an identity function, and the acquisition noise was set to zero. As depicted in ##FIG##3##Fig. 4(a)##, absorbed energy in each object [ in ##FORMU##28##(5)##] remained constant. For both the circular array and linear array, the center frequency was 15.63 MHz, and the 3-dB bandwidth was 8 MHz. ##TAB##1##Table II## summarizes all parameter values.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Fig. 1.</label><caption><p id=\"P78\">(a) One example of a commercial PAUS platform Vevo LAZR-X, Visualsonics). (b) Optical fibers are located at/near the surface of the clinical US transducer to deliver laser energy into tissue (linear transducer). The PA signal is acquired by the transducer’s piezoelectric sensors. (c) US and PA images. The main target is microvessels, or endogenous/exogenous molecules in image-guided interventions. Top images show small vessels in a human finger and bottom images show a needle insertion and gold nanorod injection into chicken breast tissue. (b) and (c) are reproduced with permission from [##REF##33397941##26##] and [##UREF##10##27##].</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Fig. 2.</label><caption><p id=\"P79\">Two processes drive PA signal acquisition. First, the optical forward process describes the generation of initial pressure derived from chromophore concentrations and the light distribution (fluence) within the 3-D medium. Second, the acoustic forward process describes the acquisition of acoustic waves originating from the initial pressure. The ultimate goal of PA imaging is to accurately quantify chromophore concentrations from acquired data. In general, two steps are required to solve this inverse problem. First, the initial pressure distribution is reconstructed by addressing the acoustic inverse problem. Then, chromophore concentrations are estimated by solving the optical inverse problem using the pressure map as input.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Fig. 3.</label><caption><p id=\"P80\">PA signal generation and detection for a simple 2-D example. Chromophores located at different in the medium generate PA waves at , and sensors located at receive them at , where denotes their propagation speed. (a) Circular sensor array surrounds the medium. (b) Linear array is at the top of the medium. The signals received from every sensor are N-shaped if the chromophores are circular with diameter , where determines the duration of the N-shape. This assumes that light attenuation within the chromophore sphere can be neglected. If the sensor receives only over a limited frequency range, the signals are bandwidth-limited N-shaped.</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Fig. 4.</label><caption><p id=\"P81\">Simulation results using standard filtered back-projection reconstruction. (a) Four example object shapes. (b)–(e) Reconstructions when the acquisition conditions are (b) circular array with full bandwidth, (c) circular array with limited bandwidth (11–19 MHz), (d) linear array with full bandwidth, and (e) linear array with limited bandwidth (11–19 MHz). Array geometry illustrated with dashed orange line for each case and all images are shown on a log-scale colormap (40-dB range).</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Fig. 5.</label><caption><p id=\"P82\">(a) FCN. The filled circle represents a neuron. One neuron takes values from neurons in the previous layer, linearly combines the numbers, performs the nonlinear operation, and passes the resultant number into the neurons in the next layer. (b) UNET. This network leverages a CNN. Each arrow indicates an operation. The pivotal layer is the convolutional one illustrated below. Inputs and outputs to this layer are multichannel maps. Input maps are convolved with small kernels (patches/filters) and the resultant maps are summed. The resultant map passes through the nonlinear function to generate one output map. The same procedure is performed for other maps.</p></caption></fig>", "<fig position=\"float\" id=\"F6\"><label>Fig. 6.</label><caption><p id=\"P83\">(a) RNN. At every time-step, an image is input. In the hidden layer, the hidden units are connected to convey information to the next or previous time-step. Trainable parameters are shared in every time-step. The output image can be produced at any time-step. (b) GAN. The generator creates a fake image to deceive the discriminator. The discriminator is a classifier to distinguish between real and fake images. (c) CGAN. The generator generates a fake image given the condition (input image). It attempts to make the image as close as the given reference as well as deceive the discriminator.</p></caption></fig>", "<fig position=\"float\" id=\"F7\"><label>Fig. 7.</label><caption><p id=\"P84\">(a) Imaging plane and sensors (linear array). The example presented here assumes four strong point-like chromophores. (b) Channel (sensor) data (raw data). Four wavefronts are shown in the data domain. (c) Transformed channel data. Data samples corresponding to the time of flight from each position are aligned along the channel axis. (d) Standard DAS imaging result (reconstructed image). All figures are reproduced with permission from [##UREF##12##37##].</p></caption></fig>", "<fig position=\"float\" id=\"F8\"><label>Fig. 8.</label><caption><p id=\"P85\">Simulation test results using (a)–(d) one circular target in the near field and (e) and (f) two circular targets in the far-field. (a) and (e) Ground truth. (b) and (f) DAS results. (c) and (g) DL results reconstructed from the DAS image. (d) and (h) DL results reconstructed from channel data directly. All figures are reproduced with permission from [##UREF##10##27##].</p></caption></fig>", "<fig position=\"float\" id=\"F9\"><label>Fig. 9.</label><caption><p id=\"P86\">Reconstruction results using (a)–(e) synthetic vascular data, (f)–(j) in vitro phantom data acquired from chicken breast tissue with two pencil leads inserted, and (k)–(o) in vivo data acquired from a human palm. (a), (f), and (k) Ground-truth or acquisition field illustrations (b), (g), and (l) DAS results. (c), (h), and (m) TR results. (d), (i), and (n) Results for UNET fed by DAS images. (e), (j), and (o) Results for the proposed DL (YNET) fed by both channel data and the DAS image. All figures are reproduced with permission from [##REF##32612929##96##].</p></caption></fig>", "<fig position=\"float\" id=\"F10\"><label>Fig. 10.</label><caption><p id=\"P87\">Reconstruction results using (a)–(d) synthetic vascular data, (e)–(h) in vitro phantom data acquired from a “W” shape wire, and (i)–(l) in vivo data acquired from a human finger. (a), (e), and (i) Ground-truth or acquisition field illustrations. (b), (f), and (j) DAS. (c), (g), and (k) Results of UNET fed by DAS images. (d), (h), and (l) Results of UNET fed by transformed channel data. All figures are reproduced with permission from [##UREF##12##37##].</p></caption></fig>", "<fig position=\"float\" id=\"F11\"><label>Fig. 11.</label><caption><p id=\"P88\">Reconstruction results using (a)–(d) synthetic circular disks data, (e)–(h) synthetic vascular data, and (i)–(l) in vivo data acquired from the mouse trunk. (a) and (e) Ground-truth. (i) US B-mode image. (b), (f), and (j) TR results. (c), (g), and (k) Results of UNET fed by TR images. (d), (h), and (l) Results of the proposed DL (WNET-GP) fed by TR images. All figures are reproduced with permission from [##UREF##54##97##].</p></caption></fig>", "<fig position=\"float\" id=\"F12\"><label>Fig. 12.</label><caption><p id=\"P89\">Reconstruction results using (a)–(c) in vitro phantom (pencil lead) data and (d)–(f) in vivo data acquired from methylene blue injected into a mouse. (a) and (d) Ground-truth or acquisition illustrations. (b) and (e) Standard image reconstruction results. (c) and (f) Results of the DL model fed by (b) and (e) images. All figures are reproduced with permission from [##UREF##55##98##].</p></caption></fig>", "<fig position=\"float\" id=\"F13\"><label>Fig. 13.</label><caption><p id=\"P90\">Reconstruction results using (a)–(d) in vitro phantom (wire) data and (e)–(h) in vivo data acquired from a human hand. Multiple frame images (DAS images) were averaged. (a) and (e) Ground-truth or acquisition illustrations. (b) and (f) Standard averaging results. (c) and (g) Results using only the CNN-based model. (d) and (h) Results using the proposed DL model combining CNN and RNN. All figures are reproduced with permission from [##UREF##56##99##].</p></caption></fig>", "<fig position=\"float\" id=\"F14\"><label>Fig. 14.</label><caption><p id=\"P91\">Reconstruction results using (a) and (b) in vitro phantom data and (c) and (d) in vivo data acquired from melanoma on a patient’s heel. The phantom is heterogeneous, where three layers had different SOSs. (a) and (c) Standard DAS images using 1540 m/s as SOS. (b) and (d) Results using the proposed DL method. All figures are reproduced with permission from [##REF##34665732##100##].</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\" orientation=\"landscape\"><label>TABLE I</label><caption><p id=\"P92\">Data Acquisition Conditions and Reconstruction Models in Reviewed Papers</p></caption><table frame=\"box\" rules=\"all\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Input Data</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Backbone DL Model</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Specialty in DL</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Loss Function</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">DL Performance<xref rid=\"TFN1\" ref-type=\"table-fn\">*</xref></th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Training Data</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">In-vivo Test Data</th><th align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Transducer (Center freq., bandwidth)</th></tr></thead><tbody><tr><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Waibel <italic toggle=\"yes\">et al.</italic> [##UREF##10##27##]</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Channel data or DAS image</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">UNET</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Skip connection is replaced by conv. layer</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">MAE(Ll)</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">PSNR(DL)-PSNR(DAS) &gt; 19.8 dB</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">In-silico Phantom</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">-</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Linear (No ref.)</td></tr><tr><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Lan <italic toggle=\"yes\">et al.</italic> [##REF##32612929##96##]</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Channel data and DAS image</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">UNET</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Both raw data and image are inputted.</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">MSE(L2)</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">PSNR(DL)-PSNR(DAS) &gt; 7.8 dB</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">In-silico Phantom</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Human palm vessels</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Linear (7MHz, 80%)</td></tr><tr><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Kim <italic toggle=\"yes\">et al.</italic> [##UREF##12##37##]</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Transformed data</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">UNET</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Transformed data are inputted</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">MSE(L2)</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">PSNR(DL)-PSNR(DAS) &gt; 6.7 dB</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">In-silico Phantom</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Human finger vessels</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Linear (15MHz, 53%)</td></tr><tr><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Vu <italic toggle=\"yes\">et al.</italic> [##UREF##54##97##]</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">TR image</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CGAN</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Conditional GAN is used</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">MSE(L2) + Wasser-stein</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">PSNR(GAN)-PSNR(UNET) &gt; 0.8 dB</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">In-silico Phantom</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mouse trunk vessels</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Linear (5MHz, 60%)</td></tr><tr><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Hariri <italic toggle=\"yes\">et al.</italic> [##UREF##55##98##]</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">DAS image</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">UNET</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Pooling step is replaced by Wavelet transform</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">MSE(L2)</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">PSNR(DL)-PSNR(DAS) &gt; 1.9 dB</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">In-vitro Phantom</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Contrast agent injected into mouse</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Linear (15MHz, No ref.)</td></tr><tr><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Anas <italic toggle=\"yes\">et al.</italic> [##UREF##56##99##]</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">DAS images (multi-frame)</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CNN+RNN</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Muti-frame images are inputted</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">MSE(L2)</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">PSNR(DL)-PSNR(DAS averaging) &gt; 5.9dB</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">In-vitro Phantom</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Human finger vessels</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Linear (No ref.)</td></tr><tr><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Jeon <italic toggle=\"yes\">et al.</italic> [##REF##34665732##100##]</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">DAS images (multi-SOS)</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">UNET</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">More links between encoder and decoder</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">MSE(L2)</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">SNR_dB(DL)-SNR_dB(DAS) &gt; 20dB</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">In-silico Phantom</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Melanoma on a human subject’s heel</td><td align=\"center\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Linear (8.5MHz, No ref.)</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T2\"><label>TABLE II</label><caption><p id=\"P94\">Equations and Parameters for Simulation</p></caption><table frame=\"box\" rules=\"all\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><tbody><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">Circular array</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Aperture length Element pitch Element numbers Radius Center frequency 3 dB bandwidth</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">125.6 mm 0.2 mm 628 20 mm 15.63 MHz 8 MHz</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">Linear array</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Aperture length Element pitch Element numbers Center frequency 3 dB bandwidth</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">25.6 mm 0.1 mm 256 15.63 MHz 8 MHz</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">Data generation</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Applied equation Absorbed energy in each object System function Noise </td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">##FORMU##28##Eqs. 5## &amp; ##FORMU##31##6## 1 (constant) Identity function 0</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Reconstruction</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Applied equation</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n##FORMU##48##Eq. 8##\n</td></tr></tbody></table></table-wrap>" ]

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id=\"M15\" display=\"inline\"><mml:mi mathvariant=\"normal\">Γ</mml:mi><mml:mo>=</mml:mo><mml:mi>β</mml:mi><mml:msup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M16\" display=\"inline\"><mml:mi>c</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M17\" display=\"inline\"><mml:mi>β</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M18\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M19\" display=\"inline\"><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M20\" display=\"inline\"><mml:mi mathvariant=\"normal\">Γ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M21\" display=\"inline\"><mml:mi mathvariant=\"normal\">Φ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo>,</mml:mo><mml:mi>λ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M22\" display=\"inline\"><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M23\" display=\"inline\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"FD4\">\n<label>(4)</label>\n<mml:math id=\"M24\" display=\"block\"><mml:mfenced separators=\"|\"><mml:mrow><mml:msup><mml:mrow><mml:mo>∇</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"M25\" display=\"inline\"><mml:mi>δ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M26\" display=\"inline\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M27\" display=\"inline\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>δ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M28\" display=\"inline\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"FD5\">\n<label>(5)</label>\n<mml:math id=\"M29\" display=\"block\"><mml:mi>p</mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant=\"normal\">Γ</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:msup><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mfenced open=\"[\" close=\"]\" separators=\"|\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">∫</mml:mo><mml:mrow><mml:mspace width=\"0.25em\"/></mml:mrow></mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mspace width=\"0.25em\"/><mml:mo>-</mml:mo><mml:mspace width=\"0.25em\"/><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:mrow></mml:mfrac><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>δ</mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mspace width=\"0.25em\"/><mml:mo>-</mml:mo><mml:mspace width=\"0.25em\"/><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"M30\" display=\"inline\"><mml:mi>J</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M31\" display=\"inline\"><mml:mi>j</mml:mi><mml:mi mathvariant=\"normal\">t</mml:mi><mml:mi mathvariant=\"normal\">h</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"FD6\">\n<label>(6)</label>\n<mml:math id=\"M32\" display=\"block\"><mml:mi>y</mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mi>ψ</mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>p</mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>n</mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mfenced></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"M33\" display=\"inline\"><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M34\" display=\"inline\"><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M35\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M36\" display=\"inline\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M37\" display=\"inline\"><mml:mo stretchy=\"false\">{</mml:mo><mml:mrow><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∣</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mspace width=\"0.25em\"/></mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>J</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M38\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M39\" display=\"inline\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M40\" display=\"inline\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M41\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M42\" display=\"inline\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M43\" display=\"inline\"><mml:mi>y</mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M44\" display=\"inline\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M45\" display=\"inline\"><mml:mi>n</mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:math></inline-formula>", "<disp-formula id=\"FD7\">\n<label>(7)</label>\n<mml:math id=\"M46\" display=\"block\"><mml:mover accent=\"true\"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>˜</mml:mo></mml:mover><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi>ϱ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">∫</mml:mo><mml:mrow><mml:mspace width=\"0.25em\"/></mml:mrow></mml:mrow><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mi>y</mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"0.25em\"/><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mi>δ</mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mspace width=\"0.25em\"/><mml:mo>-</mml:mo><mml:mspace width=\"0.25em\"/><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced><mml:mi>d</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"M47\" display=\"inline\"><mml:mi>ϱ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M48\" display=\"inline\"><mml:mo stretchy=\"false\">{</mml:mo><mml:mrow><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∣</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>J</mml:mi></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"FD8\">\n<label>(8)</label>\n<mml:math id=\"M49\" display=\"block\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo>˜</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mrow><mml:mrow><mml:mi>ϱ</mml:mi><mml:munder><mml:mo>∑</mml:mo><mml:mi>j</mml:mi></mml:munder><mml:mfrac><mml:mrow><mml:mo>∂</mml:mo><mml:mi>y</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"0.25em\"/><mml:msubsup><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mi>j</mml:mi><mml:mi>′</mml:mi></mml:msubsup></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo>−</mml:mo><mml:msubsup><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mi>j</mml:mi><mml:mi>′</mml:mi></mml:msubsup></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math>\n</disp-formula>", "<disp-formula id=\"FD9\">\n<label>(9)</label>\n<mml:math id=\"M50\" display=\"block\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo>˜</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>F</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mrow><mml:munder><mml:mo>∑</mml:mo><mml:mi>j</mml:mi></mml:munder><mml:mi>y</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msubsup><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mi>j</mml:mi><mml:mi>′</mml:mi></mml:msubsup></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo>−</mml:mo><mml:msubsup><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mi>j</mml:mi><mml:mi>′</mml:mi></mml:msubsup></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>F</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:munder><mml:mo>∑</mml:mo><mml:mi>j</mml:mi></mml:munder><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"M51\" display=\"inline\"><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M52\" display=\"inline\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo>-</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>/</mml:mo><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M53\" display=\"inline\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:mo>-</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>/</mml:mo><mml:mi>c</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M54\" display=\"inline\"><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M55\" display=\"inline\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M56\" display=\"inline\"><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mi>A</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M57\" display=\"inline\"><mml:mi>A</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M58\" display=\"inline\"><mml:mi>y</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M59\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"FD10\">\n<label>(10)</label>\n<mml:math id=\"M60\" display=\"block\"><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mo>˜</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mi mathvariant=\"normal\">a</mml:mi><mml:mi mathvariant=\"normal\">r</mml:mi><mml:mi mathvariant=\"normal\">g</mml:mi><mml:mo>⁡</mml:mo><mml:mi mathvariant=\"normal\">m</mml:mi><mml:mi mathvariant=\"normal\">i</mml:mi><mml:mi mathvariant=\"normal\">n</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:munder><mml:msubsup><mml:mrow><mml:mfenced open=\"∥\" close=\"∥\" separators=\"|\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>-</mml:mo><mml:mi>A</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>g</mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"M61\" display=\"inline\"><mml:mi>g</mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math></inline-formula>", "<disp-formula id=\"FD11\">\n<label>(11)</label>\n<mml:math id=\"M62\" display=\"block\"><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">θ</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mtext>arg</mml:mtext><mml:mspace width=\"0.25em\"/><mml:mtext>min</mml:mtext><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>J</mml:mi></mml:munderover><mml:mi>l</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"bold-italic\">x</mml:mi><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"bold-italic\">y</mml:mi><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mo>;</mml:mo><mml:mi mathvariant=\"bold-italic\">θ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"M63\" display=\"inline\"><mml:mi>J</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M64\" display=\"inline\"><mml:mi>l</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M65\" display=\"inline\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M66\" display=\"inline\"><mml:mi>j</mml:mi><mml:mi mathvariant=\"normal\">t</mml:mi><mml:mi mathvariant=\"normal\">h</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M67\" display=\"inline\"><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mi mathvariant=\"bold-italic\">ˆ</mml:mi></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>g</mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">y</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>;</mml:mo><mml:mi mathvariant=\"bold-italic\">θ</mml:mi></mml:mrow></mml:mfenced></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M68\" display=\"inline\"><mml:mi>g</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M69\" display=\"inline\"><mml:mi mathvariant=\"bold-italic\">θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M70\" display=\"inline\"><mml:mi>j</mml:mi><mml:mi mathvariant=\"normal\">t</mml:mi><mml:mi mathvariant=\"normal\">h</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M71\" display=\"inline\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">y</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M72\" display=\"inline\"><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">θ</mml:mi></mml:mrow><mml:mi mathvariant=\"bold-italic\">ˆ</mml:mi></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M73\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">K</mml:mi><mml:mi mathvariant=\"normal\">L</mml:mi></mml:mrow></mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mi>‾</mml:mi></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi><mml:mi mathvariant=\"normal\">a</mml:mi><mml:mi mathvariant=\"normal\">t</mml:mi><mml:mi mathvariant=\"normal\">a</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∥</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mtext>model</mml:mtext></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">x</mml:mi><mml:mo>∣</mml:mo><mml:mi mathvariant=\"bold-italic\">y</mml:mi><mml:mo>;</mml:mo><mml:mi mathvariant=\"bold-italic\">θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfenced></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M74\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mtext>model</mml:mtext></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">x</mml:mi><mml:mo>∣</mml:mo><mml:mi mathvariant=\"bold-italic\">y</mml:mi><mml:mo>;</mml:mo><mml:mi mathvariant=\"bold-italic\">θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M75\" display=\"inline\"><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M76\" display=\"inline\"><mml:mi mathvariant=\"bold-italic\">y</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M77\" display=\"inline\"><mml:mi mathvariant=\"bold-italic\">θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M78\" display=\"inline\"><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mi>‾</mml:mi></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi><mml:mi mathvariant=\"normal\">a</mml:mi><mml:mi mathvariant=\"normal\">t</mml:mi><mml:mi mathvariant=\"normal\">a</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M79\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mtext>data</mml:mtext></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold-italic\">x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M80\" display=\"inline\"><mml:mi>l</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"FD12\">\n<label>(12)</label>\n<mml:math id=\"M81\" display=\"block\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">h</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>l</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>φ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>l</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">W</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>l</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">h</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>l</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"M82\" display=\"inline\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">h</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>l</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M83\" display=\"inline\"><mml:mi>l</mml:mi><mml:mtext>th</mml:mtext></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M84\" display=\"inline\"><mml:mi>l</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mi mathvariant=\"normal\">t</mml:mi><mml:mi mathvariant=\"normal\">h</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M85\" display=\"inline\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">W</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>l</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M86\" display=\"inline\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M87\" display=\"inline\"><mml:mi>φ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M88\" display=\"inline\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">h</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>l</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M89\" 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[ "<boxed-text id=\"BX1\" position=\"float\"><caption><title>Highlights</title></caption><list list-type=\"bullet\" id=\"L1\"><list-item><p id=\"P95\">Integrated PA and US (PAUS) imaging holds promise for disease detection and interventions, yet clinical adoption is hindered by poor image quality from handheld US arrays.</p></list-item><list-item><p id=\"P96\">Numerous studies highlight the potential of deep learning techniques to overcome PAUS platform limitations, but challenges are the lack of performance comparisons and in vivo validations.</p></list-item><list-item><p id=\"P97\">New networks may further enhance performance by successful transfer learning from more complex virtual tissue model to real applications or unsupervised learning incorporating US data.</p></list-item></list></boxed-text>" ]

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[ "<table-wrap-foot><fn id=\"TFN1\"><label>*</label><p id=\"P93\">Improvement in PSNR/SNR with respect to the standard method in simulation/in-vitro tests</p></fn></table-wrap-foot>" ]

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[ "<p>Note: this Editorial is published in both <italic toggle=\"yes\">Philosophical Transactions A</italic> and <italic toggle=\"yes\">Philosophical Transactions B</italic>.</p>" ]

[ "<p>Earlier in 2023, the Committee on Publication Ethics (COPE) published a discussion document entitled ‘Best Practices for Editing Collections' [##UREF##0##1##]. The document was written in response to concerns that have been raised about the quality of content in themed collections, highlighted by the fact that large numbers of articles published in such issues have had to be retracted [##UREF##1##2##]. Themed collections are becoming more prevalent in the publishing industry for a number of reasons; for example, aggressive commissioning tactics by some publishers; the corruptive effect of ‘publish or perish' for researchers; and simply because increased website functionality allows publishers to collate otherwise unconnected research articles that nevertheless cover a similar topic, into a ‘special collection'. Because guest-edited collections form the basis of publications in both <italic toggle=\"yes\">Phil. Trans. A</italic> (physical sciences) and <italic toggle=\"yes\">Phil. Trans. B</italic> (biological sciences), we consider that now is an opportune moment to reflect on these journals' publishing model in the context of ongoing discussions on publishing ethics and strategies.</p>", "<p>Since the 1990s, <italic toggle=\"yes\">Phil. Trans. A</italic> and <italic toggle=\"yes\">B</italic> have used the guest-edited issue model. Some of the issues record key presentations at scientific meetings, most of which have been arranged by the Royal Society following prior review by the Society's Hooke Committee. Other issues are solicited from field leaders and high-quality scientists by members of the Editorial Boards, the Commissioning Editors or the Editors-in-Chief, and a number also result from unsolicited approaches to the journals. All themed issue proposals that are submitted to the journals via these avenues undergo extensive peer review by external reviewers and members of the journals' Editorial Boards. The discussion in the COPE document was primarily targeted towards journals that publish a mixture of article collections and single articles, with the warning that publishing a large number of guest-edited article collections can lead to concern about the independence, impartiality and credibility of the journal in the absence of strict and transparent oversight by the Editors-in-Chief, senior editorial board members and in-house publishing team. Although it is never possible to guarantee that scientific misconduct will not happen, we consider that <italic toggle=\"yes\">Phil. Trans. A</italic> and <italic toggle=\"yes\">B</italic>, with their long history of specifically publishing themed issues, already have the recommended safeguards in place and that they serve as models for how to handle the solicitation, review and publishing of article collections. More information about this can be found in a recent blog post <uri xlink:href=\"https://royalsociety.org/blog/2023/04/theme-issue-publishing-royal-society/\">https://royalsociety.org/blog/2023/04/theme-issue-publishing-royal-society/</uri>.</p>", "<p>If guest-edited collections are managed with an eye to aiding scientific discourse rather than simply filling journal pages, and are rigorously reviewed and edited, we see several advantages in their favour. They can provide cutting-edge yet balanced views of a particular field that enhance the development of ideas and understanding within the field. Equally important, they can enable early career scientists who guest edit for the journals to gain exposure as conceptual leaders in their fields. Early career scientists are often thrown in at the deep end, with little experience or support in handling editorial duties/problems. The special help that the <italic toggle=\"yes\">Phil. Trans.</italic> editorial team provides to young scientists learning the art of guest-editing ‘on the job' is a valuable feature of the journals.</p>", "<p>Over the past decade, both primary research papers and review articles have become more susceptible to peer review manipulation, financial conflicts of interest, nepotism and nefarious actors such as paper mills and citation cartels. Rapid developments in Artificial Intelligence (AI), such as ChatGPT, have added to the concerns. No journal has all the answers to these problems, so careful monitoring at all stages from inception to publication is very important. To assist guest editors in obtaining quality reviews of individual articles, <italic toggle=\"yes\">Phil. Trans. B</italic> will soon be implementing Transparent Peer Review, with reviews and the authors' responses being published at the time of publication of the article. The journals have also developed policies that allow the limited use of AI, but only where appropriate and transparently acknowledged by the authors. For the reality is that AI has become an increasingly important and versatile tool in the physical and biological sciences. Its history can be traced back over decades to the development and application of cutting-edge computational technologies in data-intensive experimental and theoretical physical and engineering science. Nowadays, AI versatility extends to providing textual content, which relies on working from existing material and is therefore open to substantial abuse, not least in respect to aspects of plagiarism. Journal policies must adapt to ensure that the peer review process is able to identify inappropriate use of these tools.</p>", "<p>The <italic toggle=\"yes\">Philosophical Transactions</italic> have a long history (since 1660) of providing a forum for scientific debate. Nowadays the need is greater than ever before for a venue that publishes high-quality collections which provide a synthesis of ideas in an increasingly complex, often interdisciplinary, scientific landscape. The problems with publication of themed issues have in many ways arisen because of the largely for-profit nature of the publishing business. The Royal Society is an independent, non-profit organization (<uri xlink:href=\"https://royalsociety.org/journals/publishing-activities/publishing-values/\">https://royalsociety.org/journals/publishing-activities/publishing-values/</uri>), enabling its focus to remain on scientific quality rather than quantity. Many fields within the physical and biological sciences view <italic toggle=\"yes\">Phil. Trans.</italic> as the favoured venue for publishing high impact volumes that report major advances and set new directions in the conceptual development of their fields. This esteem is at the root of the high reputation, and thereby continuing success, of the A and B issues of the journal and, as its distinguished history has already demonstrated, this reputation for quality will continue to underpin its future as new challenges of digital information technology in scientific publication, most notably involving AI, inevitably emerge. It is our hope that the above explanation and reaffirmation of our aspirations and procedures will sustain and enhance further the confidence of the global scientific community to continue to take advantage of the unique publication opportunities for their disciplines that <italic toggle=\"yes\">Phil. Trans.</italic> facilitates, benefiting from the compliance of the editorial policies of the journals with the recommendations of the COPE discussion document.</p>" ]

[ "<title>Data accessibility</title>", "<p>This article has no additional data.</p>", "<title>Declaration of AI use</title>", "<p>We have not used AI-assisted technologies in creating this article.</p>", "<title>Authors' contributions</title>", "<p>J.D.: writing—original draft; R.A.D.: writing—original draft.</p>", "<p>Both authors gave final approval for publication and agreed to be held accountable for the work performed therein.</p>", "<title>Conflict of interest declaration</title>", "<p>We declare we have no competing interests.</p>", "<title>Funding</title>", "<p>We received no funding for this study.</p>" ]

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[ "<title>Introduction</title>", "<p>When he first introduced the notion of a conformal boundary into the study of asymptotically empty space–times [##UREF##0##1##,##UREF##1##2##] Penrose noted that the boundary would be null, space-like or time-like according as the cosmological constant was zero, positive or negative. While most applications of the idea of a conformal boundary have been to the zero-, asymptotically Minkowskian case, there also has been work on the non-zero cases, and some of that is reviewed here. We shall concentrate on positive , which is the appropriate case for cosmology of the universe in which we live.</p>", "<p>With a space-like future conformal boundary, we may follow Friedrich [##UREF##2##3##,##UREF##3##4##] and contemplate using the boundary as a Cauchy surface with data for a conformally extended set of Einstein equations. We review this material in §3. At the other end of the universe, assuming a suitable form of Penrose’s Weyl curvature hypothesis [##UREF##4##5##] we may also contemplate rescaling an initial ‘Big Bang’ singularity for use as a Cauchy surface for another similar set of equations. This will be discussed in §4. Finally, in §5, we recall Penrose’s ‘outrageous suggestion’ [##UREF##5##6##] and contemplate a universe of successive aeons, each of which is an expansion from a rescaled Big Bang surface to a rescaled future conformal boundary which in turn provides the rescaled Big Bang of the next aeon. Now, by assumption, there is a regular conformal metric common to all aeons, while the (conformally related) physical metrics run from singularity to conformal infinity in each aeon.</p>" ]

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[ "<p>One contribution of 13 to a discussion meeting issue ‘<ext-link xlink:href=\"http://dx.doi.org/10.1098/rsta/382/2267\" ext-link-type=\"uri\">At the interface of asymptotics, conformal methods and analysis in general relativity</ext-link>’.</p>", "<p>When he first introduced the notion of a conformal boundary into the study of asymptotically empty space–times, Penrose noted that the boundary would be null, space-like or time-like according as the cosmological constant was zero, positive or negative. While most applications of the idea of a conformal boundary have been to the zero-, asymptotically Minkowskian case, there also has been work on the non-zero cases. Here, we review work with a positive , which is the appropriate case for cosmology of the universe in which we live.</p>", "<p>This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.</p>" ]

[ "<title>Conformal rescaling</title>", "<p>This section is included to fix conventions, which do vary across the literature, though the content should be familiar.</p>", "<p>Following [##UREF##1##2##] and assuming the space–time signature is and the Riemann tensor is defined by\nwe need to impose Penrose’s <italic toggle=\"yes\">asymptotic simplicity</italic>, or at least <italic toggle=\"yes\">weak asymptotic simplicity</italic> [##UREF##0##1##,##UREF##1##2##]. Following [##UREF##1##2##], we take the physical metric to be , the unphysical metric to be , and these to be related by conformal rescaling\nwith a smooth conformal factor , so that towards the conformal boundary at physical infinity, we can take . We adopt the convention that and stand for the corresponding metric convariant derivatives. As part of the asymptotic conditions we assume that has some appropriate degree of smoothness near the surface , which is the conformal boundary or , and that is finite and non-zero there. The Christoffel symbols for the two associated metric connections are related by\nwhere (and of course ).</p>", "<p>The Weyl tensors for the two metrics, with indices appropriately arranged, are equal:\nbut the Ricci tensors have a more complicated relation:\nwhence also the Ricci scalars are related by\nThe first key result follows from equation (##FORMU##24##2.5##): if the Einstein equations with cosmological constant are satisfied by the physical metric then^{<xref rid=\"FN1\" ref-type=\"fn\">1</xref>}\nwhere is the trace of the physical energy–momentum tensor. If goes to zero at the conformal boundary , either because it is zero everywhere or because matter terms are decaying ‘at infinity’ then equation (##FORMU##24##2.5##) with finiteness of and (implied by smoothness of and ) implies\nThus is time-like, space-like or null according as is negative, positive or zero, and furthermore is a suitable normal to .</p>", "<p>For the next result, from equation (##FORMU##23##2.4##), we obtain that the trace-free part of the tensor\nvanishes at , so provided the trace-free part of the physical is bounded (or, more probable, is tending to zero) at , we deduce that the trace-free part of is zero at , i.e. is <italic toggle=\"yes\">umbilic</italic>. By refining the choice of one can make extrinsically flat for non-zero or shear-free and expansion-free for zero .</p>", "<p>For the final result in this section, we note the transformation\nThe Bianchi identity gives\nwhere, assuming the Einstein equations, the Schouten tensor is\nso that, provided decays suitably towards infinity, the left-hand side in equation (##FORMU##50##2.6##) goes to zero at . We multiply equation (##FORMU##50##2.6##) by to conclude that\nIf is non-null, this at once forces to vanish at . If is null, more work is required to arrive at the same conclusion. This is a strong result which we shall see again.</p>", "<p>It is worth noting how the conformal method is particularly welcoming to a trace-free energy–momentum tensor. Given a symmetric tensor , we note that\nThus if is a physical energy–momentum tensor satisfying the conservation equation with respect to \n<italic toggle=\"yes\">and</italic> is trace-free, then is also symmetric and trace-free and satisfies the conservation equation with respect to .</p>", "<title>Data at </title>", "<p>In order to pose an initial value problem with data at , one needs a conformally extended formulation of Einstein’s equations for the unphysical metric , valid at , which implies the usual Einstein equations for the physical metric in the region . This was achieved by Friedrich [##UREF##2##3##] in the vacuum case, by starting from the identity (##FORMU##23##2.4##). If we first consider the vacuum equations with cosmological constant then the left-hand side in identity (##FORMU##23##2.4##) is\nSubstitute into identity (##FORMU##23##2.4##) and multiply by to obtain\nWhen differentiated, all the inverse powers of cancel and this becomes a non-singular set of second-order PDEs in . Next, a suitable form must be found to which existing theorems can be applied. We sketch the derivation of this form as the original papers are readily available and are straightforward to read.</p>", "<p>The method relies upon prolonging the system to reduce it to a first-order formulation by introducing more variables and to work in a tetrad formalism. Following [##UREF##6##7##], we introduce bold letters for tetrad sub and superscripts. Thus a tetrad can be written for and orthonormality in the unphysical metric is expressed by\nThe Ricci rotation coefficients for the tetrad are denoted by and are obtained by first-order equations from the tetrad. Likewise, the curvature is obtained by first-order equations from the Ricci rotation coefficients. Since equation (##FORMU##75##3.1##) has second-order derivatives of , we introduce a vector to lower the order ( was called in [##UREF##3##4##]) so these terms become . Friedrich also introduces the variable .</p>", "<p>At the end of §2, we saw that at so, assuming smoothness, we can introduce a new (smooth) quantity\nand then, using equation (##FORMU##50##2.6##), the Bianchi identity for can be written\nwhich joins the set of first-order equations.</p>", "<p>The rest of the unphysical curvature consists of the unphysical Ricci scalar, , and the trace-free part of the unphysical Ricci tensor which Friedrich incorporates in a new quantity:\nNow Friedrich writes out the first-order system for the variables . Note that does not appear in this set of variables, though it does appear in the first-order system. This is related to the residual conformal invariance in the equations: if is replaced by for positive then the variables transform among themselves, and by suitable choice of any arbitrary choice of can be made.</p>", "<p>By choosing suitable coordinate and tetrad gauge conditions, Friedrich [##UREF##2##3##] is able to extract a symmetric hyperbolic system and prove existence and uniqueness in suitable function spaces given suitable data at . The data are determined on , which can be any 3-manifold, not just as it would be for de Sitter, by choice of 3-metric, say where , and a second symmetric tensor (called in [##UREF##2##3##]) satisfying\nwhere is the Levi–Civita derivative for . The solution exists for a finite interval of the coordinate time used but, since is infinitely far away in the physical metric, this will be an infinite amount of physical time.</p>", "<p>In Friedrich’s approach, is a constant of integration. With data at (understood as ) close to data for the de Sitter metric, Friedrich’s solution may exist all the way back to a past infinity, , and beyond, possibly through several such ‘aeons’ (pre-empting Penrose’s CCC terminology) but these solutions have no convenient initial Big Bang. When they do stop, they are likely to have messy initial singularities. It is also implicit in Friedrich’s method that one could choose a Cauchy surface inside the space–time and evolve all the way forward to .</p>", "<p>In [##UREF##3##4##], Friedrich extended his method to non-vacuum solutions with a trace-free energy–momentum tensor and dealt in detail with Einstein–Yang-Mills+ metrics. These have more data at : in addition to there are a Lie-algebra-valued connection one-form which we can write , where is the Lie-algebra index, and a vector which is the electric part of the SD Yang-Mills field. Omitting the Yang-Mills index in the interest of clarity, these are subject to the rescaling freedom\nand to constraints which, following [##UREF##3##4##], we can give in a space-spinorial form: with spinor equivalents.^{<xref rid=\"FN2\" ref-type=\"fn\">2</xref>}\nwe require\nto hold on , where is the Killing form of the YM gauge group and is the Lie-algebra commutator. The second two are what is left of the YM equations.</p>", "<p>A radiation fluid source can be included in Friedrich’s picture [##UREF##7##8##], as can a massless Vlasov source [##UREF##8##9##]. Both these references use conformal methods from a Cauchy surface within the space–time and evolve all the way to . More recently, Friedrich has extended his conformal methods to cover some cases of source for which the energy–momentum tensor is not trace-free: some particular massive scalar field sources in [##UREF##9##10##], and dust [##UREF##10##11##] (both still with of course).</p>", "<p>Conformal methods are not essential for this kind of work and there is a large body of literature investigating long-time existence into the future in the presence of a positive cosmological constant. For example: [##UREF##11##12##] proves long-time existence in physical time for Einstein-Vlasov plus scalar fields with a potential plus , without conformal methods; [##UREF##12##13##–##UREF##14##15##] do this for various Einstein–Euler plus situations; and there are many earlier references, going back at least to [##UREF##15##16##], to what was called ‘the cosmological no-hair conjecture’ (or sometimes ‘theorem’). It is convenient to put all these classes in the context of a <italic toggle=\"yes\">Starobinsky expansion</italic>.</p>", "<p>The Starobinsky expansion of the space–time metric in the presence of a positive cosmological constant is [##UREF##16##17##]\nwhere\nwhere all powers may appear in except for , and the coefficient metrics are -independent. The coordinate system is defined geometrically: choose a space-like hypersurface to be far enough into the future that the normal congruence of geodesics has no conjugate points into the future. Let be proper-time on this congruence and let the space coordinates be comoving along it. Thinking of as a defining function for one sees a resemblance between (##FORMU##135##3.2##), (##FORMU##136##3.3##) and the <italic toggle=\"yes\">ambient metric construction</italic> of Riemannian geometers [##UREF##17##18##]. Using as conformal factor, we interpret as the metric of . It was shown in [##UREF##18##19##] that the metrics of Friedrich in [##UREF##2##3##,##UREF##3##4##] can all be written in this form, but has not been proved for the metrics of [##UREF##7##8##,##UREF##8##9##,##UREF##12##13##,##UREF##14##15##], though it is a convenient way to present the results.</p>", "<p>Clearly, there is in the metric form of equation (##FORMU##136##3.3##), a freedom in the choice of initial surface, which effects the change\nand requires a change of :\nThis in turn has the effect\nwhere , and this is the residual gauge freedom seen above and already noted in [##UREF##2##3##].</p>", "<p>For all the examples we consider, the coefficient has a universal expression\nwhere are, respectively, the Ricci tensor and Ricci scalar of the metric .</p>", "<p>Assuming that there is a Starobinsky expansion for the metrics of [##UREF##7##8##], one can take from [##UREF##18##19##] that data on are with non-negative , and expansions for the density and fluid velocity beginning\nsubject to the constraints of equation (##FORMU##156##3.5##) and\nThe rescaling freedom in this case is\nas is clear from expansions (##FORMU##162##3.6##).</p>", "<p>Again assuming there is a Starobinsky expansion for the Einstein–Vlasov metrics of [##UREF##8##9##], the data on become where is the non-negative distribution function (which does not change under rescaling). The constraints are that is trace-free and satisfies the divergence condition\n</p>", "<title>Penrose’s Weyl curvature hypothesis and data at the Bang</title>", "<p>The Weyl curvature hypothesis is also due to Penrose, [##UREF##4##5##]. He gives physical arguments for the initial singularity of the universe being very special as a singularity of a Lorentzian manifold, and conjectures that it is so special that while the space–time Ricci curvature is singular, the Weyl curvature is not: he conjectures that it must be finite or zero. It is not immediately clear how to make a mathematical statement of this—the metric and Ricci tensor are singular but the Weyl tensor is not—but a simple strategy for doing so is to require that there is a conformal rescaling of the physical metric so that the initial singularity becomes a regular surface in an unphysical, extended manifold. An initial singularity for which this is possible has been variously defined as <italic toggle=\"yes\">isotropic</italic> [##UREF##19##20##], <italic toggle=\"yes\">conformal</italic> [##UREF##20##21##,##UREF##21##22##] or a <italic toggle=\"yes\">conformal gauge singularity</italic> [##UREF##22##23##]. Given the relation (##FORMU##22##2.3##), if the unphysical Weyl tensor is finite, as it would be in such a setting, then so too is the physical one by (##FORMU##22##2.3##), and we have imposed the Weyl curvature hypothesis.</p>", "<p>It might be thought that the existence of such a conformal extension was a stronger assumption, even a <italic toggle=\"yes\">much</italic> stronger assumption, than finiteness of the Weyl tensor, but it is possible to show a local equivalence: given an incomplete conformal geodesic ending at the singularity, and boundedness of the components of the Weyl tensor and its derivatives up to order in a Weyl-propagated frame along , there is a conformal extension of a terminal neighbourhood of (see [##UREF##22##23##] for details).</p>", "<p>It should be noticed that the conformal rescaling envisaged here has an important difference from that in equation (##FORMU##10##2.1##): there the physical metric becomes infinitely large towards the conformal boundary, so that goes to zero at ; here, on the other hand, the physical metric becomes very small (think what happens to the volume) so is goes to infinity. For that reason, it is easier to redefine the rescaling the other way round: suppose the physical metric and unphysical metric are related by\nwith at the conformal boundary, which we will call .</p>", "<p>There is another importance difference which we may illustrate by considering FLRW metrics. Consider for simplicity the spatially flat FLRW metric\nwith a polytropic perfect fluid source, so that the energy–momentum tensor is\nThe usual range in is from 1 (dust) to 2 (stiff matter) with 4/3 picked out as radiation and having a trace-free energy–momentum tensor. The conservation equation with the metric of equation (##FORMU##185##4.2##) integrates to give\nwith a constant of integration, leaving the Friedmann equation as\nwith , to be solved. Thus\nchoosing constants of integration so that at , which is the location of the curvature singularity. Conformal rescaling as in equation (##FORMU##182##4.1##) to add the boundary should clearly be done with and this will not be differentiable at for any allowed . We can write it instead in terms of conformal time , defined via , obtaining\nThis is differentiable but with zero derivative at for , is smooth for and not smooth for larger . This behaviour of the conformal factor is quite different from that in §3, where we chose it smooth or of some finite differentiability and with non-zero derivative at the boundary.</p>", "<p>The case of a radiation fluid, when can be taken to be smooth, was considered first by Newman [##UREF##20##21##,##UREF##21##22##]. He was able to reduce the Einstein equations for an irrotational radiation fluid with a conformal gauge singularity to a first-order system in a conformal time and comoving space coordinates, of the form\nwith data . The coefficient matrices and are polynomials in the components of , which include metric and connection components and gauge and constraint quantities, and with and zero, the system is symmetric hyperbolic, so are symmetric with positive definite. Such a system would more commonly be called <italic toggle=\"yes\">Fuchsian</italic> now. There is an obvious constraint on the data, that now commonly called the <italic toggle=\"yes\">Fuchsian condition</italic>, but for his theorem, Newman requires an apparently stronger condition\nwhich in fact turns out to be no stronger. Newman also requires an eigenvalue condition, that the matrix have no positive integer eigenvalues. From this Newman [##UREF##21##22##] proves well-posedness with data just the spatial 3-metric of the data surface . As a Corollary he notes that if the initial metric is a metric of constant curvature then uniqueness of solutiion forces the solution to be FLRW: in this case, the intial Weyl tensor is finite, but should it be zero initially then it will remain zero in the evolution.</p>", "<p>Anguige &amp; Tod [##UREF##23##24##] were able to extend this result for in the range . In the extra cases, is not smooth but one none-the-less arrives at a system like equation (##FORMU##207##4.4##) and concludes as before that the initial 3-metric is all of the data—the rescaled equations are well-behaved although the conformal factor is not. In particular, one has the same Corollary: if the Weyl tensor is zero initially then it is always zero. This Corollary seemed unreasonably strong from a physical point of view, so consideration was given to massless Einstein–Vlasov solutions with a conformal gauge singularity, first with spatial homogeneity [##UREF##24##25##] and later without [##UREF##25##26##]. One still has well-posedness with data at the initial singularity but now the only datum is the initial distribution function subject to a vanishing dipole condition. This in turn determines the initial 3-metric and fundamental form but in an indirect way. Now it is possible to have the Weyl tensor zero initially but becoming non-zero later.</p>", "<p>Note we have zero in these results. One would expect the cosmological constant to have no effect near the initial singularity, just as one would expect replacing massless particles by massive ones in the Vlasov case to have negligible effect near the singularity, and this was verified at least for the spatially homogeneous case in [##UREF##26##27##].</p>", "<p>It was suggested (Anguige K and Rendall A, 2000) that one might find a bridge between the perfect fluid case and the Vlasov case, two cases for which the free data is notably different, by a consideration of the Einstein–Boltzmann equations, which might bridge the two extremes. With data at the initial singularity, this might as well be massless, and remarkably little seems to be known about the massless Boltzmann equation in curved space–time. However, in [##UREF##27##28##], it was shown that for a certain set of reasonable scattering cross sections, the isotropic and homogeneous case (i.e. FLRW) can be proved to be well-posed, and work-in-progress [##UREF##28##29##] indicates that the same set of cross-sections give a well-posed system in Bianchi type I. The hard part is the Boltzmann equation!</p>", "<p>It should be noted that the data required are very different for the Cauchy problem with data given at the initial singularity as against data given at : there is more freedom at . This is essentially because, evolving back from one may have a solution become singular but it will not have the ordered nature necessary to be a conformal gauge singularity. Conversely, evolving forward from a conformal gauge singularity in the presence of a positive one is very likely to arrive at a decent .</p>", "<title>Conformal cyclic cosmology or CCC</title>", "<p>We have seen that an expanding cosmological model with a positive is very likely to have a , and then the conformal metric will extend through it and the Weyl tensor will vanish at it. We have also seen that a simple way to impose the Weyl curvature hypothesis, of finite or zero Weyl curvature at the initial Big Bang, is to suppose that the conformal metric extends through the Bang surface. This will make the initial Weyl tensor finite rather than necessarily zero. It is hard, even impossible, to ‘cause’ the Weyl curvature hypothesis to hold <italic toggle=\"yes\">from the future</italic> but now there is a route to causing it <italic toggle=\"yes\">from the past</italic> and this is the essence of Penrose’s outrageous suggestion: suppose that the conformal metric extends through both and the Bang surface, then can become the Bang surface of a subsequent <italic toggle=\"yes\">aeon</italic>, and at it the initial Weyl tensor is necessarily zero and the Weyl curvature hypothesis holds. The physical metrics in the two aeons are different but are both conformally related to a common, non-physical ‘bridging’ metric and the Einstein equations are satisfied by both physical metrics. Each aeon is a complete universe evolving for an infinite proper time from an initial singularity, but lasting only for a finite conformal time before being followed by another aeon. There is no assumption of <italic toggle=\"yes\">periodicity</italic> but there is a conformal metric that extends through all aeons: the conformal metric can be said to be <italic toggle=\"yes\">cyclic</italic> whence Penrose’s name for this model, conformal cyclic cosmology or CCC.</p>", "<p>The simplest model of this scenario, which we will call <italic toggle=\"yes\">the toy model</italic>, is an FLRW metric with a radiation fluid as source and a positive cosmological constant. Take the metric to be\nwhere is the scale factor, is one of the three standard constant-curvature Riemannian 3-metrics, is proper-time and is conformal time, so . The energy–momentum tensor is\nwhere . From the conservation equation for this , we find that is constant. Call this constant then the only remaining independent Einstein equation is the Freedman equation which in terms of conformal time is\nwhere . We choose the solution with at , so that there is an initial singularity, then and increases monotonically^{<xref rid=\"FN3\" ref-type=\"fn\">3</xref>} with and diverges to infinity at\nwhich is finite. Now it is straightforward to see from equation (##FORMU##248##5.1##) that\nso we can make a model for CCC by using for and then replacing by for , and so on: at each passage through , is replaced by for suitable . All aeons are diffeomorphic and satisfy the Einstein equations with the same and .</p>", "<p>It is a simple matter to add a dust source to this FLRW cosmology: add a term to the right-hand side of equation (##FORMU##248##5.1##) where is a constant of integration determining the dust contribution to the density as . One immediately loses the symmetry in the Freedman equation just noted but obtains a function more closely resembling that of the observed universe. Using values of taken from observation, one can compute a more realistic and as shown in [##UREF##29##30##] one can conclude that, if is the conformal time for ‘now’ then the ratio is approximately 0.74: in other words, measured in conformal time the universe now is 74% of the way to . If we move on to the universe when it is ten times its present age (in proper time) then the fraction of conformal time remaining is just : the exponential expansion in proper time is really making itself felt, as the infinite proper time remaining is squeezed into a small amount of conformal time. For this reason, events occurring at times greater than are essentially <italic toggle=\"yes\">at</italic>\n.</p>", "<p>In a general model for CCC, we can suppose consecutive aeons have physical metrics and each conformally related to a single unphysical conformal metric by conformal factors and \nwith diverging at the common boundary and vanishing there. We assume that the product is smooth and non-zero in a neighbourhood of and then we can change the choice of so as to set^{<xref rid=\"FN4\" ref-type=\"fn\">4</xref>}\nThis can not be done if some other process fixes the choice of but otherwise it is just a gauge choice. It has the consequence that^{<xref rid=\"FN5\" ref-type=\"fn\">5</xref>}\nAs a consequence the hatted and checked Ricci tensors have a relation like that in equation (##FORMU##23##2.4##). Thus for example it won’t generally be the case that both aeons can have simple perfect fluid sources—at least one side has to have contributions like to the Ricci tensor and therefore to the energy–momentum tensor: the toy model, where both sides are in fact perfect fluids, is seen to be exceptional. It is an outstanding question what the field equations should be in CCC and a suggestion is made for them below.</p>", "<p>CCC is intended to be a model of the actual universe so it must connect with observation. That is not the principal interest of this meeting but it is worth touching on it briefly. First, CCC automatically imposes the Weyl curvature hypothesis. Next, an inevitable consequence of CCC is that the <italic toggle=\"yes\">horizon problem</italic> is dissolved: events widely separated on the sky may not have been causally connected in this aeon but they will have been if one takes account of earlier aeons. Thus we do not need inflation to solve the horizon problem and it is natural to try to do without it completely, at least near the Bang. If all that is needed physically for inflation is a period of exponential expansion, and since there will always be such a period late in any aeon, one may suppose that inflation happens <italic toggle=\"yes\">before the Bang</italic>, i.e. late in the previous aeon. In this way, the well known propensity of inflation to produce a suitable spectrum of density perturbations is retained.</p>", "<p>Massless radiation, whether gravitational or electromagnetic, moves along null geodesics and so may be expected to pass through from one aeon to the following one. Late in the previous aeon the diversity of the universe is much reduced—stars and galaxies will have gone and the universe will be populated by black holes which may be very large and will occasionally collide and merge, as well as evaporating by the Hawking process. These events will happen very close to in conformal time so the energy and radiation they generate will meet in a spherical annulus of limited radius or even a small ball. These regions of higher energy may be expected to have an effect on the mass–energy distribution on the last-scattering surface in the next aeon and therefore to give rise to characteristic imprints on the cosmic microwave background. There are in the literature claims to have detected these—see references in [##UREF##30##31##–##UREF##33##34##]—and other claims denying their significance—see e.g. references in [##UREF##34##35##–##UREF##37##38##]. Roughly speaking, the arguments are over statistical significance. There is a need for a more detailed astrophysical account of the formation and energetics of the rings and the Hawking points, and indeed for an account of the transformation of physical quantities from one aeon to the next. It seems quite likely that there is a smoothing process between the data at the Bang from the previous aeon, which will mostly arise from a distribution of evaporating super-massive black-holes, and the last-scattering-surface in the next aeon, and repeated iteration of this smoothing may explain the homogeneity of the universe. It also seems quite likely that there is a place for inflation in the period of exponential expansion at the end of the previous aeon^{<xref rid=\"FN6\" ref-type=\"fn\">6</xref>} and that this generates the density perturbations in the next aeon, but these suggestions are speculative.</p>", "<p>This article is concluded with a suggestion of what the field equations of CCC might be, based on a paper of Lübbe, [##UREF##38##39##]. Recall that a trace-free energy–momentum fits best with conformal methods, that one wants a component of the matter to have a radiation energy–momentum tensor, and that one needs some way to handle the terms in equation (##FORMU##23##2.4##), perhaps by having scalar fields in both aeons.^{<xref rid=\"FN7\" ref-type=\"fn\">7</xref>} Correcting slightly the account of [##UREF##38##39##] given in [##UREF##29##30##], suppose we have a <italic toggle=\"yes\">conformal scalar field</italic>, that is a scalar field satisfying the <italic toggle=\"yes\">conformal scalar field equation</italic>\nwhere is a constant to be chosen below. Such a scalar field has a kind of energy–momentum tensor given by\nwhere\nFor the trace and divergence of , we calculate\nso that, given the field equation , has the character of (plus or minus) a trace-free energy–momentum tensor for , conserved by virtue of the field equation.^{<xref rid=\"FN8\" ref-type=\"fn\">8</xref>} Note does not obviously satisfy any of the familiar energy conditions because of the term in . Note also that, if then are solutions (along with of course).</p>", "<p>Under conformal rescaling\nwe have\nso the conformal scalar field equation is conformally invariant, and transforms as an energy–momentum tensor should.</p>", "<p>If we regard as the energy–momentum tensor for subject to the field equation (##FORMU##301##5.3##) and include it in the Einstein equations with sources, then, choosing a sign, these can be taken to be\nwhere represents any other (extra) matter fields present, assumed trace-free.</p>", "<p>Now suppose and then is a solution of the field equation (##FORMU##301##5.3##) and we calculate from equation (##FORMU##303##5.4##)\nso the Einstein equations with these sources can be rewritten\nThis is now conformally invariant: multiply it by and look at the three terms in turn\nIf we now choose then the rescaling is\nwhich is the same as equation (##FORMU##326##5.6##) but with tildes.</p>", "<p>To make equations for CCC from this, suppose we have two conformal scalar fields, say and , and the (bridging) metric and consider the field equation\nThis is conformally invariant from what we have seen, so we can rescale to set (wherever is non-zero) giving or to set (with the corresponding caveat) giving . For the first, choose so and equation (##FORMU##334##5.7##) becomes\nwhere , while for the second choose and so that equation (##FORMU##334##5.7##) becomes\nwhere . Thus the two aeons have the same field equations, just different fields. Following the suggestion of Penrose [##UREF##5##6##], we may seek to interpret the conformal scalar fields as the dark matter in the respective aeons.</p>", "<p>If this is going to be a model of CCC, we need to blow up at a (space-like) surface and to be zero there, which can probably be arranged by choice of data (this is under investigation). We will not have unless at but this is a gauge choice we do not have to make.</p>", "<p>It remains to be seen whether CCC can be a viable theory of the universe we inhabit, but it fits very naturally into the story of conformal methods applied to cosmology.</p>" ]

[ "<title>Data accessibility</title>", "<p>This article has no additional data.</p>", "<title>Declaration of AI use</title>", "<p>I have not used AI-assisted technologies in creating this article.</p>", "<title>Authors' contributions</title>", "<p>P.T.: writing—original draft, writing—review and editing.</p>", "<title>Conflict of interest declaration</title>", "<p>I declare I have no competing interests.</p>", "<title>Funding</title>", "<p>No funding has been received for this article.</p>" ]

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display=\"block\"><mml:mrow><mml:mover><mml:mi>R</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mi>Λ</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM21\"><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM22\"><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM23\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM24\"><mml:mi>R</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM25\"><mml:mi>◻</mml:mi><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM26\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM27\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM3\"><mml:math id=\"DM8\" display=\"block\"><mml:msup><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>Θ</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>Θ</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mspace width=\"1em\"/><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mtext>at</mml:mtext></mml:mstyle><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM28\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM29\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM30\"><mml:msub><mml:mi>Θ</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM31\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM4\"><mml:math id=\"DM9\" display=\"block\"><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>Θ</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mi>Θ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>R</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM32\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM33\"><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM34\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM35\"><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>Θ</mml:mi><mml:mi>b</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM36\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM37\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM38\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM39\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM40\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM41\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M2x6\"><label>2.6</label><mml:math id=\"DM10\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mi>d</mml:mi></mml:msub><mml:msubsup><mml:mrow><mml:mover><mml:mi>C</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:msubsup><mml:mi>C</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msubsup><mml:mi>C</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mi>Υ</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230043UM5\"><mml:math id=\"DM11\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mi>d</mml:mi></mml:msub><mml:msubsup><mml:mrow><mml:mover><mml:mi>C</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>a</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>b</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230043UM6\"><mml:math id=\"DM12\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>6</mml:mn></mml:mfrac><mml:mi>Λ</mml:mi><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM42\"><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM43\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM44\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM7\"><mml:math id=\"DM13\" display=\"block\"><mml:msubsup><mml:mi>C</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mi>Θ</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mtext>at</mml:mtext></mml:mstyle><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM45\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM46\"><mml:msubsup><mml:mi>C</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM47\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM48\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM49\"><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM8\"><mml:math id=\"DM14\" display=\"block\"><mml:msup><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Θ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>Υ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM50\"><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM51\"><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM52\"><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>:=</mml:mo><mml:msup><mml:mi>Θ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM53\"><mml:mi mathvariant=\"normal\">∇</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM54\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM55\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM56\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM57\"><mml:mi>Θ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM58\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM59\"><mml:mi>Θ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM9\"><mml:math id=\"DM15\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>R</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>Λ</mml:mi><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>Λ</mml:mi><mml:msup><mml:mi>Θ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM60\"><mml:msup><mml:mi>Θ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M3x1\"><label>3.1</label><mml:math id=\"DM16\" display=\"block\"><mml:msup><mml:mi>Θ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>Θ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Θ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mi>Θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mi>Θ</mml:mi><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mi>Θ</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Θ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Θ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>c</mml:mi></mml:msup><mml:mi>Θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mi>Θ</mml:mi><mml:msup><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>c</mml:mi></mml:msup><mml:mi>Θ</mml:mi><mml:mo>+</mml:mo><mml:mi>Λ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM61\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM62\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>Θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM63\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM64\"><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow><mml:mo mathvariant=\"bold\">=</mml:mo><mml:mn mathvariant=\"bold\">0</mml:mn><mml:mo mathvariant=\"bold\">,</mml:mo><mml:mo>…</mml:mo><mml:mo mathvariant=\"bold\">,</mml:mo><mml:mn mathvariant=\"bold\">3</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM10\"><mml:math id=\"DM17\" display=\"block\"><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>η</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mtext>the Minkowski metric</mml:mtext></mml:mstyle><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM65\"><mml:msubsup><mml:mi>γ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">c</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM66\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM67\"><mml:msub><mml:mi>Σ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>:=</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM68\"><mml:msub><mml:mi>Σ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM69\"><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM70\"><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mi>Σ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">b</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM71\"><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>4</mml:mn></mml:mfrac><mml:mi>◻</mml:mi><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM72\"><mml:msubsup><mml:mi>C</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM73\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM11\"><mml:math id=\"DM18\" display=\"block\"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>:=</mml:mo><mml:msup><mml:mi>Θ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM74\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM12\"><mml:math id=\"DM19\" display=\"block\"><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msubsup><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">c</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">d</mml:mi></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM75\"><mml:mi>R</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM13\"><mml:math id=\"DM20\" display=\"block\"><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>:=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>4</mml:mn></mml:mfrac><mml:mi>R</mml:mi><mml:msub><mml:mi>η</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM76\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msubsup><mml:mi>γ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">c</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mi>Θ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>Σ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM77\"><mml:mi>R</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM78\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM79\"><mml:mrow><mml:mover><mml:mi>Θ</mml:mi><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM80\"><mml:mi>f</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM81\"><mml:mi>f</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM82\"><mml:mi>R</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM83\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM84\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM85\"><mml:msup><mml:mrow><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM86\"><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">j</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM87\"><mml:mrow><mml:mi mathvariant=\"bold\">i</mml:mi></mml:mrow><mml:mo mathvariant=\"bold\">,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">j</mml:mi></mml:mrow><mml:mo mathvariant=\"bold\">,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">k</mml:mi></mml:mrow><mml:mo>⋯</mml:mo><mml:mo mathvariant=\"bold\">=</mml:mo><mml:mn mathvariant=\"bold\">1</mml:mn><mml:mo mathvariant=\"bold\">,</mml:mo><mml:mn mathvariant=\"bold\">2</mml:mn><mml:mo mathvariant=\"bold\">,</mml:mo><mml:mn mathvariant=\"bold\">3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM88\"><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">j</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM89\"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">j</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM14\"><mml:math id=\"DM21\" display=\"block\"><mml:msup><mml:mi>a</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">j</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>=</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">i</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM90\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">i</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM91\"><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">j</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM92\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM93\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM94\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM95\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM96\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>−</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM97\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM98\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM99\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM100\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM101\"><mml:msubsup><mml:mi>A</mml:mi><mml:mi>i</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM102\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM103\"><mml:msubsup><mml:mi>E</mml:mi><mml:mi>i</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM15\"><mml:math id=\"DM22\" display=\"block\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>c</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>A</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>θ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msup><mml:mi>θ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msup><mml:mi>θ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>E</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230043UM16\"><mml:math id=\"DM23\" display=\"block\"><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mtext>As spinor fields:</mml:mtext></mml:mstyle><mml:mo> </mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>ϕ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230043UM17\"><mml:math id=\"DM24\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>κ</mml:mi><mml:mi>H</mml:mi><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>ϕ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mi>ϕ</mml:mi><mml:mrow><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mo>&gt;</mml:mo><mml:mn>2</mml:mn><mml:msubsup><mml:mi>D</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:mrow><mml:mi>C</mml:mi></mml:msubsup><mml:msub><mml:mi>α</mml:mi><mml:mrow><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>C</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msubsup><mml:mi>α</mml:mi><mml:mi>B</mml:mi><mml:mrow><mml:mi>C</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">]</mml:mo></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>ϕ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mi>ϕ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:mo>†</mml:mo></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>ϕ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msub><mml:mi>ϕ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">]</mml:mo></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM104\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM105\"><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM106\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>,</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM107\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM108\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM109\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM110\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM111\"><mml:mi>Λ</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>H</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mi>H</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M3x2\"><label>3.2</label><mml:math id=\"DM25\" display=\"block\"><mml:mi>g</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>j</mml:mi></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230043M3x3\"><label>3.3</label><mml:math id=\"DM26\" display=\"block\"><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>b</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>3</mml:mn><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM112\"><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mi>n</mml:mi><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM113\"><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM114\"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM115\"><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM116\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM117\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM118\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM119\"><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM120\"><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM121\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM122\"><mml:mi>Θ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM123\"><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM124\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM18\"><mml:math id=\"DM27\" display=\"block\"><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mrow><mml:mover><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM125\"><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM19\"><mml:math id=\"DM28\" display=\"block\"><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo stretchy=\"false\">→</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>2</mml:mn><mml:mi>H</mml:mi></mml:mrow></mml:mfrac><mml:mo> </mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>f</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mi>H</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230043M3x4\"><label>3.4</label><mml:math id=\"DM29\" display=\"block\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>c</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>θ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msup><mml:mi>θ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM126\"><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>H</mml:mi><mml:mi>f</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM127\"><mml:msub><mml:mi>b</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M3x5\"><label>3.5</label><mml:math id=\"DM30\" display=\"block\"><mml:msub><mml:mi>b</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>4</mml:mn></mml:mfrac><mml:mi>ρ</mml:mi><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM128\"><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mo> </mml:mo><mml:mi>ρ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM129\"><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM130\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM131\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM132\"><mml:mi>M</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M3x6\"><label>3.6</label><mml:math id=\"DM31\" display=\"block\"><mml:mi>μ</mml:mi><mml:mo>=</mml:mo><mml:mi>M</mml:mi><mml:mo> </mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>5</mml:mn><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo> </mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mo> </mml:mo><mml:msup><mml:mi>u</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230043M3x7\"><label>3.7</label><mml:math id=\"DM32\" display=\"block\"><mml:msup><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mi>D</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msubsup><mml:mi>c</mml:mi><mml:mi>j</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mn>64</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>9</mml:mn><mml:mi>H</mml:mi></mml:mrow></mml:mfrac><mml:mi>M</mml:mi><mml:msub><mml:mi>U</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230043UM20\"><mml:math id=\"DM33\" display=\"block\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>c</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mi>M</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>θ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msup><mml:mi>θ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msup><mml:mi>θ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:msub><mml:mi>U</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM133\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM134\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM135\"><mml:mi>f</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM136\"><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM21\"><mml:math id=\"DM34\" display=\"block\"><mml:msub><mml:mi>D</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:msubsup><mml:mi>c</mml:mi><mml:mi>i</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mn>16</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>H</mml:mi></mml:mrow></mml:mfrac><mml:mfrac><mml:mn>1</mml:mn><mml:msqrt><mml:mi>a</mml:mi></mml:msqrt></mml:mfrac><mml:mo>∫</mml:mo><mml:mi>f</mml:mi><mml:msub><mml:mi>p</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msup><mml:mi>d</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mi>p</mml:mi><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM137\"><mml:msubsup><mml:mi>C</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM138\"><mml:msubsup><mml:mrow><mml:mover><mml:mi>C</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM139\"><mml:mi>γ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM140\"><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM141\"><mml:mi>γ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM142\"><mml:msup><mml:mi>C</mml:mi><mml:mi>k</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM143\"><mml:mi>γ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM144\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM145\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM146\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM147\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM148\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M4x1\"><label>4.1</label><mml:math id=\"DM35\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>Ψ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM149\"><mml:mi>Ψ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM150\"><mml:mi>Σ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M4x2\"><label>4.2</label><mml:math id=\"DM36\" display=\"block\"><mml:mi>d</mml:mi><mml:msup><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>a</mml:mi></mml:msup><mml:mi>d</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mi>b</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230043UM22\"><mml:math id=\"DM37\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ρ</mml:mi><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi>b</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mtext> with </mml:mtext></mml:mstyle><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>γ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>ρ</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi>a</mml:mi></mml:msub><mml:mi>d</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mi>a</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>t</mml:mi><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM151\"><mml:mi>γ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM23\"><mml:math id=\"DM38\" display=\"block\"><mml:mi>ρ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>a</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>3</mml:mn><mml:mi>γ</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM152\"><mml:msub><mml:mi>ρ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM24\"><mml:math id=\"DM39\" display=\"block\"><mml:msup><mml:mrow><mml:mover><mml:mi>a</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mi>κ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mi>ρ</mml:mi><mml:msup><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM153\"><mml:mi>κ</mml:mi><mml:mo>=</mml:mo><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:mi>G</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M4x3\"><label>4.3</label><mml:math id=\"DM40\" display=\"block\"><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>t</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn><mml:mi>γ</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM154\"><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM155\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM156\"><mml:mi>Ψ</mml:mi><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mo>∼</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn><mml:mi>γ</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM157\"><mml:mi>Σ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM158\"><mml:mi>γ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM159\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM160\"><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>t</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>a</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM25\"><mml:math id=\"DM41\" display=\"block\"><mml:mi>Ψ</mml:mi><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mo>∼</mml:mo><mml:msup><mml:mi>τ</mml:mi><mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>3</mml:mn><mml:mi>γ</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM161\"><mml:mi>Σ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM162\"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>γ</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>4</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM163\"><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM164\"><mml:mi>γ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM165\"><mml:mi>Ψ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM166\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M4x4\"><label>4.4</label><mml:math id=\"DM42\" display=\"block\"><mml:msup><mml:mi>A</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msup><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>τ</mml:mi></mml:mfrac><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM167\"><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM168\"><mml:msup><mml:mi>A</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:mi>B</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM169\"><mml:mi>C</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM170\"><mml:mi>u</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM171\"><mml:mi>B</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM172\"><mml:mi>C</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM173\"><mml:msup><mml:mi>A</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM174\"><mml:msup><mml:mi>A</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM175\"><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM26\"><mml:math id=\"DM43\" display=\"block\"><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM176\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>A</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM177\"><mml:msubsup><mml:mi>h</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM178\"><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM179\"><mml:msubsup><mml:mi>h</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM180\"><mml:mi>γ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM181\"><mml:mn>1</mml:mn><mml:mo>&lt;</mml:mo><mml:mi>γ</mml:mi><mml:mo>≤</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM182\"><mml:mi>Ψ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM183\"><mml:msubsup><mml:mi>h</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM184\"><mml:msup><mml:mi>f</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM185\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM186\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM187\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM188\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM189\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM190\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM191\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM192\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM193\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM194\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM27\"><mml:math id=\"DM44\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msubsup><mml:mi>σ</mml:mi><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:msup><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msubsup><mml:mi>σ</mml:mi><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM195\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM196\"><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msubsup><mml:mi>σ</mml:mi><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM197\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM198\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM199\"><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM28\"><mml:math id=\"DM45\" display=\"block\"><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mi>ρ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>u</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>4</mml:mn></mml:mfrac><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM200\"><mml:msub><mml:mi>u</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mi>d</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mi>a</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM201\"><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM202\"><mml:mi>ρ</mml:mi><mml:msup><mml:mi>a</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM203\"><mml:mi>β</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M5x1\"><label>5.1</label><mml:math id=\"DM46\" display=\"block\"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>τ</mml:mi></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>κ</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:mi>k</mml:mi><mml:msup><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mi>a</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM204\"><mml:mi>κ</mml:mi><mml:mo>=</mml:mo><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:mi>G</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM205\"><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM206\"><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM207\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM208\"><mml:mi>a</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM209\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM29\"><mml:math id=\"DM47\" display=\"block\"><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:msubsup><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:mi>κ</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:mi>k</mml:mi><mml:msup><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mi>a</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>a</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230043UM30\"><mml:math id=\"DM48\" display=\"block\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:mi>κ</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mi>Λ</mml:mi></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM210\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM211\"><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:mi>τ</mml:mi><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mi>F</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM212\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM213\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>κ</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi>Λ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM214\"><mml:msub><mml:mi>τ</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>&lt;</mml:mo><mml:mi>τ</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>τ</mml:mi><mml:mi>F</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM215\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM216\"><mml:mi>a</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM217\"><mml:msub><mml:mi>c</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>a</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM218\"><mml:msub><mml:mi>c</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM219\"><mml:mi>ρ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM220\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM221\"><mml:mi>κ</mml:mi><mml:mi>α</mml:mi><mml:mi>a</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM222\"><mml:mi>α</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM223\"><mml:mi>α</mml:mi><mml:msup><mml:mi>a</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM224\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mtext>constant</mml:mtext></mml:mstyle><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>a</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM225\"><mml:mi>a</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM226\"><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM227\"><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM228\"><mml:msub><mml:mi>τ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM229\"><mml:msub><mml:mi>τ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:msub><mml:mi>τ</mml:mi><mml:mi>F</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM230\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM231\"><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM232\"><mml:mn>10</mml:mn><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM233\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM234\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM235\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM236\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM237\"><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM238\"><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M5x2\"><label>5.2</label><mml:math id=\"DM49\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM239\"><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM240\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM241\"><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM242\"><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM243\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM244\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM31\"><mml:math id=\"DM50\" display=\"block\"><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM245\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM32\"><mml:math id=\"DM51\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM246\"><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>Υ</mml:mi><mml:mi>b</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM247\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM248\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM249\"><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>Υ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM250\"><mml:mi>ϕ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M5x3\"><label>5.3</label><mml:math id=\"DM52\" display=\"block\"><mml:mi>Q</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>:=</mml:mo><mml:mi>◻</mml:mi><mml:mi>ϕ</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>6</mml:mn></mml:mfrac><mml:mi>R</mml:mi><mml:mi>ϕ</mml:mi><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:mi>α</mml:mi><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM251\"><mml:mi>α</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M5x4\"><label>5.4</label><mml:math id=\"DM53\" display=\"block\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>g</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mi>ϕ</mml:mi><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>α</mml:mi><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230043UM33\"><mml:math id=\"DM54\" display=\"block\"><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>6</mml:mn></mml:mfrac><mml:mi>R</mml:mi><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM252\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M5x5\"><label>5.5</label><mml:math id=\"DM55\" display=\"block\"><mml:msup><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mi>ϕ</mml:mi><mml:mi>Q</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msup><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msup><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mi>Q</mml:mi><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mi>ϕ</mml:mi><mml:msub><mml:mi>Q</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM253\"><mml:mi>Q</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM254\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM255\"><mml:mi>ϕ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM256\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM257\"><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>b</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM258\"><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mi>Λ</mml:mi><mml:mo>=</mml:mo><mml:mn>24</mml:mn><mml:mi>α</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM259\"><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM260\"><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM34\"><mml:math id=\"DM56\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230043UM35\"><mml:math id=\"DM57\" display=\"block\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mi>Q</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>Q</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM261\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM262\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM263\"><mml:mi>ϕ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM36\"><mml:math id=\"DM58\" display=\"block\"><mml:msub><mml:mi>G</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:mi>κ</mml:mi><mml:msubsup><mml:mi>T</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ex</mml:mi></mml:mrow></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:mi>Λ</mml:mi><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM264\"><mml:msubsup><mml:mi>T</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mtext>ex</mml:mtext></mml:mstyle></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM265\"><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM266\"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mi>Λ</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>6</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM267\"><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM37\"><mml:math id=\"DM59\" display=\"block\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msub><mml:mi>G</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>Λ</mml:mi><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230043M5x6\"><label>5.6</label><mml:math id=\"DM60\" display=\"block\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>κ</mml:mi><mml:msubsup><mml:mi>T</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ex</mml:mi></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM268\"><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM38\"><mml:math id=\"DM61\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mover><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>:=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>T</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ex</mml:mi></mml:mrow></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ex</mml:mi></mml:mrow></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM269\"><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM39\"><mml:math id=\"DM62\" display=\"block\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mover><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>κ</mml:mi><mml:msubsup><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ex</mml:mi></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM270\"><mml:mi>ϕ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM271\"><mml:mi>ψ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM272\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043M5x7\"><label>5.7</label><mml:math id=\"DM63\" display=\"block\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>ψ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>κ</mml:mi><mml:msubsup><mml:mi>T</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ex</mml:mi></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM273\"><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM274\"><mml:mi>ϕ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM275\"><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM276\"><mml:mi>ψ</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM277\"><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM278\"><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>,</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM279\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230043UM40\"><mml:math id=\"DM64\" display=\"block\"><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>Λ</mml:mi><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>Φ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>κ</mml:mi><mml:msubsup><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ex</mml:mi></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM280\"><mml:mi>Φ</mml:mi><mml:mo>=</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>ϕ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM281\"><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mi>ψ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM282\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi 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stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mi>Λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>κ</mml:mi><mml:msubsup><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ex</mml:mi></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM283\"><mml:mi>Ψ</mml:mi><mml:mo>=</mml:mo><mml:mi>ϕ</mml:mi><mml:msup><mml:mi>ψ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>Φ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM284\"><mml:mi>Φ</mml:mi><mml:mo>,</mml:mo><mml:mi>Ψ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM285\"><mml:mi>ϕ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM286\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM287\"><mml:mi>ψ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM288\"><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM289\"><mml:mi>ϕ</mml:mi><mml:mi>ψ</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM290\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM291\"><mml:msub><mml:mrow><mml:mover><mml:mi>G</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:mi>G</mml:mi><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>Λ</mml:mi><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM292\"><mml:mi>k</mml:mi><mml:mo>≤</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM293\"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM294\"><mml:mn>4</mml:mn><mml:mi>κ</mml:mi><mml:mi>β</mml:mi><mml:mi>Λ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>9</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM295\"><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM296\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM297\"><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM298\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM299\"><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM300\"><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM301\"><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>" ]

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[ "<fn-group><fn id=\"FN1\"><label>1</label><p>With the conventions we are using, the Einstein equations are .</p></fn><fn id=\"FN2\"><label>2</label><p>Now in abstract indices.</p></fn><fn id=\"FN3\"><label>3</label><p>We will assume that or and , to avoid recollapse.</p></fn><fn id=\"FN4\"><label>4</label><p>The minus sign is there because goes through zero at ; this equation holds where is finite, and as a limit at .</p></fn><fn id=\"FN5\"><label>5</label><p>Again this equation holds where is finite, and as a limit at .</p></fn><fn id=\"FN6\"><label>6</label><p>This was suggested in [##UREF##5##6##].</p></fn><fn id=\"FN7\"><label>7</label><p>Penrose [##UREF##5##6##] has suggested that these terms be associated with a scalar field in the later aeon, proportional to a power of , and that furthermore this field represents dark matter.</p></fn><fn id=\"FN8\"><label>8</label><p>Conformal scalars have been extensively used both in cosmology and in conformally invariant extensions of the Standard Model of particle physics; see e.g. [##UREF##39##40##–##UREF##42##43##].</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Roger Penrose suggested to discuss the asymptotic behaviour of space–times in terms of extensions of their conformal structure [##UREF##0##1##,##UREF##1##2##]. The idea is that a <italic toggle=\"yes\">physical space–time</italic>\n may admit a smooth extension where is a smooth manifold with boundary , a smooth Lorentz metric and a smooth function on , so that , and on , while on the set (referred to as the <italic toggle=\"yes\">conformal boundary</italic> or short as <italic toggle=\"yes\">Scri</italic>). It may not be easy, or not possible at all, to find for a given space–time a smooth conformal extension as above, but if it can be done, the construction provides precise and complete information about the asymptotic behaviour of the space–time in the neighbourhood of Scri.</p>", "<p>In the first 20 years following the introduction of the concept there became available related general observations, various discussions of fields and physical quantities on and near conformal boundaries, and detailed verifications of the existence of conformal boundaries in the case of a number of important exact solutions with symmetries. After becoming acquainted with the idea I tried to understand to what extent the concept applied to general solutions of Einstein’s equation\nwith cosmological constant and energy momentum tensor .</p>", "<p>The conformal extension idea will be illustrated here by a discussion of Friedmann–Lemaitre–Robertson–Walker (FLRW) models because some of their features will be important for us in the following. By these we understand space–times with manifold and metric , where is a scalar, is a three-dimensional manifold, and a -independent Riemannian metric of constant curvature on with Ricci scalar . The space–times are required to satisfy the Einstein--perfect-fluid equations with flow field , total energy density , pressure and an equation of state .</p>", "<p>To discuss the solutions in terms of a conformal representation we use the rescalings\nwith a conformal factor whose evolution is fixed by the requirements that the Ricci scalars of and satisfy throughout the solution and , on a given slice . The constant will be determined later.</p>", "<p>Then and in terms of the coordinate follows\nWith a linear equation of state where the conformal analogues of the well-known Friedmann and energy conservation equations, that represent the content of (##FORMU##24##1.1##), then read with the dot denoting ,\nWith initial data and at this implies\nand thus, <italic toggle=\"yes\">independent of the choice of</italic>\n, the conformal Friedmann equation in the form\nLet be small (that is be large) so that the right-hand side is positive for and choose the sign of the square root and the parameter so that is decreasing while is increasing. The equation can then be integrated until at some finite value of the parameter. Since and as , the hypersurface Scri = defines a boundary of the physical space–time that represents future time-like infinity with respect to the physical metric and that is space-like with respect to .</p>", "<p>That the zero of the function is given here by a finite value of the conformal coordinate helps to recognize subtle matter dependent differences in the asymptotic behaviour of as . In the vacuum case we get with , , , and whence , the solution . Then and which gives the maximally symmetric, geodesically complete, conformally flat <italic toggle=\"yes\">de Sitter solution</italic>\nIt is here not so important for us that the solution can be given explicitly but that we get precise information on the asymptotic behaviour of near Scri. The solution of the ODE above extends smoothly to Scri and in fact beyond. While the ‘physical’ de Sitter metric is defined for , which is covered by with , the cyclic function and the metric are defined and smooth for , defining a sequence of (isometric) vacuum solutions which are separated by Scri’s.</p>", "<p>When and different cases occur. If there arise smoothness and extension problems. The solutions do approach the value but only a few derivatives of the function have a finite limit as . Some solutions, such as the one obtained with for instance, admit a unique extension into a range where which is smooth where and drops smoothness as . Others, like the one obtained for , do not admit an extension beyond as a solution to the equation above.</p>", "<p>However, in the case of <italic toggle=\"yes\">pure dust, where </italic>, or in the case of <italic toggle=\"yes\">(incoherent) pure radiation, where </italic>, the solutions extend smoothly to and beyond. (The unusual word <italic toggle=\"yes\">pure</italic> is added here to distinguish these cases clearly from related ones considered later). We set for convenience , and consider the <italic toggle=\"yes\">case of pure dust with </italic> and the <italic toggle=\"yes\">case of pure radiation with </italic>. We have then and get the conformal Friedmann equation in the form\nThe global solutions and the solution manifold look in these two cases as follows.\n<italic toggle=\"yes\">Pure dust solutions</italic>. Depending on the real roots of the polynomial\nthree cases occur (assuming in the following discussions suitable parameters and signs of the square root of ).\n<list list-type=\"simple\"><list-item><label>(i)<x xml:space=\"preserve\"> </x></label><p>: has roots . There are solutions that take values in start from a Big Bang (where , resp. for a finite value of the parameter), decrease, reach their minimum (i.e. reaches a maximum), increase again and approach a Big Crunch where for a finite value of the parameter.</p><p>The solutions that take values in are cyclic, oscillating between and and passing through Scri’s on the way. These approach the de Sitter solution as .</p></list-item><list-item><label>(ii)<x xml:space=\"preserve\"> </x></label><p>: Then with simple root and double root . The solutions that take values in the domain start from a Big Bang, decrease monotonously and approach the value without ever assuming it. Similarly, starting from their minimum value the solutions which take values in are increasing, pass Scri’s, and approach asymptotically the value in both directions. Finally there is the time independent solution .</p></list-item><list-item><label>(iii)<x xml:space=\"preserve\"> </x></label><p>: has one real root . The solutions start with from a Big Bang, decrease monotonously, pass through a Scri for a finite value of the parameter, become negative, assume their minimum , increase again, pass through another Scri and approach a Big Crunch where .</p></list-item></list></p>", "<p>This solution admits a smooth cyclic extension in the following sense. Before the limit is achieved the function is defined and satisfies\nWith a redefinition of the constants this becomes the conformal Friedmann equation above with (<italic toggle=\"yes\">stiff matter equation of state</italic>) and and interchanged. The condition on above ensures that the polynomial in on the right-hand side is positive everywhere and the equation can be integrated across . Where the transformation connects to a second copy of the solution above and the process can be repeated. If the parameter is chosen so that , whence , the solution is given in terms of Jacobi’s elliptic function [##UREF##2##3##] by\nwhere , the modulus is , and the root of , that satisfies , is related to by .</p>", "<p><italic toggle=\"yes\">Pure radiation solutions</italic>. Depending on the real roots of the polynomial\nthe following cases occur.</p>", "<p>\n<list list-type=\"simple\"><list-item><label>(i)<x xml:space=\"preserve\"> </x></label><p>: Q has four simple roots . There are solutions that take values in , start from a Big Bang, achieve their minimum , increase and end in a Big Crunch. There are similar solutions that take values in .</p><p>There are cyclic solutions that take values in , oscillate between and , and pass through Scri’s on the way. These solutions approach the de Sitter solution as .</p></list-item><list-item><label>(ii)<x xml:space=\"preserve\"> </x></label><p>: Then with the two double roots . There is a strictly monotonous solution which takes values in , passes through a Scri, and approaches the values and asymptotically. There are two strictly monotonous solutions that take values in and in , respectively. The first one approaches at one end the value asymptotically and at the other end a Big Bang. The second solution is similar. Finally there are the time independent solutions .</p></list-item><list-item><label>(iii)<x xml:space=\"preserve\"> </x></label><p>: has no real root. The solution takes values in . It starts from a Big Bang, decreases with monotonously, passes through a Scri, decreases further, and reaches a Big Crunch where .</p></list-item></list></p>", "<p>Again, this solution admits a smooth cyclic extension in the following sense. With close to the Big Crunch the equation for gives\nwhich is the original equation with the roles of and swapped. It can be integrated beyond and with connected to a second copy of the solution for .</p>", "<p>If the parameter is chosen so that and is increasing with increasing near , the solution is given in terms of Jacobi’s elliptic functions by\nwhere , , , , .</p>", "<p>The ‘fine tuned’ cases (ii) will not be of interest to us in the following. We shall mainly be interested in the cases (iii) where a Big Bang in the finite past (in terms of physical time) is connected with a space-like Scri in the infinite future. These cases admit the limits which are in agreement with the 2018 results of the Planck team. We shall later focus on the ends at future time-like infinity where the solutions approach Scri.</p>", "<p>Of course, in the standard interpretation of GR only a maximal connected space–time with will be considered as a physical solution. Its conformal extension may just be considered a fun game which works because the solutions are conformally flat and extremely simple. Heeding Tolman’s admonition [##UREF##3##4##], which reads (with a slight variation) ‘ we study FLRW models primarily in order to secure definite and relatively simple mathematical problems, rather than to secure a correspondence with known reality ’ we wonder: Do any of the observations above extend to more general situations? We are in particular interested in examples where the fluid flow is not forced to be geodesic and which are not conformally flat so as to admit gravitational radiation, a concept that FLRW models allow us to talk about only in terms of approximations.</p>", "<p>Answers of any generality to the question above need global or semi-global results on suitable Cauchy problems for Einstein’s field equations. Moreover, as shown by the subtleties discussed above, they require sharp control on the asymptotic behaviour of the solutions at least at future time-like infinity. At the time when Penrose put forward his proposal no such results were available. Until the early 1980s the understanding of the Cauchy problem for Einstein’s equations was restricted to existence results local in time.</p>" ]

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[ "<title>Conformal field equations, stability results</title>", "<p>Finding initial data which develop into solutions to the Einstein equations that admit smooth conformal boundaries poses subtle problems when the cosmological constant vanishes or is negative. Because the initial slices are not compact choices have to be made about the fall-off behaviour of the data [##UREF##4##5##,##UREF##5##6##].</p>", "<p>In the same article where Einstein introduced the cosmological constant, he came to the conclusion that cosmological solutions should be spatially compact [##UREF##6##7##]. In fact, in contrast to the FLRW solutions, which are defined by ODE’s, there can be no other choice if one wishes to construct geodesically null and time-like future complete solutions to the full Einstein equations. There are no natural boundary conditions for the evolution equations if .</p>", "<p>As indicated above, the Planck team found the Universe to be constrained to be spatially flat to extremely high precision [##UREF##7##8##]. Because no restrictions are given on the ‘size’ of (in terms of the distances function defined by the prescribed metric ), we can assume , which we may wish to be simply connected, to be as large as we like so that is lying in the error margin given in [##UREF##7##8##] while still being positive.</p>", "<p>If the initial slice is compact, which will be assumed in the following, only smoothness and smallness conditions can be imposed on ‘general’ Cauchy data. The following global nonlinear stability result holds [##UREF##8##9##,##UREF##9##10##].</p>", "<p><italic toggle=\"yes\">On a slice</italic>\n\n<italic toggle=\"yes\">of the de Sitter solution to Einstein’s field equations (##FORMU##24##1.1##)</italic>\n<italic toggle=\"yes\">with</italic>\n\n<italic toggle=\"yes\">and</italic>\n\n<italic toggle=\"yes\">consider smooth Cauchy data for these equations. If these data are (in terms of suitable Sobolev norms) sufficiently close to the de Sitter data on</italic>\n, <italic toggle=\"yes\">they develop into solutions to (##FORMU##24##1.1##)</italic>\n<italic toggle=\"yes\">that are time-like and null geodesically complete and admit smooth space-like conformal boundaries at past and future time-like infinity.</italic></p>", "<p>Because the cosmological constant can be given any positive value by a conformal rescaling with a constant conformal factor, the precise value of is irrelevant here.</p>", "<p>The technical basis of this result is a remarkable feature of the Einstein equations. While they are designed to determine a metric, they can be represented in terms of a conformal factor , the conformal metric and certain tensor fields derived from them so that they imply with suitable gauge conditions equations that can be hyperbolic even where the conformal factor vanishes or becomes negative, i.e. beyond the domain where is defined [##UREF##10##11##]. We refer to these equations as <italic toggle=\"yes\">conformal Einstein equations</italic>. It should be noted that this name has subsequently also been used for conformal representations of the Einstein equations which were derived for other purposes and do not share the properties used below.</p>", "<p>Conformal de Sitter space, given above by , , , is a solution to the conformal Einstein equations that extends smoothly, as a solution to the equations and with the same expressions for the metric and conformal factor, beyond the boundaries to all of . Then on the slices . Consider Cauchy data , , for the conformal field equations on a slice with which are ‘general’ in the sense that symmetries are not necessarily imposed. If these data are sufficiently close to the conformal de Sitter data induced on , general properties of hyperbolic equations [##UREF##11##12##] guarantee that the solution , , to the conformal field equations which develop from the general data also exist on the domain and the conformal factor is negative on . The conformal field equations then ensure that there exist two hypersurfaces with , that are space-like with respect to and sandwich a domain on which . Then is the desired solution to the Einstein equations.</p>", "<p>The fact that the set of all Cauchy data on for the Einstein vacuum equations with positive cosmological constant contains an open subset (in terms of suitable Sobolev norms) of data which develop into solutions that are conformally well behaved at future and past time-like infinity shows that the existence of smooth conformal boundaries can be a fairly general feature of solutions to Einstein’s field equations. Besides a smallness condition there are no restrictions on the conformal Weyl tensor.</p>", "<p>Recovering from the technical struggles that led to this insight, I began to wonder:</p>", "<p><italic toggle=\"yes\">If the field equations ensure a smooth future evolution of</italic>\n\n<italic toggle=\"yes\">and</italic>\n\n<italic toggle=\"yes\">beyond</italic>\n, <italic toggle=\"yes\">so that they define in the future of</italic>\n\n<italic toggle=\"yes\">another ‘physical’ solution to Einstein’s field equations with metric</italic>\n, <italic toggle=\"yes\">and any gravitational radiation, represented by nonlinear perturbations of the conformal Weyl tensor, travels unimpeded across</italic>\n\n<italic toggle=\"yes\">into that domain, why should physics come to an end at the future conformal boundary</italic>\n<italic toggle=\"yes\">?</italic></p>", "<p>This behaviour may be considered as just another quirk of the field equations, that should not be taken too seriously. The history of General Relativity shows, however, that Einstein’s equations were often wiser than their solvers. Physicists made sense of the more exotic features of the solutions found by Schwarzschild and Friedmann only years after their discovery.</p>", "<p>Though it can all be found in the article referred to above, I never explicitly speculated about this in public. Being a beginner, it would hardly have been taken seriously, in particular, because I had no answers to the questions:\n<list list-type=\"simple\"><list-item><label>—<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">What happens if there is matter around?</italic></p></list-item><list-item><label>—<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">How will matter behave in the far future?</italic></p><p>Our discussion of the FLRW solutions above have shown that different matter models, exemplified there by and , may have quite diverse consequences. Moreover,</p></list-item><list-item><label>—<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">What could be the meaning of the solution ‘on the other side’ of</italic>\n<italic toggle=\"yes\">?</italic></p></list-item></list> In the stability result above the space–time defined by the metric \n<italic toggle=\"yes\">on the other side of </italic> looks like a time reversed version of the space–time end ‘on this side’: from being infinitely extended at its space sections begin to shrink. This is certainly quite different from our present idea about the beginning of a cosmological space–time. So I kept returning to the first two questions over the following years.</p>", "<p>The first stability result generalizing the one above concerns the (in four space–time dimensions) conformally invariant Maxwell- or Yang–Mills equations [##UREF##12##13##]. It shows:</p>", "<p><italic toggle=\"yes\">The nonlinear vacuum stability result outlined above generalizes to the coupled Einstein-</italic><italic toggle=\"yes\">-Maxwell–Yang–Mills equations. The perturbed solutions admit smooth conformal boundaries in the future and the past. The conformal field equations determine smooth conformal extensions of the solutions beyond these boundaries</italic>.</p>", "<p>This result establishes a pattern for analysing various other situations in which conformally covariant matter transport equations are coupled to the Einstein- equation. Christian Lübbe and Juan Valiente Kroon studied the Einstein--perfect-fluid equations with the equation of state for pure (incoherent) radiation. These matter equations have in common with the Maxwell equations that the energy–momentum tensor is trace free and the conformal matter equations have the same form as the ‘physical’ version. They show [##UREF##13##14##]:</p>", "<p>\n<italic toggle=\"yes\">The FLRW solutions with the equations of state of pure radiation and a smooth conformal boundary in the future are nonlinearly future stable in the class of all Einstein-</italic>\n\n<italic toggle=\"yes\">-perfect-fluid solutions with this equation of state. The perturbed solutions admit a smooth conformal boundary in the future and a smooth conformal extension beyond.</italic>\n</p>", "<p>A further example leading to a similar result is given by the Einstein equations coupled to the massless Vlasov matter equations [##UREF##14##15##].</p>", "<p>In the FLRW models given by (##FORMU##165##1.5##) and (##FORMU##212##1.8##) and the generalizations discussed above, forward Scri’s and backward Scri’s as well as Big Bangs and Big Crunches stand back to back in the conformal extensions. Motivated by the observations above, results on Paul Tod’s ideas about isotropic singularities [##UREF##15##16##–##UREF##18##19##], where the initial singularity is represented after a suitable conformal rescaling by a finite space-like set similar to a , and by his thoughts about the nature of entropy near the big bang, Roger Penrose proposed a cosmological model, referred to as <italic toggle=\"yes\">conformal cyclic cosmology</italic> (CCC) [##UREF##19##20##]. It considers a chain of universes where a given universe, say , develops in its future a well defined , referred to now as <italic toggle=\"yes\">the crossover surface</italic>, which is followed by another universe, , for which the crossover surface represents an isotropic singularity. The solutions (##FORMU##165##1.5##) and (##FORMU##212##1.8##) can be used to create such situations.</p>", "<p>This again gives rise to complicated questions. Strongly simplifying assumptions on and may provide situations where some kind of identification or glueing of the different ends lead to a picture as outlined above. It is not clear, however, that anything similar can be done with any degree of control if more general, conformally curved, solutions to Einstein’s equations are considered. The precise nature of the transition from the infinite future of to the beginning of is unresolved so far. It should be brokered by a mechanism which guarantees a unique extension without any interference from outside, but not necessarily preserving the time reflection invariance of hyperbolic equations. This requires a closer look at the equations and the matter models from both sides of the crossover surface, generalizing perhaps of the work initiated by Bachelot [##UREF##20##21##].</p>" ]

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[ "<p>One contribution of 13 to a discussion meeting issue ‘<ext-link xlink:href=\"http://dx.doi.org/10.1098/rsta/382/2267\" ext-link-type=\"uri\">At the interface of asymptotics, conformal methods and analysis in general relativity</ext-link>’.</p>", "<p>Smooth Cauchy data on for the Einstein--vacuum field equations with cosmological constant that are sufficiently close to de Sitter data develop into a solution that admits a smooth conformal boundary in its future. The <italic toggle=\"yes\">conformal Einstein equations</italic> determine a smooth conformal extension across that defines on ‘the other side’ again a -vacuum solution. In this article, we discuss to what extent these properties generalize to the <italic toggle=\"yes\">future asymptotic behaviour of solutions to the Einstein- equations with matter</italic>. We study Friedmann–Lemaitre–Robertson–Walker (FLRW) solutions and the Einstein- equations coupled to conformally covariant matter transport equations, to conformally privileged matter equations, and to conformally non-covariant matter equations. We present recent results on the Einstein--perfect-fluid equations with a nonlinear <italic toggle=\"yes\">asymptotic dust</italic> or <italic toggle=\"yes\">asymptotic radiation</italic> equation of state.</p>", "<p>This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.</p>" ]

[ "<title>Conformally non-covariant matter fields</title>", "<p>Finding such a mechanism, if something like it exists at all, requires among other things a sufficiently general and deep understanding of the behaviour of matter and the field equations at future time-like infinity. A number of authors analysed the future asymptotic behaviour of solutions to the Einstein--perfect fluid equations with an homentropic flow, where the entropy is constant in space and time and the equation of state can be given in the form with some suitable function . Often they assume a linear equation of state , , and the additional condition , see [##UREF##21##22##–##UREF##25##26##]. In the articles [##UREF##23##24##,##UREF##26##27##] is studied the future stability of FLRW space–times for more general classes of equations of state . All of this work was done in more conventional representations of the field equations in terms of which the questions of interest here may indeed be difficult to analyse. None of the authors above looked at these things in the way indicated above.</p>", "<p>In the following I tried to generalize the kind of analysis begun above, hoping to answer the following question:</p>", "<p><italic toggle=\"yes\">Can there be achieved</italic>\n, , , <italic toggle=\"yes\">or at least in some sense uniquely extendible conformal structures at future time-like infinity for solutions to Einstein-</italic><italic toggle=\"yes\">-matter equations with conformally non-covariant matter field equations?</italic></p>", "<title>Scalar fields</title>", "<p>There are two results pointing into that direction. In [##UREF##27##28##], H. Ringström studied the future stability of a very general class of Einstein-nonlinear scalar field systems with a scalar field equation of the form\nwhere , and an energy momentum tensor\nwith a potential of the form where is a constant and a smooth real-valued function. In [##UREF##28##29##] has been considered a special case with the following result.</p>", "<p>\n<italic toggle=\"yes\">If and , the coupled Einstein--scalar-field equations (##FORMU##24##1.1##), (##FORMU##300##3.1##) and (##FORMU##302##3.2##) imply a reduced system of conformal field equations for the unknowns , , and some tensor fields derived from them. In a suitable gauge this is hyperbolic for any sign of . Smooth Cauchy data for this system can be prescribed with on a compact -space-like 3-manifold . The development of these data backwards in time is smooth and induces on a space-like slice (in the corresponding physical space–time) smooth standard Cauchy data for the coupled Einstein--scalar-field system. For there exist (in terms of Sobolev norms) an open neighbourhood of Cauchy data on for this system so that the future development of any smooth data in this neighbourhood admit a smooth conformal extension beyond the respective future time-like infinity.</italic>\n</p>", "<p>We recall that the conformally covariant scalar operator with satisfies . With the trace of the energy–momentum tensor (##FORMU##302##3.2##) given by\nequation (##FORMU##300##3.1##) can be expressed in terms of the conformally covariant wave operator. It reads then\nwhile we get in terms of the conformal fields and the conformal representation\nEquation (##FORMU##300##3.1##) is thus not conformally covariant with our conditions but <italic toggle=\"yes\">conformally regular</italic> in the sense that there will remain no terms on the right hand side if the conditions of the theorem above are taken into account. This is a property it shares with the Einstein- vacuum field equations.</p>", "<p>The energy–momentum tensor is not trace free but satisfies\nwhere the limit above will follow only <italic toggle=\"yes\">if and can be shown to extend smoothly as </italic>. It should be mentioned that the hyperbolic reduced conformal field equations considered above as well as those considered in the following preserve the constraints implied by the conformal field equations. In the case of the present system, the proof of this fact is far from immediate.</p>", "<title>Pure dust cosmologies</title>", "<p>The flow fields of Einstein--perfect-fluid solutions with pure dust equation of state\narising from data on a compact 3-manifold where are often interpreted as representing the cosmic flow of galaxies. The field equations imply that the trace of the corresponding energy–momentum tensor satisfies\nIn the case of the conformally flat FLRW models we have seen that such solutions can develop smooth conformal boundaries. Whether this is also possible if the conformal Weyl tensor does not vanish is not obvious [##UREF##21##22##]. In [##UREF##29##30##], it has been shown:</p>", "<p><italic toggle=\"yes\">Consider a FLRW solution to the Einstein--perfect-fluid equation with pure dust equation of state arising from data on a Cauchy hypersurface . If it admits a smooth conformal boundary at future time-like infinity, then any smooth general set of Cauchy data on for the same equation which is sufficiently close (in terms of Sobolev norms) to the FLRW-data develops into a solution that is time-like and null geodesically future complete, admits a smooth conformal boundary in its infinite future, where , and extends as a smooth solution to the conformal field equations into a domain where </italic>.</p>", "<p>The unknowns and in the conformal field equations then remain bounded as and it follows that\nAssuming initial data so that , the equation for the flow vector field reduces in the case of pure dust to the equation\nand thus to an ODE. In terms of the conformal fields it takes the form\nand the term, which reflects the conformal non-covariance of the system, spreads into other equations of the conformal system.</p>", "<p>\n<italic toggle=\"yes\">The fact that the flow is geodesic combined with the particular structure of the energy momentum tensor of a perfect-fluid allows us to make in this particular case contact with and exploit some conformally invariant structure.</italic>\n</p>", "<p>Let , , satisfy the Einstein--pure-dust equations so that . If the functions and satisfy\nthe curve with and the 1-form satisfy the <italic toggle=\"yes\">conformal geodesic equations</italic>\nand\nwhere denotes the Schouten tensor of . The point of this observation is that the conformal geodesic equations are conformally invariant and the conformal geodesics are invariants of the conformal structure [##UREF##30##31##]. This was used in [##UREF##29##30##] to regularize the conformal Einstein--pure-dust system near future time-like infinity.</p>", "<p>The pure dust model is thus still <italic toggle=\"yes\">conformally privileged</italic>.</p>", "<title>Equations of state with prescribed asymptotic behaviour</title>", "<p>In the articles cited above that study the future behaviour of cosmological solutions to the Einstein--perfect-fluid equations on the basis of linear equations of state, , , no arguments are given for this choice. It appears to be rather a matter of convenience instead of being motivated by a deep understanding of its physical role. While solutions to the Einstein--perfect-fluid equations may provide good cosmological models on large scales, it seems fairly unlikely that linear equations of state represent natural requisites from the Big Bang to future time-like infinity. Models of the universe that behave at late times like a pure dust or a pure radiation FLRW model or one of their conformally curved generalizations may require transitions of the form\nwith a function that satisfy, consistent with our earlier requirements, and assumes the value or at late times. There is nothing, however, which would fix a notion of ‘late time’. The only meaningful requirement would be that these values are approximated in the limit when the space–time approaches future time-like infinity. The equation of state would then still need to recognize, however, where and when this limit will be achieved.</p>", "<p>In the cases (iii) of the FLRW models discussed above the physical density and the conformal density satisfy a relation of the form\nIn those cases we had , whence as and as . The behaviour of can thus be understood as an indicator for the approach to the infinite future or the Big Bang. In generalizing the situation we shall keep (##FORMU##382##3.3##), hoping that it will serve the indicator function at least in the far future where , but we may have to give up the relation We could think of generalizing (##FORMU##382##3.3##) by assuming to be a function. In this article we will only consider the situations near the end where and try to keep (##FORMU##382##3.3##) with . This will work well as long as can be guaranteed to stay positive and bounded as .</p>", "<p>The pure dust and the pure radiation equations of state are now generalized as follows. An <italic toggle=\"yes\">asymptotic dust</italic> equation of state is given by a function of the form\ncombined with (##FORMU##382##3.3##) where . It implies with the notation \nAn <italic toggle=\"yes\">asymptotic radiation</italic> equation of state is given by a function of the form\ncombined with (##FORMU##382##3.3##) where . It implies\nIn both cases is assumed to be a smooth function defined for all values of that satisfies\nThe positivity is required to clearly distinguish the asymptotic from the pure cases. Limits give back the pure dust and the pure radiation equations of state.</p>", "<p>The factors with positive have been included in the definitions as a simple means to control the speed at which the pure dust or the pure radiation situations is approximated as .</p>", "<p>The conditions on may appear crude but they suffice for analysing the effect of the intended modifications of the equations of state in domains where becomes small. For positive but sufficiently close to zero, the range we are interested in, the terms in curly brackets in (##FORMU##400##3.5##) and (##FORMU##403##3.7##) are positive. It follows that in the case of asymptotic dust the speed of sound is positive as and as . In the case of asymptotic radiation holds for and will remain positive if the solution can be smoothly extended into a domain where .</p>", "<p>We note that <italic toggle=\"yes\">the Cauchy problems local in time for Einstein--perfect fluids with asymptotic dust or radiation equations of state pose no problems where is sufficiently small</italic>. This follows from the results of Friedrich [##UREF##31##32##] and Friedrich &amp; Rendall [##UREF##32##33##] where only weak conditions on the equation of state are assumed.</p>", "<p>The principal parts of the matter equations are affected by the equations of state above with the consequence that <italic toggle=\"yes\">any conformal covariance or privilege is lost</italic>. Definition (##FORMU##397##3.4##) implies\nwhile definition (##FORMU##401##3.6##) gives\nIn both case if and as if \n<italic toggle=\"yes\">remains bounded in this limit.</italic> As seen below, this last condition will be met in the case of an asymptotic radiation equation of state while it is not clear whether this can be guaranteed also in the case of an asymptotic dust equationof state.</p>", "<p>The conditions on the admissible values of required above can be weakened if the equations of state above are considered in the conformal analogues of the Friedmann and energy conservation equation. In the case of asymptotic dust the system reads\nin the case of asymptotic radiation the system reads\nFor suitable and initial data these equations can be smoothly integrated across with and bounded and positive. We shall see now that the FLRW assumption gives quite a wrong impression about the similarity and simplicity of these two cases if the assumption is dropped.</p>", "<title>On the equations with an asymptotic dust equation of state</title>", "<p>To illustrate the kind of difficulties which arise in the case of an asymptotic dust equation of state we consider one of the equations contained in the complete system of conformal field equations. If the latter are written in terms of an orthonormal frame with , the connection coefficient is subject to the evolution equation\nwhere , the connection coefficients and , and the Schouten tensor obey further evolution equations. In the case of pure dust the third line is not present. In that case and the second line contains factors . Even if the third line is ignored, the arguments indicated above in the case of pure dust would not apply in the present case because the flow equation remains a partial differential equation if , . It cannot be related to the conformal geodesics equations. The most complicated term in the equation is, however, the first factor in the third line. This is not even defined in the case of pure dust. The terms can not be compensated in this equation by suitable choices of . We leave this case open.</p>", "<title>Solutions with an asymptotic radiation equation of state</title>", "<p>In the case of an asymptotic radiation equation of state the situation is quite different. Consider the equation above again. All the coefficients introduced by the radiation equation of state are well defined and approach in the limit the corresponding value in the case of pure radiation. Moreover, the terms in the equation above that contain factors come with the coefficients\nand\nBy a suitable choice of the factors thus allow us to make up for any negative power of .</p>", "<p>The situation is similar with the other equations of the conformal system. Moreover, the way the asymptotic radiation equation of state affects the principal part of the system is so that one can still extract from the complete system in a suitable gauge a reduced system that is symmetric hyperbolic irrespective of the sign of .</p>", "<p>Solutions that admit smooth conformal extensions at future time-like infinity can now be constructed from data for the conformal field equations which are given on a space-like hypersurface in the ‘physical domain’, where , or from data on the hypersurface that represents future time-like infinity. To check that the data satisfy the constraints induced on , and possibly the additional requirements implied by the assumption that on , these conditions are best expressed in terms of the unit normal to and then transformed into a frame with , whereby the evolution equations need to be used as well. Unless the field is assumed to be orthogonal to this involves some fairly tedious calculations (see [##UREF##33##34##], where the presence of a boundary requires them). Since the latter give limited insight they are skipped here. After the Cauchy problem has been solved for the given data it follows by standard arguments that the constraints and thus the complete system of conformal field equations will be solved as well (see [##UREF##29##30##,##UREF##32##33##] for detailed discussions). We can state now the following results [##UREF##34##35##].</p>", "<p>\n<italic toggle=\"yes\">The Einstein--perfect-fluid equations with an asymptotic radiation equation of state where induce in a suitable gauge a reduced system of the conformal Einstein--perfect-fluid equations that is symmetric hyperbolic irrespective of the sign of .</italic>\n</p>", "<p>\n<italic toggle=\"yes\">On a compact three-dimensional manifold one can construct smooth Cauchy data for the reduced conformal equations with , time-like future directed orthogonal to , and that satisfy the constraints induced by the conformal field equations and the special requirements on a space-like hypersurface on which .</italic>\n</p>", "<p>\n<italic toggle=\"yes\">These data determine a smooth solution to the reduced equations with hypersurface orthogonal, in the future of and in the past of . In the latter domain the solution defines a unique solution to the Einstein--perfect-fluid equations with an asymptotic radiation equation of state that is time-like geodesically future complete and for which represents a conformal boundary at the infinite time-like future.</italic>\n</p>", "<p>\n<italic toggle=\"yes\">Let be a Cauchy hypersurface for this solution in the past of and denote by the Cauchy data induced by that solution on . Any Cauchy data on for the same equations which are sufficiently close to develop into a solution that is also time-like geodesically future complete, admits a smooth conformal boundary in the future, and a smooth conformal extension beyond.</italic>\n</p>", "<p>We note that the Cauchy hypersurface above is not required to be orthogonal to and that the flow vector field comprised by the data on is not required to satisfy the condition of hypersurface orthogonality.</p>", "<title>Concluding remarks</title>", "<p>If is large, the terms involving in (##FORMU##397##3.4##) and (##FORMU##401##3.6##) may, when , look like minor perturbations of the pure dust or pure radiation equations of state. We have seen, however, that the effects of these terms are quite different in the two cases.</p>", "<p>When becomes small the term involving may in the case of the asymptotic radiation equations of state indeed be considered already for as a minor perturbation relative to the dominating first term on the right-hand side of (##FORMU##401##3.6##). This is apparently sufficient to preserve asymptotically the effects corresponding to the conformal covariance of the pure radiation equation of state.</p>", "<p>By contrast, the transition from the pure to the asymptotic dust equation of state represented by the term involving , comes with a drastic change of the principal part of the matter equations by which the conformal privilege of the pure dust case is lost completely even if is large.</p>", "<p>One could hope to simplify the analysis of this case by disentangling the two problems that are possibly interfering in the asymptotic dust case at future time-like infinity. Let the function be modified so that precisely if falls below a certain positive threshold . For the sake of discussion assume that the set defines a space-like Cauchy hypersurface with the set , on which we have an asymptotic dust equation of state, lying in its past and the set , on which we have a pure dust equation of state, lying in its future. For this picture to make sense one has to decide whether the space–time evolution extends with a sufficient degree of smoothness across the set where the change of principal part takes place which reduces the PDE for the flow field to an ODE. Since this concerns the physical domain, one could expect the answer to follow from the analysis of the fluid equations in [##UREF##31##32##,##UREF##32##33##] which treats the case and along similar lines. If the answer is positive, the problem of asymptotic smoothness at time-like infinity only concerns the fields on . This is the situation considered in [##UREF##29##30##]. The question of interest now is whether the analyses at the set and of the behaviour at time-like infinity can be combined to clarify what happens if the value of the threshold is lowered to eventually perform the limit .</p>", "<p>In the light of the preceding discussions I consider this question as particular interesting. While it may not inform us about the final nature of the matter fields at the end of our present universe it may give rise to a more subtle and appropriate definition of an asymptotic dust equation of state and will certainly give insights into the freedom allowed by the field equations and the matter equations to model possible ends at future time-like infinity.</p>" ]

[ "<title>Data accessibility</title>", "<p>This article has no additional data.</p>", "<title>Declaration of AI use</title>", "<p>I have not used AI-assisted technologies in creating this article.</p>", "<title>Authors' contributions</title>", "<p>H.F.: conceptualization, formal analysis, methodology, writing—original draft.</p>", "<title>Conflict of interest declaration</title>", "<p>I declare I have no competing interests.</p>", "<title>Funding</title>", "<p>I received no funding for this study.</p>" ]

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mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM14\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM15\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM16\"><mml:mi>M</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM17\"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mi>M</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>∪</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM18\"><mml:mi>Ω</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM19\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM20\"><mml:mrow><mml:mover><mml:mi>M</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM21\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM22\"><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM23\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>M</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM24\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044M1x1\"><label>1.1</label><mml:math id=\"DM1\" display=\"block\"><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>λ</mml:mi><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM25\"><mml:mi>λ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM26\"><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM27\"><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mo>×</mml:mo><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM28\"><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM29\"><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM30\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM31\"><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM32\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM33\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM34\"><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mtext mathvariant=\"italic\">const</mml:mtext></mml:mrow><mml:mrow><mml:mo>.</mml:mo></mml:mrow><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM35\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM36\"><mml:mi>U</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM37\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM38\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM39\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM1\"><mml:math id=\"DM2\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msup><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">→</mml:mo><mml:msup><mml:mi>U</mml:mi><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>ρ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>e</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM40\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM41\"><mml:mi>g</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM42\"><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM43\"><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM44\"><mml:mi>a</mml:mi><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM45\"><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM46\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM47\"><mml:mi>e</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM48\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>a</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM49\"><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>+</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mrow><mml:mi>t</mml:mi></mml:msubsup><mml:msup><mml:mi>a</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM2\"><mml:math id=\"DM3\" display=\"block\"><mml:mi>g</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM50\"><mml:mi>w</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM51\"><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">const</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM52\"><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM3\"><mml:math id=\"DM4\" display=\"block\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>6</mml:mn></mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:msup><mml:mfrac><mml:mi>ρ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>−</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>ρ</mml:mi><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM53\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM54\"><mml:mi>ρ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM55\"><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM4\"><mml:math id=\"DM5\" display=\"block\"><mml:mi>ρ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>Ω</mml:mi><mml:msub><mml:mi>Ω</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>−</mml:mo><mml:mi>e</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM56\"><mml:mi>e</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM5\"><mml:math id=\"DM6\" display=\"block\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>6</mml:mn></mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>Ω</mml:mi><mml:msub><mml:mi>Ω</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM57\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM58\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM59\"><mml:mi>Ω</mml:mi><mml:mo>≤</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM60\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM61\"><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM62\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM63\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM64\"><mml:msub><mml:mi>τ</mml:mi><mml:mrow><mml:mo>∗</mml:mo><mml:mo>∗</mml:mo></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM65\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM66\"><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM67\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM68\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo><mml:mo>∼</mml:mo><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM69\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM70\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM71\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM72\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM73\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM74\"><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM75\"><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM76\"><mml:msub><mml:mi>t</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM77\"><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM78\"><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM79\"><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM80\"><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>=</mml:mo><mml:mn>6</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM81\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM82\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>tan</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM83\"><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>cosh</mml:mi><mml:mo>⁡</mml:mo><mml:mi>t</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044M1x2\"><label>1.2</label><mml:math id=\"DM7\" display=\"block\"><mml:mrow><mml:mover><mml:mi>M</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>cosh</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>⁡</mml:mo><mml:mi>t</mml:mi><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM84\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM85\"><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM86\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM87\"><mml:mo>−</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo>&lt;</mml:mo><mml:mi>τ</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM88\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM89\"><mml:mi>g</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM90\"><mml:mi>τ</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM91\"><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM92\"><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>≤</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM93\"><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM94\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM95\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:msub><mml:mi>Ω</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM96\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM97\"><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>9</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM98\"><mml:mi>Ω</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM99\"><mml:mi>Ω</mml:mi><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM100\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM101\"><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>6</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM102\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM103\"><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM104\"><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM105\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM106\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM107\"><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM108\"><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM109\"><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM110\"><mml:mi>ρ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">const</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM6\"><mml:math id=\"DM8\" display=\"block\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>6</mml:mn></mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230044M1x3\"><label>1.3</label><mml:math id=\"DM9\" display=\"block\"><mml:mi>P</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>6</mml:mn></mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM111\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM112\"><mml:mi>P</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM113\"><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>&gt;</mml:mo><mml:mn>54</mml:mn><mml:mi>λ</mml:mi><mml:msubsup><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo><mml:mn>2</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM114\"><mml:mi>P</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM115\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM116\"><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo stretchy=\"false\">[</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM117\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM118\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM119\"><mml:msub><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM120\"><mml:mi>a</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM121\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM122\"><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">[</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM123\"><mml:msub><mml:mi>Ω</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM124\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM125\"><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM126\"><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>54</mml:mn><mml:mi>λ</mml:mi><mml:msubsup><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo><mml:mn>2</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM127\"><mml:mi>P</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mo>+</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>18</mml:mn></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM128\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>18</mml:mn></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM129\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>+</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM130\"><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>+</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo stretchy=\"false\">[</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM131\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>+</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM132\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM133\"><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>+</mml:mo></mml:msub><mml:mo stretchy=\"false\">[</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM134\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>+</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM135\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM136\"><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>&lt;</mml:mo><mml:mn>54</mml:mn><mml:mi>λ</mml:mi><mml:msubsup><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo><mml:mn>2</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM137\"><mml:mi>P</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM138\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM139\"><mml:mi>Ω</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM140\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM141\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM142\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM143\"><mml:mi>ω</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044M1x4\"><label>1.4</label><mml:math id=\"DM10\" display=\"block\"><mml:mn>4</mml:mn><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mn>3</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>6</mml:mn></mml:mfrac><mml:msup><mml:mi>ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mi>ω</mml:mi><mml:mn>6</mml:mn></mml:msup><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM144\"><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM145\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM146\"><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM147\"><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM148\"><mml:mi>ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM149\"><mml:mi>ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM150\"><mml:mi>ω</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM151\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM152\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM153\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM154\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>−</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM155\"><mml:mi>c</mml:mi><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044M1x5\"><label>1.5</label><mml:math id=\"DM11\" display=\"block\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:mi>Σ</mml:mi><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>c</mml:mi><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM156\"><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mi>Σ</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM157\"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mi>Σ</mml:mi><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>8</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mi>Σ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM158\"><mml:mi>Σ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:msubsup><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:msubsup><mml:mo>−</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM159\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM160\"><mml:mi>P</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM161\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mo>&lt;</mml:mo><mml:mo>−</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>6</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM162\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM163\"><mml:mi>λ</mml:mi><mml:mo>−</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:msubsup><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:msubsup><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044M1x6\"><label>1.6</label><mml:math id=\"DM12\" display=\"block\"><mml:mi>Q</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>6</mml:mn></mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM164\"><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>&gt;</mml:mo><mml:mn>16</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM165\"><mml:msub><mml:mi>Ω</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM166\"><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo stretchy=\"false\">[</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM167\"><mml:msub><mml:mi>Ω</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM168\"><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM169\"><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM170\"><mml:msub><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM171\"><mml:msub><mml:mi>Ω</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM172\"><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM173\"><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>16</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM174\"><mml:mi>Q</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM175\"><mml:msub><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:msqrt><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mrow></mml:msqrt></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM176\"><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>+</mml:mo></mml:msub><mml:mo stretchy=\"false\">[</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM177\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM178\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>+</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM179\"><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>+</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo stretchy=\"false\">[</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM180\"><mml:mo stretchy=\"false\">]</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mo stretchy=\"false\">[</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM181\"><mml:msub><mml:mi>Ω</mml:mi><mml:mo>+</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM182\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:msqrt><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:mrow></mml:msqrt></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM183\"><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>R</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>&lt;</mml:mo><mml:mn>16</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM184\"><mml:mi>Q</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM185\"><mml:mo stretchy=\"false\">]</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo stretchy=\"false\">[</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM186\"><mml:mi>Ω</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM187\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM188\"><mml:mi>ω</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM189\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044M1x7\"><label>1.7</label><mml:math id=\"DM13\" display=\"block\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mi>ω</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>6</mml:mn></mml:mfrac><mml:msup><mml:mi>ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mn>3</mml:mn></mml:mfrac><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM190\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM191\"><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM192\"><mml:mi>ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM193\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM194\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM195\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM196\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM197\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM198\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM199\"><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044M1x8\"><label>1.8</label><mml:math id=\"DM14\" display=\"block\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo 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stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM200\"><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msqrt><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:msqrt><mml:mo> </mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msqrt><mml:mi>f</mml:mi></mml:msqrt><mml:mo>+</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM201\"><mml:mi>f</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:mi>λ</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:msqrt></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM202\"><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:mi>f</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>8</mml:mn><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:msqrt></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM203\"><mml:msup><mml:mi>k</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msqrt><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:msqrt></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM204\"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>e</mml:mi><mml:mo>+</mml:mo><mml:msqrt><mml:mi>f</mml:mi></mml:msqrt><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msqrt><mml:mi>e</mml:mi><mml:msqrt><mml:mi>f</mml:mi></mml:msqrt></mml:msqrt></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM205\"><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM206\"><mml:mi>Ω</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM207\"><mml:mo>…</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM208\"><mml:mo>…</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM209\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM210\"><mml:mi>λ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM211\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM212\"><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM213\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM214\"><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM215\"><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">const</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo><mml:mo>∼</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM216\"><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM217\"><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM218\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM219\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM220\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM221\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM222\"><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM223\"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">]</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mi>π</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM224\"><mml:mi>g</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM225\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM226\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM227\"><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM228\"><mml:mi>Ω</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM229\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:mi>π</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM230\"><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM231\"><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM232\"><mml:mo>…</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM233\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM234\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mi>τ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>&lt;</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM235\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM236\"><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM237\"><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM238\"><mml:mo>…</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM239\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>τ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi>π</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM240\"><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM241\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:mi>π</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM242\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msup><mml:mo>⊂</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>τ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi>π</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM243\"><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM244\"><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM245\"><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM246\"><mml:mrow><mml:mover><mml:mi>M</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>⊂</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>τ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi>π</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM247\"><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM248\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>M</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM249\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM250\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM251\"><mml:mi>g</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM252\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM253\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM254\"><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>g</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM255\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM256\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM257\"><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM258\"><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM259\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM260\"><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>g</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM261\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM262\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM263\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM264\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM265\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM266\"><mml:mrow><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:math></inline-formula>", "<inline-formula>\n<mml:math id=\"IM267\"><mml:mi>λ</mml:mi></mml:math>\n</inline-formula>", "<inline-formula><mml:math id=\"IM268\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM269\"><mml:msub><mml:mi>U</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM270\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM271\"><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM272\"><mml:msub><mml:mi>U</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM273\"><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM274\"><mml:msub><mml:mi>U</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM275\"><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM276\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM277\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM278\"><mml:mi>w</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM279\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM280\"><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">const</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM281\"><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM282\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM283\"><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM284\"><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM285\"><mml:mo>…</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM286\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044M3x1\"><label>3.1</label><mml:math id=\"DM15\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>ϕ</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>V</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM287\"><mml:msup><mml:mi> </mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>ϕ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044M3x2\"><label>3.2</label><mml:math id=\"DM16\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:mo>−</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM288\"><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mi>μ</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM289\"><mml:mi>μ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM290\"><mml:mi>U</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM291\"><mml:mi>μ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM292\"><mml:mn>3</mml:mn><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM293\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM294\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM295\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM296\"><mml:mi>ψ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM297\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM298\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM299\"><mml:mi>g</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM300\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM301\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM302\"><mml:msub><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM303\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM304\"><mml:msub><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM305\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM306\"><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>◻</mml:mi><mml:mrow><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>6</mml:mn></mml:mfrac></mml:mstyle><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>ϕ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM307\"><mml:msub><mml:mi>◻</mml:mi><mml:mrow><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM308\"><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM7\"><mml:math id=\"DM17\" display=\"block\"><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mi>λ</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230044UM8\"><mml:math id=\"DM18\" display=\"block\"><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mi>λ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>ϕ</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>V</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mi>ϕ</mml:mi><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>6</mml:mn></mml:mfrac><mml:mi>ϕ</mml:mi><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>ϕ</mml:mi><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM309\"><mml:mi>ψ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>ϕ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM310\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM9\"><mml:math id=\"DM19\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:msub><mml:mi>ψ</mml:mi></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mi>λ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>ψ</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>V</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>ψ</mml:mi><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mi> </mml:mi><mml:mspace width=\"1em\"/><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>6</mml:mn></mml:mfrac><mml:mi>ψ</mml:mi><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>ψ</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM311\"><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM312\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM10\"><mml:math id=\"DM20\" display=\"block\"><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mi>λ</mml:mi><mml:msup><mml:mi>ψ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mtext>as </mml:mtext><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM313\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM314\"><mml:mi>ψ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM315\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM316\"><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM317\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM11\"><mml:math id=\"DM21\" display=\"block\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM318\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM319\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM12\"><mml:math id=\"DM22\" display=\"block\"><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>≠</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM320\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM321\"><mml:mi>S</mml:mi><mml:mo>∼</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM322\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM323\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM324\"><mml:mi>Ω</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM325\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM326\"><mml:mi>ρ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM327\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM13\"><mml:math id=\"DM23\" display=\"block\"><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mtext>as </mml:mtext><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM328\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM329\"><mml:msup><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM14\"><mml:math id=\"DM24\" display=\"block\"><mml:msup><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230044UM15\"><mml:math id=\"DM25\" display=\"block\"><mml:mn>0</mml:mn><mml:mo>=</mml:mo><mml:msup><mml:mi>U</mml:mi><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>g</mml:mi><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msup><mml:mi>U</mml:mi><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mi>Ω</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM330\"><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM331\"><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM332\"><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM333\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM334\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM335\"><mml:msup><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM336\"><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM337\"><mml:mi>q</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM16\"><mml:math id=\"DM26\" display=\"block\"><mml:mn>0</mml:mn><mml:mo>≠</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:msub><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>q</mml:mi><mml:mi>r</mml:mi><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mn>0</mml:mn><mml:mo>≠</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:msub><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msup><mml:mi>q</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>λ</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mi>r</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM338\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM339\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi>V</mml:mi><mml:mo>=</mml:mo><mml:mi>r</mml:mi><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM340\"><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM17\"><mml:math id=\"DM27\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>V</mml:mi></mml:msub><mml:mi>V</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>V</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>V</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mo>−</mml:mo><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>V</mml:mi><mml:mo>,</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mi>b</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">#</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230044UM18\"><mml:math id=\"DM28\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>V</mml:mi></mml:msub><mml:mi>b</mml:mi><mml:mo>−</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>V</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mi>b</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>V</mml:mi><mml:mo>,</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">#</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>V</mml:mi><mml:mo>,</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM341\"><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM342\"><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM343\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM344\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM345\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM346\"><mml:msub><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">const</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM347\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM19\"><mml:math id=\"DM29\" display=\"block\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mrow><mml:mo>∗</mml:mo><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM348\"><mml:msup><mml:mi>w</mml:mi><mml:mrow><mml:mo>∗</mml:mo><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM349\"><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mrow><mml:mo>∗</mml:mo><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≤</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM350\"><mml:msup><mml:mi>w</mml:mi><mml:mrow><mml:mo>∗</mml:mo><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM351\"><mml:msup><mml:mi>w</mml:mi><mml:mrow><mml:mo>∗</mml:mo><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM352\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM353\"><mml:mi>ρ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044M3x3\"><label>3.3</label><mml:math id=\"DM30\" display=\"block\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:msup><mml:mi>ρ</mml:mi><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM354\"><mml:mi>ρ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">const</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM355\"><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">const</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM356\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM357\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM358\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM359\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM360\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM361\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM362\"><mml:mi>ρ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mo>∗</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">const</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM363\"><mml:mi>e</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM364\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM365\"><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">const</mml:mi><mml:mo>.</mml:mo></mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM366\"><mml:mi>ρ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM367\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044M3x4\"><label>3.4</label><mml:math id=\"DM31\" display=\"block\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mspace width=\"1em\"/><mml:mtext>with some </mml:mtext><mml:mspace width=\"1em\"/><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM368\"><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM369\"><mml:msup><mml:mi> </mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044M3x5\"><label>3.5</label><mml:math id=\"DM32\" display=\"block\"><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230044M3x6\"><label>3.6</label><mml:math id=\"DM33\" display=\"block\"><mml:mrow><mml:mover><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mspace width=\"1em\"/><mml:mtext>with some </mml:mtext><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM370\"><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044M3x7\"><label>3.7</label><mml:math id=\"DM34\" display=\"block\"><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM371\"><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM372\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM20\"><mml:math id=\"DM35\" display=\"block\"><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>&lt;</mml:mo><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">const</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM373\"><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM374\"><mml:msup><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>k</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM375\"><mml:mi>k</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM376\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM377\"><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM378\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM379\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM380\"><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM381\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM382\"><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM383\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM384\"><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM385\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM386\"><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM387\"><mml:mi>Ω</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM388\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM389\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM21\"><mml:math id=\"DM36\" display=\"block\"><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:mi>w</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>ρ</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>ρ</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230044UM22\"><mml:math id=\"DM37\" display=\"block\"><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>3</mml:mn><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>ρ</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>w</mml:mi><mml:mrow><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM390\"><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM391\"><mml:mi>ρ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM392\"><mml:mrow><mml:mover><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM393\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM394\"><mml:mi>ρ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM395\"><mml:mi>k</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM23\"><mml:math id=\"DM38\" display=\"block\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>6</mml:mn></mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mi>ρ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>ρ</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230044UM24\"><mml:math id=\"DM39\" display=\"block\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>6</mml:mn></mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mfrac><mml:mi>ρ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>3</mml:mn><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>ρ</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM396\"><mml:mi>k</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM397\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM398\"><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM399\"><mml:mi>ρ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM400\"><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM401\"><mml:msub><mml:mi>e</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>U</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM402\"><mml:msub><mml:mi>f</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>U</mml:mi><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msup><mml:mo> </mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mi>a</mml:mi><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM25\"><mml:math id=\"DM40\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi>e</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mi>e</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>c</mml:mi><mml:mi>c</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mi>f</mml:mi><mml:mi>c</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>c</mml:mi><mml:mi>c</mml:mi></mml:msubsup><mml:msub><mml:mi>f</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mi> </mml:mi><mml:mspace width=\"1em\"/><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>Ω</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>b</mml:mi></mml:msup><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>Ω</mml:mi><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mi> </mml:mi><mml:mspace width=\"2em\"/><mml:mo>−</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>″</mml:mo></mml:msup><mml:mfrac><mml:mrow><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mi>w</mml:mi></mml:mrow><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mfrac><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>c</mml:mi><mml:mi>c</mml:mi></mml:msubsup><mml:msub><mml:mi>f</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>Ω</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>c</mml:mi><mml:mi>c</mml:mi></mml:msubsup><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mi>Ω</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>Ω</mml:mi><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM403\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM404\"><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>U</mml:mi><mml:mi>b</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM405\"><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>c</mml:mi><mml:mi>c</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mi>U</mml:mi><mml:mi>k</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM406\"><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM407\"><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM408\"><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM409\"><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM410\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM411\"><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM412\"><mml:mi>k</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM413\"><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM414\"><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230044UM26\"><mml:math id=\"DM41\" display=\"block\"><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mi>ρ</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230044UM27\"><mml:math id=\"DM42\" display=\"block\"><mml:msup><mml:mi>w</mml:mi><mml:mo>″</mml:mo></mml:msup><mml:mfrac><mml:mrow><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mi>w</mml:mi></mml:mrow><mml:msup><mml:mi>w</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mfrac><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>k</mml:mi></mml:msup><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mfrac><mml:mrow><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>″</mml:mo></mml:msup></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>k</mml:mi></mml:msup><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM415\"><mml:mi>k</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM416\"><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM417\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM418\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM419\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM420\"><mml:mi>Ω</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM421\"><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM422\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM423\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM424\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM425\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM426\"><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM427\"><mml:msub><mml:mi>e</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>U</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM428\"><mml:mi>U</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM429\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM430\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM431\"><mml:mi>k</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM432\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM433\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM434\"><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM435\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM436\"><mml:mi>U</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM437\"><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM438\"><mml:mo>&lt;</mml:mo><mml:mi>U</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo>&gt;&lt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM439\"><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM440\"><mml:mi>U</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM441\"><mml:mi>Ω</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM442\"><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM443\"><mml:mi>Ω</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM444\"><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM445\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM446\"><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM447\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM448\"><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM449\"><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM450\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM451\"><mml:msup><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM452\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM453\"><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM454\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM455\"><mml:mi>U</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM456\"><mml:msup><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM457\"><mml:mi>S</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM458\"><mml:mi>k</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM459\"><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM460\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM461\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM462\"><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM463\"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM464\"><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM465\"><mml:mi>k</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM466\"><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM467\"><mml:msup><mml:mi>w</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM468\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mi>ρ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM469\"><mml:msub><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>∗</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM470\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>∗</mml:mo></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM471\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>&gt;</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>∗</mml:mo></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM472\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>&lt;</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>∗</mml:mo></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM473\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>∗</mml:mo></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM474\"><mml:mi>w</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM475\"><mml:mi>w</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM476\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>&lt;</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>∗</mml:mo></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM477\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>∗</mml:mo></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM478\"><mml:msub><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>∗</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM479\"><mml:msub><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>∗</mml:mo></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>" ]

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[ "<title>Introduction</title>", "<p>Roger Penrose’s seminal idea [##UREF##0##1##,##UREF##1##2##] of using techniques from conformal geometry to study global questions in general relativity has been vastly influential. Today, these ideas permeate a number of current research programmes in mathematical physics, partial differential equations and analysis, and have contributed much to the basic language of general relativity from null infinity to Penrose diagrams. This article is an introductory article for the proceedings of a workshop of the same title, which took place 9–10th May 2023 at the Royal Society in London, aimed at providing a survey of the field 60 years after Penrose’s original work.</p>" ]

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[ "<p>One contribution of 13 to a discussion meeting issue ‘<ext-link xlink:href=\"http://dx.doi.org/10.1098/rsta/382/2267\" ext-link-type=\"uri\">At the interface of asymptotics, conformal methods and analysis in general relativity</ext-link>’.</p>", "<p>This is an introductory article for the proceedings associated with the Royal Society Hooke discussion meeting of the same title which took place in London in May 2023. We review the history of Penrose’s conformal compactification, null infinity and a number of related fundamental developments in mathematical general relativity from the last 60 years.</p>", "<p>This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.</p>" ]

[ "<title>Penrose’s conformal compactification</title>", "<p>Robert Geroch’s description in the proceedings [##UREF##2##3##] of the 1976 Cincinnati Symposium on Asymptotic Structure of Space-Time captures well an essential reason why general relativity (henceforth GR) is distinct from many other physical theories: it is that in GR, the notion of an ‘isolated system’ becomes much harder to define than in other theories. This makes GR significantly more difficult to study. For example, Newtonian gravitation is described by a potential evolving on fixed Euclidean 3-space, and a system may be said to be ‘isolated’ if the mass density vanishes outside some compact subset of this space, and decays to zero away from the compact set. In GR, however, such a separation of the fields into ‘background’ and ‘physical’ fields is much more difficult, as the spacetime metric is both; here one has to construct the evolving field and its background at the same time. A natural attempt to define an isolated system in general relativity might be via an appropriate notion of ‘asymptotic flatness’ (at least in the case of zero cosmological constant ): as one recedes ‘to infinity’, the spacetime geometry should approach Minkowski space in a suitable sense, for example that the Riemann tensor of should approach zero. However, this is mathematically awkward and somewhat non-geometric, since one is then faced with taking limits of tensor fields on manifolds, which is in general not well defined, as well as having to describe limits of differentiable structures.</p>", "<title>Null infinity</title>", "<p>Penrose’s idea of using a conformal compactification to address the problem described in the previous paragraph is particularly elegant since it in one stroke takes care of both of these difficulties. Consider the physical metric , and rescale it, , using a conformal factor , a smooth positive scalar field which approaches zero at infinity with respect to at an appropriate rate. Penrose called the <italic toggle=\"yes\">unphysical</italic> or <italic toggle=\"yes\">rescaled</italic> metric. If is chosen correctly, the limit points of all inextendible null geodesics are then brought to a finite distance, and form a submanifold called <italic toggle=\"yes\">null infinity</italic>, denoted . Asymptotic considerations with respect to are then replaced with local differential geometry in terms of and in a neighbourhood of , and decay rates in null directions with respect to may be reinterpreted in terms of regularity at with respect to .</p>", "<p>The massless fields in the physical spacetime, appropriately rescaled, then imbue with information which may be thought of as a summary of the asymptotics of the processes in the physical spacetime. Conceptually, if one then detaches from the rescaled spacetime and considers it, together with this summary of asymptotics, an object in its own right, the fields on are found to split into two distinct classes, the geometrical, or <italic toggle=\"yes\">universal</italic> fields, and the physical fields. The universal fields turn out to be, at least locally, the same no matter the original spacetime, and therefore deserve to be called universal. The physical fields, on the other hand, depend on the physical processes in the original spacetime. Penrose’s proposal therefore elegantly decouples the geometry from the physics, at least asymptotically, and for massless fields provides a way of defining an isolated system by requiring that a suitable compactification with as described above exists.</p>", "<title>Geroch conjectures from Cincinnati symposium</title>", "<p>In [##UREF##2##3##] Geroch made three conjectures, numbered Conjecture 1, Conjecture 10 and Conjecture 13 (they were interspersed with Theorems). We mention them in reverse order. Conjecture 13 concerned the uniqueness of conformal completions of initial data sets for GR. That is, consider an asymptotically Euclidean 3-manifold (for simplicity, with just one asymptotic end) with a smooth positive definite metric and a smooth symmetric tensor field , the extrinsic curvature. Consider a conformal completion of by a point , , where at one has , and . Geroch called such a pair an asymptote of . Conjecture 13 asked whether any two asymptotes and were <italic toggle=\"yes\">equivalent</italic> in the sense that there exists a conformal rescaling such that at and . This conjecture has been essentially understood through an adaptation of the method of punctures to solve the constraint equations in GR [##REF##10043484##4##–##UREF##4##6##], however, a proof does not appear to be explicitly present in the literature. We mention here also the earlier work of Ashtekar–Hansen, and their notion of AEFANSI (asymptotically empty and flat at null and spatial infinity) spacetimes [##UREF##5##7##], and the resulting notion of the Spi (spatial infinity) 4-manifold.</p>", "<p>The second of Geroch’s conjectures, Conjecture 10, asked whether in a spacetime admitting a regular null the vanishing of the (massless) Klein–Gordon or Maxwell radiation field at implies that the solution should vanish in the interior of the spacetime. This is now known to be true in some generality as a consequence of the scattering theories constructed in [##UREF##6##8##–##UREF##9##11##] and elsewhere.</p>", "<p>Finally, Conjecture 1 asked roughly whether, in a vacuum spacetime with a null , linear perturbations of Einstein’s equations with compactly supported data (on a spacelike, perhaps hyperboloidal, hypersurface ) preserve in the domain of dependence of . In essence, this amounts to the question of the stability of spacetimes to linear gravitational perturbations (which, as posed by Geroch, should have compactly supported initial data). When the background is Minkowski space, this was answered affirmatively, in the full nonlinear regime, by Friedrich [##UREF##10##12##]. The celebrated work of Christodoulou &amp; Klainerman [##UREF##11##13##] later proved the stability of Minkowski space for data on a uniformly spacelike Cauchy surface, doing away with Friedrich’s requirement that the data be prescribed on a hyperboloid (see also [##UREF##12##14##]); they obtained spacetimes with asymptotic behaviour which in general does not permit a smooth conformal compactification (see §2c for more on this). Even more general decay rates were later handled by Bieri [##UREF##13##15##] and more recently by Shen [##UREF##14##16##]. Back in the linear regime, recent work of Masaood [##UREF##15##17##,##UREF##16##18##] has handled the construction of a scattering theory for linear gravitational perturbations to a Schwarzschild black hole. A result of this kind for more general spacetimes possessing an appropriately regular null appears to still be open. Related to this is the fact that the spacetimes admitting a regular null infinity are not fully characterized, although infinite-dimensional families of such spacetimes are known [##UREF##17##19##–##UREF##20##22##].</p>", "<title>Peeling</title>", "<p>The question of what ‘appropriate’ regularity of should mean now arises naturally. In the late 1950s and early 1960s, Sachs [##UREF##21##23##,##UREF##22##24##] discovered (see also [##UREF##23##25##,##UREF##24##26##]) that in smooth asymptotically flat spacetimes a so-called peeling property was satisfied by the Weyl tensor. In essence, Sachs found that the different components of the Weyl tensor ‘peel off’ at decreasing, integer rate powers of the luminosity parameter , , as one approaches null infinity along null directions. Peeling and the history of its discovery is described beautifully in the article of Penrose in these proceedings [##UREF##25##27##], and we refer the interested reader there. The important point, however, was the observation that the peeling property of the Weyl tensor is nothing more than a consequence of the smoothness of the Weyl tensor in the conformally compactified spacetime. This was a neat correspondence. However, for reasons that will be mentioned shortly, while the picture of an isolated system defined through a conformal compactification was appealing, it was not universally accepted by all workers in the field as appropriate. In particular, questions of generality, definability of relevant physical quantities, and the applicability of the model to physical situations of interest, remained—and still do. Calculations based on the equations of motion and the radiation escaping an isolated system to infinity did not always appear to verify the Sachs peeling property, which raised doubts as to the suitability of the smoothness of the compactified picture [##UREF##26##28##,##UREF##27##29##]. Indeed, there were concerns that the conformal picture did not admit <italic toggle=\"yes\">any</italic> solutions to Einstein’s equations which contained gravitational radiation. This was put to rest by a series of results due to Corvino, Chruściel–Delay and Corvino–Schoen [##UREF##17##19##–##UREF##20##22##,##UREF##28##30##] (see also [##UREF##29##31##]), who showed that initial data for the Einstein equations could be deformed in more flexible ways than previously thought, e.g. by gluing essentially arbitrary small data in a compact region to Schwarzschildean asymptotics near spatial infinity. Friedrich’s semi-global stability result [##UREF##10##12##] with these data then produces an infinite-dimensional family of non-trivial solutions to the Einstein vacuum equations with complete and smooth conformal infinity and regular past and future timelike infinities . A more complete account of the history of peeling is given well by Friedrich in the review [##UREF##30##32##]. In general, if the decay of initial data is weaker than is required for peeling, even if the data is smooth towards spatial infinity , Friedrich [##UREF##31##33##] has shown that the resulting evolution will in general be polyhomogeneous at , a result of the degeneracy of Einstein’s equations at the points where spatial infinity meets null infinity. That quite generally the situation is no worse was recently confirmed by Hintz &amp; Vasy [##UREF##32##34##]. Work to understand these issues sharply is ongoing, and the detailed structure of the polyhomogeneities exhibited by massless scalar fields on Minkowski space is reviewed in Gasperín’s [##UREF##33##35##] contribution to this issue.</p>", "<p>The worry of mathematical genericity of peeling spacetimes being somewhat settled, the question of applicability to relevant physical situations nevertheless remains. At the time of writing, the general consensus appears to be that the smoothness assumptions in the classical definition of asymptotic simplicity are in some instances too restrictive to describe realistic physical scenarios. Kehrberger [##UREF##34##36##–##UREF##36##38##] has argued strongly that many systems of physical interest, e.g. the problem of gravitating bodies coming in from infinity, violate not only peeling, but also the weaker decay assumptions and conclusions of Christodoulou–Klainerman [##UREF##11##13##,##UREF##37##39##]. There is some suggestion, however, that the weak assumptions of Bieri [##UREF##13##15##] are sufficient to capture these situations: see the chapter by Kehrberger [##UREF##38##40##] in this issue.</p>", "<title>The BMS group</title>", "<p>One of the main motivations behind Penrose’s approach to characterizing isolated systems in GR through conformal compactifications may have been that of identifying universal structures which, in turn, could be used as tools to study the physical processes taking place in the system. One of these universal structures is <italic toggle=\"yes\">asymptotic symmetries</italic>. Generic spacetimes cannot be expected to be endowed with symmetries. In the case of an isolated system in GR, however, intuition suggests that as one recedes from the sources the geometry should become more and more Minkowski-like. This intuition, in turn, makes the notion of asymptotic symmetries plausible. The big surprise is that, in GR, the asymptotic symmetry group does not coincide with the 10-dimensional Poincaré group, but turns out to be an infinite-dimensional group, now called the Bondi–Metzner–Sachs (BMS) group, containing the Lorentz group and a distorted infinite-dimensional group of translations. As pointed out in Penrose’s [##UREF##25##27##] contribution to this volume, the BMS group first arose from considering the group of transformations which preserve the form of the line elements used in the work of Bondi and collaborators [##UREF##21##23##,##UREF##22##24##]. More geometric descriptions were found in subsequent analyses [##UREF##2##3##,##UREF##39##41##]. In this spirit, Borthwick &amp; Herfray’s [##UREF##40##42##] contribution to this volume extends the classical analysis of asymptotic symmetries in asymptotically flat spacetimes to more general classes of spacetimes. This work makes use of promising geometric notions and techniques (projective compactifications and tractor calculus) which are yet to permeate and make an impact in the analytic study of Einstein’s equations, as the conformal technique has done.</p>", "<p>In recent years, the study of the BMS group has acquired renewed interest. This has to do, to some extent, with the work by e.g. Hawking, Perry and Strominger on soft hair for black holes and its implications for the information loss paradox [##REF##27341223##43##]. The asymptotic symmetries defined by the BMS group allow the construction of an infinite number of <italic toggle=\"yes\">asymptotic charges</italic>—not all of these conserved but nevertheless satisfying balance laws between cuts of null infinity. These charges can be constructed at both past and future null infinity. A question of particular relevance in the analysis of Hawking <italic toggle=\"yes\">et al</italic>. [##REF##27341223##43##] is the way the past and future asymptotic charges are related: the past and future asymptotic charges can be identified if an <italic toggle=\"yes\">antipodal matching</italic> at spatial infinity is assumed. Friedrich’s framework of spatial infinity (see §4) provides a powerful tool to analyse the genericity of the antipodal identification of past and future null infinity and to characterize the associated class of Cauchy initial data for linear and full nonlinear GR in which this occurs. An overview of this research programme is given in Mohamed’s [##UREF##41##44##] contribution to this volume. A key outcome of this analysis is that the BMS charges are generically not well-defined unless certain regularity conditions on the initial data are assumed. When they are well defined, however, by writing the charges at the endpoints of null infinity in terms of Cauchy initial data it is possible to recover the identification of past and future charges without the need for an <italic toggle=\"yes\">a priori</italic> assumption of antipodal matching. This analysis suggests that the antipodal identification condition is, at its core, an assumption about the regularity of spatial infinity.</p>", "<title>Friedrich’s conformal Einstein equations and spatial infinity</title>", "<p>The Einstein field equations are not conformally invariant. In fact, a naive rewriting of the equations in terms of an unphysical metric conformally related to a physical metric yields equations which are formally singular at the conformal boundary. Nevertheless, it was shown by Friedrich [##UREF##42##45##,##UREF##43##46##] that there exists a larger system of PDEs, the so-called conformal Einstein field equations, which admit Einstein’s equations as a subsystem and for which standard methods of the theory of partial differential equations are applicable even at the conformal boundary. This observation opened the door to the first proofs of the global existence and nonlinear stability of de Sitter-like and Minkowski-like solutions to the Einstein field equations mentioned earlier [##UREF##10##12##]. The main idea behind these proofs is that Penrose’s compactification procedure allows one to reformulate an infinite-time existence problem as one where existence only needs to be shown for a finite amount of conformal time. In many relevant problems, it is possible to find a gauge in which the conformal Einstein field equations admit a reduction to a symmetric hyperbolic system. When this is the case, the required existence results follow from the property of Cauchy stability.^{<xref rid=\"FN1\" ref-type=\"fn\">1</xref>}</p>", "<p>Friedrich’s original analysis of the stability of the de Sitter spacetime has been generalized in many directions. In particular, it has allowed the investigation of the global existence of cosmological solutions with spatial sections of constant negative (in the conventions of Friedrich) curvature, see e.g. Minucci’s [##UREF##45##48##] contribution to this volume. Further, while Friedrich’s analysis was restricted to the vacuum case, there exist now extensions of this strategy to settings involving non-vanishing mass with trace-free energy momentum tensor, as explained in the chapter of Tod [##UREF##46##49##]. Finally, extensions to the more challenging case where the trace of the energy-momentum is non-vanishing are explored in Friedrich’s [##UREF##47##50##] article in this volume.</p>", "<p>A peculiarity of Penrose’s notion of asymptotic simplicity is that it makes no reference to spatial infinity. In the original paper [##UREF##1##2##] introducing the conformal method, Penrose himself observed that the presence of a non-vanishing ADM mass in the spacetime gives rise to a singularity of the conformal structure at the point representing spatial infinity. In the following decades, a significant amount of effort was put into trying to understand the relationship between this singular behaviour at spatial infinity and the properties of null infinity (e.g. [##UREF##2##3##]). These efforts led to the identification by Ashtekar and Hansen of minimal regularity assumptions which allow a matching between spatial and null infinity [##UREF##5##7##]. As in the case of Penrose’s asymptotically simple spacetimes, the complete classification of Ashtekar and Hansen’s class of AEFANSI spacetimes remains an open question.</p>", "<p>As mentioned in §2c, Hintz and Vasy’s proof of the nonlinear stability of Minkowski space, which uses Melrose’s techniques of geometric scattering (see §5), establishes the generic presence of polyhomogeneous asymptotics. There are clear connections between the set-up of Hintz–Vasy and that of Friedrich, and bringing these to the fore, perhaps in application to a full classification of asymptotics of spacetimes, would seem to be a promising area of research.</p>", "<title>Scattering theory</title>", "<p>The functional analytic framework of the Lax–Phillips scattering theory [##UREF##48##51##] for hyperbolic differential equations, developed in the 1960s, provides a point of departure for scattering in general relativity. Lax and Phillips deal with systems described by a group of unitary operators acting on a Hilbert space which possesses distinguished subspaces and [##UREF##49##52##]. These subspaces, to be interpreted as spaces of scattering data in the past and future, have the property that increases monotonically from the zero subspace to the whole space as varies from to , and analogously for , which decreases from to zero as varies from to . With each of , there is associated a particular spectral representation of , and the future and past representations are related by a unitary operator , the scattering matrix.</p>", "<p>Each spectral representer of the solution can be understood as a so-called asymptotic profile (i.e. the leading-order behaviour of the solution near null infinity) in the past and the future. In the early 1980s, Friedlander [##UREF##50##53##] reinterpreted the Lax–Phillips asymptotic profiles as his radiation fields [##UREF##51##54##–##UREF##53##56##], i.e. limits along null geodesics of the scattered fields multiplied by a radial coordinate . The radiation fields had been known to be simply the restrictions of the conformally rescaled fields to in Penrose’s conformal compactification. Friedlander was therefore able to give the first construction of a ‘conformal’ scattering theory for the free wave equation. In Friedlander’s picture, the property that the Lax–Phillips asymptotic profiles completely determine the solution in the interior of the spacetime became the problem of solving a characteristic Cauchy problem from . It is here that Friedlander opened the way for a geometric formulation of scattering: while the Lax–Phillips theory, being based on the use of spectral methods, was not applicable on time-dependent backgrounds, Friedlander’s picture clarified that this ought to be understood as a technical shortcoming, the relevant information for scattering being contained in the asymptotics of the physical fields. Moreover, for massless fields these asymptotics were elegantly accessible via Penrose’s conformal compactification. The combination of these ideas has become known as conformal scattering [##UREF##6##8##]. A detailed comparison of the spectral and conformal approaches to scattering, and the difficulties and advantages typically seen in each approach, is given in Nicolas’s [##UREF##54##57##] contribution to this volume.</p>", "<p>A related major development in the field of mathematical relativity has been the realization of the close connections between the ideas behind Penrose’s compactification and Melrose’s school of microlocal analysis, often called <italic toggle=\"yes\">geometric scattering</italic> [##UREF##55##58##]. Very briefly, geometric scattering attempts to develop a Fredholm theory for hyperbolic operators on appropriately (but not necessarily conformally) compactified manifolds. The methods of geometric scattering have proven a powerful tool to analyse the nonlinear stability and asymptotics of solutions to the Einstein field equations, including black hole solutions [##UREF##56##59##,##UREF##57##60##]. Applications of the techniques of geometric scattering to the study of asymptotics in general relativity are discussed in Hintz’s [##UREF##58##61##] contribution to this volume, where asymptotically de Sitter-like spacetimes are constructed from asymptotic data (prescribed at the conformal boundary). This result generalizes the classical proof by Friedrich [##UREF##10##12##] (which is restricted to four dimensions) and that of Anderson [##UREF##59##62##] (which only applies to Lorentzian manifolds with odd spatial dimensions).</p>", "<title>Black holes</title>", "<p>The development of a complete and satisfactory mathematical theory of black holes has long been one of the main aims of mathematical relativity. This subject has been made even more relevant by the direct experimental observation in 2015 of gravitational waves produced by the coalescence of a binary system of black holes [##REF##26918975##63##]. From the mathematical side, a gargantuan effort has been poured into developing a proof of the nonlinear stability of the Kerr spacetime [##UREF##60##64##–##UREF##62##66##].</p>", "<p>The notion of null infinity as describing an idealized far away observer is central to the classical definition of a black hole in the asymptotically flat setting [##UREF##63##67##,##UREF##64##68##]. While there are very precise conjectures (and proofs under certain not entirely satisfactory assumptions) regarding the uniqueness of stationary black holes, the question of the uniqueness in the cosmological setting is not well understood [##UREF##65##69##]. Conformal methods allow the study of the uniqueness of black holes in spacetimes with a positive cosmological constant by means of an asymptotic initial value problem in which initial data are prescribed at the (spacelike) conformal boundary. The contribution of Mars &amp; Peón-Nieto [##UREF##66##70##] to this volume describes an ongoing research programme to understand, from the point of view of asymptotic initial data, the sense in which the Kerr–de Sitter family of solutions is special. This programme should provide important insights into the question of the uniqueness of stationary black holes in the cosmological setting.</p>", "<p>One of the most promising sources of gravitational waves for the future generation of space-based detectors (LISA) are the so-called extreme mass-ratio inspirals (EMRI) [##UREF##67##71##]. These are black hole binary systems in which one of the components can be treated as a point source and the gravitational waves produced can be adequately treated by means of perturbation theory [##REF##30270849##72##]. As EMRIs produce a gravitational wave signal over extended periods of time, the production of wave templates to be used in their detection require very precise numerical simulations. The use of hyperboloidal foliations to numerically construct the solutions to the equations of perturbation theory provides a way of constructing the signal templates which avoids the use of problematic boundary conditions. The key idea is that null infinity provides a natural <italic toggle=\"yes\">outflow boundary</italic> on which no boundary conditions need to be prescribed. This promising application of conformal methods to the study of gravitational waves is described in Panosso Macedo’s [##UREF##68##73##] contribution to this volume.</p>", "<title>Future directions</title>", "<p>We hope that the contributions to the present volume convey to the reader the influence in current research of Penrose’s idea of studying the asymptotics of fields in general relativity using conformal geometry. To conclude, we would like to point out two directions of research which appear to us to be quite promising.</p>", "<p>The first is to further explore and develop the connections between the methods of geometric scattering and the perhaps more traditional uses of conformal methods in general relativity, e.g. using Friedrich’s conformal Einstein field equations. It is possible that a combination of the two perspectives could yield a fruitful approach to a sharp classification of decay rates of initial data and the asymptotic properties of the resulting spacetimes. In a similar vein, it would be interesting to investigate if questions of nonlinear stability of black holes might be addressed using the conformal Einstein field equations, probably in some conjunction with the ideas that are already well established within the black hole stability community.</p>", "<p>The second direction is to continue the development of numerical schemes to construct solutions to the Einstein field equations in a conformal setting: see e.g. the work by Hübner on the numerical implementation of the hyperboloidal initial value problem [##UREF##69##74##]. The value of these numerical simulations should not be underestimated and can provide valuable insights for further analytical studies. In particular, the use of conformal methods should pave the way to the numerical construction of complete scattering spacetimes which are prescribed through initial data on the whole of past null infinity—this numerical endeavour will require input from the analytic side, as described in the previous paragraph.</p>" ]

[ "<title>Acknowledgements</title>", "<p>An attempt has been made here to cover the most important historical aspects of the subject. However, due to the very large number of researchers who have contributed to the broad area of asymptotics in general relativity, some omissions have no doubt crept in. All such instances are due to limitations of space and unintentional. We thank the staff at the Royal Society for the smooth running of the workshop, particularly Annabel Sturgess for the operations on the ground and Alice Power for helping to put together this dedicated issue of the Proceedings of the Royal Society A. We also thank the many reviewers who generously gave their time to review the following chapters.</p>", "<title>Data accessibility</title>", "<p>This article has no additional data.</p>", "<title>Declaration of AI use</title>", "<p>We have not used AI-assisted technologies in creating this article.</p>", "<title>Authors' contributions</title>", "<p>G.T. and J.A.V.K.: writing—original draft, writing—review and editing.</p>", "<p>All authors gave final approval for publication and agreed to be held accountable for the work performed therein.</p>", "<title>Conflict of interest declaration</title>", "<p>This theme issue was put together by the Guest Editor team under supervision from the journal’s Editorial staff, following the Royal Society’s ethical codes and best-practice guidelines. The Guest Editor team invited contributions and handled the review process. Individual Guest Editors were not involved in assessing papers where they had a personal, professional or financial conflict of interest with the authors or the research described. Independent reviewers assessed all papers. Invitation to contribute did not guarantee inclusion.</p>", "<title>Funding</title>", "<p>No funding has been received for this article.</p>" ]

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[ "<fn-group><fn id=\"FN1\"><label>1</label><p>A similar strategy was used by Y. Choquet-Bruhat and D. Christodoulou to show the global existence of solutions to the Yang–Mills equations [##UREF##44##47##].</p></fn></fn-group>" ]

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[ "<p>One contribution of 13 to a discussion meeting issue ‘<ext-link xlink:href=\"http://dx.doi.org/10.1098/rsta/382/2267\" ext-link-type=\"uri\">At the interface of asymptotics, conformal methods and analysis in general relativity</ext-link>’.</p>", "<p>This paper describes conservation laws in general relativity (GR) dating back to the mass-energy conservation of Bondi and Sachs in the early 1960s but using 2-spinor techniques. The notion of conformal infinity is employed, and the highly original ideas of E. T. Newman are discussed in relation to twistor theory. The controversial NP constants are introduced, and their meaning is considered in a new light related to the problem of equations of motion in GR.</p>", "<p>This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.</p>" ]

[ "<title>Early work on gravitational radiation</title>", "<p>The late 1950s and early 1960s saw the beginnings of our current understanding of gravitational radiation, according to general relativity (GR), and how gravitational waves carry energy away from a system of accelerating gravitating bodies. Already in 1925 [##UREF##0##1##], H. W. Brinkmann had discovered the metric for a gravitational plane wave (subsequently commonly referred to as a <italic toggle=\"yes\">pp</italic>-wave), but his solution was not at all well known to the physicists at that time. Later considerations of gravitational plane waves, using what had seemed to be a more directly appropriate type of coordinate system, led to much confusion, as it had seemed that singularities would necessarily occur in these solutions when extended too far into the future or past. Einstein himself, in his 1937 collaboration with Rosen [##UREF##1##2##], had seemed to conclude that plane gravitational waves did not properly exist in GR, and they re-interpreted their solution as describing <italic toggle=\"yes\">cylindrical</italic> waves instead! This confusion was later greatly clarified, by Ivor Robinson, as expressed in his 1959 paper [##UREF##2##3##] with Hermann Bondi and Felix Pirani, using a coordinate patchwork. The basic underlying conundrum was later identified, in 1965, in the curious fact that a gravitational plane wave does not admit a global <italic toggle=\"yes\">Cauchy hypersurface</italic> (i.e. a spacelike three-surface that intersects every maximally extended time-like curve—or path—in the space–time). (See [##UREF##3##4##]).</p>", "<p>Although it is a normal procedure in other areas of physics to decompose the radiation field into plane-wave components, the considerations raised above suggest that such a procedure might well encounter serious difficulties in GR. Moreover, the very nonlinear nature of GR presents additional difficulties for such a procedure, and very different techniques turned out to be much more effective for providing an understanding of gravitational radiation in accordance with GR.</p>", "<p>Another concern of historical importance was the issue of whether massive bodies moving about one another solely under their mutual gravitational influences would, according to GR, actually radiate gravitationally, or even respond to the presence of a gravitational-wave radiation. Back in 1918 [##UREF##4##5##], Einstein had shown that in the linear weak-field limit of GR, a change in the mass quadrupole moment of a massive system would result in the emission of gravitational waves. However, not only was this a result directly applicable only to this linear approximation of GR, but it also did not supply an answer to how such radiation might carry away energy or influence the motions of other bodies. More importantly, non-gravitational forces might be needed in order to effect the required changes in quadrupole moment of the masses involved, so this result, albeit an important one, did not directly answer the question of whether gravitational waves might arise from purely gravitationally caused motions.</p>", "<p>Accordingly, this left open the issue, raised particularly by the Polish physicist Leopold Infeld, of whether gravitational waves would indeed be emitted if the sources were accelerating only under the influence of their own mutual gravitational interactions. Einstein's famous 1938 paper, with Infeld and Hoffmann [##UREF##5##6##], addressed the issue of how a system of gravitating bodies, considered to be point-like in some appropriate limit, would move under their mutual gravitational attractions constrained only by the GR field equations, thus going beyond Einstein's GR <italic toggle=\"yes\">assumption</italic> of geodetic motion for point-like test-particles. However, this EIH paper did not seem to give a clear overall answer to these questions. Indeed, Infeld had been of the strong opinion that bodies, interacting with each other entirely gravitationally, would not actually emit gravitational waves.</p>", "<p>It should be pointed out that there are some serious issues concerning what is specifically meant by a gravitating ‘point particle' in GR, since for a ‘positive mass point particle', we would actually have to consider a tiny black hole (or perhaps a ‘white hole’ or even possibly both at once!) rather than a point source. This is the GR version of a general issue that arises with equations of motion. In electromagnetism, for example, the motion of a charged particle in an ambient electromagnetic field should be influenced only by the <italic toggle=\"yes\">background</italic> field in which it moves, whereas the <italic toggle=\"yes\">actual</italic> field, very close to the particle itself, would be dominated by the particle's own field, rather than the background field that it is supposed to be responding to. Issues of this nature will be a main concern of the later considerations of this article.</p>", "<p>The fact that gravitational waves could actually be emitted by purely gravitating systems, despite Infeld's frequently expressed opinion, was strongly indicated by the work of Trautman in [##UREF##6##7##], (partly influenced by Jerzy Plebański). Later, Trautman joined forces with Ivor Robinson to obtain a beautiful special family of radiating vacuum solutions with gravitational waves coming radially outwards from a localized source [##UREF##7##8##,##UREF##8##9##].</p>", "<p>However, it was Hermann Bondi's work on axi-symmetric spacetimes, initially presented in outline, in <italic toggle=\"yes\">Nature</italic> in1960 [##UREF##9##10##], but then worked out in considerable detail, with his students van der Burg and Matzner, published in 1962) [##UREF##10##11##], that greatly clarified the picture. In particular, this paper identified a specific quantity <italic toggle=\"yes\">M</italic>, which could be identified as the <italic toggle=\"yes\">total retarded mass-energy</italic> of the system—where the notion or ‘retarded', as used here, will be explained shortly; see (1.1). Importantly, the value of <italic toggle=\"yes\">M</italic> was shown to be reduced, as the (retarded) time parameter <italic toggle=\"yes\">u</italic> progresses. Thus, we can take the view that part of <italic toggle=\"yes\">M</italic> is ‘carried away' by the outgoing gravitational radiation, by an amount that can be identified in a certain <italic toggle=\"yes\">positive</italic> term in the outgoing radiation field.</p>", "<p>Soon afterwards the restriction of axi-symmetry was importantly removed by Sachs [##UREF##11##12##], this work providing a clear demonstration that (positive) energy-carrying gravitational waves do indeed occur in GR, in precise accordance with Einstein's original expectations arising from his early considerations with the linear limit of GR [##UREF##4##5##]. In particular, there was provided a clear mass-energy conservation law, showing that the sum of the retarded mass-energy of the gravitating sources, together with the mass-energy carried away by the outgoing radiation, would remain constant throughout the entire process. Moreover, the contribution from the gravitational radiation, whenever present at all, would be necessarily <italic toggle=\"yes\">positive</italic>, resulting in a reduction in the total (retarded) mass-energy <italic toggle=\"yes\">M</italic> of the system whenever outgoing radiation is present.</p>", "<p>The Bondi-Sachs approach was based on a form of coordinate system in which one of the coordinates, <italic toggle=\"yes\">u</italic>, regarded as a <italic toggle=\"yes\">retarded time</italic> parameter, the constant values of which, at least sufficiently far from the sources, would describe topologically spherical surfaces moving outwards with the speed of light, so that the <italic toggle=\"yes\">u </italic>= constant hypersurfaces would be <italic toggle=\"yes\">null</italic>,\nwith null-geodesic generator lines that extend outwards to a future null infinity, as they move away from the source region. In addition to the coordinate <italic toggle=\"yes\">u</italic>, there would be an outward radial coordinate <italic toggle=\"yes\">r</italic>, that could be taken to be an <italic toggle=\"yes\">affine parameter</italic> along each null-geodesic generator (although Bondi actually chose what he referred to as a <italic toggle=\"yes\">luminosity parameter</italic> that differed slightly from an affine parameter, this difference being unimportant for our considerations here, and plays no role in the discussions that follow).</p>", "<p>The two remaining coordinates can be taken to be standard spherical polar <italic toggle=\"yes\">θ</italic> and <italic toggle=\"yes\">ϕ</italic>, which are taken to remain unchanged along the null-geodesic generators, and provide the standard two-spherical form\nin the limit of large <italic toggle=\"yes\">r</italic>, this imposing a restriction on how the affine parameter <italic toggle=\"yes\">r</italic> is chosen on the various generators of the outgoing null hypersurfaces <italic toggle=\"yes\">u </italic>= const., a matter that will be clarified in more detail in §3. This work was an especially important development, because one could identify a quantity <italic toggle=\"yes\">M</italic> that could be interpreted as the <italic toggle=\"yes\">mass</italic>(-energy) of the system at each retarded time (<italic toggle=\"yes\">u</italic> = const.) and that this mass measure <italic toggle=\"yes\">M</italic>, now regarded as a function <italic toggle=\"yes\">M</italic>(<italic toggle=\"yes\">u</italic>) of the retarded time <italic toggle=\"yes\">u</italic> so that <italic toggle=\"yes\">M</italic>(<italic toggle=\"yes\">u</italic>) is necessarily a never-increasing function of <italic toggle=\"yes\">u</italic>, which was a very satisfactory result, from the physical point of view, as it assigned a clear-cut positive value to the mass-energy carried away by gravitational radiation. The actual positivity of <italic toggle=\"yes\">M</italic> itself was a separate matter, and results of importance here were achieved by Schoen &amp; Yau [##UREF##12##13##–##UREF##14##15##], Witten [##UREF##14##15##], Ruela &amp; Tod [##UREF##15##16##], Horowitz &amp; Perry [##UREF##16##17##], Ludvigsen &amp; Vickers [##UREF##17##18##,##UREF##18##19##]).</p>", "<p>Nevertheless, at the time when Bondi first produced his important contributions to this work, he was still not completely free of Infeld's influence, making the curious tentative suggestion that certain apparently ‘non-radiative but non-static motions' that appear in his considerations might possibly represent Infeld's supposed radiation-free motions under gravity. However, the formalism was not really set up in a way that might eliminate <italic toggle=\"yes\">incoming</italic> gravitational radiation despite the expressed hope, in the paper, that such incoming waves had been excluded (see also [##UREF##19##20##]). Indeed, we shall be seeing in §3 that the ‘retarded’ (<italic toggle=\"yes\">u</italic>, <italic toggle=\"yes\">r</italic>, <italic toggle=\"yes\">θ</italic>, <italic toggle=\"yes\">ϕ</italic>) coordinate system is not at all suitable for formulating a condition expressing the absence of incoming radiation, and we may take it that Bondi's supposed ‘non-radiative but non-static motions' must actually have involved the presence of <italic toggle=\"yes\">incoming</italic> gravitational radiation. A broader perspective on this issue will be provided in §3.</p>", "<title>The 2-spinor formalism</title>", "<p>In order to express concisely the quantities occurring in the following sections, it will be convenient to incorporate the 2-spinor abstract-index formalism. Details of this, and the relevant expressions, can be found in <italic toggle=\"yes\">Spinors and Space-Time</italic> [##UREF##20##21##,##UREF##21##22##], but a very brief outline will be indicated here. In conventional notation, an expression such as\nfor example, would denote the array of components, in some coordinate system or local basis frame, of a tensor quantity <bold><italic toggle=\"yes\">T</italic></bold> where <italic toggle=\"yes\">a, b</italic>, <italic toggle=\"yes\">c</italic> range over some set of alternative values, say 1, 2, 3, … , <italic toggle=\"yes\">n.</italic> In the <italic toggle=\"yes\">abstract-index notation</italic>, on the other hand, the expression (2.1) would instead denote the entire tensor <bold><italic toggle=\"yes\">T</italic></bold> itself, but together with certain elements <italic toggle=\"yes\">a</italic>, <italic toggle=\"yes\">b</italic>, <italic toggle=\"yes\">c</italic>, taken from the tensor index labelling set\nwhere no coordinate system or local reference frame is involved. If, however, we do wish to refer to the family of components of <bold><italic toggle=\"yes\">T</italic></bold> with respect to some local reference system, then the notation\nwith bold-face upright indices, would be used (where also hybrid expressions using both types of index in the same expression are also allowed). Thus, for ordinary four-dimensional spacetime, whereas the expression (2.3) stands for an array of 64 numbers or functions, in the usual way, where the expression (2.1) stands for a single tensorial quantity.</p>", "<p>The utility of the abstract-index notation is not immediately evident, but is made particularly manifest in the <italic toggle=\"yes\">2-spinor formalism</italic>, where we can use <italic toggle=\"yes\">capital</italic> italic abstract-index letters for the two-dimensional complex spin-space at each point and <italic toggle=\"yes\">primed</italic> italic capital letters for the corresponding <italic toggle=\"yes\">complex-conjugate</italic> spin space. A particular strength of the abstract-index notation is that it allows us to exploit the power of the 2-spinor formalism in a simple and manifest way without the need to clutter expressions with translation quantities such as Infeld-van der Waerden translation symbols (see [##UREF##22##23##,##UREF##23##24##]). This is achieved by regarding each abstract spacetime index as the composite (product) of an un-primed and a primed capitalized version of the chosen tensor-index letter\nso that our tensor <bold><italic toggle=\"yes\">T</italic></bold> can be expressed in a spinor form as\nor in a hybrid form such as etc.</p>", "<p>The 2-spinor formalism provides a lot of power and flexibility that is not easy to express in the conventional tensor formalism for spacetime. For example, the vector quantity\nis easily expressed in the 2-spinor formalism, whereas its far-from-obvious expression in terms of conventional tensor notation turns out to be remarkably complicated!</p>", "<p>Since the primed and un-primed spin-spaces are algebraically different spaces (albeit complex conjugates of one another), we can, without ambiguity, commute a primed index with an un-primed one (as was implicit in the interchange of <italic toggle=\"yes\">A′</italic> with <italic toggle=\"yes\">A</italic> in (2.6)), although we must keep track of the ordering of the un-primed indices and of the primed ones separately. This is helpful for writing symmetric and skew-symmetric parts (using round brackets or square brackets, respectively). For example\nso, we can express\n</p>", "<p>Thus, we do need to keep the ordering within the system of un-primed and also within the system of primed indices, though the ordering between the two is irrelevant. Moreover, in view of the fact that spinor indices will need to be allowed to be raised or lowered, consistently with that process as applied to tensor indices, we shall need to be able to keep track of this ordering, both for the un-primed and for the primed indices. Accordingly, we need to stagger the upper and lower indices so that the necessary ordering is preserved, with a clearly determined space for each index so that it is unambiguous what this ordering is intended to be, when raised or lowered.</p>", "<p>Although the 2-spinor formalism has a considerable value in simplifying tensor expressions, where the primed and un-primed spinor indices are equal in number, it will be of importance for us that an individual <italic toggle=\"yes\">spin-vector</italic> (i.e having a single spinor index) also has a geometrical interpretation (up to a sign). This is as a future-pointing null vector (its ‘flagpole') with a null half-plane attached to it (the ‘flag' attached to it to describes the spinor's <italic toggle=\"yes\">phase</italic> [##UREF##20##21##] (##FIG##0##figure 1##).\n</p>", "<p>The ratio of the two complex components of the spin-vector provides a point on the (future) celestial sphere, but the flag-plane fixes the phase, whose sign reverses under rotation.</p>", "<p>Raising or lowering spinor indices corresponds to the same process as for tensors, this requiring the symmetric metric tensor <italic toggle=\"yes\">g_ab</italic> for lowering tensor indices and its inverse <italic toggle=\"yes\">g^{<italic toggle=\"yes\">ab</italic>}</italic> for raising them. In the case of spinors, we have <italic toggle=\"yes\">skew-symmetrical</italic> spinors and , for lowering spinor indices and and for raising them, where\n</p>", "<p>Since these epsilon quantities are skew, we have to be careful about the ordering of the indices, and we write\nand we find (being careful about the index ordering) that that and are ‘Kronecker delta' identity symbols. It should also be noted that the epsilon spinors satisfy the identities\nand the raised versions of these identities.</p>", "<p>It is often convenient to introduce a reference <italic toggle=\"yes\">spin-frame</italic> {<italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>}, } for the un-primed spin-space and {, } (usually written without the over-bars) for the complex conjugate spin-frame, for the primed spin-space. Accordingly, a typical component, in this spin-frame, of a spinor quantity would be, for example,\n</p>", "<p>It is normal to take a spin-frame to be normalized, so that\nfrom which it follows that the raising of a lower-index 1 or 1^′ on a spinor quantity becomes a raised 0 or 0′, respectively, and the raising of a lower 0 or 0′ on a spinor quantity becomes a raised 1 or 1′, respectively, but with a change of sign for each such index.</p>", "<p>A spin-frame is very closely related to a <italic toggle=\"yes\">null tetrad</italic>, which consists of two future-pointing null vectors <italic toggle=\"yes\">l^{<italic toggle=\"yes\">a</italic>}</italic> and <italic toggle=\"yes\">n^{<italic toggle=\"yes\">a</italic>}</italic> and a complex null vector <italic toggle=\"yes\">m</italic>^{<italic toggle=\"yes\">a</italic>}, which arise from a normalized spin frame according to\nso that the only two non-zero scalar products between these vectors are\n</p>", "<p>In vacuum spacetime, with zero cosmological constant (Λ = 0), the Riemann curvature tensor <italic toggle=\"yes\">R_abcd</italic> is equal to the Weyl conformal tensor <italic toggle=\"yes\">C_abcd</italic>, where it turns out that, in spinor form [##UREF##20##21##,##UREF##21##22##,##UREF##24##25##]:\nso that the curvature can be completely described by the totally symmetric spinor\n(see [##UREF##24##25##]). Moreover (as follows from the ‘fundamental theorem' that any complex polynomial in one variable is a product of linear factors), that we can express this as the symmetrized product of four single-indexed spinors\neach of which determines a null direction, as in ##FIG##0##figure 1## (via the null vectors, these four null directions being called the <italic toggle=\"yes\">principal null directions</italic>, or <italic toggle=\"yes\">PND</italic>, of the Weyl tensor (see [##UREF##24##25##]).</p>", "<p>It provides a useful classification of particular solutions of the Einstein equations to take note of the coincidences between PND [##UREF##20##21##,##UREF##21##22##]. For example, the plane waves considered in §1 (pp-waves) are all type {4}, i.e. all PND coincide, this case being referred to as <italic toggle=\"yes\">null</italic>, whereas the Schwarzschild and Kerr [##UREF##25##26##,##UREF##26##27##] solutions are all type {22} (sometimes called ‘type D'), i.e. with the PND coinciding in two separate pairs. It had been an early observation of Trautman [##UREF##6##7##] that the leading term (<italic toggle=\"yes\">r</italic>^–1-term) in a gravitationally radiating system would be <italic toggle=\"yes\">null</italic>, i.e. type {4}.</p>", "<p>Most revealing was the <italic toggle=\"yes\">peeling property</italic> discovered by Sachs [##UREF##11##12##]. What Sachs found was that not only would the leading <italic toggle=\"yes\">r</italic>⁻¹ have all its PND pointing radially outwards (null case), as Trautman had already found, but the next <italic toggle=\"yes\">r</italic>⁻² term would have <italic toggle=\"yes\">three</italic> PND pointing outwards, the <italic toggle=\"yes\">r</italic>⁻³ term would have <italic toggle=\"yes\">two</italic> PND pointing outwards, and the <italic toggle=\"yes\">r</italic>⁻⁴ term would have <italic toggle=\"yes\">one</italic> such PND. Thus, the picture is presented of the PND <italic toggle=\"yes\">peeling off</italic>, one-by-one, as we move inward from infinity (##FIG##1##figure 2##)! It should, however, be pointed out that this is what would be expected, according to Sachs's analysis, in the <italic toggle=\"yes\">general</italic> case. In special situations some of these terms might vanish, or perhaps just be more special, with more outward PND, this being understood, in the general picture presented by Sachs.\n</p>", "<p>I first heard of this ‘peeling' behaviour when I was in Syracuse, New York state in the early few months of 1961. I had been struck by its remarkable elegance, but it took several months, until after I had returned to England, before I recognized its true geometrical significance. Yet, a month or so earlier than first hearing of Sachs's remarkable result, l had attended Trautman's talk, given in Syracuse, where he showed that the leading <italic toggle=\"yes\">r</italic>^–1 term in a gravitationally radiating system would indeed be <italic toggle=\"yes\">null</italic>, i.e. type {4}. However, I was somewhat daunted by Trautman's rather complicated calculations, and I had wondered whether there might be a more geometrical way of understanding his result. I had toyed with the thought that perhaps one might simplify things by employing something like a conformal ‘inversion' to bring infinity to a finite location. However, I soon realized that even with the Schwarzschild spacetime, this would not work, because the inversion would exchange the origin point with a point at infinity, that being conformally <italic toggle=\"yes\">singula</italic>r in the Schwarzschild case—as we would now understand as the singular point <bold>i⁰</bold> in the conformal picture. This was no help at all; so that at that time (early 1961, in Syracuse) I had given up on the ‘conformal inversion' idea.</p>", "<p>However, some six months later, while back in the UK, I started to think again about Ray Sachs's very striking ‘peeling property'. I began to realize that if we move off in a <italic toggle=\"yes\">null</italic> direction rather than a spacelike one, in the conformal picture, the limits are much more favourable (mainly because we need to balance only a relatively mild <italic toggle=\"yes\">r</italic>^–1 behaviour, rather than the more serious <italic toggle=\"yes\">r</italic>^–2 divergence that we encounter in spacelike directions), and I soon realized that Sachs's peeling behaviour is simply a statement about how the <italic toggle=\"yes\">different components</italic> of the gravitational field, as defined appropriately from the Weyl spinor Ψ<italic toggle=\"yes\">_ABCD</italic>, behave at <italic toggle=\"yes\">null</italic> infinity! This realization was the key to the utility of the conformal approach to the study of gravitational radiation, and its relation to the radiation of other massless fields in relativity theory. We examine the details of this in the next section.</p>", "<title>At scri: </title>", "<p>Let us consider a general <italic toggle=\"yes\">conformal rescaling</italic> of the metric, and other relevant quantities by a smoothly varying scalar quantity Ω, known as the <italic toggle=\"yes\">conformal factor</italic>. When applied directly to the metric, we get\nwhich we accompany by\n</p>", "<p>The massless free-field equations for non-zero spin <italic toggle=\"yes\">s</italic> can be written as\nwhere <italic toggle=\"yes\">ϕ_AB</italic>_…<italic toggle=\"yes\">_E</italic> (with 2 <italic toggle=\"yes\">s</italic> indices, where <italic toggle=\"yes\">s</italic>&gt; 0 is an integer or half-integer), is totally symmetric\nEquation (3.3) is conformally invariant if <italic toggle=\"yes\">ϕ_AB</italic>_…<italic toggle=\"yes\">_E</italic> scales as a density of weight –1 under the conformal re-scaling (3.1), (3.2)\n</p>", "<p>This invariance follows from the way that the <italic toggle=\"yes\">covariant derivative</italic> operator changes under conformal rescaling. Defining\nwe find that if <italic toggle=\"yes\">κ_A</italic> and are conformal densities of respective weights <italic toggle=\"yes\">w</italic> and <italic toggle=\"yes\">w</italic>^′\nthen their covariant derivatives transform as\nwith related expressions for quantities with upper indices and/or with several such indices of either kind.</p>", "<p>The equation (3.3) encounters difficulties in conformally curved spacetime if <italic toggle=\"yes\">s </italic>&gt; 1, as we find that there are unwanted consistency conditions. However, in the particular case of <italic toggle=\"yes\">gravity</italic>, where <italic toggle=\"yes\">s </italic>= 2, we find that these consistency conditions are automatically satisfied in vacuum. Yet, we do find a curious feature with regard to conformal invariance, as we shall be seeing shortly.</p>", "<p>It is important, in what follows, to consider how null geodesics and their affine parameters behave under conformal rescaling. A <italic toggle=\"yes\">geodesic</italic> is a curve <italic toggle=\"yes\">λ</italic> that has a tangent vector <italic toggle=\"yes\">l</italic>^{<italic toggle=\"yes\">a</italic>} along it, for which we can take\nand an <italic toggle=\"yes\">affine</italic> parameter <italic toggle=\"yes\">r</italic> for <italic toggle=\"yes\">λ</italic>, associated with <italic toggle=\"yes\">l</italic>^{<italic toggle=\"yes\">a</italic>}, is then a parameter <italic toggle=\"yes\">r</italic> that satisfies\n</p>", "<p>Taking , as in (2.14), we shall also require the flag plane of <italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>} (as depicted in ##FIG##0##figure 1##, and see [##UREF##20##21##]), to be parallel propagated along <italic toggle=\"yes\">λ</italic>, which we express as\nalong <italic toggle=\"yes\">λ</italic>, this (and also (3.8<italic toggle=\"yes\">b</italic>)) being conformally invariant, by (3.8).</p>", "<p>We take note of the vacuum Bianchi identities, which when written in spinor form, using (2.16), take the form\n(see [##UREF##20##21##,##UREF##24##25##]), this having the same form as (3.3). However, when we take note of the fact that the Weyl tensor <italic toggle=\"yes\">C_abcd</italic> is actually the <italic toggle=\"yes\">conformal curvature,</italic> having the very specific conformal weight –2, we find that Ψ<italic toggle=\"yes\">_ABCD</italic> necessarily has weight 0, rather than the –1 required for the conformal invariance of (3.8). This leads us to consider that the spin-2 gravitational radiation field is really described by a symmetrical spinor quantity <italic toggle=\"yes\">ψ_ABCD</italic> which, though <italic toggle=\"yes\">equal</italic> to the Weyl curvature spinor quantity Ψ<italic toggle=\"yes\">_ABCD</italic> when referred to the physical metric <italic toggle=\"yes\">g_ab</italic>,\nhas conformal weight –1, so under conformal rescaling (3.1), (3.2),\n</p>", "<p>The significance, for us, of the quantity <italic toggle=\"yes\">ψ_ABCD</italic>, as opposed to Ψ<italic toggle=\"yes\">_ABCD</italic>, is that its conformal weighting, being that of a massless free field, allows it to retain a <italic toggle=\"yes\">finite</italic> value at <italic toggle=\"yes\">conformal infinity</italic>\n, a concept that we now turn to. An impression of this notion of is provided by ##FIG##2##figure 3##<italic toggle=\"yes\">.</italic> The future bounding hypersurface is three-dimensional (though depicted in ##FIG##2##figure 3## as two-dimensional, with one spacial direction suppressed), providing a conformally smooth future-null boundary to the spacetime , when it is itself regarded as a conformal manifold.\n</p>", "<p>Although this procedure has been found to be very broadly applicable for Bondi-Sachs-type spacetimes, questions are sometimes raised about the generality of this procedure. There are, for example, situations that one could consider as ‘physically realistic' (such as those involving a pair of slowly in-spiraling bodies) whose outgoing gravitational waves could affect the smoothness of the conformal geometry at . Moreover, in any case, one can certainly envisage all kinds of artificially introduced incoming radiation that could spoil the smooth geometry in the neighbourhood of . I am adopting the viewpoint here that all such possibilities are really irrelevant to the utility of the concept of , and that its assumed smoothness does not restrict our understanding of general gravitationally radiating systems involving their production of outgoing radiation. Moreover, at first sight one might worry that (3.11) could imply problems for remaining finite at , where Ω = 0. However, there is a theorem (see [##UREF##27##28##]), indicating that should vanish appropriately at in general situations describing outgoing gravitational radiation, thereby providing a finite at .</p>", "<p>##FIG##2##Figure 3## indeed gives us an impression of the kind of situation under consideration here. It depicts conformal infinity , which represents the totality of limiting points (<italic toggle=\"yes\">r</italic> → ∞) along the outgoing null geodesic generators of the <italic toggle=\"yes\">u </italic>= const. coordinate hypersurfaces. See (1.1). The little white flags represent the limiting location of <italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>}, using the description depicted in ##FIG##0##figure 1##, with flagpole pointing in the direction of such null geodesic. We can locally choose a spin-frame {, } at each point of and because of (3.8<italic toggle=\"yes\">c</italic>), we can demand that , this being consistent with the parallel propagation of <italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>} along <italic toggle=\"yes\">λ</italic> given by (3.8<italic toggle=\"yes\">c</italic>). The choice of is not determined in this way, but we demand that its flagpole direction (i.e. that of <italic toggle=\"yes\">n^{<italic toggle=\"yes\">a</italic>}</italic>) lies along the direction of the generator of the null hypersurface at that point.</p>", "<p>##FIG##2##Figure 3## illustrates this situation, where the little white flags represent the locations of the <italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>} spinors at , and the little black flags the corresponding locations of the spinors. The roughly horizontal wavy cross-sections, or <italic toggle=\"yes\">cuts</italic>, of , which are illustrated in ##FIG##2##figure 3##, may be considered as places where the coordinate <italic toggle=\"yes\">u</italic> takes a particular value along the cut, and the white flags' flagpoles point out orthogonally along such a cut.</p>", "<p>It should be made clear that the structure of itself is not dependent on any particular choice of family of such cuts. Any choice of cuts would be associated with a choice of <italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>} spinors (i.e. of <italic toggle=\"yes\">l</italic>^{<italic toggle=\"yes\">a</italic>} directions) there, since the spacelike two surfaces that are tangent to the cuts of there are defined as being orthogonal to the <italic toggle=\"yes\">l</italic>^{<italic toggle=\"yes\">a</italic>} directions there (being the 2-planes spanned by the real and imaginary parts of <italic toggle=\"yes\">m</italic>^{<italic toggle=\"yes\">a</italic>} there). In the Bondi-Sachs analysis, a particular restriction is made on the <italic toggle=\"yes\">u</italic> coordinate that makes the family of cuts ‘parallel' to one another in a certain sense, although this restriction will not be of particular importance to us in what follows.</p>", "<p>Nevertheless, it is worthwhile to point out the particular quantity <italic toggle=\"yes\">τ</italic> that is made to vanish for the cuts to be ‘parallel' in the Bondi-Sachs analysis, as <italic toggle=\"yes\">τ</italic> is one of the four <italic toggle=\"yes\">spin coefficients ρ</italic>, <italic toggle=\"yes\">σ</italic>, <italic toggle=\"yes\">κ</italic>, <italic toggle=\"yes\">τ</italic> [##UREF##20##21##,##UREF##28##29##], that are especially relevant to our considerations here, these being defined by\n</p>", "<p>The quantity <italic toggle=\"yes\">κ</italic> is a measure of the <italic toggle=\"yes\">curvature</italic> of null curves <italic toggle=\"yes\">λ</italic> lying along the flagpole directions of <italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>}, so that the condition <italic toggle=\"yes\">κ</italic> = 0 asserts the geodesic nature of these <italic toggle=\"yes\">λ</italic> curves, see (3.8<italic toggle=\"yes\">c</italic>), with the flag planes being taken parallel-propagated along <italic toggle=\"yes\">λ</italic>. The modulus |<italic toggle=\"yes\">κ</italic>| of the complex quantity <italic toggle=\"yes\">κ</italic> provides a measure of the curvature of <italic toggle=\"yes\">λ</italic> and arg<italic toggle=\"yes\">κ</italic> tells us the direction of this curvature in relation to <italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>}'s flag plane. The complex quantity <italic toggle=\"yes\">ρ</italic> has a <italic toggle=\"yes\">real</italic> part that describes the <italic toggle=\"yes\">convergence</italic> of the family of <italic toggle=\"yes\">λ</italic>-curves, the imaginary part describing a <italic toggle=\"yes\">rotation</italic> of the flag planes along the <italic toggle=\"yes\">λ</italic> directions (Im<italic toggle=\"yes\">ρ</italic> being taken to be zero here). The complex quantity <italic toggle=\"yes\">σ</italic> will be of particular importance to us later. It describes the <italic toggle=\"yes\">shear</italic> of the family of <italic toggle=\"yes\">λ</italic>-curves, the modulus of this complex quantity <italic toggle=\"yes\">σ</italic> describing the magnitude of this shearing and the argument (phase) describing the direction away from the <italic toggle=\"yes\">λ</italic>-curve of direction out from <italic toggle=\"yes\">λ</italic> of maximum shearing. The quantity <italic toggle=\"yes\">τ</italic> describes how <italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>} varies in a direction away from the null 3-plane element determined by <italic toggle=\"yes\">l</italic>^{<italic toggle=\"yes\">a</italic>}, so that for the ‘parallel' cuts of the Bondi-Sachs system, we indeed demand <italic toggle=\"yes\">τ</italic> = 0 all along .</p>", "<p>We now turn to the Bondi-Sachs definition of the total <italic toggle=\"yes\">mass</italic> of a system, as defined at the ‘retarded time' defined by a particular cut of . We must bear in mind that the notion of ‘mass' of relevance here is to be what we should regard as a ‘time-component' of some notion of a ‘4-vector' of some appropriate kind. For this, we need the notion of the <italic toggle=\"yes\">future celestial sphere</italic>\n, of , where each point of represents a <italic toggle=\"yes\">generator</italic> of the null 3-cylinder , these generators lying along the flagpole (i.e. <italic toggle=\"yes\">n</italic>^{<italic toggle=\"yes\">a</italic>}) directions, being defined by fixed values of <italic toggle=\"yes\">θ</italic> and <italic toggle=\"yes\">ϕ</italic> in the original metric given in (1.2). It is helpful to re-express these coordinates in terms of the single complex variable\n</p>", "<p>So that the non-reflective conformal motions of (the <italic toggle=\"yes\">Riemann sphere</italic>) are defined by bilinear transformations of <italic toggle=\"yes\">ζ</italic>:\n(See e.g. [##UREF##29##30##].) The group defined by (3.13), being the non-reflective Lorentz group, leaves the general form of the Bondi-Sachs metric unchanged. We can regard the sphere as being the space of generators of , each cut of providing a particular realization of . Accordingly, we may also regard the transformations (3.13) as acting on any particular cut itself. The notion of the direction of a <italic toggle=\"yes\">timelike vector</italic>, in relation to , would refer to the assignment of a particular unit sphere metric to , consistent with its conformal structure. This would apply, in particular, to an energy-momentum vector.</p>", "<p>In order to appreciate the Bondi-Sachs definition of energy-momentum, we need to consider the (limiting) <italic toggle=\"yes\">Weyl curvature</italic> of at <italic toggle=\"yes\">.</italic> For this, we need the values of the finite quantity , defined in (3.11), most particularly\n</p>", "<p>We also need the <italic toggle=\"yes\">shear σ</italic> of the outgoing rays, as defined in (3.12) and the important, but not quite locally defined <italic toggle=\"yes\">news function N</italic>, which can be expressed, equivalently, either as a time-integral of <italic toggle=\"yes\">ψ</italic>₄<italic toggle=\"yes\">,</italic> or as an angular integral of <italic toggle=\"yes\">ψ</italic>₃ around the cut, where we get the same answer either way even though the value of <italic toggle=\"yes\">N</italic> at a point <italic toggle=\"yes\">p</italic> is not completely determined in terms of quantities defined locally at <italic toggle=\"yes\">p</italic> (see [##UREF##30##31##], p. 427]).</p>", "<p>The Bondi-Sachs definition of the <italic toggle=\"yes\">mass M</italic> at the cut is then given by an integral over the cut of the quantity\n(with the appropriate numerical factors), and the Bondi-Sachs mass-loss formula asserts that if the integral of (3.14) is performed over a <italic toggle=\"yes\">later</italic> cut C^′ (i.e. ‘higher up', in ##FIG##2##figure 3##), then this later value is reduced from the earlier value by ()^–1 times the integral of |<italic toggle=\"yes\">N</italic>|² taken over the region of between these two cuts, with an appropriate additional contribution from the energy-momentum tensor of any ordinary matter fields that might also be present (see [##UREF##21##22##, vol. 2, p. 426]).</p>", "<title>Ezra T. Newman's approach to equations of motion</title>", "<p>The Bondi-Sachs mass-loss result was a clear landmark in understanding the contribution from the gravitational field in GR to the mass-energy of a gravitating system. Much of the early work in this area (though <italic toggle=\"yes\">not</italic> the key formula (3.15)) depended upon coordinate systems that respect the ‘parallel' (<italic toggle=\"yes\">τ </italic> = 0) asymptotic nature of the <italic toggle=\"yes\">u </italic>= const. coordinate hypersurfaces. The general study of such systems led to what is referred to as the BMS (Bondi-Metzner-Sachs) group [##UREF##10##11##,##UREF##31##32##]; see also [##UREF##32##33##]. However, it is not at all clear what physical role the <italic toggle=\"yes\">τ </italic>= 0 condition actually plays, since it is not necessary for the mass-loss result, and a physical role for the BMS group does not seem to be really compelling. Indeed, there is another approach to restricting the large coordinate freedom, for such spacetimes, which has led to some very different and highly intriguing implications. This is the <italic toggle=\"yes\">-space</italic> approach due to Ezra T. Newman [##UREF##33##34##,##UREF##34##35##].</p>", "<p>The idea of -space is to drop the ‘parallel' (<italic toggle=\"yes\">τ </italic>= 0) character of the <italic toggle=\"yes\">u </italic>= const. coordinate hypersurfaces (i.e. of the family of cuts of ) and replace it with a requirement that the asymptotic <italic toggle=\"yes\">shear σ</italic> of these hypersurfaces must vanish. At first sight, this would appear to be far too strong a requirement, since <italic toggle=\"yes\">σ</italic> is a <italic toggle=\"yes\">complex</italic> quantity, whereas the freedom per point of a cut is just <italic toggle=\"yes\">one real</italic> parameter, this freedom being simply that of sliding up or down the intersection of the cut with each individual generator of . This freedom falls far short of what would be needed for what Newman would refer to as a ‘good cut’, for which <italic toggle=\"yes\">σ</italic> would have to vanish all over the cut. Nevertheless, Newman was not to be daunted by such a blatant fact, allowing his cuts to be <italic toggle=\"yes\">complex,</italic> so that each of its points would be allowed to meet the generators of in <italic toggle=\"yes\">complex</italic> points, not necessarily real ones!</p>", "<p>This raises a number of technical issues. We are now concerned with the <italic toggle=\"yes\">complexification</italic>\n, of , rather than with itself. This, in turn, requires that be an <italic toggle=\"yes\">analytic</italic> manifold, with an analytic metric. This, in itself, is not a troublesome restriction, so long as we are not considering situations involving impulsive waves, or the like, since we can always approximate a reasonably smooth metric by an analytic one. Problems do not arise with Newman's construction provided that we need not venture too far from the real spacetime under consideration, so we can take to be analytic, with an analytic metric so that a complexification exists, which provides an open neighbourhood of , which need not extend significantly far away from itself.</p>", "<p>A more serious issue is that the ‘complexification' of a quantity that is already complex, such as <italic toggle=\"yes\">σ</italic>, is not so straightforward. In effect, we have to consider two <italic toggle=\"yes\">distinct</italic> complex quantities <italic toggle=\"yes\">σ</italic> and , where for the real spacetime , (the complex conjugate of <italic toggle=\"yes\">σ</italic>), whereas for , the complex quantities <italic toggle=\"yes\">σ</italic> and become <italic toggle=\"yes\">independent</italic> of one another. Thus, in general, we cannot expect that our notion of ‘good cut' involves making <italic toggle=\"yes\">both σ</italic> and vanish at once. Instead, we need to make a choice as to which <italic toggle=\"yes\">one</italic> of <italic toggle=\"yes\">σ</italic> or is to vanish (where the other is not required to vanish) in order to provide us with the <italic toggle=\"yes\">good cuts,</italic> the family of which is to provide us with the definition of -space, the other choice giving us the complex conjugate <italic toggle=\"yes\">-</italic>space.</p>", "<p>It turns out that <italic toggle=\"yes\">-</italic>space is indeed a 4-complex-dimensional manifold [##UREF##33##34##] and which, somewhat remarkably, turns out to have a complex metric that automatically satisfies Einstein's vacuum equations (vanishing complex Ricci tensor) [##UREF##34##35##,##UREF##35##36##]. Moreover, this construction had an important influence on twistor theory [##UREF##34##35##], leading to what is referred to as the ‘nonlinear graviton construction' [##UREF##35##36##]. This, in effect, provides us with the general solution of the ‘anti-self-dual' Einstein vacuum equations (e.g. [##UREF##36##37##]; see also [##UREF##37##38##,##UREF##38##39##], and [##UREF##39##40##]), although these are necessarily <italic toggle=\"yes\">complex</italic> solutions, or else real ones but with a different signature from the Lorentzian one needed for a physical spacetime. For gauge fields, see also [##UREF##40##41##], and [##UREF##41##42##].</p>", "<p>An application of a very different kind is Newman's discovery that numerous aspects of equations of motion for small bodies in GR (touched upon in §1) can be much more readily obtained using <italic toggle=\"yes\">-</italic>space techniques than by the standard procedures [##REF##29142499##43##,##UREF##42##44##]. This development is technically impressive, yet it perhaps lacks a convincing underlying justification for deriving equations of motion in this way. It would appear to be an important development for the theory of equations of motion to find such a justification.</p>", "<p>A central ingredient of Newman's <italic toggle=\"yes\">-</italic>space approach to equations of motion dates back to his observation that the Kerr metric for a rotating black hole can be thought of as being, in a certain sense, a displacement of the Schwarzschild metric in a <italic toggle=\"yes\">complex</italic> spatial direction. When applied to the electrically charged version of the Schwarzschild metric, namely the Reissner-Nordström metric, [##UREF##43##45##] and [##UREF##44##46##] one obtains the charged version of the Kerr metric, sometimes referred to as the Kerr-Newman metric [##UREF##26##27##]. In this sense, we can regard such a complex spatial displacement as corresponding to the acquirement of an intrinsic <italic toggle=\"yes\">angular momentum</italic>, or ‘spin'. This same general idea, when applied to the world-line of a charged particle, provides it with a magnetic moment. These complex displacements lead us to an <italic toggle=\"yes\">-</italic>space description.</p>", "<p>There is some good reason to expect that this could be related to another body of results, going back to 1965, but which have remained distinctly enigmatic ever since. These results refer to a family of quantities referred to as ‘NP constants' (Newman-Penrose constants) [##UREF##20##21##,##UREF##21##22##,##UREF##29##30##,##UREF##45##47##–##UREF##47##49##] but rarely accepted as genuine physical phenomena, since they give an impression of being at odds with accepted physical theory. In fact, this is not so, but these quantities do shed a distinctly unusual light on conventional physical expectations, seeming to suggest that certain physical quantities would have to be conserved which are certainly <italic toggle=\"yes\">not</italic> conserved, thereby evoking negative reactions from some distinguished theoreticians, ranging from serious discouragement [##UREF##48##50##] to actual disbelief [##UREF##49##51##,##UREF##50##52##].</p>", "<p>To be more explicit, for a s<italic toggle=\"yes\">tationary</italic> asymptotically flat spacetime, the gravitational NP constants can be identified as the combination of gravitational quantities given by:\nwhere the <italic toggle=\"yes\">complex</italic> dipole moment has an imaginary part that is the angular moment and there is a corresponding imaginary contribution to the quadruple moment (see [##UREF##21##22##], p. 428] and compare with~[##UREF##51##53##]). In view of such an explicit expression as (4.1), the above skeptical reactions are perhaps not surprising. For example, the constancy of the NP quantities for pure gravity implies that a stationary system, say with a non-zero quadrupole moment, but with no dipole moment or angular momentum cannot, by the emission of gravitational waves, evolve, after a finite period of time, to the same situation except with a changed quadrupole moment! The seeming contradiction with the descriptions given in §1, that gravitational waves can carry away quadrupole moment while leaving the dipole and angular momentum unchanged, is nevertheless resolved by the fact that the presence of back-scattered gravitational radiation would provide an additional contribution that would (surprisingly) prevent the state ever becoming sufficiently stationary for any contradiction with NP constancy to arise! (See [##UREF##47##49##] for a more detailed discussion of this situation.) Nevertheless, this conclusion is very non-intuitive, and I am proposing, later in this article, that a completely different light on the underlying significance of the conservation of NP quantities can be provided which could well relate to Newman's remarkable procedure for deriving equations of motion.</p>", "<p>For this, we need to re-examine what was my own initial route to anticipating an NP constancy—which seems to have been somewhat complementary to the route taken by Newman, his route perhaps having had a more direct connection with the multipole issue, which I had not anticipated. I had been concerned with a quite different question, namely the initial value problem, in Minkowski space , for the massless free-field equation (3.3)\nwhere the field <italic toggle=\"yes\">ϕ_AB</italic>_…<italic toggle=\"yes\">_E</italic> is given on an initial <italic toggle=\"yes\">null</italic> hypersurface . We wish to determine <italic toggle=\"yes\">ϕ_AB</italic>_…<italic toggle=\"yes\">_E</italic> at some arbitrary point <italic toggle=\"yes\">P</italic>, lying in the region to the future of , by performing an integral of an appropriate quantity defined by <italic toggle=\"yes\">ϕ_AB</italic>_…<italic toggle=\"yes\">_E</italic> on . It turns out that this can be achieved by performing an integral over the two-dimensional intersection\nof with the (past) light cone of <italic toggle=\"yes\">P</italic>. At each point of , we choose a spin-frame {<italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>},} where the flagpole of <italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>} points along the null generator of and that of , along the corresponding generator of , at the same point of , represented as white and black flags, respectively, at a typical point <italic toggle=\"yes\">Q</italic> of (##FIG##3##figure 4##) for the geometry involved (with white and black flags respectively, as in ##FIG##2##figure 3##). The particular scalar quantity to be integrated over , together with its surface area 2-form d, is defined in terms of the operator <italic toggle=\"yes\">ϱ</italic>₀, as given by\nwhere <italic toggle=\"yes\">n</italic> is the number of indices of <italic toggle=\"yes\">ϕ_AB</italic>_…<italic toggle=\"yes\">_E</italic> (i.e. twice the spin). Our integral to provide us with the field at <italic toggle=\"yes\">P</italic> is then simply\n\n</p>", "<p>The value of <italic toggle=\"yes\">r</italic> is the affine distance of <italic toggle=\"yes\">Q</italic> to <italic toggle=\"yes\">P</italic> where <italic toggle=\"yes\">Q</italic> is the point at which <italic toggle=\"yes\">ϕ_AB</italic>_…<italic toggle=\"yes\">_E</italic> is being examined in (4.4 and (4.5), and <italic toggle=\"yes\">r</italic> is the affine distance <italic toggle=\"yes\">QP</italic> scaled in terms of <italic toggle=\"yes\">n</italic>^{<italic toggle=\"yes\">a</italic>} = , so that we can write\n(##FIG##3##figure 4##). It might be thought that there is a topological issue of defining the directions of the flag-planes of <italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>} and consistently over the whole of , but this is not really a problem, because the integrand in (4.5) is insensitive to the phase involved in the choice of spin-frame.</p>", "<p>I refer to the expression (4.5) as the <italic toggle=\"yes\">generalized</italic>\n<italic toggle=\"yes\">Kirchhoff–d’ Adhémar formula</italic> [##UREF##52##54##,##UREF##53##55##] or the GKd formula, whose early work I was able to generalize to arbitrary spin in around 1967 [##UREF##54##56##–##UREF##56##58##].</p>", "<p>In fact, the expression (4.5) works just as well in conformally flat spacetime (see [##UREF##20##21##] p. 395), using the appropriate conformally invariant expression for the operator <italic toggle=\"yes\">ϱ</italic> of (4.4), which acts in the direction of <italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>}'s flagpole direction at the point <italic toggle=\"yes\">Q</italic> in ##FIG##2##figure 3##. The integral is to be performed at the point <italic toggle=\"yes\">P</italic>, the spin frame {<italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>}, } being taken to be parallel-propagated along each generator of from <italic toggle=\"yes\">Q</italic> to <italic toggle=\"yes\">P</italic>. This conformal invariance allows us to regard the null cone to be the future conformal infinity of an asymptotically flat spacetime \nand then becomes a particular <italic toggle=\"yes\">cut</italic> of .</p>", "<p>Now, one way of proving the GKd formula is to examine how the integrand of (4.5) might vary as the cut of t might vary as it is moved up the generators of , it being found that the integral in (4.5) actually remains constant as the cut is moved up towards <italic toggle=\"yes\">P</italic>, so that all that remains is to show that the limiting value is indeed the required value of <italic toggle=\"yes\">ϕ_AB</italic>_…<italic toggle=\"yes\">_E</italic> at <italic toggle=\"yes\">P</italic>. Such a procedure fails, in general, if the ambient spacetime manifold is conformally curved, because of troublesome curvature terms that do not vanish in general. However, what is remarkable is that if we choose to be as suggested above, then all these troublesome terms actually <italic toggle=\"yes\">vanish</italic>, so that our integral does indeed provide a quantity that is absolutely conserved without the need of a contribution such as the of |<italic toggle=\"yes\">N</italic>|² term in the Bondi-Sachs theory (referred to at the end of §3) for the loss of total mass by gravitational radiation. Instead, we now find these curiously <italic toggle=\"yes\">absolutely</italic> conserved quantities, referred to as ‘NP constants', which seem to have no <italic toggle=\"yes\">physical</italic> basis for their conservation, according to conventional theory!</p>", "<p>For a spin <italic toggle=\"yes\">s</italic> massless field, the number of independent such complex quantities would be the number of complex components of <italic toggle=\"yes\">ϕ_AB</italic>_…<italic toggle=\"yes\">_E</italic>, with 2 <italic toggle=\"yes\">s</italic> symmetrical 2-spinor indices, namely\ncomplex numbers, which provide us with 10 real numbers for gravity and 6 for electromagnetism, the two known massless fields in physics. It is striking that, moreover, in the Einstein–Maxwell theory, we retain the six quantities for the source-free electromagnetic field, but the situation is changed for the gravitational field, since the Maxwell field itself provides a source for the gravitational field. Nevertheless, we can maintain the exact conservation of 10 gravitational quantities by including a contribution involving the electromagnetic field [##UREF##57##59##].</p>", "<p>In view of all these remarkable mathematical facts, it is very hard to believe that these exactly constant quantities cannot have some kind of basic physical role to play in our actual physical world. I wish to put forward a different perspective on the quantities from previously, which might, indeed, provide such ‘NP constants' with a quite different kind of physical rationale for their constancy. To explain what I have in mind, let me first return to the interpretion of these constants given above, addressing a point not raised before in this article. First of all, where do we <italic toggle=\"yes\">locate</italic> the actual ‘spinor' <italic toggle=\"yes\">ϕ_ABCD</italic> that encapsulate the 10 NP constants in the gravitational case? If we think of the conformal picture of Minkowski space , our point <italic toggle=\"yes\">P</italic> where this <italic toggle=\"yes\">ϕ_ABCD</italic> would be located would be the ideal point i⁺, representing future timelike infinity for (see [##UREF##47##49##]). For an asymptotically flat , we might still get such a point i⁺, if all the material in that model evaporates away in radiation, but in most cases, we might consider that some material remains, so that our future timelike infinity point ‘i⁺′ m’ would be likely to be <italic toggle=\"yes\">singular</italic> (and perhaps even a singular spacelike surface). Nevertheless, our NP ‘GKd' integral would provide us with the value <italic toggle=\"yes\">ψ_ABCD</italic>, basically the <italic toggle=\"yes\">conformal curvature tensor</italic>, at this nonexistent ‘virtual point' i⁺. We appear to have to think of <italic toggle=\"yes\">ψ_ABCD</italic> as providing a kind of ‘background’ that cannot be altered by whatever the system might do in its later activity.</p>", "<p>It seems to me that this all makes much more sense if we think of the NP constants in a quite different context, namely that of <italic toggle=\"yes\">equations of motion</italic>. In that subject, we are thinking of taking a quite different kind of limit where, rather than looking at a large-scale limit of greater and greater distances away, we are more concerned with small-scale limits of tinier and tinier distances. As already mentioned in §1, equations of motion have to deal with taking an inward limit and trying to ascertain what the ‘background field' that a tiny particle might be responding to. Again, it is a question of seeking a ‘background field', but on a small, rather than a large scale. The idea is to think of the NP constants as not really applying to ‘infinity' but to what would ‘look like' infinity from the very small scale that we are concerned with. Thus the ‘’ that we are now concerned with could be thought of a fairly small distance away from the particle whose motion we are examining, but which looks more and more like the conformal infinity the closer to the particle we get. See ##FIG##4##figure 5## for a picture of what I have in mind. The scale of the picture is supposed to be small enough that we can ignore spacetime curvature's nonlinear effects, so that we can regard the GKd integral to be valid, at least in regions that are not too close to the particle that we are concerned with.\n</p>", "<p>In ##FIG##4##figure 5##, the vertical tube in the middle represents the history of such a small body, whose motion we wish to calculate, but we are concerned to eliminate the field of the body itself. We consider the point <italic toggle=\"yes\">P</italic> at the centre of the picture, lying within that body. There are four (irregular-looking) boundary loops in the picture (actually of spherical topology), and we might perform a GKd integral over any one of the four. Each would provide a version of the background field of the body (since the contribution from the body's field would not contribute, by arguments effectively provided above), but with now the point <italic toggle=\"yes\">P</italic> replacing the i^­+ of our earlier discussion. Accordingly, the upper two integrals would give the same ‘retarded' answer and the lower two, the same ‘advanced' answer. Very possibly <italic toggle=\"yes\">all</italic> are the same because all are finite, so no ‘renormalization’ is required. All this requires further study.</p>", "<p>Finally, there is a probable connection with Newman's work on his <italic toggle=\"yes\">-</italic>space approach. It seems to me that this is very likely, since both are deeply connected with twistor theory. The <italic toggle=\"yes\">-</italic>space connection has already been mentioned and the GKd integral is almost equivalent to the basic twistor contour integral for massless free fields [##UREF##55##57##,##UREF##56##58##].</p>", "<p>Thanks go to Jörg Frauendiener for help with references and to John Moussouris for financial assistance.</p>", "<p>This paper is dedicated to the memory of Ezra (Ted) Newman.</p>" ]

[ "<title>Data accessibility</title>", "<p>This article has no additional data.</p>", "<title>Declaration of AI use</title>", "<p>I have not used AI-assisted technologies in creating this article.</p>", "<title>Authors' contributions</title>", "<p>R.P.: conceptualization, writing—original draft, writing—review and editing.</p>", "<title>Conflict of interest declaration</title>", "<p>I declare I have no competing interests.</p>", "<title>Funding</title>", "<p>I received no funding for this study.</p>" ]

[ "<fig position=\"float\" id=\"RSTA20230041F1\"><label>Figure 1<x xml:space=\"preserve\">. </x></label><caption><p>A spin-vector, pictured as a future-null vector ‘flagpole', determining a point on the future-celestial (Riemann) sphere with a null half-plane ‘flag' attached, to describe (up to sign) the spinor's phase.</p></caption></fig>", "<fig position=\"float\" id=\"RSTA20230041F2\"><label>Figure 2<x xml:space=\"preserve\">. </x></label><caption><p>The Sachs peeling property.</p></caption></fig>", "<fig position=\"float\" id=\"RSTA20230041F3\"><label>Figure 3<x xml:space=\"preserve\">. </x></label><caption><p>A picture of . White flags represent <italic toggle=\"yes\">o</italic>^{<italic toggle=\"yes\">A</italic>} and black flags, . Two cuts and are indicated.</p></caption></fig>", "<fig position=\"float\" id=\"RSTA20230041F4\"><label>Figure 4<x xml:space=\"preserve\">. </x></label><caption><p>Geometry for the GKd solution of the initial value problem for a massless free field.</p></caption></fig>", "<fig position=\"float\" id=\"RSTA20230041F5\"><label>Figure 5<x xml:space=\"preserve\">. </x></label><caption><p>The G<italic toggle=\"yes\">Kd</italic> integrals surrounding the central body eliminate the field of the body to give the background field at <italic toggle=\"yes\">p</italic>.</p></caption></fig>" ]

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display=\"block\"><mml:mrow><mml:msup><mml:mi>X</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>ε</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mi>B</mml:mi></mml:msub></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mi>B</mml:mi></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>X</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:msup><mml:mi>Y</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mrow><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mrow><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>Y</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM7\"><mml:msup><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow><mml:mi>B</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow><mml:mi>B</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM8\"><mml:msup><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M2.11\"><label>2.11</label><mml:math id=\"DM13\" display=\"block\"><mml:mrow><mml:mtable columnalign=\"center center\" rowspacing=\"4pt\" columnspacing=\"1em\"><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>C</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>D</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>B</mml:mi><mml:mi>C</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mo>}</mml:mo></mml:mrow></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM9\"><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM10\"><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM11\"><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM12\"><mml:mrow><mml:msub><mml:mi>φ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>D</mml:mi><mml:mi>C</mml:mi><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>E</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M2.12\"><label>2.12</label><mml:math id=\"DM14\" display=\"block\"><mml:mrow><mml:msub><mml:mi>φ</mml:mi><mml:mrow><mml:msup><mml:mn>1011</mml:mn><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mn>0</mml:mn><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi>φ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>E</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mi>B</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mi>C</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mrow><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mrow><mml:msup><mml:mi>E</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M2.13\"><label>2.13</label><mml:math id=\"DM15\" display=\"block\"><mml:mrow><mml:msub><mml:mi>o</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:msub><mml:mi>o</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M2.14\"><label>2.14</label><mml:math id=\"DM16\" display=\"block\"><mml:mrow><mml:msup><mml:mi>l</mml:mi><mml:mi>a</mml:mi></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:msup><mml:mi>n</mml:mi><mml:mi>a</mml:mi></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:msup><mml:mi>m</mml:mi><mml:mi>a</mml:mi></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mi>m</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>a</mml:mi></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M2.15\"><label>2.15</label><mml:math id=\"DM17\" display=\"block\"><mml:mrow><mml:msub><mml:mi>l</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mi>n</mml:mi><mml:mi>a</mml:mi></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mtext>and</mml:mtext></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mi>m</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>a</mml:mi></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>1.</mml:mn></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M2.16a\"><label>2.16<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM18\" display=\"block\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M2.16b\"><label>2.16<italic toggle=\"yes\">b</italic></label><mml:math id=\"DM19\" display=\"block\"><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M2.17\"><label>2.17</label><mml:math id=\"DM20\" display=\"block\"><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mi>B</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>γ</mml:mi><mml:mi>C</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM13\"><mml:mrow><mml:msup><mml:mi>α</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:msup><mml:mi>β</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mi>β</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mtext>etc</mml:mtext></mml:mrow><mml:mo>.</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM14\"><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M3.1\"><label>3.1</label><mml:math id=\"DM21\" display=\"block\"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mtext>and</mml:mtext></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:msup><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>g</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M3.2\"><label>3.2</label><mml:math id=\"DM22\" display=\"block\"><mml:mrow><mml:mtable columnalign=\"left left left\" rowspacing=\"4pt\" columnspacing=\"1em\"><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo></mml:mtd><mml:mtd><mml:mrow><mml:msup><mml:mi>ε</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>ε</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mtext>and</mml:mtext></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo></mml:mtd><mml:mtd><mml:mrow><mml:msup><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mover><mml:mi>ε</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mo> </mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mi>ε</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mo>}</mml:mo></mml:mrow></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M3.3\"><label>3.3</label><mml:math id=\"DM23\" display=\"block\"><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi>ϕ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mo>…</mml:mo><mml:mi>E</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M3.4\"><label>3.4</label><mml:math id=\"DM24\" display=\"block\"><mml:mrow><mml:msub><mml:mi>ϕ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mo>…</mml:mo><mml:mi>E</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mo>…</mml:mo><mml:mi>E</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M3.5\"><label>3.5</label><mml:math id=\"DM25\" display=\"block\"><mml:mrow><mml:msub><mml:mi>ϕ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mo>…</mml:mo><mml:mi>E</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mo>…</mml:mo><mml:mi>E</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mo> </mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>ϕ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mo>…</mml:mo><mml:mi>E</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM15\"><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M3.6\"><label>3.6</label><mml:math id=\"DM26\" display=\"block\"><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Υ</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mtext>log</mml:mtext></mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM16\"><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M3.7\"><label>3.7</label><mml:math id=\"DM27\" display=\"block\"><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>κ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi>A</mml:mi></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>w</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>μ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:msup><mml:mi>w</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M3.8\"><label>3.8</label><mml:math id=\"DM28\" display=\"block\"><mml:mrow><mml:mtable columnalign=\"left left\" rowspacing=\"4pt\" columnspacing=\"1em\"><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi>B</mml:mi></mml:msub></mml:mrow><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mover><mml:mi>κ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mi>B</mml:mi></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>w</mml:mi></mml:msup></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>w</mml:mi><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Υ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi>B</mml:mi></mml:msub></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi>B</mml:mi></mml:msub></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Υ</mml:mi><mml:mrow><mml:mi>B</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>κ</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mtext>and</mml:mtext></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo> </mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mover><mml:mi>μ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:msup><mml:mi>w</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>w</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Υ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Υ</mml:mi><mml:mrow><mml:msup><mml:mi>B</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup><mml:mi>A</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mo>}</mml:mo></mml:mrow></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M3.8a\"><label>3.8<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM29\" display=\"block\"><mml:mrow><mml:msup><mml:mi>l</mml:mi><mml:mi>a</mml:mi></mml:msup><mml:msub><mml:mo>∇</mml:mo><mml:mi>a</mml:mi></mml:msub><mml:msup><mml:mi>l</mml:mi><mml:mi>b</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M3.8b\"><label>3.8<italic toggle=\"yes\">b</italic></label><mml:math id=\"DM30\" display=\"block\"><mml:mrow><mml:msup><mml:mi>l</mml:mi><mml:mi>a</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mn>1.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM17\"><mml:mrow><mml:msup><mml:mi>l</mml:mi><mml:mi>a</mml:mi></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M3.8c\"><label>3.8<italic toggle=\"yes\">c</italic></label><mml:math id=\"DM31\" display=\"block\"><mml:mrow><mml:msup><mml:mi>l</mml:mi><mml:mi>a</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mi>B</mml:mi></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M3.9\"><label>3.9</label><mml:math id=\"DM32\" display=\"block\"><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M3.10\"><label>3.10</label><mml:math id=\"DM33\" display=\"block\"><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M3.11\"><label>3.11</label><mml:math id=\"DM34\" display=\"block\"><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM18\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM19\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM20\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM21\"><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM22\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM23\"><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM24\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM25\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM26\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM27\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM28\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM29\"><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM30\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM31\"><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM32\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM33\"><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM34\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM35\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM36\"><mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mi>o</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi>A</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM37\"><mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mi>ι</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi>A</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM38\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM39\"><mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mi>o</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM40\"><mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mi>ι</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi>A</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM41\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM42\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM43\"><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM44\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM45\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM46\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M3.12\"><label>3.12</label><mml:math id=\"DM35\" display=\"block\"><mml:mi>κ</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:msup><mml:mn>00</mml:mn><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>o</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>ρ</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>o</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:msup><mml:mn>01</mml:mn><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>o</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mi>o</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:msup><mml:mn>11</mml:mn><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>o</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM47\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM48\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM49\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM50\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM51\"><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM52\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM53\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM54\"><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M3.12a\"><label>3.12<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM36\" display=\"block\"><mml:mi>ζ</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mtext>e</mml:mtext></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mi>cot</mml:mi><mml:mo>⁡</mml:mo><mml:mfrac><mml:mi>θ</mml:mi><mml:mn>2</mml:mn></mml:mfrac><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM55\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M3.13\"><label>3.13</label><mml:math id=\"DM37\" display=\"block\"><mml:mi>ζ</mml:mi><mml:mo stretchy=\"false\">↦</mml:mo><mml:mfrac><mml:mrow><mml:mi>α</mml:mi><mml:mi>ζ</mml:mi><mml:mo>+</mml:mo><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi><mml:mi>ζ</mml:mi><mml:mo>+</mml:mo><mml:mi>δ</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mtext>where</mml:mtext></mml:mrow><mml:mo> </mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mspace width=\"1em\"/><mml:mrow><mml:mtext>and</mml:mtext></mml:mrow><mml:mspace width=\"1em\"/><mml:mi>δ</mml:mi><mml:mo> </mml:mo><mml:mrow><mml:mtext>are complex constants</mml:mtext></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM56\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM57\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM58\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM59\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM60\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM61\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM62\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM63\"><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM64\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM65\"><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M3.14\"><label>3.14</label><mml:math id=\"DM38\" display=\"block\"><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>1111</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>1110</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mtext> and </mml:mtext></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>1100</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM66\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M3.15\"><label>3.15</label><mml:math id=\"DM39\" display=\"block\"><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo>−</mml:mo><mml:mi>σ</mml:mi><mml:mi>N</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM67\"><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:mrow><mml:mtext>G</mml:mtext></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM68\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM69\"><mml:mrow><mml:mi mathvariant=\"script\">H</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM70\"><mml:mrow><mml:mi mathvariant=\"script\">H</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM71\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM72\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM73\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM74\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"double-struck\">C</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM75\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM76\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM77\"><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM78\"><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM79\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"double-struck\">C</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM80\"><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM81\"><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM82\"><mml:mrow><mml:mover><mml:mi>σ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM83\"><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM84\"><mml:mrow><mml:mover><mml:mi>σ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mi>σ</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM85\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"double-struck\">C</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM86\"><mml:mrow><mml:mover><mml:mi>σ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM87\"><mml:mrow><mml:mover><mml:mi>σ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM88\"><mml:mrow><mml:mover><mml:mi>σ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM89\"><mml:mrow><mml:mi mathvariant=\"script\">H</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM90\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">H</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM91\"><mml:mrow><mml:mi mathvariant=\"script\">H</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM92\"><mml:mrow><mml:mi mathvariant=\"script\">H</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM93\"><mml:mrow><mml:mi mathvariant=\"script\">H</mml:mi></mml:mrow></mml:math></inline-formula>", 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display=\"block\"><mml:mrow><mml:msub><mml:mi>ϱ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>ϕ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mo>:=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:msup><mml:mn>00</mml:mn><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:msub><mml:mi>ϕ</mml:mi><mml:mrow><mml:mn>00</mml:mn><mml:mo>…</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230041M4.5\"><label>4.5</label><mml:math id=\"DM44\" 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mathvariant=\"normal\">′</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M4.6\"><label>4.6</label><mml:math id=\"DM45\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">Q</mml:mi><mml:mi mathvariant=\"bold-italic\">P</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mi>r</mml:mi><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">n</mml:mi></mml:mrow></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM111\"><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM112\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM113\"><mml:mrow><mml:msup><mml:mi>ι</mml:mi><mml:mi>A</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM114\"><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM115\"><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM116\"><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M4.7\"><label>4.7</label><mml:math id=\"DM46\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow><mml:mo>−</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM117\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM118\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM119\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM120\"><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM121\"><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM122\"><mml:mrow><mml:mi mathvariant=\"script\">J</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM123\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230041M4.8\"><label>4.8</label><mml:math id=\"DM47\" display=\"block\"><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM124\"><mml:mrow><mml:mi mathvariant=\"double-struck\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM125\"><mml:mrow><mml:mi mathvariant=\"double-struck\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM126\"><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM127\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM128\"><mml:mrow><mml:mi mathvariant=\"script\">H</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM129\"><mml:mrow><mml:mi mathvariant=\"script\">H</mml:mi></mml:mrow></mml:math></inline-formula>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>One of the main open problems in Mathematical General Relativity is that of the nonlinear stability of spacetimes. In 1986, Friedrich provided the first result concerning the global stability of the de Sitter spacetime and a semi-global stability result for the Minkowski spacetime [##UREF##0##1##,##UREF##1##2##]. These results are obtained by using the conformal Einstein field equations to reformulate Cauchy problems which are global or semi-global in time into problems which are local in time. This strategy allows us to use results obtained for quasi-linear symmetric hyperbolic systems [##UREF##2##3##,##UREF##3##4##] to prove the existence of solutions which are suitably close to known reference spacetimes. More results [##UREF##4##5##–##UREF##11##12##] using the <italic toggle=\"yes\">conformal Einstein field equations</italic> show that these equations are a powerful tool for the analysis of the stability of spacetimes. They provide a system of field equations for geometric objects defined on a four-dimensional Lorentzian manifold , the so-called <italic toggle=\"yes\">unphysical spacetime</italic>, which is conformally related to a spacetime , the so-called <italic toggle=\"yes\">physical spacetime</italic>, satisfying the Einstein field equations. The conformal Einstein field equations constitute a system of differential conditions on the curvature tensors with respect to the Levi–Civita connection of and the conformal factor .</p>", "<p>A problem one encounters when discussing the conformal structure of spacetimes by means of these equations is that of the gauge freedom. In the original formulation of the conformal Einstein field equations [##UREF##12##13##] the gauge is fixed by means of <italic toggle=\"yes\">gauge source functions</italic>. An alternative approach of gauge fixing is by exploiting the properties of a congruence of curves which are invariants of the conformal structure. These curves are known as <italic toggle=\"yes\">conformal geodesics</italic> and they have been originally introduced as a tool for the local analysis of the structure of conformally rescaled spacetimes [##UREF##13##14##]. Using this gauge allows us to define a <italic toggle=\"yes\">conformal Gaussian gauge system</italic> in which coordinates are propagated along conformal geodesics. To combine this gauge choice with the conformal Einstein field equations, it is necessary to make use of a more general version of the latter, the <italic toggle=\"yes\">extended conformal Einstein field equations</italic>. These equations contain a larger gauge freedom due to the use of a <italic toggle=\"yes\">Weyl connection</italic>. This is a torsion-free connection which provides a transport equation along the conformal geodesics preserving conformally orthonormal frames and the causal nature of their vectors.</p>", "<p>One of the advantages of the <italic toggle=\"yes\">conformal Gaussian gauge system</italic> is that it gives an <italic toggle=\"yes\">a priori</italic> knowledge of the structure of the conformal boundary of the spacetime. This aspect is used to obtain an alternative proof of the semi-global nonlinear stability of the Minkowski spacetime and of the global nonlinear stability of the de Sitter spacetime by Lübbe and Valiente Kroon [##UREF##7##8##]. In [##UREF##14##15##] the results obtained in [##UREF##0##1##,##UREF##7##8##] are generalized to de Sitter-like spacetimes with compact spatial sections of negative scalar curvature. The existence and stability result follows from explicit calculations and the requirement that the data are close to de Sitter-like data. The success of this approach in the analysis of the global properties of asymptotically simple spacetimes leads to the question of whether a similar strategy can be used to study the evolution of black hole spacetimes. A first step in this direction is made in [##UREF##15##16##] where certain aspects of the conformal structure of the sub-extremal Schwarzschild–de Sitter spacetime are analysed in order to adapt techniques from the asymptotically simple setting to the black hole case. More precisely, since this solution can be studied by means of the extended conformal Einstein field equations—see [##UREF##16##17##]. These equations are used to obtain a result concerning the evolution of the region of this spacetime which is bounded by the Cosmological horizon known as the <italic toggle=\"yes\">Cosmological region</italic>. In particular, in analogy to the de Sitter-like case, the Cosmological region has an asymptotic region admitting a smooth conformal extension with a space-like conformal boundary and there exists a conformal representation in which the induced 3-metric on the conformal boundary is homogeneous. Thus, it is possible to integrate the extended conformal field equations along single conformal geodesics—see [##UREF##17##18##,##UREF##18##19##].</p>", "<p>In this review article, the discussion of the construction of a conformal Gaussian gauge system leading to a hyperbolic reduction of the conformal Einstein field equations in the de Sitter-like case [##UREF##14##15##] and the sub-extremal Schwarzschild–de Sitter case [##UREF##15##16##] is revisited and presented in a coherent and contiguous way.</p>", "<title>Notations and conventions</title>", "<p>The signature convention for Lorentzian spacetime metrics will be . In this article, the abstract index notation is used. Accordingly, the lowercase Latin indices will denote spacetime abstract tensor indices and will be used as spacetime frame indices taking the values . In this way, given a basis a generic tensor is denoted by while its components in the given basis are denoted by . The Greek indices denote spacetime coordinate indices while the indices denote spatial coordinate indices. An index-free notation is used where convenient. Given a 1-form and a vector , the action of on is denoted by . The <italic toggle=\"yes\">musical isomorphisms</italic>\n and are used to denote the contravariant version of and the covariant version of with respect to a given Lorentzian metric . This notation can be extended to tensors of higher rank.</p>", "<p>The conventions for the curvature tensors are fixed by the relation\n</p>" ]

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[ "<title>Standard results for symmetric hyperbolic systems</title>", "<p>The nature of the conformal Einstein field equations requires the <italic toggle=\"yes\">hyperbolic reduction</italic> of the conformal evolution equations to discuss the existence and asymptotic properties of their solutions. In this case, the existence of a conformal Gaussian gauge system allows a hyperbolic reduction of the conformal evolution equations as a quasi-linear symmetric hyperbolic system. The purpose of this section is to provide a brief summary of the known technical results for quasi-linear symmetric hyperbolic systems that will be used in the analysis of the evolution of the spacetimes under consideration.</p>", "<title>Kato’s theorem on symmetric hyperbolic systems</title>", "<p>Kato’s theory is concerned with symmetric hyperbolic systems in which the unknown is regarded as a -valued function over where is a Hilbert space. The Hilbert space can be real or complex and infinite dimensional. In this article, we are interested in the case where and . In this case, the symmetric hyperbolic system becomes a <italic toggle=\"yes\">standard</italic> partial differential equation.</p>", "<p>Given a -dimensional symmetric hyperbolic quasi-linear systems of the form\nfor , , , and initial conditions\nIn Kato’s theory, the coefficients and are nonlinear operators depending on sending -valued functions over into -matrix valued functions on . Similarly, is a nonlinear operator depending on sending -valued functions on into -valued functions on .</p>", "<p>Consider , the space of -vector valued functions over such that their entries have finite Sobolev norm of order . Let be a bounded open subset of . Writing\none has that for fixed and \nand\n</p>", "<p>Now, let denote the sets of smooth functions of compact support from to . Given any non-zero not identically zero, then belongs to the <italic toggle=\"yes\">uniformly local Sobolev spaces</italic>\n if and only if\nIn the following, for fixed and , the coefficients are functions from to whereas is a function from to . In Kato’s terminology this is equivalent to requiring that is a function from to and from to . Accordingly, one has the following reformulation of Theorem II in [##UREF##3##4##]:</p>", "<title>Theorem 3.1.</title>", "<p><italic toggle=\"yes\">Let</italic>\n\n<italic toggle=\"yes\">be a positive integer such that</italic>\n. <italic toggle=\"yes\">Let</italic>\n, \n<italic toggle=\"yes\">and</italic>\n\n<italic toggle=\"yes\">as above with</italic>\n. <italic toggle=\"yes\">Assume that the following conditions hold</italic>:\n<list list-type=\"simple\"><list-item><label>(i)<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">The components</italic>\n (<italic toggle=\"yes\">respectively</italic>, ) <italic toggle=\"yes\">are bounded in the</italic>\n-<italic toggle=\"yes\">norm</italic> (<italic toggle=\"yes\">respectively</italic>\n-<italic toggle=\"yes\">norm</italic>) <italic toggle=\"yes\">for</italic>\n, <italic toggle=\"yes\">uniformly in</italic>\n.</p></list-item><list-item><label>(ii)<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">For each</italic>\n, <italic toggle=\"yes\">the map</italic>\n\n<italic toggle=\"yes\">is uniformly Lipschitz continuous on</italic>\n\n<italic toggle=\"yes\">from the</italic>\n-<italic toggle=\"yes\">norm to the</italic>\n-<italic toggle=\"yes\">norm, uniformly in</italic>\n. <italic toggle=\"yes\">Similarly, the map</italic>\n\n<italic toggle=\"yes\">is Lipschitz continuous from the</italic>\n-<italic toggle=\"yes\">norm to the</italic>\n-<italic toggle=\"yes\">norm, again uniformly in</italic>\n.</p></list-item><list-item><label>(iii)<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">The map</italic>\n\n<italic toggle=\"yes\">is Lipschitz continuous on</italic>\n\n<italic toggle=\"yes\">from the</italic>\n-<italic toggle=\"yes\">norm to the</italic>\n-<italic toggle=\"yes\">norm, uniformly in</italic>\n.</p></list-item><list-item><label>(iv)<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">The maps</italic>\n\n<italic toggle=\"yes\">are continuous in the</italic>\n-<italic toggle=\"yes\">norm for each</italic>\n. <italic toggle=\"yes\">Similarly, the map</italic>\n\n<italic toggle=\"yes\">is continuous in the</italic>\n-<italic toggle=\"yes\">norm for each</italic>\n.</p></list-item><list-item><label>(v)<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">The map</italic>\n\n<italic toggle=\"yes\">is Lipschitz-continuous on</italic>\n\n<italic toggle=\"yes\">in the</italic>\n-<italic toggle=\"yes\">norm, uniformly for</italic>\n.</p></list-item><list-item><label>(vi)<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">For each</italic>\n\n<italic toggle=\"yes\">the matrix-valued functions</italic>\n\n<italic toggle=\"yes\">are symmetric for each</italic>\n.</p></list-item><list-item><label>(vii)<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">The matrix</italic>\n\n<italic toggle=\"yes\">is positive definite with eigenvalues larger that, say, 1 for each</italic>\n\n<italic toggle=\"yes\">and each</italic>\n.</p></list-item><list-item><label>(viii)<x xml:space=\"preserve\"> </x></label><p>.</p></list-item></list>\n<italic toggle=\"yes\">Then there is a unique solution</italic>\n\n<italic toggle=\"yes\">to</italic> (##FORMU##185##3.1##) <italic toggle=\"yes\">and</italic> (##FORMU##189##3.2##) <italic toggle=\"yes\">defined on</italic>\n\n<italic toggle=\"yes\">where</italic>\n\n<italic toggle=\"yes\">such that</italic>\n<italic toggle=\"yes\">where</italic>\n\n<italic toggle=\"yes\">can be chosen common to all initial conditions</italic>\n\n<italic toggle=\"yes\">in a suitably small condition of a given point in</italic>\n.</p>", "<p>Since the conditions of the above theorem are hard to verify, Kato provides sufficient conditions for these requirements to be satisfied:</p>", "<title>Theorem 3.2.</title>", "<p><italic toggle=\"yes\">Suppose that</italic>\n. <italic toggle=\"yes\">Let</italic>\n\n<italic toggle=\"yes\">be the subset of</italic>\n\n<italic toggle=\"yes\">consisting of pairs</italic>\n\n<italic toggle=\"yes\">such that</italic>\n<italic toggle=\"yes\">where</italic>\n\n<italic toggle=\"yes\">and</italic>\n\n<italic toggle=\"yes\">are fixed. Let</italic>\n<italic toggle=\"yes\">and</italic>\n<italic toggle=\"yes\">where</italic>\n\n<italic toggle=\"yes\">denotes the set of</italic> ()-<italic toggle=\"yes\">matrix valued functions over</italic>\n\n<italic toggle=\"yes\">with the properties</italic>\n<list list-type=\"simple\"><list-item><label>(a)<x xml:space=\"preserve\"> </x></label><p>,</p></list-item><list-item><label>(b)<x xml:space=\"preserve\"> </x></label><p>,</p></list-item><list-item><label>(c)<x xml:space=\"preserve\"> </x></label><p>,</p></list-item><list-item><label>(d)<x xml:space=\"preserve\"> </x></label><p>,</p></list-item></list>\n<italic toggle=\"yes\">where</italic>\n\n<italic toggle=\"yes\">and the sets</italic>\n and \n<italic toggle=\"yes\">denote the spaces of functions having derivatives up to the</italic>\n<italic toggle=\"yes\">th order which are continuous and bounded in the supremum norm. Then conditions (i)–(v) in theorem 3.1 are satisfied by</italic>\n, \n<italic toggle=\"yes\">provided that</italic>\n\n<italic toggle=\"yes\">is chosen as a ball in</italic>\n\n<italic toggle=\"yes\">with</italic>\n\n<italic toggle=\"yes\">as centre and a sufficiently small radius</italic>\n. <italic toggle=\"yes\">In addition, (ix) is satisfied if (a) is assumed to hold with</italic>\n\n<italic toggle=\"yes\">replaced by</italic>\n.</p>", "<title>Existence and stability result for symmetric hyperbolic systems on manifolds with compact spatial sections</title>", "<p>The result contained in theorem 3.1 can be extended to Cauchy problems for symmetric hyperbolic system with data prescribed on compact three-dimensional manifolds.</p>", "<p>Given a compact three-dimensional manifold , there exists a finite cover consisting of open sets such that . On each of the open sets it is possible to introduce coordinates which allow one to identify with open subsets . As is assumed to be a smooth manifold, the coordinate patches can be chosen so that the change of coordinates on intersecting sets is smooth. The initial data are a smooth function on and can be restricted to a particular open set . The restriction , in local coordinates , can be regarded as a function . Now, assuming that is bounded with smooth boundary , it is possible to extend to a function by means of the following proposition:</p>", "<title>Proposition 3.3.</title>", "<p><italic toggle=\"yes\">Let</italic>\n\n<italic toggle=\"yes\">be a bounded region with smooth boundary</italic>\n. <italic toggle=\"yes\">Then there exists a linear operator</italic>\n<italic toggle=\"yes\">such that for each</italic>\n:\n<list list-type=\"simple\"><list-item><label>(i)<x xml:space=\"preserve\"> </x></label><p>\n<italic toggle=\"yes\">almost everywhere in</italic>\n;</p></list-item><list-item><label>(ii)<x xml:space=\"preserve\"> </x></label><p>\n<italic toggle=\"yes\">has support in an open bounded set</italic>\n\n<italic toggle=\"yes\">with</italic>\n;</p></list-item><list-item><label>(iii)<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">There exists a constant</italic>\n\n<italic toggle=\"yes\">depending only on</italic>\n\n<italic toggle=\"yes\">and</italic>\n\n<italic toggle=\"yes\">such that</italic>\n</p></list-item></list>\n<italic toggle=\"yes\">The</italic>\n-<italic toggle=\"yes\">valued function</italic>\n\n<italic toggle=\"yes\">is an</italic>\n<italic toggle=\"yes\">extension</italic>\n<italic toggle=\"yes\">of</italic>\n\n<italic toggle=\"yes\">to</italic>\n.</p>", "<p>— see [##UREF##10##11##,##UREF##20##21##].</p>", "<p>Using these extensions, it is possible to define the Sobolev norm\nNow, for each of the one can formulate an initial value problem of the form\nand\nIf this system satisfies the conditions of theorem 3.1 the theory implies existence, uniqueness and stability. However, this theorem only applies to settings in which the spatial sections are diffeomorphic to . One makes use of standard results on causality theory implying that\nwhere denotes the causal future of —see e.g. [##UREF##10##11##], theorem 14.1. Accordingly, the value of on is determined only by the data on . Then the solution on is independent of the particular extension being used. Hence, one can speak of a solution on a domain . Since the manifold is smooth and as a consequence of uniqueness, it follows that given two solutions and defined, respectively, on intersecting domains and they must coincide on . Proceeding in the same manner over the whole finite cover of and since the compactness of ensures the existence of a minimum non-zero existence time for the whole of the domains , then there is a unique solution on with which is constructed by patching together the localized solutions defined, respectively, on the domains . This result is summarized by the following theorem [##UREF##10##11##]:</p>", "<title>Theorem 3.4.</title>", "<p><italic toggle=\"yes\">Consider the Cauchy problem for a quasi-linear symmetric hyperbolic system</italic>\n<italic toggle=\"yes\">and</italic>\n<italic toggle=\"yes\">with data on an orientable compact three-dimensional manifold</italic>\n. <italic toggle=\"yes\">If there is</italic>\n\n<italic toggle=\"yes\">such that</italic>\n\n<italic toggle=\"yes\">is positive definite with lower bound</italic>\n\n<italic toggle=\"yes\">for all</italic>\n, <italic toggle=\"yes\">then</italic>:\n<list list-type=\"simple\"><list-item><label>(i)<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">There exists</italic>\n\n<italic toggle=\"yes\">and a unique solution</italic>\n\n<italic toggle=\"yes\">to the Cauchy problem defined on</italic>\n\n<italic toggle=\"yes\">such that</italic>\n<italic toggle=\"yes\">Moreover</italic>, \n<italic toggle=\"yes\">is positive definite with lower bound</italic>\n\n<italic toggle=\"yes\">for all</italic>\n.</p></list-item><list-item><label>(ii)<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">There exists</italic>\n\n<italic toggle=\"yes\">such that one common existence time</italic>\n\n<italic toggle=\"yes\">can be chosen for all initial conditions in the open ball</italic>\n\n<italic toggle=\"yes\">and such that</italic>\n.</p></list-item><list-item><label>(iii)<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">If the solution</italic>\n\n<italic toggle=\"yes\">with initial data</italic>\n\n<italic toggle=\"yes\">exists on</italic>\n\n<italic toggle=\"yes\">for some</italic>\n, <italic toggle=\"yes\">then the solutions to all initial conditions in</italic>\n\n<italic toggle=\"yes\">exists on</italic>\n\n<italic toggle=\"yes\">if</italic>\n\n<italic toggle=\"yes\">is sufficiently small</italic>.</p></list-item><list-item><label>(iv)<x xml:space=\"preserve\"> </x></label><p><italic toggle=\"yes\">If</italic>\n\n<italic toggle=\"yes\">and</italic>\n\n<italic toggle=\"yes\">are chosen as in</italic>\n\n<italic toggle=\"yes\">and one has a sequence</italic>\n\n<italic toggle=\"yes\">such that</italic>\n<italic toggle=\"yes\">then for the solutions</italic>\n\n<italic toggle=\"yes\">with</italic>\n\n<italic toggle=\"yes\">with it holds that</italic>\n<italic toggle=\"yes\">uniformly in</italic>\n.</p></list-item></list></p>" ]

[]

[ "<title>Conclusion</title>", "<p>This review article provides a discussion based on [##UREF##14##15##,##UREF##15##16##] describing how the extended conformal Einstein field equations and a gauge adapted to the conformal geodesics can be used to study the evolution of vacuum spacetimes with positive Cosmological constant. In the de Sitter-like case, this analysis identifies a class of spacetimes for which it is possible to prove nonlinear stability and the existence of a regular conformal representation. More precisely, a class of de Sitter-like spacetimes is identified which can be conformally embedded into a portion of a cylinder whose spatial sections have negative scalar curvature. The conformal embedding is realized by means of a conformal factor which depends quadratically on the affine parameter of the conformal geodesics and this parameter is also used as a time coordinate for the physical metric. This result led to the idea that this technique could be adapted to black hole type of spacetimes. When adapting the strategy to the Cosmological region of the Schwarzschild–de Sitter spacetime, one encounters several difficulties. Whether in the de Sitter-like case, the explicit form of the unphysical metric is known. In the Schwarzschild–de Sitter case the explicit form of the unphysical metric is not known. This means that, in the former case, one can solve the conformal Einstein constraint equations and recast the unphysical spacetime as a solution to the conformal Einstein field equations. Moreover, one can construct a conformal Gaussian gauge system and show how the main and subsidiary evolution systems can be recasted as quasi-linear symmetric hyperbolic systems. In the latter case, since the unphysical metric is not known, one analyses the behaviour of a congruence of -conformal geodesics to use their properties to construct the conformal Gaussian gauge system. The background solution is obtained by computing the components of the curvature tensors in the conformal Levi–Civita connection and then using the transformation laws to recast them as solutions to the extended conformal Einstein field equations. The main evolution system is of the same form as for the de Sitter-like spacetime with the difference that for the Schwarzschild–de Sitter spacetime whereas the magnetic part of the rescaled Weyl tensor vanishes, the electric part of this tensor is non-vanishing. As a consequence, the main evolution system contains extra terms involving . Nonetheless, once recasted as a symmetric hyperbolic system its structural form is the same as for the de Sitter-like spacetime. For what concerns the subsidiary evolution system, it is the same as for the de Sitter-like spacetime. Another difference is that the sub-extremal Schwarzschild–de Sitter spacetime expressed in terms of a conformal Gaussian gauge system gives rise to a solution to the extended conformal Einstein field equations on the region . This region corresponds to the future domain of dependence of a portion of the initial hypersurface with topology where . Thus, one needs to recast this solution as a solution with compact spatial sections in order to use theorem 3.4 to prove its existence, uniqueness and stability. As a result, one shows that it is possible to construct solutions to the vacuum Einstein field equations in this region containing a portion of the asymptotic region and which are nonlinear perturbations of the exact Schwarzschild–de Sitter spacetime. Crucially, although the spacetimes constructed have an infinite extent to the future, they exclude the <italic toggle=\"yes\">asymptotic points</italic>\n and . From the analysis of the asymptotic initial value problem in [##UREF##16##17##], it is known that these points contain singularities of the conformal structure. Thus, they cannot be dealt with by the approach discussed in the article. In order to have a complete statement on the nonlinear stability of the Cosmological region it is necessary to address the asymptotic points. Moreover, since the initial hypersurfaces considered in the article are space-like and the evolution doesn’t include the Cosmological horizon . A complete statement should also include the case in which . This suggests reformulating the existence and stability results in [##UREF##15##16##] in terms of a characteristic initial value problem with data prescribed on Cosmological horizons. Again, to avoid the singularities of the conformal structure, the characteristic data has to be prescribed away from the asymptotic points. Alternatively, one could consider datasets which become exactly Schwarzschild–de Sitter near the asymptotic points. The associated evolution problem by means of a generalization of the methods used in [##UREF##25##26##] should allow us to reach any suitable hypersurface with constant .</p>" ]

[ "<p>One contribution of 13 to a discussion meeting issue ‘<ext-link xlink:href=\"http://dx.doi.org/10.1098/rsta/382/2267\" ext-link-type=\"uri\">At the interface of asymptotics, conformal methods and analysis in general relativity</ext-link>’.</p>", "<p>This article provides a discussion on the construction of conformal Gaussian gauge systems to study the evolution of solutions to the Einstein field equations with positive Cosmological constant. This is done by means of a gauge based on the properties of conformal geodesics. The use of this gauge, combined with the extended conformal Einstein field equations, yields evolution equations in the form of a symmetric hyperbolic system for which standard Cauchy stability results can be employed. This strategy is used to study the global properties of de Sitter-like spacetimes with constant negative scalar curvature. It is then adapted to study the evolution of the Schwarzschild–de Sitter spacetime in the static region near the conformal boundary. This review is based on Minucci <italic toggle=\"yes\">et al</italic>. 2021 <italic toggle=\"yes\">Class. Quantum Grav</italic>. <bold>38</bold>, 145026. (<ext-link xlink:href=\"http://dx.doi.org/10.1088/1361-6382/ac0356\" ext-link-type=\"uri\">doi:10.1088/1361-6382/ac0356</ext-link>) and Minucci <italic toggle=\"yes\">et al</italic>. 2023 <italic toggle=\"yes\">Class. Quantum Grav</italic>. <bold>40</bold>, 145005. (<ext-link xlink:href=\"http://dx.doi.org/10.1088/1361-6382/acdb3f\" ext-link-type=\"uri\">doi:10.1088/1361-6382/acdb3f</ext-link>).</p>", "<p>This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.</p>" ]

[ "<title>Tools of conformal geometry</title>", "<p>The purpose of this section is to provide a brief summary of the technical tools of conformal geometry that will be used in the analysis of the evolution of the spacetimes under consideration.</p>", "<p>Let be a vacuum spacetime satisfying the Einstein field equations with positive Cosmological constant\nand let denote an unphysical Lorentzian metric conformally related to via the relation\nwith a suitable conformal factor. The Levi–Civita connections of the metrics and are denoted by and , respectively. The set of points for which is called the <italic toggle=\"yes\">conformal boundary</italic>.</p>", "<title>Weyl connections</title>", "<p>A Weyl connection is a torsion-free connection such that\nIt follows from the above that the connections and are related to each other by\nwhere is a fixed smooth covector and is an arbitrary vector. Given that\none has that\nIn the following, it will be convenient to define\n</p>", "<title>The frame version of the extended conformal Einstein field equations</title>", "<p>The <italic toggle=\"yes\">extended conformal Einstein field equations</italic> constitute a conformal representation of the vacuum Einstein field equations written in terms of <italic toggle=\"yes\">Weyl connections</italic>—see [##UREF##19##20##]. These equations are formally regular at the conformal boundary. Moreover, a solution to the extended conformal equations implies, in turn, a solution to the vacuum Einstein field equations away from the conformal boundary.</p>", "<p>Let , denote a -orthogonal frame with associated coframe . Thus, one has that\nThe frame formulation of the <italic toggle=\"yes\">extended conformal Einstein field equations</italic> is obtained by defining the following <italic toggle=\"yes\">zero-quantities</italic>:\n\n\n\nwhere the components of the <italic toggle=\"yes\">geometric curvature</italic>\n and the <italic toggle=\"yes\">algebraic curvature</italic>\n are given, respectively, by\nand\nIn terms of the zero-quantities (2.4<italic toggle=\"yes\">a</italic>–), the <italic toggle=\"yes\">extended conformal Einstein field equations</italic> are given by the conditions\nIn the above equations, the fields and are regarded as <italic toggle=\"yes\">conformal gauge fields</italic> which are determined by gauge conditions. These conditions will be determined through conformal geodesics—see §2d. In order to account for this, we write down the gauge equations as\n\n\nThen, the conditions\nwill be called the <italic toggle=\"yes\">supplementary conditions</italic>. They play a role in relating the Einstein field equations to the extended conformal Einstein field equations and also in the propagation of the constraints.</p>", "<p>The correspondence between the Einstein field equations and the extended conformal Einstein field equations is given by the following—see proposition 8.3 in [##UREF##10##11##]:</p>", "<title>Proposition 2.1.</title>", "<p><italic toggle=\"yes\">Let</italic>\n<italic toggle=\"yes\">denote a solution to the extended conformal Einstein field equations</italic> (##FORMU##62##2.5##) <italic toggle=\"yes\">for some choice of the conformal gauge fields</italic>\n\n<italic toggle=\"yes\">satisfying the supplementary conditions (##FORMU##68##2.7##). Furthermore, suppose that</italic>\n<italic toggle=\"yes\">on an open subset</italic>\n. <italic toggle=\"yes\">Then the metric</italic>\n<italic toggle=\"yes\">is a solution to the Einstein field equations (##FORMU##28##2.1##) on</italic>\n.</p>", "<title>The conformal Einstein constraint equations</title>", "<p>The <italic toggle=\"yes\">conformal Einstein constraint equations</italic> are intrinsic equations implied by the standard conformal Einstein field equations on a space-like hypersurface.</p>", "<p>Let denote a space-like hypersurface in an unphysical spacetime and let denote a -orthonormal frame adapted to . Now, let be the <italic toggle=\"yes\">intrinsic metric</italic> of with associated Levi–Civita connection . Since is the directional covariant derivative acting on hypersurface-defined objects. The conformal constraint equations in the vacuum case are given by—see [##UREF##10##11##]:\n\n\n\n\n\n\n\n\n\nwith the understanding that\nand where\nMoreover, denotes the restriction of the spacetime conformal factor to and is the normal component of the gradient of . The field denotes the components of the Schouten tensor of the induced metric on . The fields and correspond, respectively, to the electric and magnetic parts of the rescaled Weyl tensor. The scalar denotes the <italic toggle=\"yes\">Friedrich scalar</italic> defined as\nwith the Ricci scalar of the metric . Finally, denotes the spatial components of the Schouten tensor of .</p>", "<title>Conformal geodesics</title>", "<p>The extended conformal Einstein field equations, being expressed in terms of a Weyl connection, contain a larger gauge freedom than the standard conformal Einstein field equations. In this case, the gauge used to analyse the evolution of the spacetimes under consideration is based on the properties of <italic toggle=\"yes\">conformal geodesics</italic>.</p>", "<title>Basic definitions</title>", "<p>A <bold><italic toggle=\"yes\">conformal geodesic</italic></bold> on a spacetime is a pair consisting of a curve and a covector along satisfying the equations\nand\nwhere denotes the <italic toggle=\"yes\">Schouten tensor</italic> of the Levi–Civita connection . A vector is said to be <italic toggle=\"yes\">Weyl propagated</italic> if along it satisfies the equation\n</p>", "<p>The <italic toggle=\"yes\">conformal geodesic equations</italic> (2.9<italic toggle=\"yes\">a</italic>) and (2.9<italic toggle=\"yes\">b</italic>) are conformally invariant, in the sense that if a solution is obtained with one representative of the conformal class of metrics, then a simple transformation of the solution exists to map it into a solution with respect to another representative. Hence, these curves are coming solely from the <italic toggle=\"yes\">conformal structure</italic>.</p>", "<p>A congruence of conformal geodesics can be used to single out a metric out of the equivalence class of conformally related metrics . This is due to the following property:</p>", "<title>Proposition 2.2.</title>", "<p><italic toggle=\"yes\">Let</italic>\n\n<italic toggle=\"yes\">denote a vacuum spacetime with positive Cosmological constant. Suppose that</italic>\n\n<italic toggle=\"yes\">is a solution to the conformal geodesic equations (2.9a</italic>,) <italic toggle=\"yes\">and that</italic>\n\n<italic toggle=\"yes\">is a</italic>\n-<italic toggle=\"yes\">orthonormal frame propagated along the curve according to equation (##FORMU##123##2.10##). If</italic>\n\n<italic toggle=\"yes\">satisfies</italic>\n<italic toggle=\"yes\">then one has that</italic>\n<italic toggle=\"yes\">where the coefficients</italic>\n<italic toggle=\"yes\">are constant along the conformal geodesic and are subject to the constraints</italic>\n</p>", "<p>A proof of this result can be found in [##UREF##10##11##].</p>", "<p>Thus, if a spacetime can be covered by a non-intersecting congruence of conformal geodesics, then the location of the conformal boundary is known <italic toggle=\"yes\">a priori</italic> in terms of data at a fiduciary initial hypersurface .</p>", "<title>The -adapted conformal geodesic equations</title>", "<p>As a consequence of the normalization condition (##FORMU##132##2.11##), the parameter is the -proper time of the curve . In some computations it is more convenient to consider a parametrization in terms of a -proper time of the curve as it allows us to work directly with the physical metric. To this end, consider the parameter transformation given by\nwith inverse . Now, consider\nin equations (2.9<italic toggle=\"yes\">a</italic>,) so that one obtains the following -<bold>adapted equations for the conformal geodesics</bold>:\nand\nwith . For a vacuum spacetime with Cosmological constant one has that\n</p>", "<title>A conformal Gaussian gauge system</title>", "<p>The use of the extended conformal Einstein field equations and a gauge adapted to a congruence of conformal geodesics allows the construction of <italic toggle=\"yes\">conformal Gaussian gauge systems</italic>. To construct a <italic toggle=\"yes\">conformal Gaussian gauge system</italic>, one considers a region of the spacetime which is covered by a non-intersecting congruence of conformal geodesics and proceeds as follows:\n<list list-type=\"simple\"><list-item><label>—<x xml:space=\"preserve\"> </x></label><p>The property of the congruence of conformal geodesics stated by proposition 2.2 allows one to single out a <italic toggle=\"yes\">canonical representative</italic>\n of the conformal class with an explicitly known conformal factor as given by the formula (##FORMU##133##2.12##).</p></list-item><list-item><label>—<x xml:space=\"preserve\"> </x></label><p>Let denote a -orthonormal frame which is Weyl propagated along the conformal geodesics. To every congruence of conformal geodesics one can associate a Weyl connection by setting . It follows that for this connection one has\nThis gauge choice is supplemented by choosing the parameter of the conformal geodesics as the time coordinate so that\n</p></list-item><list-item><label>—<x xml:space=\"preserve\"> </x></label><p>Since the initial data for the congruence of conformal geodesics is prescribed on a fiduciary space-like hypersurface . On one can choose some local coordinates . These coordinates can be extended off by requiring them to remain constant along the conformal geodesic which intersects at the point with coordinates . The spacetime coordinates obtained in this way are known as <italic toggle=\"yes\">conformal Gaussian coordinates</italic>.</p></list-item></list> The collection of conformal factor , Weyl propagated frame and coordinates is known as a <italic toggle=\"yes\">conformal Gaussian gauge system</italic>.</p>", "<p>One of the advantages of this procedure is that one has an <italic toggle=\"yes\">a priori</italic> knowledge of the location of the conformal boundary. This is in contrast with other conformal gauges in which the conformal factor is an unknown. The use of a conformal Gaussian gauge system leads to a particularly simple system of conformal evolution equations. The evolution of all the geometric unknowns is either fixed by the gauge or given by transport equations along the congruence of conformal geodesics. Moreover, whether the standard Gaussian gauge, adapted to metric geodesics, is known to develop caustics. However, by looking at the geodesic deviation equation for the conformal geodesics, extra terms appear involving the 1-form that counter the contribution from the curvature. Thus, these curves are less likely to develop caustics. Besides, since in [##UREF##17##18##], Friedrich shows that there exists a congruence of conformal geodesics in the Schwarzschild spacetime that yields a semi-global frame that is regular up to and beyond future null infinity. The use of a gauge adapted to these curves is a more convenient choice.</p>", "<title>de Sitter-like spacetimes</title>", "<p>In this section, we discuss the evolution of de Sitter-like spacetimes which can be conformally embedded into a portion of a cylinder whose sections have negative scalar curvature as in [##UREF##14##15##]. The conformal embedding is realized by means of a conformal factor which depends on the affine parameter of the conformal geodesics.</p>", "<title>Basic properties</title>", "<p>A de Sitter-like spacetime is a solution to the vacuum Einstein field equations with positive Cosmological constant (##FORMU##28##2.1##) given by and\nwhere is a positive definite Riemannian metric over a compact manifold with constant negative curvature. The Riemann curvature tensor of is given by\nIn particular, by setting ,^{<xref rid=\"FN1\" ref-type=\"fn\">1</xref>} it follows from the above expressions that\nMoreover, since\nit follows that\nA spacetime of the form given by will be known as a <italic toggle=\"yes\">background solution</italic>.</p>", "<title>Metric geodesics as conformal geodesics</title>", "<p>The analysis of the metric geodesics on with , where is a proportionality function, by means of the geodesic equation\nand the metric (##FORMU##413##4.1##) shows that is constant along the integral curves of . Hence, without loss of generality one can set so that the curves\nare non-intersecting time-like -geodesics over . These curves can be recasted as conformal geodesics by means of a reparametrization and a 1-form given by the Ansatz\nThe resulting pair with\ndescribes a congruence of non-intersecting time-like conformal geodesics on the background spacetime .</p>", "<title>The conformal factor associated to the congruence of conformal geodesics</title>", "<p>The parameter introduced in the previous section is used as a new time coordinate in the metric (##FORMU##413##4.1##) so that\nThis metric is singular at . This line element suggests the introduction of a new unphysical metric via the relation\nso that\nis well defined for with . The <italic toggle=\"yes\">spatial metric</italic>\n is conformally related to via\nwith associated Levi–Civita connection to be denoted by , whereas is the Levi–Civita connection of the metric . The integral curves of the vector field are geodesics of the metric given by equation (##FORMU##446##4.4##). Moreover, since is a closed 1-form the Weyl connection is, in fact, a Levi–Civita connection which coincides with .</p>", "<p>A Penrose–Carter diagram of conformal representation of the background solution described by the metric (##FORMU##446##4.4##) is given in ##FIG##0##figure 1##.\n</p>", "<title>The background spacetime as a solution to the conformal Einstein field equations</title>", "<p>The <italic toggle=\"yes\">unphysical spacetime</italic>\n is recast as a solution to the conformal Einstein field equations. This construction is done using an adapted frame formalism.</p>", "<title>The frame</title>", "<p>Let , , denote a -orthonormal frame over with associated cobasis . Accordingly, one has that\nso that\nThe above frame is used to introduce a -orthonormal frame with associated cobasis so that . This is done by setting\nand\nso that\n</p>", "<title>The connection coefficients</title>", "<p>The connection coefficients of the Levi–Civita connection with respect to the frame are defined through the relations\nSimilarly, for the connection coefficients of the Levi–Civita connection with respect to the frame one has that\nUsing these relations, it follows that the only non-vanishing connection coefficients are\nwhere denote the components of the <italic toggle=\"yes\">Weingarten tensor</italic>.</p>", "<p>Thus, all the connection coefficients are smooth over .</p>", "<title>Conformal fields</title>", "<p>The components of the conformal fields appearing in the extended conformal Einstein field equations are obtained by solving the conformal Einstein constraints discussed in §2c.</p>", "<p>This is done by means of an adapted frame with and by making the identification in equations (2.8<italic toggle=\"yes\">a</italic>–. The analysis of these equations gives\nand\nThus, all the fields are regular up to the conformal boundary and the metric (##FORMU##446##4.4##) is conformally flat.</p>", "<title>Evolution equations</title>", "<p>In this section, we discuss the evolution system associated with the extended conformal Einstein equations (##FORMU##62##2.5##) written in terms of a conformal Gaussian system. In addition, we also discuss the subsidiary evolution system satisfied by the zero-quantities associated to the field equations, (2.4a–), and the supplementary zero-quantities (2.6<italic toggle=\"yes\">a</italic>–).</p>", "<title>The conformal Gaussian gauge</title>", "<p>To obtain suitable evolution equations for the conformal fields a <italic toggle=\"yes\">conformal Gaussian gauge</italic> is used. More precisely, it is assumed that a region is covered by a congruence of non-intersecting conformal geodesics. Then, by choosing\nfor , , proposition 2.2 gives the conformal factor\nalong the curves of the congruence. The choice of initial data for the conformal factor is associated to a congruence that leaves orthogonally a fiduciary initial hypersurface with . Since the conformal factor given by equation (##FORMU##499##4.7##) does not depend on the initial data for the evolution equations it can be regarded as valid not only for the background solution but also for its perturbations. The choice has the consequence that the Weyl connection is just the conformal Levi–Civita connection on the initial hypersurface .</p>", "<p>Along the congruence of conformal geodesics one considers a -orthogonal frame which is Weyl-propagated and such that . The Weyl connection associated to the congruence then satisfies\nBy choosing the parameter, of the conformal geodesics as time coordinate one gets the additional gauge condition\nOn we choose some local coordinates . These coordinates can be extended off the initial hypersurface so that the coordinates thus obtained are <italic toggle=\"yes\">conformal Gaussian coordinates</italic>.</p>", "<title>Structural properties of the evolution and subsidiary equations</title>", "<p>In the conformal Gaussian gauge, the various fields associated with the extended vacuum conformal Einstein field equations satisfy the evolution equations\n\n\n\n\n\nLetting , , and denote, respectively, the independent components of the coefficients of the frame, the connection coefficients, the Schouten tensor of the Weyl connection and the rescaled Weyl tensor and setting, for convenience, with and one has the following:</p>", "<title>Lemma 4.1.</title>", "<p><italic toggle=\"yes\">The extended conformal Einstein field equations (##FORMU##62##2.5##) expressed in terms of a conformal Gaussian gauge imply that the evolution equations (4.8a)–(4.8f) can be written as a symmetric hyperbolic system for the components</italic>\n\n<italic toggle=\"yes\">of the form</italic>\n<italic toggle=\"yes\">and</italic>\n<italic toggle=\"yes\">where</italic>\n\n<italic toggle=\"yes\">is the unit matrix</italic>, \n<italic toggle=\"yes\">is a constant matrix</italic>, \n<italic toggle=\"yes\">is a smooth matrix-valued function</italic>, \n<italic toggle=\"yes\">is a smooth matrix-valued function of the coordinates</italic>, \n<italic toggle=\"yes\">are Hermitian matrices depending smoothly on the frame coefficients and</italic>\n\n<italic toggle=\"yes\">is a smooth matrix-valued function of the connection coefficients</italic>.</p>", "<p>Regarding the subsidiary evolution system, it follows from the system\n\n\n\n\n\nthat the zero-quantities , , , , , and satisfy, if the conformal evolution equations (4.8<italic toggle=\"yes\">a</italic>–<italic toggle=\"yes\">e</italic>) hold, a symmetric hyperbolic system which is homogeneous in the zero-quantities. More precisely, upon defining , these equations can be recasted as a symmetric hyperbolic system of the form\nwhere . The particular situation in which all the zero-quantities vanish identically gives rise to the subsidiary evolution system.</p>", "<p>Since the spacetime has compact spatial sections there is no need to care about boundaries and the associated difficulties of constraint violations advecting from them, governed through the subsidiary system—see [##UREF##21##22##,##UREF##22##23##].</p>", "<title>The perturbative argument</title>", "<p>In the following, we look for solutions to the system (4.9<italic toggle=\"yes\">a</italic>,) of the form\nwhere is the solution to the conformal evolution equations (4.8<italic toggle=\"yes\">a</italic>–) implied by a background solution, while denotes a small perturbation. Accordingly, one can set\nand\nNow, on the initial surface , described by the condition , one has that being the exact de Sitter-like solution. As the conformal factor and the covector are universal, it follows that\nSubstituting (4.12<italic toggle=\"yes\">a</italic>,) into equations (4.9<italic toggle=\"yes\">a</italic>,) and upon defining the following matrices:\nand\nwhere\nit is possible to write the evolution equations for as a quasi-linear symmetric hyperbolic system\nExistence and stability results for the solution to the initial value problem for the system (##FORMU##858##5.15##) with data follow from known results for symmetric hyperbolic systems over . More precisely, it is observed that:\n<list list-type=\"simple\"><list-item><label>(a)<x xml:space=\"preserve\"> </x></label><p>The matrices are positive definite and depend linearly on the solution with coefficients which are constant.</p></list-item><list-item><label>(b)<x xml:space=\"preserve\"> </x></label><p>The dependence of on is at most quadratic: there are linear and quadratic terms for the connection coefficients; linear terms for the components of the Schouten tensor. The explicit dependence on comes from the conformal factor and the covector —this dependence is smooth.</p></list-item><list-item><label>(c)<x xml:space=\"preserve\"> </x></label><p>The connection coefficients and the components of the Schouten tensor of the background solution are smooth functions () of .</p></list-item><list-item><label>(d)<x xml:space=\"preserve\"> </x></label><p>The dependence of the frame coefficients of the background solution is smooth () on for with .</p></list-item></list></p>", "<p>It follows from the above observations that the conditions of theorem (3.4) are satisfied, thus ensuring existence, uniqueness and Cauchy stability for the solution .</p>", "<p>The existence and Cauchy stability of the solution to the initial value problem for the original conformal evolution problem\nand\nfollows from the fact that satisfies the same properties as and then it exists in the same solution manifold and with the same regularity properties, existence and uniqueness.</p>", "<title>A solution to the Einstein field equations</title>", "<p>In this section, we discuss the connection between the solution to the conformal evolution systems and the actual solution to the Einstein field equations.</p>", "<p>From the discussion in §4e((ii)), it follows that the independent components of the zero-quantities satisfy the symmetric hyperbolic system (##FORMU##551##4.11##). Then, a solution to the initial value problem\nand\nis given by . Moreover, from theorem 3.4 it follows that this is the unique solution. Thus, the zero-quantities must vanish on . This result is summarized by the following</p>", "<title>Proposition 4.2 (Propagation of the constraints).</title>", "<p><italic toggle=\"yes\">Let</italic>\n\n<italic toggle=\"yes\">denote initial data for the conformal evolution equations on a 3-manifold</italic>\n\n<italic toggle=\"yes\">such that</italic>\n<italic toggle=\"yes\">and</italic>\n<italic toggle=\"yes\">then the solution</italic>\n\n<italic toggle=\"yes\">to the conformal evolution equations implies a</italic>\n\n<italic toggle=\"yes\">solution</italic>\n\n<italic toggle=\"yes\">to the extended conformal field equations on</italic>\n.</p>", "<p>Now, given the propagation of the constraints, proposition 4.2, and proposition 2.1 it follows that the metric obtained from the solution to the conformal evolution equations implies a solution to the vacuum Einstein field equations with .</p>", "<p>The main result of this discussion is contained in the following theorem</p>", "<title>Theorem 4.3.</title>", "<p><italic toggle=\"yes\">Let</italic>\n\n<italic toggle=\"yes\">denote smooth initial data for the conformal evolution equations satisfying the conformal constraint equations on a hypersurface</italic>\n. <italic toggle=\"yes\">Then, there exists</italic>\n\n<italic toggle=\"yes\">such that if</italic>\n<italic toggle=\"yes\">then there exists a unique</italic>\n\n<italic toggle=\"yes\">solution</italic>\n\n<italic toggle=\"yes\">to the vacuum Einstein field equation with positive Cosmological constant over</italic>\n\n<italic toggle=\"yes\">for</italic>\n\n<italic toggle=\"yes\">whose restriction to</italic>\n\n<italic toggle=\"yes\">implies the initial data</italic>\n. <italic toggle=\"yes\">Moreover, the solution</italic>\n\n<italic toggle=\"yes\">remains suitably close to the background solution</italic>\n.</p>", "<title>Schwarzschild–de Sitter spacetimes</title>", "<p>In this section, the behaviour of the conformal geodesics in the Cosmological region of the sub-extremal Schwarzschild–de Sitter spacetime is discussed. The aim of this analysis is to adapt the technique described in the de Sitter-like setting and valid, in general, for asymptotically simple spacetimes to the black hole case as presented in [##UREF##15##16##].</p>", "<title>Basic properties</title>", "<p>The <italic toggle=\"yes\">Schwarzschild–de Sitter spacetime</italic>\n is a spherically symmetric solution to the vacuum Einstein field equations with positive Cosmological constant (##FORMU##28##2.1##) with and line element given in <italic toggle=\"yes\">standard coordinates</italic>\n by\nwhere\ndenotes the standard metric on . The coordinates take the range\nThis line element can be rescaled so that\nwhere\nIn our conventions, , and are dimensionless quantities.</p>", "<title>Horizons and global structure</title>", "<p>The location of the horizons of the Schwarzschild–de Sitter spacetime follows from the analysis of the zeros of the function in the line element (##FORMU##628##5.2##).</p>", "<p>Since , the function can be factorized as\nwhere and are, in general, distinct positive roots of and is a negative root. Moreover, one has that\nThe root corresponds to a black hole type of horizon and to a Cosmological de Sitter-like type of horizon. Using Cardano’s formula for cubic equations, we have\n\n\nwhere the parameter is defined through the relation\nThe sub-extremal case is characterized by and , describing a black hole in a Cosmological setting. The Penrose–Carter diagram of the sub-extremal Schwarzschild–de Sitter is well known—see ##FIG##1##figure 2##.\n</p>", "<title>Construction of a conformal Gaussian gauge in the Cosmological region</title>", "<p>This study begins with the qualitative analysis of the behaviour of the conformal geodesics of the Schwarzschild–de Sitter spacetime prescribed in terms of data on hypersurfaces of constant in the Cosmological region.</p>", "<title>Basic set-up</title>", "<p>It is assumed that\ncorresponding to the Cosmological region of the Schwarzschild–de Sitter spacetime. Given a fixed , denotes the space-like hypersurface of constant in this region. Points on are described in terms of the coordinates .</p>", "<p>In order to fix the congruence of conformal geodesics, the value of the conformal factor over is chosen so that\nThe second condition implies that the resulting conformal factor will have a time reflection symmetry with respect to . Then it is required that\nThe latter, in turn, implies that\nThese conditions give rise to a congruence of conformal geodesics which acts trivially in the angular directions. Accordingly, the analysis of these curves is effectively given by the metric\nFinally, in order to exclude the asymptotic points and , it is defined\nwhere the constant is assumed large enough so that .</p>", "<title>Analysis of the behaviour of the conformal geodesics</title>", "<p>The congruence of conformal geodesics prescribed by the initial data (##FORMU##669##5.5##) is such that , so that after reparametrization reduces to a congruence of metric geodesics. Thus, the geodesic equations imply that\nwhere is a constant. Evaluating at one readily finds that\nwith . Moreover, since the units normal to and are parallel to each other then .</p>", "<p>In order to study the behaviour of these curves and obtain simpler expressions, it is set and . It follows then from proposition (2.2) that the conformal factor is\nNow, since the relation between the physical proper time and the unphysical proper time is obtained from equation (##FORMU##145##2.13##) so that\nthen\nand since this congruence of conformal geodesics is reparametrized as metric geodesics, it will reach the conformal boundary orthogonally [##UREF##13##14##]. Now, since the dependence of the physical proper time on is given by\nwhich can be written in terms of elliptic functions (e.g. [##UREF##23##24##]), it follows from the general theory of elliptic functions that is an analytic function of its arguments. Moreover, one has that\nAccordingly, the curves escape to infinity in an infinite amount of physical proper time. Using the reparametrization formulae (##FORMU##690##5.9##) the latter corresponds to a finite amount of unphysical proper time.</p>", "<title>Analysis of the behaviour of the conformal deviation equation</title>", "<p>In [##UREF##17##18##] (see also [##UREF##18##19##]), it has been shown that for congruences of conformal geodesics in spherically symmetric spacetimes the behaviour of the deviation vector of the congruence can be understood by considering the evolution of a scalar satisfying the equation\nwhere denotes the Levi–Civita covariant derivative of and denotes the Ricci scalar of . If does not vanish, then the congruence is non-intersecting.</p>", "<p>Since in the present case one has and , it follows that the evolution equation (##FORMU##698##5.10##) takes the form\nSince this setting and , it follows that\nBy solving this last differential equation and reverting to , one has that\nwhich is non-vanishing in the limit . Thus, we have the following Proposition</p>", "<title>Proposition 5.1.</title>", "<p><italic toggle=\"yes\">The congruence of conformal geodesics given by the initial conditions (##FORMU##669##5.5##) leaving the initial hypersurface</italic>\n\n<italic toggle=\"yes\">reach the conformal boundary</italic>\n\n<italic toggle=\"yes\">without developing caustics</italic>.</p>", "<p>This behaviour of the conformal geodesics is shown in ##FIG##2##figure 3##.\n</p>", "<title>Conformal Gaussian coordinates in the sub-extremal Schwarzschild–de Sitter spacetime</title>", "<p>The congruence of conformal geodesics defined by the initial conditions (##FORMU##669##5.5##) is used to construct a <italic toggle=\"yes\">conformal Gaussian coordinate system</italic> in a domain in the chronological future of containing a portion of the conformal boundary . This analysis is carried out by considering the coordinate in terms of which the line element (##FORMU##628##5.2##) takes the form\nwhere\nThe above expression suggest defining an <italic toggle=\"yes\">unphysical metric</italic>\n via\nMore precisely, one has\nNow, let denote the Cosmological region of the Schwarzschild–de Sitter spacetime—that is\nMoreover, denote by the conformal representation of defined by the conformal factor defined by the non-singular congruence of conformal geodesics. Let , for one has that in terms of these coordinates\nwhere with .</p>", "<p>The conformal geodesics defined by the initial conditions (##FORMU##669##5.5##) define a map which is analytic in the parameters . This map is invertible since the Jacobian of the transformation is non-zero for the given value of the parameters. The inverse map\ngives the transformation from the <italic toggle=\"yes\">standard Schwarzschild coordinates</italic>\n into the <italic toggle=\"yes\">conformal Gaussian coordinates</italic>\n. This result is summarized by the following</p>", "<title>Proposition 5.2.</title>", "<p><italic toggle=\"yes\">The congruence of conformal geodesics on</italic>\n\n<italic toggle=\"yes\">defined by the initial conditions on</italic>\n\n<italic toggle=\"yes\">given by (##FORMU##669##5.5##) induce a conformal Gaussian coordinate system over</italic>\n\n<italic toggle=\"yes\">which is related to the standard coordinates</italic>\n\n<italic toggle=\"yes\">via a map which is analytic</italic>.</p>", "<title>The background spacetime as a solution to the conformal Einstein field equations</title>", "<p>The Schwarzschild–de Sitter spacetime in the region\nis cast as a solution to the extended conformal Einstein field equations by means of a Weyl propagated frame.</p>", "<title>The frame</title>", "<p>Since the congruence of conformal geodesics implied by the initial data (##FORMU##669##5.5##) satisfies , the Weyl propagation equation (##FORMU##123##2.10##) reduces to the usual parallel propagation equation. Given the spherical symmetry of the Schwarzschild–de Sitter spacetime, the discussion of a frame adapted to the symmetry of the spacetime can be carried out by considering the two-dimensional Lorentzian metric (##FORMU##670##5.6##). The <italic toggle=\"yes\">time leg</italic> of the frame is set as so that\nwhere . Now, upon defining\nwith , one has that the <italic toggle=\"yes\">radial leg</italic> of the frame is given by\nThe Weyl propagated frame is completed by choosing <italic toggle=\"yes\">two arbitrary orthonormal vectors</italic>\n and spanning the tangent space of and defining the vectors on by constantly extending the value of the associated coefficients along the conformal geodesics. This analysis leads to the following result</p>", "<title>Proposition 5.3.</title>", "<p><italic toggle=\"yes\">Let</italic>\n\n<italic toggle=\"yes\">denote the vector tangent to the conformal geodesics defined by the initial data (##FORMU##669##5.5##) and let</italic>\n\n<italic toggle=\"yes\">be an arbitrary orthonormal pair of vectors spanning the tangent bundle of</italic>\n. <italic toggle=\"yes\">Then the frame</italic>\n\n<italic toggle=\"yes\">obtained by the procedure described in the previous paragraph is a</italic>\n-<italic toggle=\"yes\">orthonormal Weyl propagated frame. The frame depends analytically on the unphysical proper time</italic>\n and the initial position \n<italic toggle=\"yes\">of the curve</italic>.</p>", "<title>The Weyl connection</title>", "<p>The connection coefficients associated to a conformal Gaussian gauge are made up of two pieces: the 1-form defining the Weyl connection and the Levi–Civita connection of the metric .</p>", "<p>The congruence of conformal geodesics discussed in §5c arises from initial data chosen so that the curves with tangent given by satisfy the standard (affine) geodesic equation. Consequently, the (spatial) 1-form vanishes. Now, since , by observing equation (##FORMU##677##5.7##) for with and by introducing , it follows that\nThen, from the conformal transformation rule\nand by recalling that , it follows that vanishes at . However, away from the conformal boundary.</p>", "<title>The connection coefficients</title>", "<p>Since the coordinates and connection coefficients associated with the physical connection are not well adapted to a discussion near the conformal boundary we resort to the unphysical Levi–Civita connection to compute .</p>", "<p>The connection coefficients are defined through the relation\nThe only non-vanishing Christoffel symbols are given by\nThese coefficients are analytic at . Since a contraction with the coefficients of the frame does not change this, it follows that the Weyl connection coefficients are smooth functions of the coordinates used in the conformal Gaussian gauge on the future of the fiduciary initial hypersurface up to and beyond the conformal boundary.</p>", "<title>The rescaled Weyl tensor</title>", "<p>Given a time-like vector, the components of the rescaled Weyl tensor can be encoded in the electric and magnetic parts relative to the given vector. For the vector these are given by\nwhere denotes the Hodge dual of . A computation using the package <monospace>xAct</monospace> for <monospace>Mathematica</monospace> [##UREF##24##25##] readily gives that the only non-zero components of the electric part are given by\nwhile the magnetic part vanishes identically. These expressions are regular at —by disregarding the coordinate singularity due to the use of spherical coordinates. The smoothness of the components of the Weyl tensor is retained when contracted with the coefficients of the frame .</p>", "<title>The Schouten tensor</title>", "<p>A similar computer algebra calculation shows that the non-zero components of the Schouten tensor of the metric are given by\nThe above expressions are analytic on —in particular at and by disregarding the coordinate singularity on the angular components. To obtain the components of the Schouten tensor associated with the Weyl connection we make use of the transformation rule\nThe smoothness of has already been established in §5d((ii)). Thus, the components of with respect to the Weyl propagated frame are regular on .</p>", "<title>Construction of a background solution with compact spatial sections</title>", "<p>From the previous discussion, it follows that the sub-extremal Schwarzschild–de Sitter spacetime expressed in terms of a conformal Gaussian gauge system gives rise to a solution to the extended conformal Einstein field equations on the region . Since has the topology of where is an open interval, the spacetime arising from will have spatial sections with the same topology. As part of the perturbative argument is based on the general theory of symmetric hyperbolic systems as given in theorem 3.4 it is convenient to consider solutions with compact spatial sections.</p>", "<p>This construction is based on the observation that the Killing vector in the Cosmological region of the spacetime is space-like. Thus, given a fixed , the hypersurface defined by the condition has a translational invariance. Now, one identifies the time-like hypersurfaces and generated, respectively, by the future-directed geodesics emanating from at the points with and to obtain a smooth spacetime manifold with compact spatial sections—see ##FIG##3##figure 4##. The metric on induces a metric on which, on an abuse of notation, is denoted again by . Since the initial conditions defining the congruence of conformal geodesics of §5c have translational invariance, the resulting curves also have this property. Accordingly, the congruence of conformal geodesics on induces a non-intersecting congruence of conformal geodesics on . Thus, the solution to the extended conformal Einstein field equations in a conformal Gaussian gauge implies a similar solution over the manifold denoted by . The initial data induced by on will be denoted by .\n</p>", "<title>Structural properties of the evolution and subsidiary equations</title>", "<p>The conformal Gaussian gauge system leads to a <italic toggle=\"yes\">hyperbolic reduction</italic> of the extended conformal Einstein field equation (##FORMU##62##2.5##). The particular form of the resulting evolution equations is not required in this analysis, only general structural properties.</p>", "<p>The extended conformal Einstein field equations (##FORMU##62##2.5##) expressed in terms of a conformal Gaussian gauge imply evolution equations in the form of a symmetric hyperbolic system for the components and as in lemma 4.1. Now, since the evolution equations hold, the independent components of the zero-quantities\nnot determined by either the evolution equations or the gauge conditions satisfy a symmetric hyperbolic system which is homogeneous in the zero-quantities. As a result, if the zero-quantities vanish on a fiduciary space-like hypersurface , then they also vanish on the domain of dependence [##UREF##2##3##].</p>", "<title>The perturbative argument</title>", "<p>Let and denotes the <italic toggle=\"yes\">background solution</italic> being a solution to the evolution equations arising from the initial data prescribed on . Solutions to the evolution equations which can be regarded as a perturbation of the background solution are constructed by introducing a perturbative argument\nwith being a small perturbation. This means, in particular, that one can write\nThe components of , and are our unknowns. Making use of the decomposition (##FORMU##847##5.13##) and exploiting that is a solution to the conformal evolution equations one obtains the equations\nand\nNow, it is convenient to define\nand\nwhere\ndenote, respectively, expressions which are quadratic, linear and constant terms in the unknowns.</p>", "<p>In terms of the above expressions, it is possible to rewrite the system (5.14<italic toggle=\"yes\">a</italic>,) in the more concise form\nThis is a quasi-linear symmetric hyperbolic system for which theorem 3.4 can be applied to obtain an existence and stability result for a Cauchy problem with initial data .</p>", "<title>A solution to the Einstein field equations</title>", "<p>The evolution equations (5.14<italic toggle=\"yes\">a</italic>,) imply the same subsidiary system as for de Sitter-like spacetimes. Thus, the <italic toggle=\"yes\">propagation of the constraints</italic> follows from the same argument—see proposition 3.1 §4g. Given the propagation of the constraints and proposition 2.1, one has the metric obtained from the solution to the conformal evolution equations implies a solution to the vacuum Einstein field equations with positive Cosmological constant on .</p>", "<p>The main result of this discussion is contained in the following theorem:</p>", "<title>Theorem 5.4.</title>", "<p><italic toggle=\"yes\">Let</italic>\n\n<italic toggle=\"yes\">denote smooth initial data for the conformal evolution equations satisfying the conformal constraint equations on a hypersurface</italic>\n. <italic toggle=\"yes\">There exists</italic>\n\n<italic toggle=\"yes\">such that if</italic>\n<italic toggle=\"yes\">then there exists a unique</italic>\n\n<italic toggle=\"yes\">solution</italic>\n\n<italic toggle=\"yes\">to the vacuum Einstein field equation with positive Cosmological constant over</italic>\n\n<italic toggle=\"yes\">for</italic>\n\n<italic toggle=\"yes\">whose restriction to</italic>\n\n<italic toggle=\"yes\">implies the initial data</italic>\n. <italic toggle=\"yes\">Moreover, the solution</italic>\n\n<italic toggle=\"yes\">remains suitably close to the background solution</italic>\n.</p>", "<p>In particular, the resulting spacetime is a nonlinear perturbation of the sub-extremal Schwarzschild–de Sitter spacetime on a portion of the Cosmological region of the background solution which contains a portion of the asymptotic region.</p>" ]

[ "<title>Data accessibility</title>", "<p>This article has no additional data.</p>", "<title>Declaration of AI use</title>", "<p>We have not used AI-assisted technologies in creating this article.</p>", "<title>Authors' contributions</title>", "<p>M.M.: conceptualization, formal analysis, investigation, writing—original draft, writing—review and editing.</p>", "<title>Conflict of interest declaration</title>", "<p>We declare we have no competing interests.</p>", "<title>Funding</title>", "<p>No funding has been received for this article.</p>" ]

[ "<fig position=\"float\" id=\"RSTA20230040F1\"><label>Figure 1<x xml:space=\"preserve\">. </x></label><caption><p>Penrose–Carter diagram of the background solution. The conformal representation discussed in the main text has compact spatial sections of negative scalar curvature. The vertical lines and correspond to axes of symmetry. The solution has a singularity in the past and a space-like future conformal boundary. Hence, in our discussion we only consider future evolution of the initial hypersurface .</p></caption></fig>", "<fig position=\"float\" id=\"RSTA20230040F2\"><label>Figure 2<x xml:space=\"preserve\">. </x></label><caption><p>Penrose–Carter diagram of the sub-extremal Schwarzschild–de Sitter spacetime. The serrated line denotes the location of the singularity; the continuous black line denotes the conformal boundary; the dashed line shows the location of the black hole and Cosmological horizons denoted by and respectively. As described in the main text, these horizons are located at and . The excluded points and where the singularity seems to meet the conformal boundary correspond to asymptotic regions of the spacetime that does not belong to the singularity nor the conformal boundary.</p></caption></fig>", "<fig position=\"float\" id=\"RSTA20230040F3\"><label>Figure 3<x xml:space=\"preserve\">. </x></label><caption><p>The conformal geodesics are plotted on the Penrose–Carter diagram of the Cosmological region of the sub-extremal Schwarzschild–de Sitter spacetime. The purple line represents the initial hypersurface corresponding to . The red lines represent conformal geodesics with constant time leaving this initial hypersurface.</p></caption></fig>", "<fig position=\"float\" id=\"RSTA20230040F4\"><label>Figure 4<x xml:space=\"preserve\">. </x></label><caption><p>The red curves identify the time-like hypersurfaces and . The resulting spacetime manifold has compact spatial sections, , with the topology of .</p></caption></fig>" ]

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id=\"IM39\"><mml:msub><mml:mi>f</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM40\"><mml:msup><mml:mi>v</mml:mi><mml:mi>a</mml:mi></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM4\"><mml:math id=\"DM6\" display=\"block\"><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msup><mml:mi>v</mml:mi><mml:mi>b</mml:mi></mml:msup><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi>a</mml:mi></mml:msub><mml:msup><mml:mi>v</mml:mi><mml:mi>b</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi>b</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ξ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mi>v</mml:mi><mml:mi>c</mml:mi></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM5\"><mml:math id=\"DM7\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>a</mml:mi></mml:msub><mml:msup><mml:mi>v</mml:mi><mml:mi>b</mml:mi></mml:msup><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi>a</mml:mi></mml:msub><mml:msup><mml:mi>v</mml:mi><mml:mi>b</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi>b</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>β</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:msup><mml:mi>v</mml:mi><mml:mi>c</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mi>β</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mi>Ξ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mi>Ξ</mml:mi><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x3\"><label>2.3</label><mml:math id=\"DM8\" display=\"block\"><mml:msub><mml:mi>d</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>≡</mml:mo><mml:mi>Ξ</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mi>Ξ</mml:mi><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM41\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM42\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:mn mathvariant=\"bold\">3</mml:mn></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM43\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM44\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ω</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msup><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM6\"><mml:math id=\"DM9\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>η</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ω</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x4a\"><label>2.4<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM10\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:mover><mml:mi>Σ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x4b\"><label>2.4<italic toggle=\"yes\">b</italic></label><mml:math id=\"DM11\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mrow><mml:mover><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x4c\"><label>2.4<italic toggle=\"yes\">c</italic></label><mml:math id=\"DM12\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mi>d</mml:mi><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x4d\"><label>2.4<italic toggle=\"yes\">d</italic></label><mml:math id=\"DM13\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/><mml:mspace width=\"1em\"/></mml:mtd><mml:mtd><mml:mi>Λ</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mi>d</mml:mi><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM45\"><mml:mrow><mml:mover><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM46\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM7\"><mml:math id=\"DM14\" display=\"block\"><mml:mrow><mml:mover><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msub></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM8\"><mml:math id=\"DM15\" display=\"block\"><mml:mrow><mml:mover><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mi>Ξ</mml:mi><mml:mrow><mml:mover><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM47\"><mml:mi>d</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M2x5\"><label>2.5</label><mml:math id=\"DM16\" display=\"block\"><mml:mrow><mml:mover><mml:mi>Σ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mi>Λ</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM48\"><mml:mi>Ξ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM49\"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M2x6a\"><label>2.6<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM17\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>Ξ</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mi>Ξ</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x6b\"><label>2.6<italic toggle=\"yes\">b</italic></label><mml:math id=\"DM18\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi>γ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ξ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msup><mml:mi>Ξ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>6</mml:mn></mml:mfrac><mml:mi>λ</mml:mi><mml:msup><mml:mi>Ξ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>η</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x6c\"><label>2.6<italic toggle=\"yes\">c</italic></label><mml:math id=\"DM19\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:msub><mml:mi>ς</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x7\"><label>2.7</label><mml:math id=\"DM20\" display=\"block\"><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mi>γ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mi>ς</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM9\"><mml:math id=\"DM21\" display=\"block\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msup><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM50\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ξ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM10\"><mml:math id=\"DM22\" display=\"block\"><mml:mi>Ξ</mml:mi><mml:mo>≠</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mo movablelimits=\"true\" form=\"prefix\">det</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>η</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo>⊗</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM51\"><mml:mrow><mml:mi mathvariant=\"script\">U</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM11\"><mml:math id=\"DM23\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>Ξ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>η</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ω</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msup><mml:mo>⊗</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ω</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msup></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM52\"><mml:mrow><mml:mi mathvariant=\"script\">U</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM53\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM54\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM55\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM56\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM57\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM58\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">h</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM59\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM60\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">D</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM61\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi>D</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M2x8a\"><label>2.8<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM24\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mi>Σ</mml:mi><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>Ω</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x8b\"><label>2.8<italic toggle=\"yes\">b</italic></label><mml:math id=\"DM25\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:mi>Σ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msub><mml:mi>Ω</mml:mi><mml:mo>−</mml:mo><mml:mi>Ω</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x8c\"><label>2.8<italic toggle=\"yes\">c</italic></label><mml:math id=\"DM26\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:mi>Σ</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msup><mml:mi>Ω</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x8d\"><label>2.8<italic toggle=\"yes\">d</italic></label><mml:math id=\"DM27\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>Σ</mml:mi><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">l</mml:mi></mml:mrow></mml:msup><mml:mi>Ω</mml:mi><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x8e\"><label>2.8<italic toggle=\"yes\">e</italic></label><mml:math id=\"DM28\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">l</mml:mi></mml:mrow></mml:msup><mml:mi>Ω</mml:mi><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x8f\"><label>2.8<italic toggle=\"yes\">f</italic></label><mml:math id=\"DM29\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x8g\"><label>2.8<italic toggle=\"yes\">g</italic></label><mml:math id=\"DM30\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>χ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x8h\"><label>2.8<italic toggle=\"yes\">h</italic></label><mml:math id=\"DM31\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>6</mml:mn><mml:mi>Ω</mml:mi><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>Σ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msub><mml:mi>Ω</mml:mi><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msup><mml:mi>Ω</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x8i\"><label>2.8<italic toggle=\"yes\">i</italic></label><mml:math id=\"DM32\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>Ω</mml:mi><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x8j\"><label>2.8<italic toggle=\"yes\">j</italic></label><mml:math id=\"DM33\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:msub><mml:mi>l</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>Ω</mml:mi><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>χ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>χ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi 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id=\"IM68\"><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM69\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM70\"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM71\"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM72\"><mml:mi>s</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM14\"><mml:math id=\"DM36\" display=\"block\"><mml:mi>s</mml:mi><mml:mo>≡</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>4</mml:mn></mml:mfrac><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msup><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>a</mml:mi></mml:msup><mml:mi>Ξ</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>24</mml:mn></mml:mfrac><mml:mi>R</mml:mi><mml:mi>Ξ</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM73\"><mml:mi>R</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM74\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM75\"><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM76\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM77\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM78\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM79\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM80\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM81\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M2x9a\"><label>2.9<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM37\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mi mathvariant=\"normal\">♯</mml:mi></mml:msup></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x9b\"><label>2.9<italic toggle=\"yes\">b</italic></label><mml:math id=\"DM38\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi mathvariant=\"normal\">♯</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi mathvariant=\"normal\">♭</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">L</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM82\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">L</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM83\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">∇</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM84\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">v</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM85\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M2x10\"><label>2.10</label><mml:math id=\"DM39\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">v</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">v</mml:mi></mml:mrow><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>−</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">v</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">v</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mi mathvariant=\"normal\">♯</mml:mi></mml:msup><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM86\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM87\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM88\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM89\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM90\"><mml:mi>b</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM91\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM92\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM93\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M2x11\"><label>2.11</label><mml:math id=\"DM40\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>Θ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x12\"><label>2.12</label><mml:math id=\"DM41\" display=\"block\"><mml:mi>Θ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>Θ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>Θ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mrow><mml:mover><mml:mi>Θ</mml:mi><mml:mo>¨</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM15\"><mml:math id=\"DM42\" display=\"block\"><mml:msub><mml:mi>Θ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mrow><mml:mo>≡</mml:mo></mml:mrow><mml:mi>Θ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>Θ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mrow><mml:mo>≡</mml:mo></mml:mrow><mml:mrow><mml:mover><mml:mi>Θ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>Θ</mml:mi><mml:mo>¨</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mrow><mml:mo>≡</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>Θ</mml:mi><mml:mo>¨</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM16\"><mml:math id=\"DM43\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>Θ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:msub><mml:mi>Θ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mi>Θ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi>Θ</mml:mi><mml:mo>¨</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi mathvariant=\"normal\">♯</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>6</mml:mn></mml:mfrac><mml:mi>λ</mml:mi><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM94\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM95\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM96\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM97\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM98\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM99\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM100\"><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM101\"><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM102\"><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M2x13\"><label>2.13</label><mml:math id=\"DM44\" display=\"block\"><mml:mfrac><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>τ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>Θ</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mtext>so that </mml:mtext><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:mrow><mml:mrow><mml:mi>τ</mml:mi></mml:mrow></mml:msubsup><mml:mfrac><mml:mtext>ds</mml:mtext><mml:mrow><mml:mi>Θ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mtext>s</mml:mtext><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM103\"><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M2x14\"><label>2.14</label><mml:math id=\"DM45\" display=\"block\"><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>Θ</mml:mi><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">x</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mi>ϖ</mml:mi><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mi mathvariant=\"normal\">♭</mml:mi></mml:msup><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mi>ϖ</mml:mi><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM104\"><mml:mi>b</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM105\"><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M2x15a\"><label>2.15<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM46\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi mathvariant=\"normal\">♯</mml:mi></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M2x15b\"><label>2.15<italic toggle=\"yes\">b</italic></label><mml:math id=\"DM47\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>β</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:msup><mml:mrow/><mml:mi mathvariant=\"normal\">♭</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">L</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo>,</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">L</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi><mml:mi mathvariant=\"normal\">♭</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM106\"><mml:msup><mml:mrow><mml:mover><mml:mi>β</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mi mathvariant=\"normal\">♯</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM17\"><mml:math id=\"DM48\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">L</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>6</mml:mn></mml:mfrac><mml:mi>λ</mml:mi><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM107\"><mml:mrow><mml:mi mathvariant=\"script\">U</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM108\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM109\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM110\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM111\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM112\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM113\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM114\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM115\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>a</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM116\"><mml:msub><mml:mi>f</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM18\"><mml:math id=\"DM49\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM117\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM19\"><mml:math id=\"DM50\" display=\"block\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">∂</mml:mi></mml:mrow><mml:mi>τ</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM118\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM119\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM120\"><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM121\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM122\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM123\"><mml:mi>p</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM124\"><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM125\"><mml:mover><mml:mi>x</mml:mi><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM126\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM127\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM128\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM129\"><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM130\"><mml:mrow><mml:mi mathvariant=\"script\">P</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM131\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>m</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM132\"><mml:mrow><mml:mi mathvariant=\"script\">P</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM133\"><mml:mrow><mml:mi mathvariant=\"script\">P</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM134\"><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM135\"><mml:mi>N</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M3x1\"><label>3.1</label><mml:math id=\"DM51\" display=\"block\"><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM136\"><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>t</mml:mi><mml:mo>≤</mml:mo><mml:mi>T</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM137\"><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM138\"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M3x2\"><label>3.2</label><mml:math id=\"DM52\" display=\"block\"><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM139\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM140\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM141\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM142\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM143\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM144\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo>×</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM145\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM146\"><mml:mrow><mml:mi mathvariant=\"bold\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM147\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM148\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM149\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM150\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM151\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM152\"><mml:msup><mml:mi>H</mml:mi><mml:mi>s</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM153\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM154\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM155\"><mml:mi>s</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM156\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM157\"><mml:msup><mml:mi>H</mml:mi><mml:mi>s</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM20\"><mml:math id=\"DM53\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mi>μ</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>a</mml:mi><mml:msup><mml:mrow/><mml:mi>μ</mml:mi></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:msub><mml:mi>b</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mi>μ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM158\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM159\"><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM21\"><mml:math id=\"DM54\" display=\"block\"><mml:mi>a</mml:mi><mml:msup><mml:mrow/><mml:mi>μ</mml:mi></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>:</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>m</mml:mi></mml:msup><mml:mo stretchy=\"false\">→</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM22\"><mml:math id=\"DM55\" display=\"block\"><mml:msub><mml:mi>b</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>:</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>m</mml:mi></mml:msup><mml:mo stretchy=\"false\">→</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM160\"><mml:msubsup><mml:mi>C</mml:mi><mml:mn>0</mml:mn><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM161\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM162\"><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM163\"><mml:mi>ϕ</mml:mi><mml:mo>∈</mml:mo><mml:msubsup><mml:mi>C</mml:mi><mml:mn>0</mml:mn><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM164\"><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM165\"><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi>u</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:msubsup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM23\"><mml:math id=\"DM56\" display=\"block\"><mml:munder><mml:mo movablelimits=\"true\" form=\"prefix\">sup</mml:mo><mml:mrow><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:munder><mml:mo>∥</mml:mo><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:msub><mml:mo>∥</mml:mo><mml:mi>s</mml:mi></mml:msub><mml:mo>&lt;</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>y</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≡</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>y</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>−</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM166\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM167\"><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM168\"><mml:msubsup><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mi>μ</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM169\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM170\"><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi>u</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM171\"><mml:msub><mml:mi>b</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM172\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM173\"><mml:msup><mml:mi>H</mml:mi><mml:mi>s</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM174\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mi>μ</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM175\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM176\"><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi>u</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">P</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM177\"><mml:mrow><mml:mi mathvariant=\"bold\">B</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM178\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM179\"><mml:msup><mml:mi>H</mml:mi><mml:mi>s</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">P</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM180\"><mml:mi>s</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM181\"><mml:mi>s</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>3</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>=</mml:mo><mml:mn>5</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM182\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mi>μ</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM183\"><mml:mrow><mml:mi mathvariant=\"bold\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM184\"><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM185\"><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>t</mml:mi><mml:mo>≤</mml:mo><mml:mi>T</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM186\"><mml:msubsup><mml:mi>a</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mi>μ</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM187\"><mml:msub><mml:mi>b</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM188\"><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi>u</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM189\"><mml:msup><mml:mi>H</mml:mi><mml:mi>s</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM190\"><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM191\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM192\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM193\"><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">↦</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM194\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM195\"><mml:msup><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM196\"><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi>u</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM197\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM198\"><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM199\"><mml:msup><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM200\"><mml:msup><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM201\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM202\"><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">↦</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM203\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM204\"><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM205\"><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi>u</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM206\"><mml:mi>t</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM207\"><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">↦</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM208\"><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi>u</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM209\"><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM210\"><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM211\"><mml:msup><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM212\"><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM213\"><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">↦</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM214\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM215\"><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi>u</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM216\"><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM217\"><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM218\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mi>μ</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM219\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>m</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM220\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM221\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM222\"><mml:mrow><mml:mi mathvariant=\"bold\">v</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM223\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM224\"><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM225\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM226\"><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>≤</mml:mo><mml:mi>T</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM24\"><mml:math id=\"DM57\" display=\"block\"><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>∈</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>;</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>∪</mml:mo><mml:msup><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msup><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>;</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM227\"><mml:msup><mml:mi>T</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM228\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM229\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM230\"><mml:mi>s</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>3</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>=</mml:mo><mml:mn>5</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM231\"><mml:mi>Ω</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM232\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM233\"><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:munder><mml:mi>v</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM25\"><mml:math id=\"DM58\" display=\"block\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>v</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>&lt;</mml:mo><mml:mi>ω</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM234\"><mml:mi>ω</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM235\"><mml:msub><mml:mi>v</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mi>s</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⊂</mml:mo><mml:msup><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM26\"><mml:math id=\"DM59\" display=\"block\"><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mi>μ</mml:mi></mml:msup><mml:mo>:</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>×</mml:mo><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">⟶</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM27\"><mml:math id=\"DM60\" display=\"block\"><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo>:</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>×</mml:mo><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">⟶</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM236\"><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM237\"><mml:mi>N</mml:mi><mml:mo>×</mml:mo><mml:mi>N</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM238\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM239\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo>∈</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>;</mml:mo><mml:msubsup><mml:mi>C</mml:mi><mml:mi>b</mml:mi><mml:mi>s</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM240\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo>∈</mml:mo><mml:mi>L</mml:mi><mml:mi>i</mml:mi><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>;</mml:mo><mml:msubsup><mml:mi>C</mml:mi><mml:mi>b</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM241\"><mml:mrow><mml:mi mathvariant=\"bold\">B</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>;</mml:mo><mml:msubsup><mml:mi>C</mml:mi><mml:mi>b</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM242\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">B</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:msup><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>;</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mi>s</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>∩</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>;</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM243\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">B</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≡</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM244\"><mml:msubsup><mml:mi>C</mml:mi><mml:mi>b</mml:mi><mml:mi>r</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM245\"><mml:msubsup><mml:mi>C</mml:mi><mml:mi>b</mml:mi><mml:mi>r</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Ω</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM246\"><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM247\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mi>μ</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM248\"><mml:mrow><mml:mi mathvariant=\"bold\">B</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM249\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM250\"><mml:msup><mml:mi>H</mml:mi><mml:mi>s</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM251\"><mml:msub><mml:mi>v</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM252\"><mml:msub><mml:mi>R</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM253\"><mml:mi>s</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM254\"><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM255\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM256\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mi>M</mml:mi></mml:msub><mml:mo>⊂</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM257\"><mml:msubsup><mml:mo>∪</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>M</mml:mi></mml:msubsup><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:msub><mml:mrow/><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM258\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:msub><mml:mrow/><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM259\"><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:msub><mml:mrow/><mml:mi>i</mml:mi></mml:msub><mml:mo>≡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:msup><mml:mrow/><mml:mi>α</mml:mi></mml:msup><mml:msub><mml:mrow/><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM260\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:msub><mml:mrow/><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM261\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>⊂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM262\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM263\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:msub><mml:mrow/><mml:mo>⋆</mml:mo></mml:msub><mml:mo>:</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM264\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM265\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:msub><mml:mrow/><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM266\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi>i</mml:mi><mml:mo>⋆</mml:mo></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM267\"><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM268\"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>⋆</mml:mo></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM269\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>⊂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM270\"><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM271\"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>⋆</mml:mo></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM272\"><mml:mrow><mml:mi mathvariant=\"script\">E</mml:mi></mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>⋆</mml:mo></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">→</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM273\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>⊂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM274\"><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM28\"><mml:math id=\"DM61\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"script\">E</mml:mi></mml:mrow><mml:mo>:</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⟶</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM275\"><mml:mi mathvariant=\"bold-italic\">u</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM276\"><mml:mrow><mml:mi mathvariant=\"script\">E</mml:mi></mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM277\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM278\"><mml:mrow><mml:mi mathvariant=\"script\">E</mml:mi></mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM279\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM280\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>⊂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM281\"><mml:mi>C</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM282\"><mml:mrow><mml:mi mathvariant=\"script\">U</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM283\"><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM29\"><mml:math id=\"DM62\" display=\"block\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">E</mml:mi></mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>≤</mml:mo><mml:mi>C</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM284\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM285\"><mml:mrow><mml:mi mathvariant=\"script\">E</mml:mi></mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM286\"><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM287\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM30\"><mml:math id=\"DM63\" display=\"block\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>M</mml:mi></mml:munderover><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi>i</mml:mi><mml:mo>⋆</mml:mo></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM288\"><mml:mrow><mml:mi mathvariant=\"script\">E</mml:mi></mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>⋆</mml:mo></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM31\"><mml:math id=\"DM64\" display=\"block\"><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM32\"><mml:math id=\"DM65\" display=\"block\"><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">E</mml:mi></mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>⋆</mml:mo></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∈</mml:mo><mml:mi>H</mml:mi><mml:msup><mml:mrow/><mml:mi>m</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:msup><mml:mrow/><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">for</mml:mi></mml:mrow><mml:mo> </mml:mo><mml:mi>m</mml:mi><mml:mo>≥</mml:mo><mml:mn>4.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM289\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM33\"><mml:math id=\"DM66\" display=\"block\"><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∩</mml:mo><mml:msup><mml:mi>I</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>∖</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">∅</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM290\"><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM291\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM292\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM293\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">D</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>≡</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM294\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM295\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">D</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM296\"><mml:mrow><mml:mi mathvariant=\"script\">E</mml:mi></mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>⋆</mml:mo></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM297\"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM298\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">D</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>⊂</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM299\"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM300\"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:mi>j</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM301\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">D</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM302\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">D</mml:mi></mml:mrow><mml:mi>j</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM303\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">D</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>⋂</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">D</mml:mi></mml:mrow><mml:mi>j</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM304\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM305\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM306\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">D</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM307\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM308\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>×</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM309\"><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:munder><mml:mo movablelimits=\"true\" form=\"prefix\">min</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>M</mml:mi></mml:mrow></mml:munder><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM310\"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:mi>M</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM311\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">D</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">D</mml:mi></mml:mrow><mml:mi>M</mml:mi></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM34\"><mml:math id=\"DM67\" display=\"block\"><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM35\"><mml:math id=\"DM68\" display=\"block\"><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∈</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">for</mml:mi></mml:mrow><mml:mo> </mml:mo><mml:mi>m</mml:mi><mml:mo>≥</mml:mo><mml:mn>4</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM312\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM313\"><mml:mi>δ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM314\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM315\"><mml:mi>δ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM316\"><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM317\"><mml:mi>T</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM318\"><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM319\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>×</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM36\"><mml:math id=\"DM69\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>∈</mml:mo><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>×</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM320\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM321\"><mml:mi>δ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM322\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>×</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM323\"><mml:mi>ϵ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM324\"><mml:mi>T</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM325\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mi>ϵ</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM326\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mi>ϵ</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⊂</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM327\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM328\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM329\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM330\"><mml:mi>T</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM331\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mi>ϵ</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM332\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM333\"><mml:mi>ϵ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM334\"><mml:mi>ϵ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM335\"><mml:mi>T</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM336\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>i</mml:mi><mml:mi>i</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM337\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>⋆</mml:mo><mml:mi>n</mml:mi></mml:msubsup><mml:mo>∈</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mi>ϵ</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM37\"><mml:math id=\"DM70\" display=\"block\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>⋆</mml:mo><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">⟶</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">as</mml:mi></mml:mrow><mml:mo> </mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM338\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM339\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≡</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>⋆</mml:mo><mml:mi>n</mml:mi></mml:msubsup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM38\"><mml:math id=\"DM71\" display=\"block\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>−</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">⟶</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">as</mml:mi></mml:mrow><mml:mo> </mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM340\"><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM341\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM342\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mo>×</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M4x1\"><label>4.1</label><mml:math id=\"DM72\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>t</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>sinh</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>⁡</mml:mo><mml:mi>t</mml:mi><mml:mo> </mml:mo><mml:mrow><mml:mover><mml:mi>γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM343\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">γ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM344\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM345\"><mml:msup><mml:mi>r</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">γ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM346\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">γ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM39\"><mml:math id=\"DM73\" display=\"block\"><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>k</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi>γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi>γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM347\"><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM40\"><mml:math id=\"DM74\" display=\"block\"><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mi>γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>6.</mml:mn></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM41\"><mml:math id=\"DM75\" display=\"block\"><mml:mrow><mml:mover><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>12</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x2\"><label>4.2</label><mml:math id=\"DM76\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mrow><mml:mover><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM348\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM349\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM350\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM351\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>≡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>s</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>α</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">∂</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM352\"><mml:mi>α</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM42\"><mml:math id=\"DM77\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM353\"><mml:mi>α</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM354\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">∂</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM355\"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM43\"><mml:math id=\"DM78\" display=\"block\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>⋆</mml:mo></mml:msub><mml:mrow><mml:mo>∈</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM356\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM357\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM358\"><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">↦</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM359\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM44\"><mml:math id=\"DM79\" display=\"block\"><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">β</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">′</mml:mi><mml:mi mathvariant=\"normal\">♭</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>t</mml:mi><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM360\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM45\"><mml:math id=\"DM80\" display=\"block\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo> </mml:mo><mml:mtext>arctanh</mml:mtext><mml:mo> </mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>τ</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>τ</mml:mi><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mo> </mml:mo><mml:mi>τ</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM361\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM362\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M4x3\"><label>4.3</label><mml:math id=\"DM81\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\">(</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>τ</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>τ</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">γ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM363\"><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM364\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM46\"><mml:math id=\"DM82\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>Θ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">with</mml:mi></mml:mrow><mml:mo> </mml:mo><mml:mi>Θ</mml:mi><mml:mo>≡</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x4\"><label>4.4</label><mml:math id=\"DM83\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>τ</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>τ</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM365\"><mml:mi>τ</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mrow><mml:mi>τ</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM366\"><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM367\"><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">h</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM368\"><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM47\"><mml:math id=\"DM84\" display=\"block\"><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">h</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>≡</mml:mo><mml:msup><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM369\"><mml:mrow><mml:mover><mml:mi>D</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM370\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"fraktur\">D</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM371\"><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM372\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">∂</mml:mi></mml:mrow><mml:mi>τ</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM373\"><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">g</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM374\"><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">β</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM375\"><mml:mrow><mml:mi mathvariant=\"bold\">∇</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM376\"><mml:msub><mml:mi>Γ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM377\"><mml:msub><mml:mi>Γ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM378\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM379\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM380\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM381\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM382\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">γ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM383\"><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM384\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">α</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msup><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM48\"><mml:math id=\"DM85\" display=\"block\"><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">c</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">c</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">α</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">c</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM49\"><mml:math id=\"DM86\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">γ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mrow><mml:mi 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mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM387\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">ω</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msup><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM388\"><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">ω</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi 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mathvariant=\"bold-italic\">ω</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>τ</mml:mi><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">α</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM52\"><mml:math id=\"DM89\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>η</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">ω</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msup><mml:mo>⊗</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">ω</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM389\"><mml:msub><mml:mrow><mml:mover><mml:mi>γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM390\"><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo mathvariant=\"bold\">˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM391\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">c</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM53\"><mml:math id=\"DM90\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>D</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">c</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi 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mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">α</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>D</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">c</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM392\"><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM393\"><mml:mrow><mml:mover><mml:mi mathvariant=\"bold\">∇</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM394\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">e</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM54\"><mml:math id=\"DM91\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">e</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">e</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">ω</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">e</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x5\"><label>4.5</label><mml:math id=\"DM92\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>τ</mml:mi></mml:mfrac><mml:msub><mml:mrow><mml:mover><mml:mi>γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>τ</mml:mi></mml:mfrac><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>τ</mml:mi></mml:mfrac><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM395\"><mml:msub><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM396\"><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>×</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM397\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">∂</mml:mi></mml:mrow><mml:mi>τ</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM398\"><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">↦</mml:mo><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM399\"><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M4x6a\"><label>4.6<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM93\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>D</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>Σ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">n</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>τ</mml:mi><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>s</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x6b\"><label>4.6<italic toggle=\"yes\">b</italic></label><mml:math id=\"DM94\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>d</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mo>∗</mml:mo></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>d</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM400\"><mml:mi>d</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM401\"><mml:mi>c</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM402\"><mml:mrow><mml:mi mathvariant=\"script\">U</mml:mi></mml:mrow><mml:mo>⊂</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM55\"><mml:math id=\"DM95\" display=\"block\"><mml:msub><mml:mi>Θ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>Θ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>Θ</mml:mi><mml:mo>¨</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM403\"><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM404\"><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M4x7\"><label>4.7</label><mml:math id=\"DM96\" display=\"block\"><mml:mi>Θ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM405\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM406\"><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM407\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM408\"><mml:msub><mml:mrow><mml:mover><mml:mi>Θ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM409\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM410\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM411\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM412\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">τ</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM413\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>a</mml:mi></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM56\"><mml:math id=\"DM97\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">τ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">L</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM414\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM57\"><mml:math id=\"DM98\" display=\"block\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">∂</mml:mi></mml:mrow><mml:mi>τ</mml:mi></mml:msub><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mi>e</mml:mi><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mi>μ</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mrow/><mml:mi>μ</mml:mi></mml:msup><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM415\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM416\"><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM417\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M4x8a\"><label>4.8<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM99\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mi>ν</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mi>ν</mml:mi></mml:msup><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x8b\"><label>4.8<italic toggle=\"yes\">b</italic></label><mml:math id=\"DM100\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x8c\"><label>4.8<italic toggle=\"yes\">c</italic></label><mml:math id=\"DM101\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x8d\"><label>4.8<italic toggle=\"yes\">d</italic></label><mml:math id=\"DM102\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>Ξ</mml:mi><mml:mrow><mml:mover><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mi>c</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:mrow></mml:msub><mml:msup><mml:mi>g</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x8e\"><label>4.8<italic toggle=\"yes\">e</italic></label><mml:math id=\"DM103\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mi>ϵ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mi>d</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mi>ϵ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:mrow></mml:msubsup><mml:msup><mml:mi>d</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>χ</mml:mi><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>χ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x8f\"><label>4.8<italic toggle=\"yes\">f</italic></label><mml:math id=\"DM104\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:msup><mml:mi>d</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msup><mml:mi>ϵ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>a</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:msub><mml:mi>ϵ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>χ</mml:mi><mml:msup><mml:mi>d</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msubsup><mml:mi>χ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mi>d</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM418\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM419\"><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM420\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">L</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM421\"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM422\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow></mml:mrow><mml:mo>≡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM423\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow></mml:mrow><mml:mo>≡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">L</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM424\"><mml:mrow><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:mrow><mml:mo>≡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM425\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M4x9a\"><label>4.9<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM105\" display=\"block\"><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">K</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">Q</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">L</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x9b\"><label>4.9<italic toggle=\"yes\">b</italic></label><mml:math id=\"DM106\" display=\"block\"><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">A</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">A</mml:mi></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM426\"><mml:mrow><mml:mi mathvariant=\"bold\">I</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM427\"><mml:mrow><mml:mi mathvariant=\"bold\">K</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM428\"><mml:mrow><mml:mi mathvariant=\"bold\">Q</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM429\"><mml:mrow><mml:mi mathvariant=\"bold\">L</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM430\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mi>μ</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM431\"><mml:mrow><mml:mi mathvariant=\"bold\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M4x10a\"><label>4.10<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM107\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>Σ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>Σ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>Σ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mi> </mml:mi><mml:mspace width=\"1em\"/><mml:mo>−</mml:mo><mml:mrow><mml:mover><mml:mi>Σ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>Θ</mml:mi><mml:mi>ϵ</mml:mi><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mi>ϵ</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">h</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:mi>Λ</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">h</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x10b\"><label>4.10<italic toggle=\"yes\">c</italic></label><mml:math id=\"DM108\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:mi> </mml:mi><mml:mspace width=\"1em\"/><mml:mo>+</mml:mo><mml:mi>ϵ</mml:mi><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msup><mml:mi>δ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:msup><mml:mi>d</mml:mi><mml:msup><mml:mrow/><mml:mrow><mml:mo>∗</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mi>S</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mi> </mml:mi><mml:mspace width=\"1em\"/><mml:mo>−</mml:mo><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msub><mml:mi>d</mml:mi><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msub><mml:mi>d</mml:mi><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x10c\"><label>4.10<italic toggle=\"yes\">e</italic></label><mml:math id=\"DM109\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:mi> </mml:mi><mml:mspace width=\"1em\"/><mml:mo>+</mml:mo><mml:mi>Θ</mml:mi><mml:mi>γ</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mi>d</mml:mi><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>Θ</mml:mi><mml:mi>γ</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mi>d</mml:mi><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>ϵ</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:mi>ϵ</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">h</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:mi>Λ</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">h</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>Ω</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:mi>d</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:mi>d</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mi> </mml:mi><mml:mspace width=\"1em\"/><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mover><mml:mi>Σ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msub><mml:mi>d</mml:mi><mml:msup><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>ς</mml:mi><mml:msup><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:mi>d</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi>χ</mml:mi><mml:mi>Ω</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x10d\"><label>4.10<italic toggle=\"yes\">g</italic></label><mml:math id=\"DM110\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>γ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msub><mml:mo>;</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x10e\"><label>4.10<italic toggle=\"yes\">h</italic></label><mml:math id=\"DM111\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>γ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msub><mml:mi>γ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>γ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>γ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>η</mml:mi><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>β</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>γ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mi>λ</mml:mi><mml:msup><mml:mi>Θ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x10f\"><label>4.10<italic toggle=\"yes\">i</italic></label><mml:math id=\"DM112\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>ς</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mi>ς</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mi>ς</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mover><mml:mi>Σ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">j</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">k</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM432\"><mml:msub><mml:mrow><mml:mover><mml:mi>Σ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM433\"><mml:msup><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM434\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM435\"><mml:msub><mml:mrow><mml:mover><mml:mi>Λ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM436\"><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM437\"><mml:msub><mml:mi>γ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM438\"><mml:msub><mml:mi>ς</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM439\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">X</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>≡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Σ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mi>Λ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mi>ς</mml:mi></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M4x11\"><label>4.11</label><mml:math id=\"DM113\" display=\"block\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">X</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">H</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">X</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM440\"><mml:mrow><mml:mi mathvariant=\"bold\">H</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM441\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM442\"><mml:mi>b</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM58\"><mml:math id=\"DM114\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM443\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM444\"><mml:mi>f</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM445\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M4x12a\"><label>4.12<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM115\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"2em\"/><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M4x12b\"><label>4.12<italic toggle=\"yes\">b</italic></label><mml:math id=\"DM116\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"2em\"/><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM446\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM447\"><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM448\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM449\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM450\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM59\"><mml:math id=\"DM117\" display=\"block\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">K</mml:mtext></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">Q</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM451\"><mml:mi>b</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM452\"><mml:mi>b</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM60\"><mml:math id=\"DM118\" display=\"block\"><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mtable rowspacing=\"4pt\" columnspacing=\"1em\"><mml:mtr><mml:mtd><mml:mrow><mml:mtext mathvariant=\"bold\">I</mml:mtext></mml:mrow></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mrow><mml:mtext mathvariant=\"bold\">I</mml:mtext></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mtable rowspacing=\"4pt\" columnspacing=\"1em\"><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM61\"><mml:math id=\"DM119\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≡</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">Q</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">L</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">K</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM62\"><mml:math id=\"DM120\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">Q</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mtable rowspacing=\"4pt\" columnspacing=\"1em\"><mml:mtr><mml:mtd><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">Q</mml:mtext></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">L</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mtable rowspacing=\"4pt\" columnspacing=\"1em\"><mml:mtr><mml:mtd><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">Q</mml:mtext></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">Q</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mtext mathvariant=\"bold\">L</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">K</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mtable rowspacing=\"4pt\" columnspacing=\"1em\"><mml:mtr><mml:mtd><mml:mrow><mml:mtext mathvariant=\"bold\">K</mml:mtext></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM453\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M4x13\"><label>4.13</label><mml:math id=\"DM121\" display=\"block\"><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM454\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM455\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM456\"><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">A</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>μ</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo>⋆</mml:mo></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM457\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM458\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">B</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM459\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM460\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM461\"><mml:msub><mml:mi>d</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM462\"><mml:msup><mml:mi>C</mml:mi><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM463\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM464\"><mml:msup><mml:mi>C</mml:mi><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM465\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM466\"><mml:mi>τ</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mstyle><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM467\"><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM468\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM63\"><mml:math id=\"DM122\" display=\"block\"><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM64\"><mml:math id=\"DM123\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mrow><mml:mo>∈</mml:mo></mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">for</mml:mi></mml:mrow><mml:mo> </mml:mo><mml:mi>m</mml:mi><mml:mo>≥</mml:mo><mml:mn>4</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM469\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM470\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM471\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">X</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>≡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Σ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mi>Λ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mi>ς</mml:mi></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM65\"><mml:math id=\"DM124\" display=\"block\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">X</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">H</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">X</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM66\"><mml:math id=\"DM125\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">X</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM472\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">X</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM473\"><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">[</mml:mo></mml:mrow><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM474\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM475\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM67\"><mml:math id=\"DM126\" display=\"block\"><mml:mrow><mml:mover><mml:mi>Σ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>Ξ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mrow/><mml:mi>c</mml:mi></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>Λ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM68\"><mml:math id=\"DM127\" display=\"block\"><mml:mi>δ</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>γ</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mi>ς</mml:mi><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM476\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM477\"><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM478\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM479\"><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">[</mml:mo></mml:mrow><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM480\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>Θ</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM481\"><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM482\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM483\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM484\"><mml:mi>ε</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM69\"><mml:math id=\"DM128\" display=\"block\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>&lt;</mml:mo><mml:mi>ε</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>m</mml:mi><mml:mo>≥</mml:mo><mml:mn>4</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM485\"><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM486\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM487\"><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM488\"><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM489\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM490\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM491\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM492\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM493\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM494\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM495\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>φ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M5x1\"><label>5.1</label><mml:math id=\"DM129\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi></mml:mrow><mml:mi>r</mml:mi></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>t</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi></mml:mrow><mml:mi>r</mml:mi></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>r</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi mathvariant=\"bold-italic\">σ</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM70\"><mml:math id=\"DM130\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">σ</mml:mi></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>θ</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>θ</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>sin</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>⁡</mml:mo><mml:mi>θ</mml:mi><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>φ</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>φ</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM496\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM497\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>φ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM71\"><mml:math id=\"DM131\" display=\"block\"><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>r</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>θ</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mi>φ</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M5x2\"><label>5.2</label><mml:math id=\"DM132\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>t</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>r</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi mathvariant=\"bold-italic\">σ</mml:mi><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM72\"><mml:math id=\"DM133\" display=\"block\"><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mfrac><mml:mi>M</mml:mi><mml:mi>r</mml:mi></mml:mfrac><mml:mo>−</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mi>M</mml:mi><mml:mo>≡</mml:mo><mml:mn>2</mml:mn><mml:mi>m</mml:mi><mml:msqrt><mml:mfrac><mml:mi>λ</mml:mi><mml:mn>3</mml:mn></mml:mfrac></mml:msqrt><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM498\"><mml:mi>M</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM499\"><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM500\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM501\"><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM502\"><mml:mi>λ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM503\"><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM73\"><mml:math id=\"DM134\" display=\"block\"><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>r</mml:mi></mml:mfrac><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM504\"><mml:msub><mml:mi>r</mml:mi><mml:mi>b</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM505\"><mml:msub><mml:mi>r</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM506\"><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM507\"><mml:msub><mml:mi>r</mml:mi><mml:mo>−</mml:mo></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM74\"><mml:math id=\"DM135\" display=\"block\"><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mi>r</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM508\"><mml:msub><mml:mi>r</mml:mi><mml:mi>b</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM509\"><mml:msub><mml:mi>r</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M5x3a\"><label>5.3<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM136\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi>r</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt></mml:mfrac><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>ϕ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M5x3b\"><label>5.3<italic toggle=\"yes\">b</italic></label><mml:math id=\"DM137\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi>r</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>ϕ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt><mml:mi>sin</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>ϕ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M5x3c\"><label>5.3<italic toggle=\"yes\">c</italic></label><mml:math id=\"DM138\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:msub><mml:mi>r</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>ϕ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt><mml:mi>sin</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>ϕ</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM510\"><mml:mi>ϕ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M5x4\"><label>5.4</label><mml:math id=\"DM139\" display=\"block\"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>ϕ</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mfrac><mml:mi>π</mml:mi><mml:mn>2</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM511\"><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:mi>M</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>2</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM512\"><mml:mi>ϕ</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM513\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">H</mml:mi></mml:mrow><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM514\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">H</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM515\"><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM516\"><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM517\"><mml:mrow><mml:mi mathvariant=\"script\">Q</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM518\"><mml:mrow><mml:msup><mml:mi mathvariant=\"script\">Q</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM519\"><mml:mi>r</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM75\"><mml:math id=\"DM140\" display=\"block\"><mml:msub><mml:mi>r</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>&lt;</mml:mo><mml:mi>r</mml:mi><mml:mo>&lt;</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM520\"><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM521\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM522\"><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM523\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM524\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>φ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM525\"><mml:msub><mml:mi>Θ</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM526\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM76\"><mml:math id=\"DM141\" display=\"block\"><mml:msub><mml:mi>Θ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>Θ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM527\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM77\"><mml:math id=\"DM142\" display=\"block\"><mml:msubsup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo>⊥</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>β</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>Θ</mml:mi><mml:mo>⋆</mml:mo><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msub><mml:mi>Θ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M5x5\"><label>5.5</label><mml:math id=\"DM143\" display=\"block\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mspace width=\"1em\"/><mml:mi>t</mml:mi><mml:msup><mml:mrow/><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mrow/><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msqrt><mml:msub><mml:mi>D</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:msqrt></mml:mfrac><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>r</mml:mi><mml:msup><mml:mrow/><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mrow/><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>β</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>⋆</mml:mo></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>β</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo>⋆</mml:mo></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M5x6\"><label>5.6</label><mml:math id=\"DM144\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">ℓ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>t</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>r</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>r</mml:mi><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM528\"><mml:mrow><mml:mi mathvariant=\"script\">Q</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM529\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">Q</mml:mi></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM78\"><mml:math id=\"DM145\" display=\"block\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>∙</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo> </mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo> </mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mo>∙</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mo>∙</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM530\"><mml:msub><mml:mi>t</mml:mi><mml:mo>∙</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM531\"><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>∙</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∩</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup><mml:mo>≠</mml:mo><mml:mi>∅</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM532\"><mml:msup><mml:mi>β</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M5x7\"><label>5.7</label><mml:math id=\"DM146\" display=\"block\"><mml:msup><mml:mi>r</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msqrt><mml:msup><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:msqrt><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msup><mml:mrow/><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>′</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM533\"><mml:mi>γ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM534\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM79\"><mml:math id=\"DM147\" display=\"block\"><mml:msubsup><mml:mi>t</mml:mi><mml:mo>⋆</mml:mo><mml:mo>′</mml:mo></mml:msubsup><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>γ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM535\"><mml:msub><mml:mi>D</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM536\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM537\"><mml:msubsup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo><mml:mo>′</mml:mo></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM538\"><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM539\"><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM540\"><mml:msub><mml:mi>τ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M5x8\"><label>5.8</label><mml:math id=\"DM148\" display=\"block\"><mml:mi>Θ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>4</mml:mn></mml:mfrac><mml:msup><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM541\"><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM542\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M5x9\"><label>5.9</label><mml:math id=\"DM149\" display=\"block\"><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">arctanh</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mrow><mml:mi mathvariant=\"normal\">tanh</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM80\"><mml:math id=\"DM150\" display=\"block\"><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>±</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">as</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>±</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM543\"><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM544\"><mml:mi>r</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM81\"><mml:math id=\"DM151\" display=\"block\"><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:msqrt><mml:mfrac><mml:mrow><mml:mover><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mo>−</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac></mml:msqrt><mml:mo> </mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mrow><mml:mover><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM545\"><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM82\"><mml:math id=\"DM152\" display=\"block\"><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mspace width=\"1em\"/><mml:mtext>as</mml:mtext><mml:mspace width=\"1em\"/><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM546\"><mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M5x10\"><label>5.10</label><mml:math id=\"DM153\" display=\"block\"><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:msub><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:msub><mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">ℓ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">z</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:mrow></mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM547\"><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM548\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">ℓ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM549\"><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">ℓ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM550\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">ℓ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM551\"><mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM552\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM553\"><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">ℓ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mrow><mml:msubsup><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM83\"><mml:math id=\"DM154\" display=\"block\"><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mi>M</mml:mi><mml:msup><mml:mi>r</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>r</mml:mi><mml:mo>≡</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM554\"><mml:mi>r</mml:mi><mml:mo>≥</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>&gt;</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM555\"><mml:mi>ω</mml:mi><mml:mo>≡</mml:mo><mml:mi>Θ</mml:mi><mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM84\"><mml:math id=\"DM155\" display=\"block\"><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:msup><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>r</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:msub><mml:mi>ρ</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:mfrac><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msup><mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM556\"><mml:mi>ω</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM85\"><mml:math id=\"DM156\" display=\"block\"><mml:mi>ω</mml:mi><mml:mo>≥</mml:mo><mml:mfrac><mml:msub><mml:mi>r</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:msub><mml:mi>ρ</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:msup><mml:mi>τ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mn>4</mml:mn></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM557\"><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>±</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM558\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM559\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM560\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM561\"><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM562\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>∙</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM563\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM564\"><mml:mi>z</mml:mi><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM86\"><mml:math id=\"DM157\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msup><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo> </mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>t</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>z</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>z</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant=\"bold-italic\">σ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM87\"><mml:math id=\"DM158\" display=\"block\"><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≡</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>z</mml:mi></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM565\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM88\"><mml:math id=\"DM159\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>Ξ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>Ξ</mml:mi><mml:mo>≡</mml:mo><mml:mi>z</mml:mi><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M5x11\"><label>5.11</label><mml:math id=\"DM160\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>t</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>z</mml:mi><mml:mo>⊗</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>z</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant=\"bold-italic\">σ</mml:mi><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM566\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi>S</mml:mi><mml:mi>d</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mi>I</mml:mi></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM89\"><mml:math id=\"DM161\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi>S</mml:mi><mml:mi>d</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mi>I</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo> </mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo> </mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>&gt;</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM567\"><mml:mi>S</mml:mi><mml:mi>d</mml:mi><mml:msub><mml:mi>S</mml:mi><mml:mi>I</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM568\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi>S</mml:mi><mml:mi>d</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mo>~</mml:mo></mml:mover></mml:mrow><mml:mi>I</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM569\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM570\"><mml:mi>z</mml:mi><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM571\"><mml:mi>z</mml:mi><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M5x12\"><label>5.12</label><mml:math id=\"DM162\" display=\"block\"><mml:mi>S</mml:mi><mml:mi>d</mml:mi><mml:msub><mml:mi>S</mml:mi><mml:mi>I</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mo>×</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo> </mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo> </mml:mo><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≤</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM572\"><mml:msub><mml:mi>z</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM573\"><mml:msub><mml:mi>r</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>&gt;</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM574\"><mml:mi>ψ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM575\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM90\"><mml:math id=\"DM163\" display=\"block\"><mml:msup><mml:mi>ψ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>:</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>×</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mo>∙</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mo>∙</mml:mo></mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>×</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mo>∙</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mo>∙</mml:mo></mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">↦</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM576\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>φ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM577\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>φ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM578\"><mml:mi>S</mml:mi><mml:mi>d</mml:mi><mml:msub><mml:mi>S</mml:mi><mml:mi>I</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM579\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM580\"><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>∙</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM581\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM91\"><mml:math id=\"DM164\" display=\"block\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo>∙</mml:mo></mml:msub><mml:mrow><mml:mo>≡</mml:mo></mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>×</mml:mo><mml:mo 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mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>Θ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM584\"><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">∂</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">∂</mml:mi></mml:mrow><mml:mi>r</mml:mi></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM93\"><mml:math id=\"DM166\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">ω</mml:mi></mml:mrow><mml:mo>≡</mml:mo><mml:msub><mml:mi>ϵ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">ℓ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo>,</mml:mo><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM585\"><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">ω</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM94\"><mml:math id=\"DM167\" display=\"block\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>Θ</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ω</mml:mi></mml:mrow><mml:mi mathvariant=\"normal\">♯</mml:mi></mml:msup><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM586\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM587\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">2</mml:mn></mml:mrow><mml:mo>⋆</mml:mo></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM588\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">3</mml:mn></mml:mrow><mml:mo>⋆</mml:mo></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM589\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM590\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">3</mml:mn></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM591\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo>∙</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM592\"><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM593\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">2</mml:mn></mml:mrow><mml:mo>⋆</mml:mo></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">3</mml:mn></mml:mrow><mml:mo>⋆</mml:mo></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM594\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM595\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">3</mml:mn></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM596\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM597\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM598\"><mml:msub><mml:mi>t</mml:mi><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM599\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM600\"><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM601\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM602\"><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">∂</mml:mi></mml:mrow><mml:mi>r</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM603\"><mml:msup><mml:mi>r</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM604\"><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM605\"><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM95\"><mml:math id=\"DM168\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>≈</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>z</mml:mi></mml:mfrac><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>z</mml:mi><mml:mspace width=\"1em\"/><mml:mtext>for</mml:mtext><mml:mo> </mml:mo><mml:mi>z</mml:mi><mml:mo>≈</mml:mo><mml:mn>0.</mml:mn></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM96\"><mml:math id=\"DM169\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mi>Ξ</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">d</mml:mtext></mml:mrow><mml:mi>Ξ</mml:mi></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM606\"><mml:mi>Ξ</mml:mi><mml:mo>=</mml:mo><mml:mi>z</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM607\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM608\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM609\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">β</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM610\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">∇</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM611\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">∇</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM612\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">∇</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM613\"><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM97\"><mml:math id=\"DM170\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM614\"><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>μ</mml:mi></mml:msub><mml:msup><mml:mrow/><mml:mi>ν</mml:mi></mml:msup><mml:msub><mml:mrow/><mml:mi>λ</mml:mi></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM98\"><mml:math id=\"DM171\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:msup><mml:mrow/><mml:mi>t</mml:mi></mml:msup><mml:msub><mml:mrow/><mml:mi>z</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>z</mml:mi></mml:msub><mml:msup><mml:mrow/><mml:mi>z</mml:mi></mml:msup><mml:msub><mml:mrow/><mml:mi>z</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>M</mml:mi><mml:mi>z</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>M</mml:mi><mml:mi>z</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:msup><mml:mrow/><mml:mi>z</mml:mi></mml:msup><mml:msub><mml:mrow/><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>z</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>M</mml:mi><mml:mi>z</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>M</mml:mi><mml:mi>z</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>φ</mml:mi></mml:msub><mml:msup><mml:mrow/><mml:mi>θ</mml:mi></mml:msup><mml:msub><mml:mrow/><mml:mi>φ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mi>cos</mml:mi><mml:mo>⁡</mml:mo><mml:mi>θ</mml:mi><mml:mi>sin</mml:mi><mml:mo>⁡</mml:mo><mml:mi>θ</mml:mi><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>θ</mml:mi></mml:msub><mml:msup><mml:mrow/><mml:mi>φ</mml:mi></mml:msup><mml:msub><mml:mrow/><mml:mi>φ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>cot</mml:mi><mml:mo>⁡</mml:mo><mml:mi>θ</mml:mi><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM615\"><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM616\"><mml:msub><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM617\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM618\"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM619\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM99\"><mml:math id=\"DM172\" display=\"block\"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mi>b</mml:mi></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msup><mml:mi>d</mml:mi><mml:mrow><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi>a</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:msup><mml:mrow/><mml:mo>∗</mml:mo></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mover><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn mathvariant=\"bold\">0</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mrow/><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM620\"><mml:mi>d</mml:mi><mml:msup><mml:mrow/><mml:mo>∗</mml:mo></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM621\"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>c</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM100\"><mml:math id=\"DM173\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mi>M</mml:mi><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:msup><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>M</mml:mi><mml:mi>z</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mi>M</mml:mi><mml:mn>2</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>φ</mml:mi><mml:mi>φ</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mi>M</mml:mi><mml:mn>2</mml:mn></mml:mfrac><mml:msup><mml:mi>sin</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>⁡</mml:mo><mml:mi>θ</mml:mi><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM622\"><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM623\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM624\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM101\"><mml:math id=\"DM174\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:mi>z</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>M</mml:mi><mml:mi>z</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>z</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:mi>z</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>M</mml:mi><mml:mi>z</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>M</mml:mi><mml:mi>z</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>φ</mml:mi><mml:mi>φ</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msup><mml:mi>sin</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>⁡</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>M</mml:mi><mml:mi>z</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM625\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo>∙</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM626\"><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM627\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">∇</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM102\"><mml:math id=\"DM175\" display=\"block\"><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mrow><mml:mover><mml:mi>β</mml:mi><mml:mo 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stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM630\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM631\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo>∙</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM632\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo>∙</mml:mo></mml:msub><mml:mo>⊂</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi 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mathvariant=\"script\">T</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>t</mml:mi><mml:mo>∙</mml:mo></mml:msub></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM642\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">T</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi>t</mml:mi><mml:mo>∙</mml:mo></mml:msub></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM643\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM644\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>t</mml:mi><mml:mo>∙</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM645\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>t</mml:mi><mml:mo>∙</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM646\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo>∙</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM647\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM648\"><mml:mi>S</mml:mi><mml:mi>d</mml:mi><mml:msub><mml:mi>S</mml:mi><mml:mi>I</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM649\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo 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mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM663\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow></mml:mrow><mml:mo>≡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">L</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM664\"><mml:mrow><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:mrow><mml:mo>≡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mrow><mml:mi 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mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow/><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">d</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mrow><mml:mover><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mi>Λ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">c</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mi>γ</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mi>ς</mml:mi><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">a</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">b</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM665\"><mml:msub><mml:mrow><mml:mrow><mml:mover><mml:mi mathvariant=\"script\">S</mml:mi><mml:mo mathvariant=\"script\" stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM666\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>≡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">v</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM667\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM668\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM669\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM104\"><mml:math id=\"DM177\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM670\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M5x13\"><label>5.13</label><mml:math id=\"DM178\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM671\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM672\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM673\"><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM674\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">u</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M5x14a\"><label>5.14<italic toggle=\"yes\">a</italic></label><mml:math id=\"DM179\" display=\"block\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">K</mml:mtext></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">Q</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">Q</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">L</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">L</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040M5x14b\"><label>5.14<italic toggle=\"yes\">b</italic></label><mml:math id=\"DM180\" display=\"block\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">I</mml:mtext></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>.</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM105\"><mml:math id=\"DM181\" display=\"block\"><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mtable rowspacing=\"4pt\" columnspacing=\"1em\"><mml:mtr><mml:mtd><mml:mrow><mml:mtext mathvariant=\"bold\">I</mml:mtext></mml:mrow></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mrow><mml:mtext mathvariant=\"bold\">I</mml:mtext></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mtable rowspacing=\"4pt\" columnspacing=\"1em\"><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:msup><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">e</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM106\"><mml:math id=\"DM182\" display=\"block\"><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≡</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">Q</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">L</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">K</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo></mml:math></disp-formula>", "<disp-formula id=\"RSTA20230040UM107\"><mml:math id=\"DM183\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">Q</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mtable rowspacing=\"4pt\" columnspacing=\"1em\"><mml:mtr><mml:mtd><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">Q</mml:mtext></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">L</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mtable rowspacing=\"4pt\" columnspacing=\"1em\"><mml:mtr><mml:mtd><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mtext mathvariant=\"bold\">Q</mml:mtext></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">Q</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">Γ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:mtd><mml:mtd><mml:mrow><mml:mtext mathvariant=\"bold\">L</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">L</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">K</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mtable rowspacing=\"4pt\" columnspacing=\"1em\"><mml:mtr><mml:mtd><mml:mrow><mml:mtext mathvariant=\"bold\">K</mml:mtext></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">υ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow></mml:mtd><mml:mtd><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mn>0</mml:mn></mml:mtd><mml:mtd><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mi>Γ</mml:mi><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM675\"><mml:mi>b</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040M5x15\"><label>5.15</label><mml:math id=\"DM184\" display=\"block\"><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>τ</mml:mi></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">A</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">B</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>τ</mml:mi><mml:mo>,</mml:mo><mml:munder><mml:mi>x</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM676\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM677\"><mml:mi>b</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM678\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>Θ</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">g</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM679\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>≡</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>∙</mml:mo></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM680\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM681\"><mml:msub><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM682\"><mml:mi>ε</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230040UM108\"><mml:math id=\"DM185\" display=\"block\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mtext mathvariant=\"bold\">u</mml:mtext></mml:mrow><mml:mo>˘</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>&lt;</mml:mo><mml:mi>ε</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>m</mml:mi><mml:mo>≥</mml:mo><mml:mn>4</mml:mn></mml:math></disp-formula>", "<inline-formula><mml:math id=\"IM683\"><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM684\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM685\"><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>×</mml:mo><mml:msub><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM686\"><mml:msub><mml:mrow><mml:mover><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM687\"><mml:msub><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM688\"><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM689\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM690\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold\">u</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM691\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM692\"><mml:mi>Θ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM693\"><mml:mi>τ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM694\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM695\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM696\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM697\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM698\"><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow><mml:mo stretchy=\"false\">~</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM699\"><mml:mrow><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mo>˚</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM700\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mo>∙</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM701\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mo>∙</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM702\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM703\"><mml:mi>I</mml:mi><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM704\"><mml:mi>I</mml:mi><mml:mo>⊂</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM705\"><mml:mrow><mml:mi mathvariant=\"script\">Q</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM706\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">Q</mml:mi></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM707\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mo>⋆</mml:mo></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM708\"><mml:msub><mml:mi>r</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM709\"><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM710\"><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM711\"><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM712\"><mml:mi>λ</mml:mi></mml:math></inline-formula>" ]

[]

[]

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[ "<fn-group><fn id=\"FN1\"><label>1</label><p>The value for the Cosmological constant is conventional and set for convenience. This analysis can be carried out for any other positive value of .</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>In this paper, we physically motivate and give an overview of mathematical constructions of solutions to the linearized Einstein vacuum equations around Schwarzschild with the following features:\n<list list-type=\"simple\"><list-item><label>(i)<x xml:space=\"preserve\"> </x></label><p>They describe the far field region of a system of infalling masses coming from the infinite past and following approximately hyperbolic Keplerian orbits.</p></list-item><list-item><label>(ii)<x xml:space=\"preserve\"> </x></label><p>They have no incoming radiation from past null infinity.</p></list-item><list-item><label>(iii)<x xml:space=\"preserve\"> </x></label><p>They violate peeling near past null infinity: near , the extremal component of the Weyl tensor (a.k.a. ) decays like rather than .</p></list-item><list-item><label>(iv)<x xml:space=\"preserve\"> </x></label><p>They violate peeling near future null infinity: near , the other extremal component of the Weyl tensor, (a.k.a. ), decays like rather than . In particular, the constructed class of spacetimes does not admit a smooth null infinity/conformal compactification.</p></list-item><list-item><label>(v)<x xml:space=\"preserve\"> </x></label><p>They decay slower towards spacelike infinity than assumed in most stability works. Both and decay like along , as opposed to the -decay rate assumed in [##UREF##0##1##].</p></list-item></list>\nIn the list above, the bullet points (ii) and (iii) play the role of <italic toggle=\"yes\">scattering data assumptions</italic>, from which we will construct a (unique) <italic toggle=\"yes\">scattering solution</italic>. These scattering data assumptions, in turn, are motivated by the physical considerations involved in (i). Finally, (iv) and (v) play the role of theorems that we prove as we analyse asymptotic properties of this scattering solution.</p>", "<p>The detailed construction and asymptotic analysis of these solutions will be presented in the follow-up paper [##UREF##1##2##], see also [##UREF##2##3##]. The purpose of the present paper is to give a rough outline of the main problems, ideas and results of the construction, as well as a discussion of the physical motivation for the construction.</p>", "<p>The remainder of this introduction will provide some background and context for the general problem studied in the series of papers <italic toggle=\"yes\">The Case Against Smooth Null Infinity</italic>.</p>", "<title>Isolated systems</title>", "<p>The desire to study systems in isolation is ubiquitous in physics: given a physical process, we first identify which parts of its surroundings and interactions we can neglect. We then seek a mathematical model that treats the process as independent of those surroundings and interactions. Finally, we attempt to study the resulting model—the <italic toggle=\"yes\">isolated system</italic>—using suitable mathematical methods, hoping that its predictions will accurately approximate the actual physical process.</p>", "<p>Consider, for instance, the motion of the Earth and Sun around each other within Newtonian theory. A simple way to study this in isolation is to disregard all other matter in the universe, i.e. to model the surroundings of the Earth and Sun by <italic toggle=\"yes\">vacuum</italic>, and to only consider (Newtonian) gravitational interactions between the Earth and Sun. By taking into account further interactions with other planets, etc., the resulting predictions can be made more and more accurate; but the conceptual approach to <italic toggle=\"yes\">isolate</italic> the system remains the same.</p>", "<p>Two reasons why this conceptual approach is simple within Newtonian theory are that Newtonian theory assumes that matter moves on a fixed geometric background, and that the theory has a simple and unambiguous realization of vacuum.</p>", "<p>In general relativity (GR), however, the situation is different. The Einstein equations,\n\nare equations for the geometry of an a priori unknown spacetime, and, even when we set the matter part as well as the cosmological constant to 0 (so as to ignore cosmological effects), the remaining Einstein vacuum equations,\n\nstill form a highly complicated system of geometric PDEs with a large space of solutions, the dynamical degrees of freedom of these solutions being carried by gravitational radiation which, if focused enough, can even create black holes [##UREF##3##4##]. Intuitively, we want to exclude such events from occurring and expect that the gravitational radiation of isolated systems should disperse at large distances; we want the spacetime to asymptote to the Minkowski spacetime in some very weak sense. Making this precise requires a thorough understanding of the asymptotic structure of gravitational radiation, and, indeed, notions of isolated systems in general relativity are often masked behind expressions such as ‘asymptotic flatness’.</p>", "<p>In the next two subsections, we will give an overview of (and some commentary on) some of the historically most influential works and approaches to this problem, and explain some shortcomings of these approaches. These approaches model isolated systems based on what can be called <italic toggle=\"yes\">the asymptotic problem</italic>. We will then go over certain arguments against these approaches, which also take into account what can be called the <italic toggle=\"yes\">generation problem</italic> and the <italic toggle=\"yes\">propagation problem</italic> of gravitational radiation.</p>", "<p>We will also argue that the matter of how to model isolated systems is not only relevant from an epistemological point of view, but that it has very concrete implications for important mathematical problems and even astrophysical effects such as gravitational wave tails.</p>", "<title>The works of Bondi <italic toggle=\"yes\">et al.</italic> and the peeling property</title>", "<p>In a series of papers by Bondi <italic toggle=\"yes\">et al.</italic> [##UREF##4##5##–##UREF##7##8##], the authors introduced various ideas whose influence on future works concerning gravitational radiation is difficult to overestimate. While the initial project had the aim of understanding conditions that guarantee that a given asymptotic region of spacetime only features outgoing radiation, its scope quickly widened; the authors essentially opened up a new field of study, centred around the question: to what degree can we understand aspects of spacetime by only analysing its possible asymptotic behaviour, i.e. by its behaviour near infinity? For instance, can we deduce from soft arguments certain symmetries and regularity properties that gravitational radiation or spacetimes should satisfy, without studying the actual ‘interior’ of spacetime? And can this potentially give us insights a posteriori into the interior of spacetimes? (Note that several modern approaches to quantum gravity ask exactly this question, with countless conjectured correspondences between the bulk and the boundary of spacetime.)</p>", "<p>To go back to the concrete question at hand, the authors argued in [##UREF##5##6##,##UREF##6##7##] that the gravitational field of (vacuum) spacetimes with no incoming radiation should, by analogy to the behaviour of linearized gravity around Minkowski (cf. §2), admit power series expansions in along <italic toggle=\"yes\">outgoing</italic> null geodesics, denoting the luminosity parameter along these geodesics. Operating under this assumption, a system of coordinates (Bondi coordinates) was constructed, and it was cautiously suggested in [##UREF##8##9##] that the notion of an isolated system/asymptotic flatness might be captured by the existence of such Bondi coordinates. It was further shown that the Weyl curvature tensor ‘peels’ towards [##UREF##5##6##,##UREF##7##8##,##UREF##9##10##,##UREF##10##11##], a statement which in Newman–Penrose notation can be written as\n\nwhere denote the Newman–Penrose scalars [##UREF##9##10##] ( and are defined in equation (##FORMU##101##2.1##) of the present paper), and are some functions independent of .^{<xref rid=\"FN1\" ref-type=\"fn\">1</xref>}</p>", "<p>At the technical level, this peeling property greatly simplifies the Einstein equations, transforming them into a set of a few relatively simple evolutionary equations and a set of algebraic identities. Its imposition, therefore, led to a stark increase in activity and progress in studying the previously almost impenetrable Einstein equations.</p>", "<p>At the physical level, however, there was debate from early on whether the peeling property captures the no incoming radiation condition, or whether it is even logically related to it (it only is for gravity linearized around the Minkowski spacetime), and several works [##UREF##12##13##–##UREF##15##16##] considered more general expansions of the Weyl curvature tensor than equation (##FORMU##25##1.3##). The approach of <italic toggle=\"yes\">imposing certain asymptotic behaviour</italic> and then studying the consequences, however, remained popular ever since. Furthermore, many of the early discoveries that came out of the general framework set up by Bondi [##UREF##4##5##], such as the Bondi mass-loss formula, or the BMS symmetry group at infinity [##UREF##6##7##–##UREF##8##9##,##UREF##16##17##], were soon understood to be somewhat robust with respect to a broad class of violations of peeling.</p>", "<title>Penrose’s smooth conformal compactification</title>", "<p>Only very briefly after the appearance of Sachs [##UREF##5##6##], a concept of pristine geometric elegance and clarity that, in particular, captures the peeling property discussed above was introduced by Penrose [##UREF##17##18##,##UREF##18##19##]. This concept gave a new, very concrete proposal for modelling isolated systems; we paraphrase it here:</p>", "<p>An isolated astrophysical system should asymptotically become empty and approach the Minkowski spacetime. The Minkowski spacetime has the property that it can be conformally compactified and, upon compactification, admits smooth conformal null boundaries called future and past null infinity ( and ), respectively. The hope is now that spacetimes describing isolated astrophysical processes should share this property. Precisely this idea is captured by the definition of <italic toggle=\"yes\">asymptotic simplicity</italic>: without going into details, a physical spacetime is called asymptotically simple if there exists another (unphysical) manifold and a conformal factor such that the conformally rescaled spacetime isometrically embeds into the unphysical manifold, and can be smoothly extended to its null boundaries. This property is also referred to as the spacetime possessing <italic toggle=\"yes\">a smooth null infinity</italic>.</p>", "<p>Owing to the conformal properties of the Weyl tensor, spacetimes with a smooth null infinity satisfy the peeling property of equation (##FORMU##25##1.3##) near . Similarly, along ingoing null geodesics near , they satisfy\n\nNote that the original argument relating equation (##FORMU##25##1.3##) to the absence of incoming radiation would similarly relate equation (##FORMU##35##1.4##) to the absence of outgoing radiation.</p>", "<p>Since asymptotic simplicity defines a class of <italic toggle=\"yes\">global</italic> spacetimes, it has played a crucial role in understanding various global aspects of spacetimes, for instance related to black holes (which in most textbooks are defined using asymptotic simplicity).^{<xref rid=\"FN2\" ref-type=\"fn\">2</xref>}</p>", "<p>Again, the conceptual approach of asymptotic simplicity is to understand aspects of spacetime by certain considerations concerning their asymptotic behaviour, which in turn are motivated by soft/formal considerations, that is to say, considerations that do not directly take into account the physical structure of the system under consideration.</p>", "<p>Nonetheless, even though there was not yet a direct connection established between the generation of gravitational waves by some physical system and their asymptotic behaviour as posited by peeling or asymptotic simplicity, and even though the conformal irregularity of spatial infinity in the presence of non-trivial ADM mass was later understood to cause trouble to the simultaneous regularity of <italic toggle=\"yes\">both</italic> future <italic toggle=\"yes\">and</italic> past null infinity [##UREF##19##20##,##UREF##20##21##], there was still a class of physical spacetimes for which it was relatively clear that they would possess a smooth null infinity, namely <italic toggle=\"yes\">past-stationary spacetimes</italic> (cf. ##FIG##0##figure 1##). Such spacetimes, sometimes referred to as <italic toggle=\"yes\">Bondi bomb spacetimes</italic>, are modelled to be exactly stationary up until some sudden, violent explosion of gravitational waves (which is triggered by something so weak that it is neglected within the model). It is relatively straightforward to see that such a spacetime satisfies peeling at least until the retarded time at which the bomb explodes. Furthermore, by propagation of regularity arguments, the spacetimes will then continue to satisfy peeling for any finite retarded time [##UREF##21##22##,##UREF##22##23##].\n</p>", "<p>From the mathematical perspective, this Bondi bomb scenario corresponds to the study of <italic toggle=\"yes\">forward</italic> Cauchy problems where the initial data are assumed to be compactly supported. (Note that Bondi bomb spacetimes as depicted in ##FIG##0##figure 1##, evolving from stationary to non-stationary spacetimes cannot arise from backwards evolution from Cauchy data for standard matter models; one would have to manually insert the explosion of the bomb into the matter model!)</p>", "<title>The case against smooth conformal compactification</title>", "<p>The previous two subsections discussed the <italic toggle=\"yes\">asymptotic problem of gravitational radiation</italic> on relatively abstract grounds. We will now, in addition, take into account the <italic toggle=\"yes\">generation</italic> (what is the structure of gravitational radiation <italic toggle=\"yes\">generated</italic> by a physical process?) and the <italic toggle=\"yes\">propagation problem</italic> (how does this radiation <italic toggle=\"yes\">propagate</italic> through spacetime?) of gravitational radiation. Of course, as long as the generation problem is studied at <italic toggle=\"yes\">finite</italic> time, one inevitably has to make <italic toggle=\"yes\">a choice</italic> for the asymptotic behaviour of radiation along that time slice. On the other hand, if we study the generation and propagation problem in the <italic toggle=\"yes\">infinite past</italic>, we will obtain a dynamical prediction on the asymptotic behaviour of gravitational radiation!</p>", "<p>One of the most basic physical systems that any notion of isolated system should be able to describe is the two-body problem. Let us, for instance, think about two masses approaching each other from the infinite past. If these masses move at non-relativistic relative speeds, then Newtonian theory provides a reasonable approximation for the movement of these masses. For similar reasons that the hydrogen atom is classically unstable, we are then forced to consider orbits which are unbound in the infinite past, i.e. either approximately parabolic or hyperbolic Keplerian orbits [##UREF##23##24##].</p>", "<p>Next, given this set-up, we can attempt to understand the generation of gravitational radiation by these masses coming in from the infinite past using post-Newtonian approximations for gravity linearized around Minkowski. It then turns out, as was already understood very early on [##UREF##24##25##,##UREF##25##26##], that the quadrupolar radiation generated in this way fails to satisfy peeling near (equation (##FORMU##35##1.4##)): The study of the <italic toggle=\"yes\">generation problem</italic> shows that the asymptotic structure of gravitational radiation as posited by peeling or asymptotic simplicity fails, and it <italic toggle=\"yes\">constructively provides a different prediction for this asymptotic structure</italic>. A brief reconstruction of this argument along with a generalization to higher multipoles is provided in §2 of the present paper.</p>", "<p>However, the same early arguments, by extending the assumed validity of the post-Minkowskian predictions all the way to infinite advanced times, still found that the future null infinity of such spacetimes should be smooth, as they were still made within the framework of linearized gravity around Minkowski. The general idea that peeling fails near but holds true near was somewhat substantiated by the evidence of Schmidt &amp; Stewart [##UREF##19##20##] and Porrill &amp; Stewart [##UREF##20##21##] concerning the incompatibility between peeling at and at .</p>", "<p>New light on the matter was shed by Damour in 1986 [##UREF##26##27##] (see also [##UREF##27##28##]): in addition to taking into account the generation problem (to find that peeling fails near past null infinity), Damour also analysed the <italic toggle=\"yes\">propagation problem</italic> of gravitational radiation by perturbatively studying a subset of the equations of linearized gravity around Schwarzschild and propagating the asymptotics near past infinity towards . The result of this preliminary analysis was that peeling also fails near future null infinity—the rigorous and complete analysis of this propagation problem is the topic of the present paper.</p>", "<p>Further influential heuristics against smooth conformal compactification were put forth by Christodoulou in 2002 [##UREF##28##29##]. To this date, his argument remains the only argument treating the full nonlinear Einstein equations without symmetry: Christodoulou suggests that any solution to (##FORMU##21##1.2##) arising from suitable initial data must necessarily violate peeling near provided that there is no incoming radiation from and provided that the limit of the ingoing shear along (a.k.a. the News function) decays as predicted by Einstein’s quadrupole formula for a system of infalling masses following approximately hyperbolic Keplerian orbits in the infinite past.</p>", "<p>Note that this counterargument against previous notions of isolated systems can be interpreted to be itself based upon a different prescription of modelling isolated systems, the assumption being that isolated systems can be modelled by spacetimes arising from the initial data assumed in [##UREF##0##1##] (cf. the definition of C–K compatible initial data in remark 1.1 of Kehrberger [##UREF##29##30##]). In particular, these spacetimes are assumed to have certain decay towards spatial infinity .</p>", "<p>The mathematically rigorous study of the case against smooth null infinity was then initiated in a series of papers of which the present one is the fourth. While the previous approaches towards modelling isolated systems either imposed certain asymptotic behaviour on gravitational radiation at some finite time, be it towards (peeling, Bondi coordinates…) or towards (as in [##UREF##0##1##]), or by setting it to 0 as in the Bondi bomb spacetimes, the approach towards modelling isolated systems taken in this series is to let them arise as solutions to <italic toggle=\"yes\">scattering problems in the infinite past</italic>. One of the current goals of this series of papers is the following: we ultimately want to set up a scattering problem for the nonlinear Einstein vacuum equations where the scattering data are informed by the post-Newtonian prediction (the <italic toggle=\"yes\">generation</italic> problem^{<xref rid=\"FN3\" ref-type=\"fn\">3</xref>}) for a system of masses coming in from the infinite past; and we then want to solve this scattering problem (i.e. understand the <italic toggle=\"yes\">propagation</italic> problem) in order to understand the asymptotic structure of gravitational radiation near past <italic toggle=\"yes\">and</italic> future null infinity (##FIG##1##figure 2##).\n</p>", "<p>As a precursor to this problem, we will first study the corresponding problem for the equations of linearized gravity around Schwarzschild. The purpose of the current paper is to give an overview over this problem, its full and lengthy mathematical treatment being contained in follow-up work. We hope that we will be able to convey to the reader many of the main ideas (as the mechanism behind the failure of peeling is remarkably simple to describe) and results, and to also elucidate various points that we only touched upon in the exposition above.</p>", "<p>Since the series was initiated by an overview of Christodoulou’s argument against peeling, we also reflect upon this argument, see §8.</p>", "<title>But why does this matter?</title>", "<p>Before we move on to the main body of the paper, we want to put some effort into outlining a few reasons why the issue of null infinity being smooth or irregular is relevant.</p>", "<p>First, there is the epistemological level: if we have a widely used and accepted concept that aims to model isolated systems, such as that of asymptotic simplicity, we should critically examine its justifications and shortcomings, and understand what kind of physics can, or cannot, be captured by the concept. To be quite concrete, we will see in the main body of the paper that a spacetime can, somewhat ironically, essentially only be expected to have a smooth future null infinity if there is <italic toggle=\"yes\">no outgoing radiation</italic> (or a fine-tuned mix of outgoing and incoming radiation) in the infinite past; any radiation that reaches spatial infinity will generically lead to an irregular future null infinity. Thus, operating under the assumption of a smooth null infinity excludes any physical scenario where there is radiation in the infinite past and effectively only leaves us with various types of Bondi bomb scenarios.^{<xref rid=\"FN4\" ref-type=\"fn\">4</xref>}</p>", "<p>Second, there is the physical level: any physicist used to making approximations wherever possible should now feel inclined to point out that, from a measurement perspective, the violent outburst of gravitational radiation during, say, the merger of two black holes, should completely dominate the radiation that the black holes emitted in the ‘infinite’ past (when they were ‘infinitely far away from each other’), effectively also turning it into a Bondi bomb scenario. However, as was observed in [##UREF##30##31##,##UREF##31##32##] (see also [##UREF##32##33##]), such intuitive arguments must necessarily become incorrect eventually: no matter the strength of any outbursts of gravitational radiation at finite time, if one measures for sufficiently long times, the late-time tails of gravitational radiation will be dominated by the effects of the irregularity of (which in turn arises from the radiation in the infinite past). This is because the late-time decay rates associated with radiation coming from the infinite past and radiation coming from some finite-time explosion are different, and the decay rate of the latter is faster (see also §7). Conversely, quantitatively studying the effects of gravitational radiation in the infinite past is exactly what allows one to control the error made by setting this very radiation to vanish. The methods and results sketched in this paper can thus also be seen as tools that can be used to try and justify the assumption of past-stationarity.</p>", "<p>Third, there is the mathematical level. The largest part of mathematical problems, e.g. concerning the stability or instability of explicit solutions such as the Minkowski spacetime, singular/regular behaviour in the interior of black holes, or the study of precise asymptotics of gravitational waves, are studied from the perspective of the Cauchy problem: initial data are posed on an asymptotically flat (terminating at ) or an asymptotically null (terminating at ) hypersurface. One is thus forced to make an assumption on the asymptotic decay behaviour of the initial data set. Since there is no a priori way to understand the asymptotic decay of an initial dataset, any choice for this decay might thus be thought of as an ad hoc choice of modelling a class of isolated systems. Conversely, the approach taken in this paper towards modelling isolated systems taken here dynamically produces certain decay behaviour towards or towards , and will, therefore, affect all of the mathematical problems listed above.</p>", "<p>For instance, the well-known and much studied Price’s law (see e.g. [##UREF##33##34##–##UREF##40##41##]) governing the late-time behaviour of gravitational radiation has mostly been studied under the assumption of compactly supported initial data (##FIG##0##figure 1##). It has also been studied under the slightly weaker assumption of initial data satisfying peeling and recently, in [##UREF##30##31##,##UREF##31##32##], under assumptions that violate peeling, and it was shown that this crucially changes the late-time behaviour, cf. our comment two paragraphs above. But since the precise late-time behaviour of gravitational radiation in the exterior of black holes affects the regularity in the interior of black holes, this, too, is influenced by the regularity of (see e.g. [##UREF##41##42##–##UREF##44##45##]).</p>", "<p>The situation is similar for stability studies of explicit solutions to equation (##FORMU##21##1.2##): let us recall the monumental work on the stability of Minkowski spacetime [##UREF##0##1##]. In this work, Christodoulou and Klainerman consider perturbations of the Minkowski initial data that satisfy inter alia ( denoting the metric on the initial data surface )), and show that the corresponding solutions disperse and asymptotically approach the Minkowski spacetime with quantitative rates. Since these quantitative rates were slower than predicted by equation (##FORMU##25##1.3##), the results of Christodoulou &amp; Klainerman [##UREF##0##1##] were, in particular, incorrectly interpreted by some to disprove peeling, even though [##UREF##0##1##] only showed that there are certainly classes of spacetimes with decay behaviour towards weaker than peeling—the question whether peeling is ‘true’ or not can only be answered by taking into account physical arguments. And indeed, as we will see in the present paper, the initial decay is itself too strong to model certain physically relevant perturbations—the present work constructs spacetimes which, restricted to , asymptotically decay like\n\nOut of all the various proofs of stability of the Minkowski spacetime floating about in the literature (e.g. [##UREF##0##1##,##UREF##45##46##–##UREF##50##51##] and references therein), only Bieri’s class of perturbations [##UREF##51##52##] and that of [##UREF##52##53##] is large enough to allow for such slow decay—all other proofs assume in particular that !</p>", "<p>Somewhat surprisingly, even though we must thus conclude that most stability works starting from a Cauchy hypersurface make assumptions too strong to model a physically relevant class of perturbations, we will see that these very perturbations exhibit sufficient decay towards so as to still be compatible with most stability works that start from an asymptotically null initial data hypersurface [##UREF##53##54##–##UREF##55##56##] (see §8).^{<xref rid=\"FN5\" ref-type=\"fn\">5</xref>}</p>", "<title>Structure of the remainder of the paper</title>", "<p>In §2, we give an outline of the post-Newtonian analysis of the <italic toggle=\"yes\">generation problem</italic> for a system of infalling masses coming from the infinite past.</p>", "<p>In §3, we describe how this post-Newtonian analysis informs the mathematical set-up of a scattering problem for the linearized Einstein vacuum equations around Schwarzschild, i.e. the mathematical set-up for the <italic toggle=\"yes\">propagation problem</italic>.</p>", "<p>In §§4 and 5, we describe how this scattering problem is solved, and we study the asymptotic properties of the Weyl curvature tensor of the scattering solution. In particular, we give a quick and intuitive sketch of why peeling fails in the way it does.</p>", "<p>In §6, starting from the asymptotics of the Weyl tensor, we sketch how to derive the asymptotics of the remainder of the system and comment on various interesting properties that the resulting solutions have. We briefly discuss consequences of our results on late-time asymptotics in §7 and conclude in §8.</p>" ]

[]

[ "<title>The main result</title>", "<p>We already formulate a rough version of one of the main results of Kehrberger &amp; Masaood [##UREF##1##2##].</p>", "<title>Theorem 3.3.</title>", "<p><italic toggle=\"yes\">There exists a unique solution to</italic>\n<bold>LGS</bold>\n<italic toggle=\"yes\">attaining the scattering data of definition 3.2. This solution satisfies, throughout</italic>\n:\n\n<italic toggle=\"yes\">Here</italic>, \n<italic toggle=\"yes\">for some algebraic constants</italic>\n.</p>", "<p><italic toggle=\"yes\">Moreover, the solution satisfies, as</italic>\n,\n</p>", "<p>The proof of this result will be sketched in §§4 and 5, and further commentary on the result will be provided in the sections afterwards.</p>" ]

[]

[ "<title>Conclusion</title>", "<p>Clearly, the results (##FORMU##318##3.6##) and (##FORMU##438##5.8##), which show that near (in complete agreement with [##UREF##26##27##]), violate the peeling property (##FORMU##25##1.3##) for (recall that is equivalent to ). As opposed to the violation of peeling near , which is derived using post-Newtonian approximations around Minkowski (§2), the violation of peeling near is caused by the mass term in the Schwarzschild metric (in the post-Minkowskian picture, this effect would only be seen at order in ).</p>", "<p>While our results thus rule out that the constructed spacetimes admit a smooth (or ) or admit Bondi coordinates near , both of these concepts can still be made sense of provided they are sufficiently weakened: in the case of Bondi coordinates, it suffices to drop condition (iii) from Sachs [##UREF##7##8##]. To be precise, the -term in eqn. (4.1) of Sachs [##UREF##7##8##], which captures the fall-off of and in eqn. (2.9) of Sachs [##UREF##7##8##], does <italic toggle=\"yes\">not</italic> vanish, it carries physical information. See [##UREF##73##75##] for the computation of these terms if . The non-vanishing of this term then generates a in the expansion of , a term in the expansion of , etc.</p>", "<p>In the case of Penrose’s asymptotic simplicity, the assumption on the regularity of the compactification needs to be weakened significantly. A non-trivial problem of interest to many would be to understand the optimal regularity with which a conformal boundary can be still be attached if .^{<xref rid=\"FN8\" ref-type=\"fn\">8</xref>}</p>", "<p>Another notion of asymptotic flatness mentioned in the introduction is that of Christodoulou &amp; Klainerman [##UREF##0##1##], which demands the spacetime to feature certain decay towards spatial infinity (along ). The demanded decay in particular implies that would decay like towards . Now, inspection of equation (##FORMU##318##3.6##) or equation (##FORMU##412##5.5##) immediately shows that along . In other words, the class of spacetimes constructed in [##UREF##0##1##] decays too fast to capture the kinds of physics we are describing in this paper. Furthermore, we can see that the interpretation of Christodoulou's argument [##UREF##28##29##] given in (the paragraphs above equation (2.4)) of [##UREF##29##30##] does not hold, at least at the linearised level: For, if we set the coefficient to vanish in definition 3.2, this will neither affect the no incoming radiation condition on , nor the decay rate of the energy loss along , see equation (##FORMU##598##6.2##). Moreover, the induced data along will decay fast enough for the framework of Christodoulou &amp; Klainerman [##UREF##0##1##] to apply. But at the same time, will now decay like towards by equation (##FORMU##318##3.6##), from which one easily shows that decays like . But this is in contradiction with the argument, which states that ; cf. eqn. (1.10) of Kehrberger [##UREF##29##30##]. See also [##UREF##1##2##] for further discussion.</p>", "<p>Finally, a notion of asymptotic flatness that does admit our constructed spacetimes is that of Dafermos <italic toggle=\"yes\">et al.</italic> [##UREF##53##54##]. Indeed, as we said in §6, the constructed spacetime is ‘extendable to ’ in the sense of def. 3.4 of Holzegel [##UREF##70##72##] for any ; the parameter measuring decay towards .</p>", "<p>The fact that our constructions decay slower near than demanded by Christodoulou &amp; Klainerman [##UREF##0##1##] but still decay sufficiently fast near is somewhat surprising as the spacetimes constructed in [##UREF##0##1##] are generically only ‘extendable to ’ for . This remarkable improvement in decay towards compared to decay near is related to our constructions not having incoming radiation from and will be further discussed elsewhere.</p>" ]

[ "<p>One contribution of 13 to a discussion meeting issue ‘<ext-link xlink:href=\"http://dx.doi.org/10.1098/rsta/382/2267\" ext-link-type=\"uri\">At the interface of asymptotics, conformal methods and analysis in general relativity</ext-link>’.</p>", "<p>This paper is the fourth in a series dedicated to the mathematically rigorous asymptotic analysis of gravitational radiation under astrophysically realistic set-ups. It provides an overview of the physical ideas involved in setting up the mathematical problem, the mathematical challenges that need to be overcome once the problem is posed, as well as the main new results we will obtain in upcoming work. From the physical perspective, this includes a discussion of how post-Newtonian theory provides a prediction on the gravitational radiation emitted by infalling masses from the infinite past <italic toggle=\"yes\">in the intermediate zone</italic>, i.e. up to some finite advanced time. From the mathematical perspective, we then take this prediction, together with the condition that there be no incoming radiation from , as a starting point to set up a scattering problem for the linearized Einstein vacuum equations around Schwarzschild and near spacelike infinity, and we outline how to solve this scattering problem and obtain the asymptotic properties of the scattering solution near and . The full mathematical details will be presented in the sequel to this paper.</p>", "<p>This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.</p>" ]

[ "<title>The physical set-up for the generation problem</title>", "<p>We here sketch the physical set-up that will inform our mathematical construction. This section does not aim or claim to be rigorous, instead, we want to give a ‘quick and dirty’ justification for the assumptions we will make for our mathematical scattering set-up in §3. Some conceptual difficulties are mentioned in §3(c).</p>", "<title>The physical picture that we want to describe…</title>", "<p>…is that of a system of two infalling masses (with negligible internal structure) whose trajectories approach hyperbolic (or parabolic) Keplerian orbits in the infinite past. More generally, the contents of this section apply to systems of infalling masses from the infinite past whose relative velocities approach constant, non-relativistic values. In addition, we prohibit incoming radiation from . By virtue of these masses moving slowly and the separation between them becoming large, we expect perturbations around the Newtonian theory to provide us with a good prediction on the gravitational radiation generated by this system.</p>", "<p>The plan is now as follows: we enclose the system by an ingoing null cone from , truncated at some finite retarded time since we only care about the behaviour of the system in the distant past. Up until this null cone (we denote this region <italic toggle=\"yes\">the intermediate region</italic>), we assume the validity of a suitable post-Newtonian framework in order to understand the <italic toggle=\"yes\">generation problem</italic>, in particular, we use this framework to get a prediction on the behaviour of gravitational radiation along towards (##FIG##2##figure 3##).\n</p>", "<title>The multipolar post-Minkowskian expansion and the post-Newtonian prediction</title>", "<p>The generation of gravitational waves by Newtonian or post-Newtonian sources has been studied in early references, see, for instance, [##UREF##57##58##,##UREF##58##59##] (and [##REF##28179846##60##] for a more recent review). The physical picture of two weakly interacting masses in the infinite past as described in §2(a) was first studied to leading order in the post-Newtonian expansion parameter by Walker &amp; Will [##UREF##23##24##,##UREF##25##26##], denoting the characteristic speed of the system and the speed of light. This expansion corresponds to an expansion that only takes into account the quadrupole moment of gravitational radiation, higher multipole moments being of higher order in . The relevant results can be read off from eqns. (61), (62) in [##UREF##25##26##]. In particular (the reason why we single out the following results will be made clear in §3), the following asymptotic behaviour was found along the incoming Minkowskian null cones: was found to decay like towards , was found to decay like (violating peeling, equation (##FORMU##35##1.4##)), and the ingoing shear along the null cones, , was found to decay like . In each case, the coefficient of the relevant decay rate is related to (time derivatives of) the quadrupole moment of the system, more precise expressions will be given below.</p>", "<p>In order to obtain a more complete picture, we also want to understand the asymptotic behaviour of higher multipoles; for this, we consult Thorne’s work on multipole expansions in general relativity [##UREF##59##61##] (see also the appendix of Sachs [##UREF##5##6##] or Pirani’s lecture notes [##UREF##60##62##]).</p>", "<p>We provide a short summary of the computations in [##UREF##59##61##]: we will temporarily work in Cartesian Minkowskian coordinates , and we will take Latin indices to only range from 1 to 3. First, Thorne considers linearized perturbations around the Minkowski metric . By imposing de Donder gauge, the linearized Einstein vacuum equations then reduce to Minkowskian wave equations for the Cartesian components of the trace-reversed metric perturbation , and the general <italic toggle=\"yes\">outgoing wave</italic> solution for , assuming the coordinates to be mass-centred, is then given in eqns. (8.13) of Thorne [##UREF##59##61##] in terms of an expansion into multipoles and , the radiation field corresponding to this solution being given by eqn. (4.8) of Thorne [##UREF##59##61##].</p>", "<p>Let now denote the usual spherical polar coordinates, introduce the double null coordinates , , define the null tetrad , , , and then define the Newman–Penrose scalars\n\nwhere denotes the linearized Weyl tensor of . From the form of Thorne's metric (8.13), we then compute (this generalizes [##UREF##61##63##,##UREF##62##64##]), for \n\nwhere the and are related to and via Thorne’s (4.7) and and denote the electric and magnetic transverse traceless tensor spherical harmonics defined in Thorne’s eqn. (2.30). (See Thorne’s eqn. (2.38) for the relation to spin-weighted harmonics.)</p>", "<p>By now considering slow-motion, weak internal gravity sources in the ‘near zone’ of spacetime, by using the outgoing Minkowskian Green’s function and by then matching it to the vacuum expression for Thorne's metric (8.13), it is derived that, to leading order in , the moments and are given by the Newtonian mass and current multipole moments (eqn. (5.27) of Thorne [##UREF##59##61##]):\n\nIn the formulae above, denotes the Levi-Civita symbol, and denote the Newtonian mass and momentum density and is the magnetic vector spherical harmonic (defined in Thorne’s eqn. (2.18)).</p>", "<p>With these computations at hand, we can now extend the results of Walker &amp; Will [##UREF##25##26##] for the setting of two pointlike particles moving along approximately hyperbolic Keplerian orbits beyond the mass quadrupole approximation. For such orbits, the leading-order behaviour of the relative position vector of the particles is given by as (see [##UREF##63##65##] for a similar statement for particles). Here, the coefficient corresponds to the asymptotic relative velocity of the orbit, and by Newton’s equations of motion.</p>", "<p>If we insert this asymptotic behaviour for into the multipole moments (##FORMU##115##2.3##), we find that and , where is of magnitude , and where . We conclude from this and equation (##FORMU##105##2.2##) that, as we approach , all -modes of the Newman–Penrose scalars decay like:\n\nwhere is computed from (and ), is computed from (and ) and, up to complex conjugation and -dependent multiple, . In the above, we used that near . It is also straightforward to compute that all higher -modes of the ingoing shear decay as for (cf. eqn. (63) of Walker &amp; Will [##UREF##25##26##]), i.e. attains a finite, nonzero limit at .</p>", "<p>A similar story can be told for parabolic orbits, where . The behaviour of the modes of and near is then given by . By contrast to hyperbolic orbits, since parabolic orbits do not grow linearly in time, higher -modes will now decay faster towards : .</p>", "<title>Comments and conceptual problems</title>", "<p>We conclude with some comments. First, note that equation (##FORMU##105##2.2##) implies that near , independently of the structure of the multipole moments. Historically, this fact was used as justification for the peeling property of equation (##FORMU##25##1.3##), e.g. in [##UREF##9##10##]. Similarly, the fact that outgoing wave solutions to the linearized field equations around Minkowski have an expansion in powers of (see Thorne’s metric (8.13)) motivated the set-up of Bondi coordinates.</p>", "<p>What we will see in the present note, however, is that Thorne’s expression for the metric perturbation (8.13) breaks down at late advanced times, i.e. near , owing to the effects of the Schwarzschild term . In fact, taking into account this Schwarzschild term will also lead to corrections in Einstein’s classical quadrupole formula along (eqn. (4.18) of Thorne [##UREF##59##61##]). Of course, one might then wonder whether it is already too much to assume the validity of the framework all the way until the null cone , rather than, say, just assuming it to hold up until some timelike cylinder along which as in [##UREF##26##27##], see ##FIG##2##figure 3##. Let us here just mention that the propagation of radiation from such a timelike cylinder to the null cone was studied at the level of scalar waves in [##UREF##29##30##,##UREF##64##66##] on Schwarzschild: the result of these studies is that the leading-order decay towards is the same along the cylinder and along .</p>", "<p>Second, the system that we consider, growing linearly in time, has the feature that all higher-order corrections within the post-Newtonian expansion (see §V.D of Thorne [##UREF##59##61##]), even though they are of smaller magnitude in , feature the same decay towards , in just the same way as all electric angular modes of or feature the same decay towards . Of course, one could interpret the approximate statements (##FORMU##115##2.3##) as the leading-order result for metric perturbations on fixed angular frequency. If we really wanted to do consistent perturbation theory, however, then writing down the leading-order expressions (##FORMU##115##2.3##) for the th electric multipole moment means that we should also consider higher-order post-Newtonian expansions of all lower electric multipoles (i.e. expand the th multipole up to orders) along with higher-order post-Minkowskian expansions around , etc.</p>", "<p>Last, while we do not expect that the effects of higher-order perturbations will change the decay rates of equation (##FORMU##136##2.4##) (they will certainly change the coefficients), it would certainly be an interesting problem to understand this in more detail, cf. footnote 3.</p>", "<title>The mathematical set-up for the propagation problem</title>", "<p>In the previous section, we gave a rough sketch of the <italic toggle=\"yes\">generation problem</italic> of gravitational radiation in a post-Newtonian framework. We now describe how we convert the information from §2, which was perturbatively obtained for linearized gravity around Minkowski, into a mathematically formulated scattering set-up for linearized gravity around Schwarzschild.</p>", "<p>Studying linearized gravity around Schwarzschild will on the one hand provide us with a clear understanding of the failure of peeling owing to the presence of mass near spatial infinity. On the other hand, we will set up the problem in such a way as to serve as a strong foundation for the study of the actual, nonlinear Einstein vacuum equations.</p>", "<title>The system of linearized gravity around Schwarzschild <bold>(LGS)</bold></title>", "<p>Consider the Schwarzschild metric in the familiar form . We introduce the double null coordinates , , where , so that the metric takes the form\n\nHere, , and denotes the standard metric on , with . (In the following, capital Latin indices will always refer to indices on the sphere and range from 1 to 2.) We will restrict our attention to the very exterior of spacetime, that is to say, we will work on the manifold (##FIG##3##figure 4##); and we will work with the ON frame\n\n</p>", "<p>The (double null) system of linearized gravity around Schwarzschild is obtained by considering a general one-parameter family of Einstein metrics on such that :\n\nand by then linearizing the arising Einstein and Bianchi equations in (i.e. by writing , etc. and only keeping terms of order ) (see [##UREF##53##54##] for details).</p>", "<p>The result is a coupled system of 10 hyperbolic equations for the linearized curvature coefficients—the linearized Bianchi equations—together with a system of around 30 transport and elliptic equations for the linearized metric and connection coefficients. We will refer to this system as linearized gravity around Schwarzschild <bold>LGS</bold>.</p>", "<p>Examples for linearized metric components are the linearized lapse and the linearized metric on the spheres , which is split up into its trace and its tracefree part, .</p>", "<p>Examples for linearized connection coefficients are the linearization of the ingoing and outgoing null expansions, denoted and , respectively, and of the ingoing and outgoing null shears: and . The work [##UREF##53##54##] employs the Christodoulou–Klainerman formalism. For those familiar with the very closely related Newman–Penrose formalism, the ingoing and outgoing null expansions (or shears) are called and (or and ), respectively.</p>", "<p>Examples for linearized curvature coefficients are the linearized Gauss curvature on the spheres, as well as the extremal curvature components and :\n</p>", "<p>These are the real-valued symmetric tracefree two-tensor analogues to and : identifying , , , we have .</p>", "<p>Examples for equations of <bold>LGS</bold> are given by (writing , etc.)\n</p>", "<title>The pure gauge solutions and the linearized Kerr solutions of <bold>LGS</bold></title>", "<p>Central to the understanding of <bold>LGS</bold> is the existence of two classes of explicit solutions to it. The first class of these consists of the <italic toggle=\"yes\">linearized Kerr solutions</italic>: these are produced by linearizing a nearby Kerr metric in double null coordinates around Schwarzschild, or by linearizing a Schwarzschlid solution with nearby mass around the original Schwarzschild.</p>", "<p>The other class of solutions is given by the <italic toggle=\"yes\">pure gauge solutions</italic>. These are generated by linearizing nonlinear coordinate transformations of the double null coordinates that preserve the double null form of the metric (##FORMU##198##3.3##).</p>", "<p>It turns out that the linearized Kerr solutions are entirely supported on spherical harmonics with . Conversely, any solution supported on is given by a linear combination of such linearized Kerr and pure gauge solutions. In other words, the -part of <bold>LGS</bold> does not carry any dynamics. In what follows, we will, therefore, exclusively discuss solutions supported on .</p>", "<p>In view of the existence of these solutions, it is very helpful to extract quantities from the system <bold>LGS</bold> that are invariant under addition of pure gauge or Kerr solutions. Examples for such <italic toggle=\"yes\">gauge invariant</italic> quantities are and (i.e. and ). In fact, any solution that has both and (or and ) vanishing is given by a combination of linearized Kerr and pure gauge solutions (see [##UREF##53##54##,##UREF##65##67##] for details).</p>", "<title>Scattering theory for <bold>LGS</bold>: the seed scattering data</title>", "<p>The quantitative stability of <bold>LGS</bold> was first shown in [##UREF##53##54##] (and provided the central ingredient to the more recent nonlinear stability of Schwarzschild proof [##UREF##54##55##]), and a global scattering theory to <bold>LGS</bold> was written down in [##UREF##66##68##].</p>", "<p>In the present paper, we will be concerned with the <italic toggle=\"yes\">semi-global scattering theory</italic> for <bold>LGS</bold>: given appropriate data on the truncated ingoing null cone (whose future end sphere we will denote as and whose past limiting sphere we will denote as , see ##FIG##3##figure 4##) coming from together with data along to the future of this cone, we want to construct the unique solution to <bold>LGS</bold> ‘restricting’ to these data in the limit.</p>", "<p>We now describe what scattering data consist of, and which parts of these scattering data carry physical information. As the -part <bold>LGS</bold> is non-dynamical, we will only present the discussion for data and solutions supported on .</p>", "<title>Definition 3.1.</title>", "<p>An \n<italic toggle=\"yes\">seed scattering dataset</italic> consists of the following prescribed quantities: along , prescribe and . Along (to the future of ), prescribe , and the radiation field of the outgoing null shear (a.k.a. the News tensor). Finally, prescribe on the values of as well as the ingoing null expansion , and prescribe on the weighted outgoing null expansion together with the Gauss curvature .</p>", "<p>From these seed scattering data, one can construct all other quantities (e.g. or ) along and , and then construct the unique solution to <bold>LGS</bold> in the entire domain of dependence that restricts to these data (in the limit). A sketch of this is given in §4.</p>", "<title>Bondi normalization of the seed scattering data</title>", "<p>We now discuss which part of the seed data carry physical meaning. For instance, while along carries radiative physical information, along (<italic toggle=\"yes\">without</italic> the extra -weight) only describes information related to the choice of angular coordinates at infinity that can always be made to vanish by a gauge transformation. More generally, in [##UREF##1##2##], it is shown that, upon adding pure gauge solutions, any seed scattering dataset can be Bondi normalized at ; this means:\n<list list-type=\"simple\"><list-item><label>—<x xml:space=\"preserve\"> </x></label><p> and along can be set to zero, and</p></list-item><list-item><label>—<x xml:space=\"preserve\"> </x></label><p> and on can be set to zero.</p></list-item></list>\nTogether, the two bullet points imply that and vanish along all of , meaning that the spheres at are the standard round spheres. This normalization captures the spirit of the original Bondi coordinates, except that it does not make any assumptions on -expansions. Henceforth, we will always assume our seed data to be Bondi normalized.</p>", "<p>The Bondi normalization as described above still leaves us with some remaining gauge freedom which, for example, contains the group of BMS transformations on , see [##UREF##66##68##] for details. The remaining gauge freedom can also be used to set along and to 0. Doing this, we see that the physical content of the seed scattering data is carried by , as well as .</p>", "<title>Seed scattering data describing the exterior of the -body problem</title>", "<p>Having understood what constitutes scattering data for <bold>LGS</bold>, we now want to pose scattering data as motivated by the considerations in §2. What we would like to do is to just take the statements (##FORMU##136##2.4##) and demand that our seed scattering data are such that the predicted asymptotics for and near are satisfied. But of course, the results of §2 concern linearized gravity around Minkowski and, as such, do not translate directly into results for linearized gravity around Schwarzschild. Complications in relating the two include the logarithmic divergence of the null cones in Schwarzschild compared to Minkowski, metric perturbations around Schwarzschild being different objects from metric perturbations around Minkowski, as well as changes to the treatment of the near zone generating the relationship to the Newtonian theory. As these complications are quite severe, we will not attempt to treat them, but instead just hope that all of these changes are subleading in terms of decay towards . On the basis of this hope, we formulate the following definition.</p>", "<title>Definition 3.2.</title>", "<p>A Bondi normalized seed scattering dataset is said to describe <italic toggle=\"yes\">the exterior of a system of infalling masses following approximately hyperbolic orbits</italic> if the following conditions are satisfied for all :\n<list list-type=\"simple\"><list-item><label>(I)<x xml:space=\"preserve\"> </x></label><p> along satisfies for some independent of , and the limit along , which is finite by the second of equations (##FORMU##224##3.5##), is non-vanishing.</p></list-item><list-item><label>(II)<x xml:space=\"preserve\"> </x></label><p>The limit is finite and non-vanishing.</p></list-item></list>\nFinally, we say that such data also satisfy the <italic toggle=\"yes\">no incoming radiation condition from </italic> if additionally the News tensor vanishes along .</p>", "<p>In the above definition, denotes the projection of on to the electric and magnetic tensor harmonics and introduced earlier (below equation (##FORMU##105##2.2##)). We will occasionally use an overline to denote magnetic conjugation: given , denotes .</p>", "<p>The assumptions (I) and (II) are directly motivated by equation (##FORMU##136##2.4##) and the corresponding result for (below (##FORMU##136##2.4##)), whereas the last condition of definition 3.2 realizes the physical requirement of excluding incoming radiation from , it is the linearized version of demanding that the Bondi mass along be constant.</p>", "<p>It is, of course, possible to formulate the conditions of definition 3.2 directly in terms of the seed data—see [##UREF##1##2##] (in fact, for (I), the reader can see this from equations (##FORMU##224##3.5##)).</p>", "<title>Solving the scattering problem I: constructing the unique solution</title>", "<p>In §3, we outlined the mathematical set-up of the scattering problem for <bold>LGS</bold> and related this set-up to the generation of gravitational waves discussed in §2. We will now sketch the general construction of scattering solutions to <bold>LGS</bold>.</p>", "<title>The Teukolsky equations and the Regge–Wheeler equation</title>", "<p>The system <bold>LGS</bold> may at first seem quite intractable, on the one hand owing to its size (it consists of around 40 equations for 19 unknowns), on the other hand owing to the large space of pure gauge solutions. A key realization in unlocking <bold>LGS</bold>, bypassing both difficulties, is that the gauge invariant quantities and each satisfy a decoupled wave equation known as the Teukolsky equation, which, in components (), reads:\n\nHere, denotes the Laplacian on acting on a two-tensor, its eigenvalues are , . Henceforth, we will by or always mean the derivative of the components of (i.e. ), whereas, by , we will mean the components of acting on a two-tensor (i.e. ).</p>", "<p>Despite the large body of literature on linear wave equations on black-hole backgrounds, a robust analysis of the Teukolsky equation for a long time seemed inaccessible owing to the first-order terms that appear when writing equation (##FORMU##326##4.1##) in standard wave form (). Triumph over this difficulty was achieved in [##UREF##53##54##] by exploiting the following observation: the first-order terms in equation (##FORMU##326##4.1##) disappear for certain weighted higher-order derivatives of . Indeed, if denotes either or , then equation (##FORMU##326##4.1##) implies\n\nEquation (##FORMU##343##4.2##), known as the Regge–Wheeler equation, is nothing but the tensorial linear wave equation with a positive potential and, as such, can be treated using well-established methods. In particular, equation (##FORMU##343##4.2##) admits a conserved energy: if denotes the canonical energy momentum tensor associated with (##FORMU##343##4.2##), then .</p>", "<p>To give context for the relevance of equation (##FORMU##343##4.2##) within <bold>LGS</bold>, let us summarize the strategy of the proof of stability of <bold>LGS</bold> in [##UREF##53##54##]. First, the authors prove stability estimates for using wave equation methods for equation (##FORMU##343##4.2##). Then, they achieve from these estimates control over the extremal curvature components and . Finally, they use the control over and , together with delicate considerations of gauge, to derive estimates for the entire remainder of <bold>LGS</bold>.</p>", "<title>Scattering theory for the Regge–Wheeler equation</title>", "<p>The construction of a scattering theory for <bold>LGS</bold> follows a pattern similar to that of the proof of stability for <bold>LGS</bold>: given a seed scattering dataset (definition 3.1), we first derive expressions for the induced along .</p>", "<p>We then temporarily ignore the rest of the system and only focus on the construction of the unique solution to equation (##FORMU##343##4.2##) restricting to these data. This is done in the following way: we consider a sequence of finite characteristic initial value problems (as depicted in ##FIG##4##figure 5##), where the data are posed on an outgoing null cone at finite retarded time such that these data approach the scattering data along in the limit . The unique existence of solutions to these finite characteristic initial value problems is ensured by standard theorems. We finally exploit the fact that equation (##FORMU##343##4.2##) admits a conserved energy to show uniform-in- estimates for these solutions, from which we deduce that the sequence of finite solutions constructed this way converges to a unique limiting solution—the scattering solution. In fact, the conserved energy associated with equation (##FORMU##343##4.2##) defines <italic toggle=\"yes\">unitary</italic> Hilbert space isomorphisms between spaces of scattering data along and spaces of restrictions of the scattering solution to any Cauchy hypersurface in . For details, see [##UREF##1##2##,##UREF##67##69##].\n</p>", "<title>Scattering theory for the remainder of <bold>LGS</bold></title>", "<p>At this point, we have a scattering solution to equation (##FORMU##343##4.2##) that restricts correctly to the data along induced by the seed scattering data. We can now construct the remaining quantities of <bold>LGS</bold> by systemically defining other quantities in terms of the seed scattering data and the solution . For instance, by integrating from , and by computing the value of along from the seed scattering data, we can define in all of . Similarly, we define .</p>", "<p>Owing to the high degree of redundancy in the system <bold>LGS</bold> (there are more equations than unknowns), this procedure is somewhat subtle. Let us illustrate the difficulty: the system <bold>LGS</bold> includes equations for the , the and angular derivatives for , one of them being written down in equations (##FORMU##224##3.5##). We could thus choose to define as the solution to the third of equations (##FORMU##224##3.5##) (assuming we have already defined ), but we might then have difficulties proving that the other equations for are also satisfied. Overcoming this problem requires being very careful about the order in which to define the various quantities and a precise understanding of the structure of the various equations (see [##UREF##1##2##,##UREF##66##68##] for details).</p>", "<p>We finish this section by remarking that the clean split that can be done for the linear problem, namely to first do a limiting argument to construct a solution to equation (##FORMU##343##4.2##) and to then construct all the remaining quantities from this limiting solution , will probably be no longer possible in the full, nonlinear theory. Instead, one will have to construct a sequence of finite solutions to the entire system, and then show that this entire sequence converges.</p>", "<title>Solving the scattering problem II: asymptotic analysis of solutions to the Teukolsky equations</title>", "<p>So far, we have written down scattering data along that capture the radiation emitted by an -body system moving along approximately hyperbolic orbits in the infinite past (definition 3.2), and we have established that one can associate a unique scattering solution to these data (§4). With this solution obtained, we shall now sketch how to obtain asymptotic properties of the solution, focusing first on the extremal Weyl components and .</p>", "<title>The induced scattering data for and </title>", "<p>Since both and satisfy decoupled equations, we can also decouple the presentation of their analysis from the rest of the system. For this, we merely need to write down the scattering data for and that are induced by seed scattering data as in definition 3.2. We will henceforth use the notation\n</p>", "<title>Lemma 5.1.</title>", "<p><italic toggle=\"yes\">Given seed data satisfying definition 3.2, the induced data for</italic>\n\n<italic toggle=\"yes\">are given by</italic>\n\n<italic toggle=\"yes\">for any</italic>\n.</p>", "<p><italic toggle=\"yes\">Similarly, the induced data for</italic>\n\n<italic toggle=\"yes\">are given by</italic>\n\n<italic toggle=\"yes\">as well as the values on the spheres</italic>\n, \n<italic toggle=\"yes\">of the first two transversal derivatives</italic>:\n\n<italic toggle=\"yes\">Both</italic>\n\n<italic toggle=\"yes\">and</italic>\n\n<italic toggle=\"yes\">are independent of</italic>\n\n<italic toggle=\"yes\">and their precise values will not matter for this paper</italic>.</p>", "<p>The information from this lemma will suffice to derive the asymptotics for and . Note that, even though and satisfy decoupled evolution equations, they are coupled at the level of the data. This is a manifestation of the Teukolsky–Starobinsky identities relating and [##UREF##68##70##].</p>", "<title>A sketch of the problem for the lowest angular mode </title>", "<p>Recall that the elliptic operator in equation (##FORMU##326##4.1##) has eigenvalues , with . In particular, for , , and equation (##FORMU##326##4.1##) simplifies to\n\nThe simple yet crucial observation now is that the RHS of equation (##FORMU##412##5.5##) has an extra decay of compared to the LHS, even though the LHS only contains two derivatives. Thus, if we have any preliminary global decay estimate of , then we can insert this estimate into the RHS of equation (##FORMU##412##5.5##) and integrate the equation in and to improve this estimate.</p>", "<p>To illustrate this, let us assume that for some . We then have that We now integrate this from in , where vanishes by equations (##FORMU##390##5.2##). The result is that . Finally, we integrate this from in to obtain that . Since is bounded by according to equations (##FORMU##390##5.2##), we can divide by to improve the original estimate to and iterate to improve further.</p>", "<p>In practice, the conservation of the energy associated to equation (##FORMU##343##4.2##) gives us the initial estimate , and iteratively inserting this estimate into equation (##FORMU##412##5.5##) gives us the estimate\n\nWe now insert the last estimate into (##FORMU##412##5.5##) one last time, writing \n\nwhere the final equality follows from integrating by parts (; the last term being the dominant term near ). By integrating again in , we finally deduce that\n\nUsing this procedure, we can find higher and higher order terms. For instance, the next-to leading-order term in , namely , adds the term to the expansion above. This proves equation (##FORMU##318##3.6##) for .</p>", "<title>A sketch for higher angular modes </title>", "<p>For , we crucially exploited the vanishing of in equation (##FORMU##326##4.1##). This made the RHS of equation (##FORMU##326##4.1##) decay three powers faster in than the LHS, allowing us to iteratively improve any estimate that we insert into the RHS.</p>", "<p>For , it looks like this cannot be done, since the RHS now only decays two powers faster, which no longer yields an improvement after integrating in and in . This hurdle is, similarly to how we descended from the Teukolsky equations to the Regge–Wheeler equations, overcome by commuting: denote , and, abusing notation, . A tedious computation, starting from equation (##FORMU##326##4.1##), gives\n\nwhere . But if is supported on angular frequency , then the first line on the RHS of equation (##FORMU##452##5.9##) vanishes, so we have\n\nThe first term on the RHS now features an weight multiplying , whereas the second term features an -weight that multiplies instead of . But since , decays one power faster compared to , so we can view this term to also feature three extra powers in decay compared to the LHS of equation (##FORMU##456##5.10##).</p>", "<p>At this point, for , we can more or less mimic the argument presented in §5(b) for to obtain an estimate for .</p>", "<p>First, we compute the transversal derivatives , , along by inductively integrating equation (##FORMU##452##5.9##) from , where for all boundary terms vanish. The result of this is that (and for ), with .</p>", "<p>Second, we perform an iterative argument as for to produce a global estimate of the form\n\nThird and finally, we integrate this estimate times from :\n\nThe computations involved are now getting more and more involved, but it is relatively straight-forward to see that the nested integral in the second line produces a leading-order term , whereas the sum in the first line clearly remains bounded as along —this sum exactly corresponds to the sum in the first line in equation (##FORMU##318##3.6##). This proves equation (##FORMU##318##3.6##).</p>", "<title>A sketch for higher angular modes </title>", "<p>Compared to , the main difference in the case is that, with the decay rate , the RHS of equation (##FORMU##326##4.1##) decays like , so we cannot integrate it from along —instead, we have to integrate it from in order to compute the transversal derivative along . This is why we had to specify along as part of the seed data.</p>", "<p>The commuted equation, (##FORMU##452##5.9##) with , can then be integrated from , but the boundary term picked up from , is a constant by equation (##FORMU##396##5.4##).</p>", "<p>At this point, all higher-order transversal derivatives along can be computed as for —all further boundary terms vanish at by equation (##FORMU##393##5.3##). In particular, we find that , so the leading-order decay of higher-order transversal derivatives along is not dictated by the leading-order decay of itself (i.e. it does not depend on ).</p>", "<p>We can now, in exact analogy to equation (##FORMU##478##5.11##), show that\n\nfrom which we derive an expression for by again applying the formula (##FORMU##481##5.12##):\n\nThe leading-order logarithmic behaviour is entirely governed by , instead of the coefficient that governs the leading-order decay of towards , the reason being that this behaviour is determined by higher-order transversal derivatives along .^{<xref rid=\"FN6\" ref-type=\"fn\">6</xref>}</p>", "<p>However, if we compute the expression for the radiation field of , namely , the coefficient only enters at Schwarzschildean order, that is to say: it does not contribute if . A very lengthy computation produces the second statement of theorem 3.3, (##FORMU##322##3.7##) (see [##UREF##1##2##] for details).</p>", "<title>The summing of the -modes</title>", "<p>The method we used to find the asymptotics for higher angular modes comes at a high price: commuting with up to times generates a large amount of error terms that, although they all decay faster towards infinity (i.e. in ), individually grow extremely fast in . In particular, if we write , the bounds for the error terms in the estimates above are not summable even if the initial data are smooth.</p>", "<p>We expect that this is only an artefact of the method that we use, and that there is a way to sum the estimates above. The investigation of this expectation is ongoing work [##UREF##69##71##]. Very roughly speaking, we hope to tackle the issue in four steps:\n<list list-type=\"simple\"><list-item><label>(1)<x xml:space=\"preserve\"> </x></label><p>Consider the initial data for : . We write the solution as a sum of the solution arising from the initial data (without the -term) and the difference .</p></list-item><list-item><label>(2)<x xml:space=\"preserve\"> </x></label><p>A robust energy estimate on the difference then proves that in .</p></list-item><list-item><label>(3)<x xml:space=\"preserve\"> </x></label><p>Next, we consider . For this solution, we establish a persistence of polyhomogeneity result. To explain: loosely speaking, a function is said to be polyhomogeneous near a boundary if it has a series expansion in , with a countable set of numbers , , in a robust, square-integrable sense. What we will show is that, since the initial data for are polyhomogeneous near , will also be polyhomogeneous near and near . This complements the result of [##UREF##48##49##].</p></list-item><list-item><label>(4)<x xml:space=\"preserve\"> </x></label><p>We can finally determine the coefficients of the polyhomogeneous expansion of by decomposing into angular frequencies and using the methods of §5—since we have already established that is polyhomogeneous, we no longer need to worry about issues of summability at this stage.</p></list-item></list></p>", "<title>Solving the scattering problem III: asymptotic analysis of the remainder of <bold>LGS</bold></title>", "<p>Equipped with asymptotic expressions for and , we are now in a position to derive the asymptotic behaviour of all other quantities contained in <bold>LGS</bold>. For instance, asymptotic expressions for , and follow from integrating the equations (##FORMU##224##3.5##). To obtain the asymptotic behaviour for the remaining Weyl curvature quantities , , one needs to integrate the Bianchi identities; the result of this is that near (recall that is equivalent to ) implies that , denoted in [##UREF##53##54##], goes like near (as predicted by Christodoulou [##UREF##28##29##]), whereas the peeling rates (##FORMU##25##1.3##) hold true for , and .</p>", "<p>In order to obtain asymptotic expressions for all the other quantities contained in <bold>LGS</bold>, we would of course have to introduce <bold>LGS</bold> in much more detail. For the scope of the current paper, suffice it to say that the constructed solution can be proved to be <italic toggle=\"yes\">extendable to for any </italic> in the sense of Def. 3.4 of Holzegel [##UREF##70##72##] (which also features a compact introduction of <bold>LGS</bold>). We now move on to highlighting a few other points of particular interest.</p>", "<title>Bondi normalizing in addition to </title>", "<p>Provided we have Bondi normalized our seed scattering data at , i.e. made the metric perturbations and , as well as and vanish at , it is natural to ask whether we can Bondi normalize the resulting solution along as well. The answer is affirmative: without going into too much detail, starting from Bondi normalized seed scattering data and our results for and , we derive that the limits and vanish automatically. On the other hand, we find that the limits and exist and decay in . It is now possible to add a pure gauge solution that simultaneously kills off both of these terms. Since this gauge solution decays in , it does not affect the Bondi normalization of .</p>", "<title>Corrections to the quadrupole formula: along </title>", "<p>In the nonlinear theory, the rate of change of energy along is given by the Bondi mass-loss formula [##UREF##4##5##]: , denoting the average over and denoting the News function at . Of course, the linearization of this formula is zero, but we can still define the linearized energy loss to be\n\nAn expression for can be inferred via integration of the second of equations (##FORMU##224##3.5##) and our asymptotic expression (##FORMU##322##3.7##) for :\n\nFor and , this can be seen to exactly coincide with Einstein’s original quadrupole formula, though perhaps in a slightly less familiar form: equation (##FORMU##598##6.2##) relates the limiting behaviour near (governed by and according to definition 3.2) to the limiting behaviour along as . The full relation to the quadrupole formula is then established by relating the coefficients and to the Newtonian quadrupole moments (##FORMU##115##2.3##) as described in §2.</p>", "<p>More generally, equation (##FORMU##598##6.2##) represents the linear corrections to (the multipole generalization of) this formula when the linearization is done around Schwarzschild instead of Minkowski.^{<xref rid=\"FN7\" ref-type=\"fn\">7</xref>} Since the present work is meant as a precursor to the treatment of the full Einstein vacuum equations, we hope to eventually make the propagation part of this formula, i.e. going from to , precise. Establishing a full, nonlinear relation between the coefficients and and the physical multipole moments of the described matter distribution would be an exciting project on its own, cf. footnote 3.</p>", "<title>The antipodal matching condition</title>", "<p>Since the analysis produces an arbitrarily precise understanding of the asymptotic behaviour of solutions near , it is also relevant for several questions concerning relations between the past limit point of and the future limit point of . To give an example, we can prove the following about the magnetic parts of the News functions:\n\nthis is closely related to Strominger’s antipodal matching condition. See also [##UREF##66##68##].</p>", "<title>The question of late-time asymptotics</title>", "<p>The entirety of the previous sections was concerned with the asymptotic behaviour near infinity restricted to . It turns out, however, that this, in a certain sense, entirely fixes the possible asymptotics at late times, i.e. as , owing to the existence of certain conserved, modified Newman–Penrose charges along . This is discussed in detail in [##UREF##30##31##] at the level of the scalar wave equation, together with a heuristic principle to relate solutions of the Teukolsky equation to scalar waves (§6 of Gajic &amp; Kehrberger [##UREF##30##31##]). Unfortunately, a small mistake crept into the latter heuristics: in the equation above (##FORMU##595##6.1##), it was assumed that the global asymptotic behaviour of (a.k.a. —we now drop the ) would be governed by its decay towards , so was taken to be 3. But as we stressed in §5(d), it is the decay of two transversal derivatives of near that determines the logarithmic behaviour near , and the value of should instead be 2.</p>", "<p>Thus, if we extend, for instance, the data along all the way to the future event horizon, we end up with the picture for global asymptotics shown in ##FIG##5##figure 6##.\n</p>" ]

[ "<title>Data accessibility</title>", "<p>This article has no additional data.</p>", "<title>Declaration of AI use</title>", "<p>We have not used AI-assisted technologies in creating this article.</p>", "<title>Authors' contributions</title>", "<p>L.K.: writing—original draft.</p>", "<title>Conflict of interest declaration</title>", "<p>I declare I have no competing interests.</p>", "<title>Funding</title>", "<p>I received no funding for this study.</p>" ]

[ "<fig position=\"float\" id=\"RSTA20230039F1\"><label>Figure 1<x xml:space=\"preserve\">. </x></label><caption><p>Depiction of a ‘Bondi bomb’ spacetime, i.e. a spacetime that is stationary in the past. The often made assumption of compactly supported initial data corresponds to precisely such a scenario.</p></caption></fig>", "<fig position=\"float\" id=\"RSTA20230039F2\"><label>Figure 2<x xml:space=\"preserve\">. </x></label><caption><p>Schematic depiction of the generation problem (I), the propagation problem (II) and the asymptotic problem (III). If problems (I) and (II) are studied in the <italic toggle=\"yes\">infinite past</italic>, they determine the asymptotic behaviour (III) near . To study them at <italic toggle=\"yes\">finite</italic> time, one instead needs to make an ad hoc assumption about this asymptotic behaviour near .</p></caption></fig>", "<fig position=\"float\" id=\"RSTA20230039F3\"><label>Figure 3<x xml:space=\"preserve\">. </x></label><caption><p>We apply perturbative methods to study the generation of gravitational waves up until some null cone . The region beyond will later be studied rigorously by taking the behaviour along as given.</p></caption></fig>", "<fig position=\"float\" id=\"RSTA20230039F4\"><label>Figure 4<x xml:space=\"preserve\">. </x></label><caption><p>We rigorously study the propagation problem to the future of the null cone by formulating a scattering problem for <bold>LGS</bold> in . We always think of the scattering data along as capturing the radiation of some physical system, cf. ##FIG##2##figure 3##.\n</p></caption></fig>", "<fig position=\"float\" id=\"RSTA20230039F5\"><label>Figure 5<x xml:space=\"preserve\">. </x></label><caption><p>The limiting procedure that produces the scattering solution.\n</p></caption></fig>", "<fig position=\"float\" id=\"RSTA20230039F6\"><label>Figure 6<x xml:space=\"preserve\">. </x></label><caption><p>We can extend the scattering data along all the way to the future event horizon in order to obtain the asymptotics near future timelike infinity. These turn out to be completely fixed by the asymptotics at early times, i.e. they are independent of how we extend the data along to the future. We note here that, if the spacetime were stationary in the past, then the late-time asymptotics would be three powers faster [##UREF##30##31##,##UREF##39##40##].</p></caption></fig>" ]

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id=\"IM18\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM19\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM20\"><mml:mi>o</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>7</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M1x1\"><label>1.1</label><mml:math id=\"DM1\" display=\"block\"><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>R</mml:mi><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>Λ</mml:mi><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math>\n</disp-formula>", "<disp-formula id=\"RSTA20230039M1x2\"><label>1.2</label><mml:math id=\"DM2\" display=\"block\"><mml:msub><mml:mi>R</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM21\"><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM22\"><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM23\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M1x3\"><label>1.3</label><mml:math id=\"DM3\" display=\"block\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>Ψ</mml:mi><mml:mi>i</mml:mi><mml:mn>0</mml:mn></mml:msubsup><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>5</mml:mn><mml:mo>+</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>6</mml:mn><mml:mo>+</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM24\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM25\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM26\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM27\"><mml:msubsup><mml:mi>Ψ</mml:mi><mml:mi>i</mml:mi><mml:mn>0</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM28\"><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM29\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM30\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>−</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM31\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM32\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M1x4\"><label>1.4</label><mml:math id=\"DM4\" display=\"block\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mo>−</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>Ψ</mml:mi><mml:mi>i</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mrow><mml:mtext>past</mml:mtext></mml:mrow></mml:mrow></mml:msubsup><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>5</mml:mn><mml:mo>+</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>6</mml:mn><mml:mo>+</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mn>4.</mml:mn></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM33\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM34\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM35\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM36\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM37\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM38\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM39\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM40\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM41\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM42\"><mml:mi>N</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM43\"><mml:msup><mml:mi>i</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM44\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM45\"><mml:msup><mml:mi>i</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM46\"><mml:mi>N</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM47\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM48\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM49\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM50\"><mml:msup><mml:mi>i</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM51\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM52\"><mml:msup><mml:mi>i</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM53\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM54\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM55\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>o</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM56\"><mml:mi>g</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM57\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM58\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM59\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>o</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM60\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M1x5\"><label>1.5</label><mml:math id=\"DM5\" display=\"block\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi></mml:mrow><mml:mi>r</mml:mi></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM61\"><mml:mi>g</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>δ</mml:mi><mml:mo>+</mml:mo><mml:mi>o</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM62\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM63\"><mml:mi>N</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM64\"><mml:mi>N</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM65\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM66\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM67\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM68\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM69\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM70\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM71\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM72\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM73\"><mml:mi>v</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>c</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM74\"><mml:mi>v</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM75\"><mml:mi>c</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM76\"><mml:mi>v</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>c</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM77\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM78\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM79\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM80\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM81\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM82\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM83\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM84\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM85\"><mml:msubsup><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>η</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM86\"><mml:msub><mml:mi>η</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM87\"><mml:msubsup><mml:mi>g</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:mo>−</mml:mo><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mstyle><mml:msub><mml:mi>η</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mi>η</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:msup><mml:msubsup><mml:mi>g</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mi>β</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM88\"><mml:msup><mml:mi>g</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM89\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM90\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM91\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>φ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM92\"><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM93\"><mml:mi>v</mml:mi><mml:mo>=</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM94\"><mml:mi>l</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM95\"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM96\"><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mo>±</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt><mml:mi>r</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>θ</mml:mi></mml:msub><mml:mo>±</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>φ</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi>sin</mml:mi><mml:mo>⁡</mml:mo><mml:mi>θ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M2x1\"><label>2.1</label><mml:math id=\"DM6\" display=\"block\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>W</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:msup><mml:mi>l</mml:mi><mml:mi>μ</mml:mi></mml:msup><mml:msubsup><mml:mi>m</mml:mi><mml:mn>2</mml:mn><mml:mi>ν</mml:mi></mml:msubsup><mml:msup><mml:mi>l</mml:mi><mml:mi>ρ</mml:mi></mml:msup><mml:msubsup><mml:mi>m</mml:mi><mml:mn>2</mml:mn><mml:mi>σ</mml:mi></mml:msubsup><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>W</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>ρ</mml:mi><mml:mi>σ</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:msup><mml:mi>n</mml:mi><mml:mi>μ</mml:mi></mml:msup><mml:msubsup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mi>ν</mml:mi></mml:msubsup><mml:msup><mml:mi>n</mml:mi><mml:mi>ρ</mml:mi></mml:msup><mml:msubsup><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mi>σ</mml:mi></mml:msubsup><mml:mo>,</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM97\"><mml:msup><mml:mi>W</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM98\"><mml:mi>g</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM99\"><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M2x2\"><label>2.2</label><mml:math id=\"DM7\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:msub><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:munderover><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>s</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:msubsup><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msubsup><mml:mi>T</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mtext>E</mml:mtext></mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:mfrac><mml:mi>s</mml:mi><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>s</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msubsup><mml:mi>T</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mtext>B</mml:mtext></mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msup></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:msubsup><mml:mi>m</mml:mi><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msubsup><mml:msubsup><mml:mi>m</mml:mi><mml:mi>s</mml:mi><mml:mi>j</mml:mi></mml:msubsup><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM100\"><mml:msub><mml:mi>I</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM101\"><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM102\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>ℓ</mml:mi></mml:msub></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM103\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM104\"><mml:msubsup><mml:mi>T</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mtext>E</mml:mtext></mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM105\"><mml:msubsup><mml:mi>T</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mtext>B</mml:mtext></mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM106\"><mml:mi>v</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>c</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM107\"><mml:msub><mml:mi>I</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM108\"><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M2x3\"><label>2.3</label><mml:math id=\"DM8\" display=\"block\"><mml:mrow><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:msub><mml:mi>I</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>16</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:msqrt><mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow></mml:mfrac></mml:msqrt><mml:mo>∫</mml:mo><mml:mi>ρ</mml:mi><mml:mo>⋅</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>Y</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi><mml:mo>∗</mml:mo></mml:mrow></mml:msup><mml:mrow/><mml:msup><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mi>x</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>32</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:msqrt><mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac></mml:msqrt><mml:mo>∫</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>ϵ</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>p</mml:mi><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>x</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>⋅</mml:mo><mml:mi>ρ</mml:mi><mml:msub><mml:mi>v</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⋅</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>Y</mml:mi><mml:mi>j</mml:mi><mml:mrow><mml:mrow><mml:mtext>B</mml:mtext></mml:mrow><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi><mml:mo>∗</mml:mo></mml:mrow></mml:msubsup><mml:mrow/><mml:msup><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mi>x</mml:mi><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mo>}</mml:mo></mml:mrow></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM109\"><mml:mi>ϵ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM110\"><mml:mi>ρ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM111\"><mml:mi>ρ</mml:mi><mml:msub><mml:mi>v</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM112\"><mml:msup><mml:mi>Y</mml:mi><mml:mrow><mml:mrow><mml:mtext>B</mml:mtext></mml:mrow><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM113\"><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mrow><mml:mtext>rel</mml:mtext></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM114\"><mml:msubsup><mml:mi>x</mml:mi><mml:mrow><mml:mrow><mml:mtext>rel</mml:mtext></mml:mrow></mml:mrow><mml:mi>i</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>a</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mi>b</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM115\"><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM116\"><mml:mi>N</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM117\"><mml:mi>v</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM118\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>a</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>∼</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>v</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM119\"><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mrow><mml:mtext>rel</mml:mtext></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM120\"><mml:msub><mml:mi>I</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mi>I</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup><mml:msup><mml:mi>u</mml:mi><mml:mi>ℓ</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:msubsup><mml:mi>I</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:msup><mml:mi>u</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM121\"><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup><mml:msup><mml:mi>u</mml:mi><mml:mi>ℓ</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:msup><mml:mi>u</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM122\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mi>I</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM123\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>v</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>c</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>ℓ</mml:mi></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM124\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>∼</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>v</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>c</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM125\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM126\"><mml:mi>ℓ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM127\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>±</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>±</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>u</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>u</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>∓</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM128\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>±</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M2x4\"><label>2.4</label><mml:math id=\"DM9\" display=\"block\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:msubsup><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:msubsup><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:msup><mml:mi>r</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mfrac><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msub><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:msubsup><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mfrac><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM129\"><mml:msubsup><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM130\"><mml:msubsup><mml:mi>I</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM131\"><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM132\"><mml:msubsup><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM133\"><mml:msubsup><mml:mi>I</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM134\"><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM135\"><mml:mi>ℓ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM136\"><mml:msubsup><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:mo>∼</mml:mo><mml:msubsup><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM137\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>∼</mml:mo><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM138\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM139\"><mml:mi>ℓ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM140\"><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM141\"><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>λ</mml:mi><mml:mi>ℓ</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM142\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM143\"><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mrow><mml:mtext>rel</mml:mtext></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∼</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM144\"><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM145\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM146\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM147\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM148\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>11</mml:mn></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM149\"><mml:mi>ℓ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM150\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM151\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>±</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:mo>≲</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:mo>−</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM152\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM153\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM154\"><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM155\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM156\"><mml:mi>M</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM157\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM158\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM159\"><mml:mrow><mml:mi mathvariant=\"script\">T</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM160\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>∼</mml:mo><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM161\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM162\"><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM163\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM164\"><mml:mi>v</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>c</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM165\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>−</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM166\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM167\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM168\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>−</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM169\"><mml:mi>ℓ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM170\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM171\"><mml:mi>n</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM172\"><mml:mi>η</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM173\"><mml:msub><mml:mi>g</mml:mi><mml:mi>M</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:msup><mml:mi>θ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>sin</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>⁡</mml:mo><mml:mi>θ</mml:mi><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:msup><mml:mi>φ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM174\"><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM175\"><mml:mi>v</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM176\"><mml:msup><mml:mi>r</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M3x1\"><label>3.1</label><mml:math id=\"DM10\" display=\"block\"><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:mi>u</mml:mi><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:mi>v</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:msup><mml:mi>θ</mml:mi><mml:mi>A</mml:mi></mml:msup><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:msup><mml:mi>θ</mml:mi><mml:mi>B</mml:mi></mml:msup><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM177\"><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM178\"><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM179\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM180\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>θ</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>θ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>φ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM181\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mi>M</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mi>u</mml:mi></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>v</mml:mi></mml:msub><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M3x2\"><label>3.2</label><mml:math id=\"DM11\" display=\"block\"><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>r</mml:mi></mml:mfrac><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>θ</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>r</mml:mi><mml:mi>sin</mml:mi><mml:mo>⁡</mml:mo><mml:mi>θ</mml:mi></mml:mrow></mml:mfrac><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mrow><mml:mi>φ</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>Ω</mml:mi></mml:mfrac><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>Ω</mml:mi></mml:mfrac><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM182\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM183\"><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM184\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM185\"><mml:mi mathvariant=\"bold-italic\">g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ε</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM186\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mi>M</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM187\"><mml:mi mathvariant=\"bold-italic\">g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>M</mml:mi></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M3x3\"><label>3.3</label><mml:math id=\"DM12\" display=\"block\"><mml:mi mathvariant=\"bold-italic\">g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϵ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi mathvariant=\"bold-italic\">Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ε</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:mi>u</mml:mi><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:mi>v</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">g</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ε</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:msup><mml:mi>θ</mml:mi><mml:mi>A</mml:mi></mml:msup><mml:mo>−</mml:mo><mml:msup><mml:mi mathvariant=\"bold-italic\">b</mml:mi><mml:mi>A</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϵ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:msup><mml:mi>θ</mml:mi><mml:mi>B</mml:mi></mml:msup><mml:mo>−</mml:mo><mml:msup><mml:mi mathvariant=\"bold-italic\">b</mml:mi><mml:mi>B</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ϵ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM188\"><mml:mi>ε</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM189\"><mml:mi mathvariant=\"bold-italic\">Ω</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:msqrt><mml:mo>+</mml:mo><mml:mi>ε</mml:mi><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM190\"><mml:mi>ε</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM191\"><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM192\"><mml:mover><mml:mrow><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM193\"><mml:mover><mml:mrow><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>+</mml:mo><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mstyle><mml:mrow><mml:mrow><mml:mtext>tr</mml:mtext></mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:mrow><mml:mo>⋅</mml:mo><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM194\"><mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">tr</mml:mi></mml:mrow><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM195\"><mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">tr</mml:mi></mml:mrow><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM196\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM197\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM198\"><mml:mi>μ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM199\"><mml:mi>ρ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM200\"><mml:mi>λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM201\"><mml:mi>σ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM202\"><mml:mover><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM203\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM204\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M3x4\"><label>3.4</label><mml:math id=\"DM13\" display=\"block\"><mml:mrow><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mover><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>⁡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mover><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>⁡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mo>}</mml:mo></mml:mrow></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM205\"><mml:mover><mml:mrow><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM206\"><mml:mover><mml:mrow><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM207\"><mml:mi>l</mml:mi><mml:mo>=</mml:mo><mml:mi>Ω</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM208\"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>e</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM209\"><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mo>±</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>±</mml:mo><mml:mi>i</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM210\"><mml:munderover><mml:mrow><mml:mi>Ψ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo>∓</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:munderover><mml:mo>=</mml:mo><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>±</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mo>±</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mo>±</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM211\"><mml:munderover><mml:mrow><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:munderover><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>⁡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M3x5\"><label>3.5</label><mml:math id=\"DM14\" display=\"block\"><mml:mrow><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>Ω</mml:mi><mml:mover><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mover><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mover><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mover><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mover><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mo>}</mml:mo></mml:mrow></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM212\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>θ</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>θ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM213\"><mml:mi>ℓ</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM214\"><mml:mi>ℓ</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM215\"><mml:mi>ℓ</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM216\"><mml:mi>ℓ</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM217\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM218\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM219\"><mml:mover><mml:mrow><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM220\"><mml:mover><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM221\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM222\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM223\"><mml:mover><mml:mrow><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM224\"><mml:mover><mml:mrow><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM225\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mo>∩</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>v</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM226\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM227\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM228\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM229\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM230\"><mml:mi>ℓ</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM231\"><mml:mi>ℓ</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM232\"><mml:mi>ℓ</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM233\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM234\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM235\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM236\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM237\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM238\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM239\"><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM240\"><mml:mi>r</mml:mi><mml:mo>⋅</mml:mo><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM241\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM242\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM243\"><mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">tr</mml:mi></mml:mrow><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM244\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM245\"><mml:mi>r</mml:mi><mml:mo>⋅</mml:mo><mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">tr</mml:mi></mml:mrow><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM246\"><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mover><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM247\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM248\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM249\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM250\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM251\"><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM252\"><mml:mi>r</mml:mi><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM253\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM254\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM255\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM256\"><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM257\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM258\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM259\"><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM260\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM261\"><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mover><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM262\"><mml:mi>r</mml:mi><mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">tr</mml:mi></mml:mrow><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM263\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM264\"><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mover><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM265\"><mml:mi>r</mml:mi><mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">tr</mml:mi></mml:mrow><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM266\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM267\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM268\"><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM269\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM270\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM271\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM272\"><mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">tr</mml:mi></mml:mrow><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM273\"><mml:mi>r</mml:mi><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM274\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM275\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM276\"><mml:mi>N</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM277\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM278\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM279\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM280\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM281\"><mml:mi>N</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM282\"><mml:mi>ℓ</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM283\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM284\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM285\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:mo>−</mml:mo><mml:mi>ϵ</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM286\"><mml:msub><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM287\"><mml:mi>u</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM288\"><mml:munder><mml:mo movablelimits=\"true\" form=\"prefix\">lim</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:munder><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:munderover><mml:mrow><mml:mrow><mml:mover><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:munderover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM289\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM290\"><mml:munder><mml:mo movablelimits=\"true\" form=\"prefix\">lim</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:munder><mml:msup><mml:mi>r</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM291\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM292\"><mml:mi>r</mml:mi><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM293\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM294\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM295\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM296\"><mml:msup><mml:mi>T</mml:mi><mml:mrow><mml:mrow><mml:mtext>E2</mml:mtext></mml:mrow><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM297\"><mml:msup><mml:mi>T</mml:mi><mml:mrow><mml:mrow><mml:mtext>B2</mml:mtext></mml:mrow><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM298\"><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mtext>E</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mtext>B</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM299\"><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM300\"><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mtext>E</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mtext>B</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM301\"><mml:msub><mml:mi>λ</mml:mi><mml:mi>ℓ</mml:mi></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM302\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM303\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM304\"><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mo>∩</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M3x6\"><label>3.6</label><mml:math id=\"DM15\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:msubsup><mml:mi>α</mml:mi><mml:mi>ℓ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:munderover><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow></mml:mrow><mml:mi>r</mml:mi></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>⋅</mml:mo><mml:mfrac><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mi>r</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mi> </mml:mi><mml:mspace width=\"1em\"/><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>M</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mfrac><mml:mrow><mml:msub><mml:mover><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mi>log</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>⁡</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup></mml:mfrac><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>M</mml:mi><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM305\"><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM306\"><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM307\"><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M3x7\"><label>3.7</label><mml:math id=\"DM16\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:munder><mml:mo movablelimits=\"true\" form=\"prefix\">lim</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mtext>const</mml:mtext></mml:mrow><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:munder><mml:mi>r</mml:mi><mml:msubsup><mml:mi>α</mml:mi><mml:mi>ℓ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:mn>2</mml:mn><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:msub><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:msub><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo><mml:mi>ϵ</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM308\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM309\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM310\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M4x1\"><label>4.1</label><mml:math id=\"DM17\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>s</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mfrac><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>s</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mi> </mml:mi><mml:mspace width=\"1em\"/><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mfrac><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>s</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>A</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:mn>2.</mml:mn></mml:mtd></mml:mtr></mml:mtable></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM311\"><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM312\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM313\"><mml:mo>−</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM314\"><mml:mi>ℓ</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow><mml:mrow><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM315\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM316\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM317\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM318\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM319\"><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:mover><mml:mo>⁡</mml:mo><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM320\"><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM321\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:mover><mml:mo>⁡</mml:mo><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM322\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mo>⋯</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM323\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM324\"><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Φ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM325\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM326\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M4x2\"><label>4.2</label><mml:math id=\"DM18\" display=\"block\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Φ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>=</mml:mo><mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo stretchy=\"false\">(</mml:mo><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:mover><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Φ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>6</mml:mn><mml:mi>M</mml:mi><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mfrac><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Φ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM327\"><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM328\"><mml:msup><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>μ</mml:mi></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>ν</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM329\"><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Φ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM330\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM331\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM332\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM333\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM334\"><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Φ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:msup><mml:mi>Ω</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM335\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM336\"><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi>n</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM337\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM338\"><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM339\"><mml:mi>n</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM340\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM341\"><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM342\"><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Φ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:msup><mml:mi>Ω</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM343\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM344\"><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Φ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM345\"><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Φ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM346\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM347\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:msup><mml:mi>Ω</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM348\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM349\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:msup><mml:mi>Ω</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM350\"><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM351\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM352\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo>−</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM353\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo>−</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM354\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM355\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM356\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM357\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM358\"><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Φ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM359\"><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Φ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM360\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM361\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM362\"><mml:mi>N</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM363\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM364\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM365\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM366\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM367\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM368\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM369\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM370\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x1\"><label>5.1</label><mml:math id=\"DM19\" display=\"block\"><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>:=</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM371\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x2\"><label>5.2</label><mml:math id=\"DM20\" display=\"block\"><mml:mrow><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:mrow><mml:msubsup><mml:mi>α</mml:mi><mml:mi>ℓ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mi>ℓ</mml:mi></mml:msub><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>6</mml:mn></mml:mfrac><mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:msub><mml:mover><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mi>ℓ</mml:mi></mml:msub><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mi>ℓ</mml:mi></mml:msub><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mo>−</mml:mo><mml:mi>ϵ</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:msubsup><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi><mml:mi>n</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mo>}</mml:mo></mml:mrow></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM372\"><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow><mml:mrow><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM373\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x3\"><label>5.3</label><mml:math id=\"DM21\" display=\"block\"><mml:mrow><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mi>ℓ</mml:mi></mml:msub><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mo>−</mml:mo><mml:mi>ϵ</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mspace width=\"2em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"2em\"/></mml:mtd><mml:mtd><mml:msubsup><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi><mml:mi>n</mml:mi></mml:msubsup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mi mathvariant=\"normal\">∀</mml:mi><mml:mo> </mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow><mml:mrow><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable><mml:mo>}</mml:mo></mml:mrow></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM374\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM375\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x4\"><label>5.4</label><mml:math id=\"DM22\" display=\"block\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:msubsup><mml:mi>α</mml:mi><mml:mi>ℓ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mo>′</mml:mo></mml:msubsup><mml:mspace width=\"1em\"/><mml:mrow><mml:mi mathvariant=\"normal\">and</mml:mi></mml:mrow><mml:mspace width=\"1em\"/><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:msubsup><mml:mi>α</mml:mi><mml:mi>ℓ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mi>ℓ</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM376\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM377\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mo>′</mml:mo></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM378\"><mml:mi>u</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM379\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM380\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM381\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM382\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM383\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM384\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM385\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM386\"><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mi>s</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM387\"><mml:mo>−</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM388\"><mml:mi>ℓ</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow><mml:mrow><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM389\"><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM390\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x5\"><label>5.5</label><mml:math id=\"DM23\" display=\"block\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Ω</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mn>30</mml:mn><mml:mi>M</mml:mi><mml:msup><mml:mi>Ω</mml:mi><mml:mn>6</mml:mn></mml:msup></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>7</mml:mn></mml:msup></mml:mfrac><mml:mo>⋅</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM391\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM392\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM393\"><mml:mi>u</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM394\"><mml:mi>v</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM395\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>≲</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM396\"><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM397\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Ω</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>≲</mml:mo><mml:mi>M</mml:mi><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM398\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM399\"><mml:mi>u</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM400\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM401\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>≲</mml:mo><mml:mi>M</mml:mi><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>3</mml:mn><mml:mo>−</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM402\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM403\"><mml:mi>v</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM404\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi>M</mml:mi><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mo>−</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM405\"><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM406\"><mml:msup><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM407\"><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM408\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>≲</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM409\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>≲</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x6\"><label>5.6</label><mml:math id=\"DM24\" display=\"block\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi>M</mml:mi><mml:mo>⋅</mml:mo><mml:mi>r</mml:mi><mml:mo> </mml:mo><mml:mo stretchy=\"false\">⟹</mml:mo><mml:mo> </mml:mo><mml:mspace width=\"1em\"/><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>M</mml:mi><mml:mo>⋅</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>⋅</mml:mo><mml:msup><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM410\"><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x7\"><label>5.7</label><mml:math id=\"DM25\" display=\"block\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>30</mml:mn><mml:mi>M</mml:mi><mml:mo>⋅</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow><mml:mi>u</mml:mi></mml:msubsup><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>u</mml:mi><mml:mrow><mml:mo>′</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>7</mml:mn></mml:msup></mml:mfrac><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:msup><mml:mi>u</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>M</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mi>M</mml:mi><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM411\"><mml:msup><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>7</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>6</mml:mn></mml:mfrac></mml:mstyle><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>6</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>15</mml:mn></mml:mfrac></mml:mstyle><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>15</mml:mn></mml:mfrac></mml:mstyle><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM412\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM413\"><mml:mi>v</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x8\"><label>5.8</label><mml:math id=\"DM26\" display=\"block\"><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>M</mml:mi><mml:mo>⋅</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>M</mml:mi><mml:mo>⋅</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM414\"><mml:msup><mml:mi>r</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM415\"><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:msub><mml:mover><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM416\"><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:msub><mml:mover><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msup><mml:mi>log</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>⁡</mml:mo><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM417\"><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM418\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM419\"><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM420\"><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mi>s</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM421\"><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM422\"><mml:mi>ℓ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM423\"><mml:mi>u</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM424\"><mml:mi>v</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM425\"><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msup><mml:mo>:=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>s</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM426\"><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:mover><mml:mo>⁡</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msup><mml:mo>:=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>s</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:mover><mml:mo>⁡</mml:mo><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x9\"><label>5.9</label><mml:math id=\"DM27\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow></mml:mrow><mml:mo>∘</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mi> </mml:mi><mml:mspace width=\"1em\"/><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msubsup><mml:mi>c</mml:mi><mml:mi>N</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM427\"><mml:msubsup><mml:mi>c</mml:mi><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM428\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM429\"><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mi>N</mml:mi></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x10\"><label>5.10</label><mml:math id=\"DM28\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd/><mml:mtd><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:msubsup><mml:mi>c</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM430\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM431\"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM432\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM433\"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM434\"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM435\"><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mo>∫</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msup><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:mi>v</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM436\"><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM437\"><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM438\"><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM439\"><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM440\"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM441\"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM442\"><mml:mi>N</mml:mi><mml:mo>≤</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM443\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM444\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM445\"><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM446\"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mi>ℓ</mml:mi></mml:msub><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM447\"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mi>N</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM448\"><mml:mi>N</mml:mi><mml:mo>≤</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM449\"><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>ℓ</mml:mi></mml:msup><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>ℓ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>!</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM450\"><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x11\"><label>5.11</label><mml:math id=\"DM29\" display=\"block\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mi>M</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>ℓ</mml:mi></mml:msup><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>M</mml:mi><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow></mml:mrow><mml:mi>r</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:msub><mml:mover><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>r</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM451\"><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM452\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x12\"><label>5.12</label><mml:math id=\"DM30\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>s</mml:mi><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>r</mml:mi></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mi>i</mml:mi></mml:msup></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd><mml:mi> </mml:mi><mml:mspace width=\"1em\"/><mml:mo>+</mml:mo><mml:munder><mml:mrow><mml:munder><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mi>v</mml:mi><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mi>v</mml:mi></mml:msubsup><mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>⋯</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mrow><mml:mi>v</mml:mi><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:msubsup><mml:mfrac><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac></mml:mrow><mml:mo>⏟</mml:mo></mml:munder></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo> </mml:mo><mml:mtext>integrals</mml:mtext></mml:mrow></mml:munder><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>Ω</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:mo>)</mml:mo></mml:mrow><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub><mml:mo>⋯</mml:mo><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM453\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM454\"><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM455\"><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mtext>const</mml:mtext></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM456\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM457\"><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM458\"><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM459\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM460\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM461\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM462\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM463\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM464\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM465\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM466\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM467\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM468\"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM469\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM470\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM471\"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mi>ℓ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mi>ℓ</mml:mi></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM472\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM473\"><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM474\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM475\"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mi>ℓ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mi>ℓ</mml:mi></mml:msub><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM476\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM477\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM478\"><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x13\"><label>5.13</label><mml:math id=\"DM31\" display=\"block\"><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>M</mml:mi><mml:msub><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:mo>⋅</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM479\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M5x14\"><label>5.14</label><mml:math id=\"DM32\" display=\"block\"><mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:mi>r</mml:mi><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:mtd><mml:mtd><mml:mi> </mml:mi><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>i</mml:mi><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>r</mml:mi></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mi>i</mml:mi></mml:msup><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:msub><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo>−</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>6</mml:mn><mml:msup><mml:mi>r</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mi>r</mml:mi><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM480\"><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM481\"><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM482\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM483\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM484\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM485\"><mml:msubsup><mml:mi>α</mml:mi><mml:mi>ℓ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM486\"><mml:munder><mml:mo movablelimits=\"true\" form=\"prefix\">lim</mml:mo><mml:mrow><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:munder><mml:mi>r</mml:mi><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM487\"><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM488\"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM489\"><mml:mi>ℓ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM490\"><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Ω</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi mathvariant=\"normal\">∂</mml:mi><mml:mi>v</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM491\"><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM492\"><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM493\"><mml:mi>ℓ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM494\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:munderover><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM495\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM496\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:mo>−</mml:mo><mml:mi>ϵ</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM497\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM498\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mrow><mml:mtext>phg</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM499\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mrow><mml:mtext>phg</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mi>r</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:msubsup><mml:mi>r</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM500\"><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM501\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM502\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM503\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:mo>−</mml:mo><mml:mi>ϵ</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM504\"><mml:msup><mml:mi>D</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow><mml:mo>∪</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM505\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mrow><mml:mtext>phg</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM506\"><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM507\"><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>log</mml:mi><mml:mi>q</mml:mi></mml:msup><mml:mo>⁡</mml:mo><mml:mi>x</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM508\"><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM509\"><mml:mi>q</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM510\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mrow><mml:mtext>phg</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM511\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM512\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mrow><mml:mtext>phg</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM513\"><mml:msup><mml:mi>i</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM514\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM515\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM516\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mrow><mml:mtext>phg</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM517\"><mml:msubsup><mml:mi>α</mml:mi><mml:mrow><mml:mrow><mml:mtext>phg</mml:mtext></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM518\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM519\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM520\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM521\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM522\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM523\"><mml:mover><mml:mrow><mml:msub><mml:mi>Ψ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM524\"><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM525\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo>∼</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM526\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM527\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM528\"><mml:mover><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM529\"><mml:mover><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM530\"><mml:mover><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM531\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM532\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM533\"><mml:mover><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM534\"><mml:mover><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM535\"><mml:mover><mml:mrow><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM536\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM537\"><mml:mi>s</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM538\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM539\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM540\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM541\"><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM542\"><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM543\"><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mover><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM544\"><mml:mi>r</mml:mi><mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">tr</mml:mi></mml:mrow><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM545\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM546\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM547\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM548\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM549\"><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mover><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM550\"><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mrow><mml:mpadded width=\"0\"><mml:mtext>⧸</mml:mtext></mml:mpadded></mml:mrow><mml:mrow><mml:mi>g</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM551\"><mml:mover><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM552\"><mml:mi>r</mml:mi><mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">tr</mml:mi></mml:mrow><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM553\"><mml:mi>u</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM554\"><mml:mi>u</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM555\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM556\"><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:mi>E</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:mi>u</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM557\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM558\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM559\"><mml:mrow><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mrow/><mml:mrow><mml:mtext>d</mml:mtext></mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>4</mml:mn></mml:mfrac></mml:mstyle><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:munder><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">χ</mml:mi><mml:mo mathvariant=\"bold\" stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo mathvariant=\"bold\">_</mml:mo></mml:munder><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM560\"><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mo>⋅</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM561\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM562\"><mml:mi mathvariant=\"bold-italic\">r</mml:mi><mml:munder><mml:mrow><mml:mover><mml:mi mathvariant=\"bold-italic\">χ</mml:mi><mml:mo mathvariant=\"bold\" stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo mathvariant=\"bold\">_</mml:mo></mml:munder><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM563\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M6x1\"><label>6.1</label><mml:math id=\"DM33\" display=\"block\"><mml:mstyle displaystyle=\"true\" scriptlevel=\"0\"><mml:mrow><mml:mfrac><mml:mrow><mml:mover><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo>:=</mml:mo><mml:mo>−</mml:mo><mml:mstyle displaystyle=\"true\" scriptlevel=\"0\"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>16</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mi mathvariant=\"double-struck\">S</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>r</mml:mi><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>⁡</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow></mml:mrow><mml:mo>+</mml:mo></mml:msup></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mi>sin</mml:mi><mml:mo>⁡</mml:mo><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>θ</mml:mi><mml:mrow><mml:mi mathvariant=\"normal\">d</mml:mi></mml:mrow><mml:mi>φ</mml:mi><mml:mo>.</mml:mo></mml:mstyle></mml:mstyle></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM564\"><mml:mi>r</mml:mi><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM565\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M6x2\"><label>6.2</label><mml:math id=\"DM34\" display=\"block\"><mml:munder><mml:mrow><mml:mo form=\"prefix\">lim</mml:mo></mml:mrow><mml:mrow><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">const</mml:mi></mml:mrow><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:munder><mml:mo>⁡</mml:mo><mml:mi>r</mml:mi><mml:munderover><mml:mrow><mml:mrow><mml:mover><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mi>ℓ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:munderover><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mstyle displaystyle=\"true\" scriptlevel=\"0\"><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:mrow><mml:mi>ℓ</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac></mml:mrow><mml:mo>+</mml:mo><mml:mstyle displaystyle=\"true\" scriptlevel=\"0\"><mml:mrow><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:msub><mml:mrow><mml:mover><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mo accent=\"false\">¯</mml:mo></mml:mover></mml:mrow><mml:mi>ℓ</mml:mi></mml:msub><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:mstyle></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:mo>−</mml:mo><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM566\"><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM567\"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM568\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM569\"><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM570\"><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM571\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM572\"><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM573\"><mml:msub><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM574\"><mml:msub><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM575\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM576\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM577\"><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM578\"><mml:munder><mml:mrow><mml:mi mathvariant=\"script\">B</mml:mi></mml:mrow><mml:mo>_</mml:mo></mml:munder></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM579\"><mml:msup><mml:mi>i</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM580\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM581\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<disp-formula id=\"RSTA20230039M6x3\"><label>6.3</label><mml:math id=\"DM35\" display=\"block\"><mml:munder><mml:mrow><mml:mo form=\"prefix\">lim</mml:mo></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:munder><mml:mo>⁡</mml:mo><mml:munder><mml:mrow><mml:mo form=\"prefix\">lim</mml:mo></mml:mrow><mml:mrow><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:munder><mml:mo>⁡</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msubsup><mml:mrow><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:munder><mml:mi>χ</mml:mi><mml:mo>_</mml:mo></mml:munder><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">B</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>ℓ</mml:mi></mml:msup><mml:munder><mml:mrow><mml:mo form=\"prefix\">lim</mml:mo></mml:mrow><mml:mrow><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:munder><mml:mo>⁡</mml:mo><mml:munder><mml:mrow><mml:mo form=\"prefix\">lim</mml:mo></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:mrow></mml:munder><mml:mo>⁡</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msubsup><mml:mrow><mml:mover><mml:mrow><mml:mrow><mml:mover><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">B</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"IM582\"><mml:mi>u</mml:mi><mml:mo>≤</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM583\"><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"normal\">∞</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM584\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM585\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM586\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM587\"><mml:mover><mml:mrow/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM588\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM589\"><mml:msub><mml:mi>p</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM590\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM591\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM592\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM593\"><mml:msub><mml:mi>p</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM594\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM595\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM596\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM597\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo>∼</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM598\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM599\"><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM600\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM601\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM602\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM603\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM604\"><mml:msup><mml:mi>ϵ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM605\"><mml:mi>g</mml:mi><mml:mo>=</mml:mo><mml:mi>η</mml:mi><mml:mo>+</mml:mo><mml:mi>ϵ</mml:mi><mml:msup><mml:mi>g</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>ϵ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>g</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>ϵ</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM606\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM607\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM608\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM609\"><mml:mi>b</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM610\"><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM611\"><mml:mi>γ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM612\"><mml:mi>δ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM613\"><mml:msub><mml:mi>Ψ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>∼</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM614\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM615\"><mml:mi>U</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM616\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM617\"><mml:mi>V</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:mo>+</mml:mo><mml:mo>⋯</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM618\"><mml:msup><mml:mi>Ψ</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM619\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM620\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM621\"><mml:mi>o</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mn>7</mml:mn></mml:mrow><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM622\"><mml:msup><mml:mi>i</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM623\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup><mml:mo>∼</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM624\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM625\"><mml:mrow><mml:mi mathvariant=\"script\">A</mml:mi></mml:mrow></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM626\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM627\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM628\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM629\"><mml:msup><mml:mi>α</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM630\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>log</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>⁡</mml:mo><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM631\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM632\"><mml:mover><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM633\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM634\"><mml:mover><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mover><mml:mo>∼</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>r</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM635\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM636\"><mml:mi>s</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM637\"><mml:mi>s</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM638\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM639\"><mml:msup><mml:mi>i</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM640\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM641\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM642\"><mml:mi>s</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM643\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM644\"><mml:msup><mml:mi>i</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM645\"><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM646\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM647\"><mml:mi>Λ</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM648\"><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM649\"><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>", "<inline-formula><mml:math id=\"IM650\"><mml:mi>α</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>" ]

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[ "<fn-group><fn id=\"FN1\"><label>1</label><p>For an overview over the field before the advent of the ideas discussed in the present section, we refer to the lecture notes [##UREF##11##12##] (from 1958) and references therein.</p></fn><fn id=\"FN2\"><label>2</label><p>We note in passing that, owing to another condition in the definition of asymptotic simplicity, not even the Schwarzschild spacetime is asymptotically simple owing to the presence of trapped null geodesics, but this is of no further relevance here.</p></fn><fn id=\"FN3\"><label>3</label><p>A longer-term goal that we will not address here is the development of a rigorous understanding of the <italic toggle=\"yes\">generation problem</italic>.</p></fn><fn id=\"FN4\"><label>4</label><p>This entire discussion of course take place under the pretence that can reasonably be set to 0 and that the astrophysical process under consideration starts in the infinite past. The question of how to formalize the limiting process where is sent to 0 is yet a completely different story, and we do not dare to touch upon it here.</p></fn><fn id=\"FN5\"><label>5</label><p>With these last two paragraphs loosely relating the decay behaviour towards spatial infinity to that towards future null infinity, we feel that it may be helpful to point out that research in the direction of finding conditions on Cauchy data near spatial infinity that are necessary or sufficient to ensure peeling near null infinity as pioneered by Friedrich [##UREF##56##57##], while intriguing in its own right, generally, just like asymptotic simplicity itself, suffers from the same conceptual difficulty of establishing the physical meaning of these conditions and is hence not directly related to the research programme outlined in this article.</p></fn><fn id=\"FN6\"><label>6</label><p>This important detail was overlooked in the brief outlook given in [##UREF##30##31##], which, therefore, made an incorrect prediction on .</p></fn><fn id=\"FN7\"><label>7</label><p>In parts of the literature, such corrections arising from backscattering off of the curvature of the spacetime are referred to as ‘tail effects’, see [##UREF##71##73##,##UREF##72##74##] and references therein.</p></fn><fn id=\"FN8\"><label>8</label><p>Based on the results of Hintz &amp; Vasy [##UREF##48##49##] (see remark 8.2 therein), one may expect this regularity to be for any . Note, however, that not even the question of the minimal conformal regularity that ensures equation (##FORMU##25##1.3##) is resolved, cf. [##UREF##74##76##] and §2 of Friedrich [##UREF##75##77##].</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p id=\"Par7\">Vaccines are the most powerful pharmaceutical tool to prevent infections with SARS-CoV-2 and combat the COVID-19 pandemic. Fast vaccine uptake by as many individuals as possible saves lives, people’s health, and livelihoods. Despite large-scale vaccine roll-out campaigns, many countries, most prominently in Europe, have experienced a rise in case numbers in the late summer and early fall of 2021 and reported effective reproduction numbers above one for an extended period of time^{##UREF##0##1##}. This means that on average, every infected person infected more than one other person, thus causing exponentially rising incidences^{##UREF##1##2##}. Since the beginning of this pandemic, such resurgences have, in part, been mitigated by harsh non-pharmaceutical interventions (NPIs) such as lockdowns or curfews that limit the population’s contacts, thereby decreasing the effective reproduction number and relieving overburdened public health systems^{##REF##33199859##3##,##REF##32269067##4##}. Measures that affect large parts of the general population over a long period of time can have devastating effects, such as increasing social inequality and domestic violence, detrimental impacts on mental health, or economic disruptions^{##REF##32581299##5##–##UREF##4##9##}. Such harsh restrictions should therefore be considered a last resort of pandemic control.</p>", "<p id=\"Par8\">During the onset of the fourth COVID-19 wave in Germany, many hospitals and intensive care units (ICUs) were operating at maximum capacity or were projected to do so at a later point^{##UREF##5##10##}. In the four weeks between Oct 11, 2021, and Nov 7, 2021, Germany’s central public health institute, the Robert Koch Institute (RKI) reported 250,552 new symptomatic infections in individuals with known vaccination status, 90,471 of which were attributed to vaccinated individuals, i.e. 36% were symptomatic breakthrough cases (41% in age groups eligible for vaccination)^{##UREF##6##11##,##UREF##7##12##}. During this time, the average vaccination rate in different age groups [0,12), [12,18), [18,60), and 60+ were 0%, 40.1%, 72.4%, and 85.1%, respectively, leading to 0%, 4.8%, 41.6%, and 61.9% of new cases being classified as symptomatic breakthrough cases within the respective age groups^{##UREF##6##11##}, Table ##TAB##0##1##. Simultaneously, the effective reproduction number remained at a relatively stable value of (under the assumption of a generation time of four days)^{##UREF##8##13##}.</p>", "<p id=\"Par9\">Given that breakthrough cases are a challenge both for communication and vaccine acceptance^{##REF##35214710##14##} and that harsh NPIs may be illegitimate for vaccinated individuals, the above situation raises two important questions: How much does the unvaccinated population contribute to the infection dynamics despite being in the minority? And could targeted NPIs aiming at reducing the contacts of unvaccinated individuals temporarily and sufficiently suppress the infection dynamics such that harsh, large-scale NPIs could be avoided?</p>", "<p id=\"Par10\">To address these questions, we establish the contribution matrix approach, a theoretical concept derived from the next-generation matrix framework^{##UREF##9##15##}. The contribution matrix quantifies the contributions to caused by the infection pathways from un-/vaccinated individuals to other un-/vaccinated individuals, considering the age and contact structure of the population, vaccination rates, as well as expected vaccine efficacies regarding susceptibility and transmission reductions, respectively. In its general form, it quantifies the contributions made by any combination of two subpopulations.</p>", "<p id=\"Par11\">Based on this approach, we estimate that in October 2021, around 32%–51% (depending on vaccine efficacy) of the effective reproduction number was caused by unvaccinated individuals infecting other unvaccinated individuals (see Fig. ##FIG##0##1##). Since unvaccinated individuals have a higher probability of suffering from severe disease^{##REF##33306989##16##–##UREF##11##18##}, this contribution is the major factor that drove the public health system into a crisis characterized by hospitals and ICUs reaching maximum capacity. In contrast, we estimate that only 15%–18% of the reproduction number were attributable to vaccinated individuals infecting unvaccinated individuals. In October 2021, about 65% of the German population was fully vaccinated, implying that the majority of the overall population contributed little to the amplification of the crisis. In total, we estimate that the vaccinated population contributed 24%–39% to while the unvaccinated population contributed the remaining 61%–76%, despite the fact that unvaccinated individuals have been in the minority in Germany. 9%–21% of new infections would be caused by vaccinated individuals infecting other vaccinated people. In total, we estimate that unvaccinated individuals were involved in 8–9 out of 10 new infections, either as infecting, acquiring infection, or both.</p>", "<p id=\"Par12\">We further argue that regarding the situation in the fall of 2021, the unvaccinated would have had to reduce their transmissibility two to three times as strongly as the vaccinated in order for the system to reach (and hence containment of the infection wave), if the burden of contact reductions were to be distributed between the two subpopulations according to their respective contributions. Moreover, decreasing mixing between individuals of distinct vaccination status can decrease . Ultimately, a higher vaccine uptake would have led to less unvaccinated being involved in infections, which can not only decrease , but is critical for relieving an overburdened public health system, as they are more likely to suffer from severe disease. Combinations of these interventions that address mainly the unvaccinated might have rendered the dynamics subcritical.</p>" ]

[ "<title>Methods</title>", "<title>Mathematical framework</title>", "<p id=\"Par13\">We use a population-structured compartmental infectious disease model that captures a variety of aspects regarding vaccination against COVID-19 (see Supplementary Methods, Sec. ##SUPPL##1##1.1.1##). The model’s dynamics are fully described by the next-generation matrix <italic>K</italic>_{<italic>j</italic><italic>i</italic>} of small domain (see Supplementary Methods, Sec. ##SUPPL##1##1.1.2##), which quantifies the average number of offspring in group <italic>j</italic> caused by a single infectious individual in group <italic>i</italic>^{##UREF##9##15##}. Here, the index <italic>i</italic> (or <italic>j</italic>, respectively) refers to the subpopulation that is determined by a respective age group and the vaccination status within that group, thus yielding two subpopulations per age group. In the regime of small outbreaks (relative to the total population size), the ordinary differential equations governing the epidemic growth can be linearized, with the dynamics being determined by <italic>K</italic>_{<italic>j</italic><italic>i</italic>}, such that the generational growth of the number of infected individuals in group <italic>i</italic> followsThe incidence approaches the eigenstate <italic>y</italic>_{<italic>i</italic>} of <italic>K</italic>_{<italic>j</italic><italic>i</italic>} that corresponds its spectral radius, which in turn is equal to the effective reproduction number^{##UREF##9##15##}. Hence, the entries of the normalized eigenvector contain the relative frequency of newly infected individuals in age/vaccination group <italic>i</italic>.</p>", "<p id=\"Par14\">Consequently, the number of <italic>j</italic>-offspring caused by <italic>i</italic>-individuals in a dynamical system defined by <italic>K</italic>_{<italic>j</italic><italic>i</italic>} is given by the contribution matrixSumming over all matrix elements of <italic>C</italic>_{<italic>j</italic><italic>i</italic>} yields the effective reproduction number (see Supplementary Methods, Sec. ##SUPPL##1##1.1.1##)). A single matrix element <italic>C</italic>_{<italic>j</italic><italic>i</italic>} can thus be considered the contribution of the <italic>i</italic> → <italic>j</italic> infection pathway to the reproduction number (a derivation of the concept and an operational definition of <italic>C</italic>_{<italic>j</italic><italic>i</italic>} can be found in the Supplementary Methods, Sec. ##SUPPL##1##1.1.1##–##SUPPL##1##1.1.2## and Sec. ##SUPPL##1##1.2.4##), respectively). The normalized contribution matrix <italic>C</italic>_{<italic>j</italic><italic>i</italic>}/ gives the relative contributions of <italic>i</italic> → <italic>j</italic> infections towards (and consequently, towards the total number of new infections).</p>", "<p id=\"Par15\">We derive explicit equations for the contributions of un-/vaccinated individuals in the homogeneous case, i.e. ignoring age structure (see Supplementary Methods, Sec. ##SUPPL##1##1.2.3##). These contributions arewhere <italic>v</italic> is the vaccine uptake, <italic>s</italic> is the susceptibility reduction after vaccination, is the adjusted transmissibility reduction (i.e. it contains the relative increase of the recovery rate after a breakthrough infection <italic>b</italic> and viral shedding reduction <italic>r</italic>), is the base transmissibility of unvaccinated infecteds, and is the base transmissibility of vaccinated infecteds (both of which quantify differences in behavior in the respective groups). The total effective reproduction number is given by</p>", "<title>Model structure, parameters, and scenarios</title>", "<p id=\"Par16\">In the full model, we construct the next-generation matrix of small domain (see Supplementary Methods, Eq. (##SUPPL##1##S3##)) based on the following observations, assumptions, and estimates: We structure the population into four age groups [0,12) (children), [12,18) (adolescents), [18,60) (adults), and 60+ (elderly). Contact numbers between those age groups and subpopulation sizes were constructed based on the POLYMOD (2005) data set^{##REF##18366252##19##,##UREF##12##20##} using the ‘socialmixr’ software package^{##UREF##13##21##} (see Supplementary Methods, Sec. ##SUPPL##1##1.2.1##). Since vaccine efficacy was, at the time of writing, estimated only for the status “fully vaccinated” in Germany without distinguishing between different vaccines, we solely distinguish between “unvaccinated” and “vaccinated” individuals in the model, regardless of the make of the received doses (note that by the fall of 2021, a total number of four vaccine types was available in Germany, i.e., Spikevax (Moderna), Ad26.COV2.S (Janssen), Vaxzevria (AstraZeneca), and Comirnaty (BioNTech/Pfizer) with the latter being by far the most used^{##UREF##14##22##}). Following the example of Scholz et al.^{##UREF##15##23##}, we further assume that children and adolescents have reduced susceptibility to the virus and a reduced base transmissibility if infected, as was observed in Germany, Israel, and Greece^{##REF##32975186##24##–##UREF##17##26##}. In the discussed time frame, 14.7%, 9.4%, 60.2%, and 15.7% of new cases can be attributed to the respective age groups [0,12), [12,18), [18,60), and 60+^{##UREF##7##12##}. In order to match this distribution approximately, we calibrate the base susceptibility (i. e. susceptibility without vaccination) and infectiousness of our model by assuming that children are 72% as susceptible and 63% as infectious as adults (72% and 81% for adolescents), which is larger than what was observed for the wild type^{##UREF##16##25##,##UREF##17##26##}, see Supplementary Methods, Sec. ##SUPPL##1##1.2##. However, since the B.1.617.2 variant (Delta) that was predominant in Germany in October/November 2021 was generally observed to be more infectious than the wild type^{##UREF##18##27##}, such an increase is plausible. Note that in principle, heterogeneous ascertainment may lead to a distribution of detected cases that is skewed towards the adult population, as children and adolescents may have higher probability of suffering from an asymptomatic infection^{##UREF##19##28##} and thus are less likely to be detected via symptom-based testing strategies. Yet, by the fall of 2021, Germany made regular screening via rapid antigen tests mandatory in schools across the country, potentially lowering the level of under-ascertainment in these age groups^{##UREF##20##29##}. Nevertheless, we test how our results change by assuming children and adolescents are as susceptible as adults in a sensitivity analysis (see Supplementary Methods, Sec. ##SUPPL##1##1.3.3##). Additionally, note that we ignore the number of recovered individuals. Until Oct 10, 2021, about 4.3 million infections were reported in Germany^{##UREF##8##13##}, 74% of which likely received a vaccination^{##UREF##21##30##–##UREF##23##32##} and are therefore considered as vaccinated in our analysis. With an under-ascertainment ratio of about 1.8^{##UREF##24##33##}, we estimate that the total number of non-vaccinated recovered individuals was on the order of 2.4% of the population in Germany at the time, and therefore negligible in our analysis.</p>", "<p id=\"Par17\">In Germany, an estimated average vaccine efficacy of 72% against symptomatic COVID-19 in adults and the elderly was found for cases reported between Oct 11, 2021 and Nov 7, 2021^{##UREF##6##11##}. Vaccine efficacy for adolescents was not reported due to the respective data being potentially unreliable (low number of cases). Because these efficacies were computed for symptomatic cases, we use their values as a “high efficacy” scenario regarding vaccine efficacy in our analysis, because unreported and/or asymptomatic breakthrough infections might lower the estimated efficacies (see Supplementary Methods, Sec. ##SUPPL##1##1.2.5##). However, note that these observed 72% vaccine efficacy are in line with an estimated population-wide vaccine efficacy against infection based on vaccination time series and waning immunity data that was published in a meta-review by the WHO^{##UREF##25##34##,##REF##35202601##35##}. In order to obtain breakthrough infection rates in adolescents on the order of observed symptomatic breakthrough cases we assume a vaccine efficacy of <italic>s</italic> = 92% for adolescents. Despite being comparably large, this value seems justified considering that adolescents have been made eligible to receive a vaccine in Germany only shortly prior to the study period, and a high vaccine efficacy against infection with the Delta variant has been reported for this age group^{##UREF##26##36##}. Regarding the infectiousness of individuals suffering from breakthrough infections, viral load of vaccinated patients suffering from symptomatic COVID-19 was reported to be at the same level as of those unvaccinated^{##REF##34826623##37##,##UREF##27##38##}. Another study from the UK found decreased infectiousness in breakthrough infections^{##REF##34726481##39##}. Considering both these results, we assume a conservative transmission reduction of <italic>r</italic> = 10% for breakthrough infections. In agreement with the literature^{##REF##34826623##37##,##REF##34192428##40##} we further consider that the average infectious period of breakthrough infections is shorter than for unvaccinated individuals and assume a 50% increase in recovery rate for the vaccinated, amounting to an average infectious period that is 2/3 as long as that of unvaccinated infecteds (<italic>b</italic> = 3/2) (see Supplementary Methods, Sec. ##SUPPL##1##1.2.2##). Such an increased recovery rate can also be caused by deliberate behavior. As individuals that are not opposed to vaccination typically adhere to protection measures more consistently^{##UREF##28##41##}, behavioral changes following a breakthrough infection might further decrease the effective infectious period. Note that together with a decreased duration of infection <italic>b</italic> = 3/2, the adjusted transmission reduction reads , which is lower than a 63% reduction that was observed for household transmissions of the Delta variant between infected vaccinated and susceptible unvaccinated individuals in the Netherlands in August and September 2021, close to our observational period^{##REF##34738514##42##}. As this reduction was observed to wane over time^{##UREF##29##43##}, is a reasonable assumption.</p>", "<p id=\"Par18\">In a second, “medium efficacy” scenario, we consider that vaccine efficacies against infection are in the range of 50%–60%, i.e. lower than the observed value against symptomatic COVID-19, and lower than vaccine efficacies reported in the UK for the Comirnaty (BioNTech/Pfizer) vaccine^{##UREF##30##44##}, considering that partial immunity might have waned over time^{##REF##34650248##45##}. Since vaccine efficacy is expected to decrease with age^{##REF##34650248##45##,##REF##34192737##46##}, we assume an efficacy against infection of <italic>s</italic> = 60% for adolescents and adults as well as <italic>s</italic> = 50% for the elderly (see Supplementary Methods, Sec. ##SUPPL##1##1.2.4##).</p>", "<p id=\"Par19\">Finally, we also discuss a “low efficacy” scenario where the susceptibility reduction is assumed to be much lower than the observed efficacy against symptomatic COVID-19, namely 50% for adolescents and adults, and 40% for the elderly (see Supplementary Methods, Sec. ##SUPPL##1##1.2.5##).</p>", "<p id=\"Par20\">To summarize the main scenarios, for the “high efficacy” the vaccination efficacy <italic>s</italic> for adolescents, adults, and elderly is assumed to be 92%, 72%, 72%, in the “medium efficacy” scenario 60%, 60%, 50%, and in the “low efficacy” scenario 50%, 50%, 40%, respectively.</p>", "<p id=\"Par21\">Based on these considerations we compute the respective full-model next generation matrices <italic>K</italic>_{<italic>j</italic><italic>i</italic>} and numerically find the normalized population eigenvectors corresponding to the respective and the contribution matrices <italic>C</italic>_{<italic>j</italic><italic>i</italic>}, which we further reduce to the two-dimensional vaccination status space by summing over the respective contributions of age groups (see Supplementary Methods, Eq. (##SUPPL##1##S2##)).</p>", "<title>Reporting summary</title>", "<p id=\"Par22\">Further information on research design is available in the ##SUPPL##3##Nature Research Reporting Summary## linked to this article.</p>" ]

[ "<title>Results</title>", "<p id=\"Par23\">As a first model validation we find that for the high efficacy scenario the relative size of breakthrough infections within age groups eligible for vaccination is in good agreement with the share of reported symptomatic breakthrough cases (Table ##TAB##0##1##), albeit being slightly larger than reported values, mirroring the fact that the official number of breakthrough infections is likely affected by underreporting^{##UREF##6##11##} and that the number of infections will be larger than the number of symptomatic breakthrough cases.</p>", "<p id=\"Par24\">For all scenarios, we find that the largest entry in the contribution matrix is given by the unvaccinated → unvaccinated infection pathway, with a 51.4% (high efficacy), 38.1% (medium efficacy) and 31.6% (low efficacy) contribution respectively, see Tables ##TAB##1##2##, ##TAB##2##3##, ##TAB##3##4## and Fig. ##FIG##0##1##. Most noteworthy, these numbers represent the largest contributions although the unvaccinated population is smaller than the vaccinated one. Moreover, the total contribution of the unvaccinated population to the effective reproduction number is 75.9%, 66.6%, and 61.1% for the high, medium, and low efficacy scenarios, respectively. In total, the unvaccinated population plays a role in 91.1% (high), 84% (medium), or 79.3% (low efficacy) of cases—either as infecting, acquiring infection, or both.</p>", "<p id=\"Par25\">Since vaccine efficacy is expected to decrease with age and time passed after vaccination^{##REF##34650248##45##}, we test how our results for the “medium efficacy” scenario change when assuming a more pessimistic susceptibility reduction of 40% for the elderly while keeping 60% for all other age groups (see Supplementary Methods, Sec. ##SUPPL##1##1.3.5##). We find that our results do not change substantially (see Supplementary Table ##SUPPL##1##3##), which can be attributed to the fact that the elderly generally have a lower contact behavior than other age groups.</p>", "<p id=\"Par26\">In order to test the validity of the homogeneous approach, we further use Eqs. (##FORMU##16##3##)–(##FORMU##19##6##)) to compute the contribution matrix with <italic>v</italic> = 65%, <italic>s</italic> = 72%, <italic>r</italic> = 10%, and <italic>b</italic> = 3/2, assuming . We find relative contributions of , , , , hence being in good agreement with the results of the age-structured model (cf. Tab. ##TAB##1##2##), showing that Eqs. (##FORMU##16##3##)–(##FORMU##19##6##)) can be used to estimate the order of magnitude of the contributions by the respective infection pathways. We expect this approximation to lose its validity for situations in which model assumptions become even more heterogeneous (e.g. strong differences in contact structure between age groups, vaccine uptakes per age group, or vaccine efficacy per age group).</p>", "<p id=\"Par27\">During the period of time when vaccine efficacies were measured^{##UREF##6##11##}, the reproduction number in Germany was reported to be at a relatively stable value of ^{##UREF##8##13##}. In order to achieve temporary epidemic control, it is necessary to reach a value of for a substantial amount of time^{##UREF##1##2##}. We therefore study how the effective reproduction number would change if the transmissibility of unvaccinated individuals would be reduced. This could, for instance, be achieved by strict enforcement of contact rules regarding unvaccinated individuals at private and public gatherings that were partially in place in Germany^{##UREF##31##47##}. For our analysis we gauge <italic>K</italic>_{<italic>j</italic><italic>i</italic>} such that for either of the base scenarios and then individually scale the transmissibility of the vaccinated and unvaccinated to obtain those values at which the critical value is attained, Fig. ##FIG##1##2##a. We find that a transmission reduction of 22%–27% in the unvaccinated population would suffice to reach without the need for any further restrictions. In contrast, NPIs that would affect both, vaccinated and unvaccinated to the same degree, would need to cause more than 17% of transmissibility reduction across the entire population to achieve epidemic control. For completeness and to put numbers in perspective one may also consider the unlikely scenario where NPIs are only in place for the vaccinated population yielding a required transmissibility reduction of 43%–73% in that group to achieve epidemic control, highlighting that vaccinated individuals would have to decrease their transmissibility less strongly than unvaccinated individuals for a distribution of the burden of contact reductions that corresponds to their respective contributions. The way to reach in the plane spanned by NPI-based transmissibility reductions in both respective subpopulations that acknowledges these contributions with appropriate weighting is given by the linear function that is perpendicular to the isoclines shown in Fig. ##FIG##1##2##a. Using the fact that the homogeneous model given by Eqs. (##FORMU##16##3##)–(##FORMU##19##6##) yields acceptable approximations to the full model, we use Eq. (##FORMU##23##7##) to derive the slope of this function (see Supplementary Methods, Sec. ##SUPPL##1##1.3.2##). This quantity has to be read as “if the unvaccinated population reduces its transmissibility by 10%, the vaccinated population has to reduce its transmissibility by <italic>χ</italic> × 10% in order for the system to quickly approach ”. With <italic>v</italic> = 65%, <italic>s</italic> = 72%, for the “high efficacy” scenario, as well as <italic>s</italic> = 60% for the “medium” and <italic>s</italic> = 50% for the “low efficacy” scenario, we find <italic>χ</italic> = 0.31, <italic>χ</italic> = 0.45, and <italic>χ</italic> = 0.55, respectively, which suggests that in order to adequately distribute the burden of further transmissibility reductions between the respective subpopulations, unvaccinated individuals would have to reduce their transmissibility two to three times as strongly as the vaccinated population.</p>", "<p id=\"Par28\">We further test the robustness of our results regarding vaccine efficacy by varying an age-independent vaccine efficacy against infection that ranges from <italic>s</italic> = 100% to <italic>s</italic> = 0%, (i) leaving <italic>r</italic> = 10% and <italic>b</italic> = 3/2 unchanged as an optimistic estimation and (ii) proportionally scaling <italic>r</italic> = <italic>s</italic>/10 and <italic>b</italic> = <italic>s</italic>/2 + 1 as a pessimistic estimation, while assuming vaccine uptake as reported in the Methods section (see Supplementary Methods, Sec. ##SUPPL##1##1.3.1##). We find a monotonic decrease of breakthrough infections from non-zero values for <italic>s</italic> = 0% to zero for <italic>s</italic> = 100%. Notably, we find that as long as vaccine efficacies do not drop below 22% (optimistic) or 41% (pessimistic), the majority of new cases remains to be caused by the minority of the population, which are the unvaccinated (see Fig. ##FIG##2##3## and the results for an additional “very low efficacy” scenario in Supplementary Methods Sec. ##SUPPL##1##1.3.7## as well as Supplementary Table ##SUPPL##1##6##).</p>", "<p id=\"Par29\">Next, we also account for the fact that the infectiousness of children and adolescents has been a matter of debate^{##REF##32975186##24##–##UREF##17##26##,##UREF##32##48##,##UREF##33##49##}. While for all analyses presented above we assumed reduced infectiousness for those respective age groups compared to adults and elderly, we now assume (as an upper limit) that children and adolescents are as infectious as adults (see Supplementary Methods, Sec. ##SUPPL##1##1.3.3##). This generally leads to higher contributions by unvaccinated individuals to the overall share of infections since they represent by far the majority in these age groups. We find that the unvaccinated in this scenario cause 76%–85% of all new infections for the “medium” and “high” scenario, respectively (see Supplementary Tables ##SUPPL##1##1## and ##SUPPL##1##2##) which is substantially larger than the 67%–76% obtained when susceptibility and infectiousness in children and adolescents is reduced (see again Fig. ##FIG##0##1##a, b and Tabs. ##TAB##1##2##, ##TAB##2##3##).</p>", "<p id=\"Par30\">Moreover, we test how our results change if the assumption of homogeneous mixing between vaccinated and unvaccinated individuals is no longer met. This captures the likely scenario that vaccinated and unvaccinated populations are more prone to meet individuals of similar vaccination status rather than opposing vaccination status either due to homophily^{##UREF##34##50##–##REF##24079377##52##} or deliberate non-pharmaceutical interventions, such as limiting access to public gatherings, immune shielding^{##REF##32382154##53##}, or social distancing informed by serological testing^{##REF##33397941##54##}. We conceptualize this process by scaling the off-diagonal matrix elements indicating offspring caused by vaccinated infecting unvaccinated individuals and vice versa with a constant factor <italic>m</italic> ∈ [0, 1] such that <italic>m</italic> = 1 refers to our base scenario of homogeneous mixing between the two groups, Fig. ##FIG##1##2##b and Supplementary Methods Sec. ##SUPPL##1##1.3.4##. As expected, we find that the relative contribution to made by the unvaccinated increases monotonically with decreasing <italic>m</italic> (inset of Fig. ##FIG##1##2##b). In case the system was, in fact, already in a state of heterogeneous mixing during the observational period, this implies that our main results shown in Fig. ##FIG##0##1## present lower bounds of the contribution made by unvaccinated individuals. If mixing was decreased by additional NPIs that reduce contacts between unvaccinated and vaccinated individuals, the absolute value of decreases with decreasing mixing <italic>m</italic>. This illustrates the efficacy such NPIs would have towards mitigation, assuming that the reduced inter-group contacts are not balanced by increased intra-group contact numbers. In the latter case, an increased number of contacts between unvaccinated individuals could even lead to an increase in , potentially worsening the situation.</p>", "<p id=\"Par31\">Ultimately, we investigate how different the situation would have been if vaccine uptake was higher than 65% in the fall of 2021. To this end, we choose the “medium efficacy” scenario, but increase the respective vaccine uptake for adolescents, adults, and elderly to 90% each, leading to an 80% uptake in the total population, Fig. ##FIG##1##2##c and Supplementary Methods Sec. ##SUPPL##1##1.3.6##. In this case, the effective reproduction number would be lowered to a value of instead of , implying epidemic control. Because more people would be vaccinated, both the relative and absolute contributions of vaccinated individuals to would increase. Yet, the most important differences to the base scenario of <italic>v</italic> = 65% are the respective reductions of the absolute contributions of unvaccinated individuals, which would decrease from (i) <italic>C</italic>_{<italic>u</italic>←<italic>u</italic>} + <italic>C</italic>_{<italic>v</italic>←<italic>u</italic>} = 0.8 to <italic>C</italic>_{<italic>u</italic>←<italic>u</italic>} + <italic>C</italic>_{<italic>v</italic>←<italic>u</italic>} = 0.37 for infections caused and (ii) from <italic>C</italic>_{<italic>u</italic>←<italic>u</italic>} + <italic>C</italic>_{<italic>u</italic>←<italic>v</italic>} = 0.67 to <italic>C</italic>_{<italic>u</italic>←<italic>u</italic>} + <italic>C</italic>_{<italic>u</italic>←<italic>v</italic>} = 0.3 for becoming infected, both more than halved (see Supplementary Tables ##SUPPL##1##4## and ##SUPPL##1##5##). Because unvaccinated infecteds have a much higher probability of suffering from severe disease and being hospitalized, such a reduction can be substantial for relieving an overburdened public health system.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par32\">After vaccine rollout programs in Germany have slowed down in late summer, incidences were rising to unprecedented levels in the fall of 2021, with hospitals and ICUs having reached maximum capacity. As about 41% of reported cases aged 12 or above were recorded as breakthrough infections in October 2021, two questions naturally arise: (i) How much were the vaccinated still contributing to the infection dynamics and (ii) how need NPIs to be targeted and calibrated to help achieve temporal epidemic control?</p>", "<p id=\"Par33\">Here, we developed a model-based framework that allows for quantifying the contributions of different infection pathways between and across vaccinated and unvaccinated groups towards the effective reproduction number . Based on this framework and reasonable assumptions regarding vaccine efficacy, we conclude that about 61%–76% percent of the effective reproduction number were caused by unvaccinated individuals, with 32%–51% of its value determined by unvaccinated individuals infecting other unvaccinated individuals. Depending on the assumed efficacy scenario, 34%–50% of the infections are expected to be breakthrough infections. Although these numbers might seem comparatively large at first glance, such results that focus solely on the presence or absence of an infection (not the severity) are to be expected^{##REF##8225751##55##}. Our study highlights the importance of analyzing the limited contribution these breakthrough cases make towards the overall infection dynamics, especially in relation to the size of the respective vaccinated/unvaccinated subpopulations. Additionally, such proportions of breakthrough infections are not necessarily indicative of a potential burden to the public health system, as all vaccines against COVID-19 have been reported to substantially reduce the risk of a severe course of the disease^{##UREF##6##11##,##REF##33306989##16##–##UREF##11##18##}.</p>", "<p id=\"Par34\">We further showed that targeted NPIs that would decrease the transmissibility of unvaccinated individuals by 22%–27% could have suppressed epidemic growth reaching , under the assumption that vaccinated individuals would continue to behave as before, i.e., with no additional NPIs in place for this respective group. Yet, it is questionable how well NPIs can be targeted towards single subpopulations, both for ethical and pragmatic reasons. We found that for NPIs that would affect both unvaccinated and vaccinated individuals, those that reduce the transmissibility of the unvaccinated two to three times as strongly as the vaccinated population would reduce in the most efficient manner.</p>", "<p id=\"Par35\">Our assumptions regarding vaccine efficacy against transmission (effective transmissibility reduction) were lower than values observed in the Netherlands^{##REF##24079377##52##}. Assuming that the efficacy is of larger value would further increase the contributions of unvaccinated individuals towards the infection dynamics. Similarly, if children and adolescents were found to be as susceptible and infectious as adults, the contributions made by the unvaccinated subpopulation would be of larger value as well.</p>", "<p id=\"Par36\">The analyses performed here represent model-based estimations that are limited by data quality and a large number of parameters that have to be estimated based on available empirical results. This includes epidemiological data as well as contact data from the POLYMOD study, which is already over 15 years old and might therefore inaccurately portray the mixing behavior of the German population at the time of writing. A further limiting factor is that the under-ascertainment of breakthrough infections might be larger than accounted for, as vaccinated infecteds experiencing mild symptoms might not be as likely to have their infection reported, thus leading to a potential overestimation of vaccine efficacy. Yet, vaccinated individuals might have increased their contact behavior compared to unvaccinated individuals, a behavorial change that compensates for the vaccine-induced lowered individual risk of infection. Because vaccine efficacies were estimated using Farrington’s method^{##UREF##6##11##,##REF##8225751##55##}, such a relative increase in contact behavior of vaccinated individuals could lead to an underestimation of the true vaccine efficacy, thus potentially balancing a hypothetical inequality in ascertainment. Due to such uncertainties, future empirical studies, e.g. using contact tracing data, will be necessary to confirm or refute our claims.</p>", "<p id=\"Par37\">While we consider population mixing across age groups, we also implicitly assume homogeneous mixing between vaccinated and unvaccinated individuals in our base scenarios. Yet, the intention to vaccinate has been shown to follow rules of social contagion, rendering it likely that vaccinated and unvaccinated individuals mix less across groups^{##UREF##36##56##}. We showed that, in this case, the contribution of unvaccinated individuals to would be of even larger magnitude. NPIs that reduced contacts between both subpopulations (i.e. reduced mixing) would lead to a decrease in , as long as these reductions are not balanced by an increase in contacts among unvaccinated individuals, in which case might even increase, highlighting the necessity for well-targeted measures. We want to stress that one should be careful, however, not to overinterpret this result as explicit advice for future NPIs to increase segregation between the vaccinated and unvaccinated. Indeed, other research shows that after measures that restricted access to shopping and leisure activities only for the unvaccinated, societal polarization was high^{##UREF##36##56##}. While this may reduce mixing, it creates other, potentially worse societal problems. Our analysis does not account for any psychological or socio-cultural consequences of such policies or recommendations^{##REF##32382153##57##} and, as always, recommendations should be weighed against potential risks.</p>", "<p id=\"Par38\">Finally, an increased vaccine uptake would increase both the relative and absolute contributions that the vaccinated population makes towards while similarly decreasing the effective reproduction number’s absolute value, potentially leading to temporary epidemic control under the assumption of unchanged behavior. In light of the slow growth of vaccine uptake in Germany after the summer 2021^{##UREF##14##22##} and low intention to vaccinate among those that are unvaccinated^{##UREF##28##41##}, such an increase in uptake, however, seems unlikely to be achieved.</p>", "<p id=\"Par39\">We furthermore stress that our results are estimations made for the comparatively short period between October 11, 2021 and November 7, 2021. As vaccine efficacy against infection has been reported to decrease with time, fast and wide-spread booster vaccination is a crucial measure to avoid an increasing reproduction number and a potentially worsening public health crisis. Also, the spread of immune escape variants may change the situation.</p>", "<p id=\"Par40\">In summary, our results suggest that a minority of the population (i.e., the unvaccinated) contributed a substantial part to the infection dynamics, thus making them the primary driver of the public health crisis in Germany during the fourth wave of the COVID-19 pandemic and presumably also in other countries that were experiencing similar dynamics. We also show that this effect can be compensated through targeted NPIs that effectively lower the transmissibility of infected, yet unvaccinated, individuals. Hence, our study further underlines the importance of vaccines as a pharmaceutical intervention regarding epidemic control and highlights the importance of increasing vaccine uptake, e.g. through campaigning or low-threshold offers, wherever possible, in order to achieve efficient and long-term epidemic control and preventing an overload of public health systems.</p>" ]

[]

[ "<title>Background</title>", "<p id=\"Par1\">While the majority of the German population was fully vaccinated at the time (about 65%), COVID-19 incidence started growing exponentially in October 2021 with about 41% of recorded new symptomatic cases aged twelve or above being symptomatic breakthrough infections, presumably also contributing to the dynamics. So far, it remained elusive how significant this contribution was and whether targeted non-pharmaceutical interventions (NPIs) may have stopped the amplification of the crisis.</p>", "<title>Methods</title>", "<p id=\"Par2\">We develop and introduce a contribution matrix approach based on the next-generation matrix of a population-structured compartmental infectious disease model to derive contributions of respective inter- and intragroup infection pathways of unvaccinated and vaccinated subpopulations to the effective reproduction number and new infections, considering empirical data of vaccine efficacies against infection and transmission.</p>", "<title>Results</title>", "<p id=\"Par3\">Here we show that about 61%–76% of all new infections were caused by unvaccinated individuals and only 24%–39% were caused by the vaccinated. Furthermore, 32%–51% of new infections were likely caused by unvaccinated infecting other unvaccinated. Decreasing the transmissibility of the unvaccinated by, e. g. targeted NPIs, causes a steeper decrease in the effective reproduction number than decreasing the transmissibility of vaccinated individuals, potentially leading to temporary epidemic control. Reducing contacts between vaccinated and unvaccinated individuals serves to decrease in a similar manner as increasing vaccine uptake.</p>", "<title>Conclusions</title>", "<p id=\"Par4\">A minority of the German population—the unvaccinated—is assumed to have caused the majority of new infections in the fall of 2021 in Germany. Our results highlight the importance of combined measures, such as vaccination campaigns and targeted contact reductions to achieve temporary epidemic control.</p>", "<title>Plain language summary</title>", "<p id=\"Par5\">With about 65% of its citizens vaccinated at the time, Germany experienced a large wave of COVID-19 in the fall of 2021, regionally overburdening the healthcare system. We are interested in how much this crisis was driven by infections in vaccinated versus unvaccinated people. We use a mathematical model to show that transmission of the disease during this period was largely driven by the unvaccinated population, despite representing a smaller proportion of the overall population. Our results suggest that higher vaccine uptake, reduced mixing between vaccinated and unvaccinated people, and targeted contact-reduction measures would have been effective measures to control spread at the time. These findings may have implications for how we manage future waves of COVID-19 or other diseases.</p>", "<p id=\"Par6\">Maier et al. develop a mathematical model to examine the contributions of vaccinated vs. unvaccinated populations to the wave of SARS-CoV-2 infections in Germany in autumn 2021. They report that the unvaccinated population were the main drivers of transmission and that targeted non-pharmaceutical interventions would likely have mitigated this.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s43856-022-00176-7.</p>", "<title>Acknowledgements</title>", "<p>We thank L. E. Sander for valuable comments. B. F. M. is financially supported by the Joachim Herz Stiftung as an <italic>Add-On Fellow for Interdisciplinary Life Science</italic>.</p>", "<title>Author contributions</title>", "<p>B.F.M. developed the initial research idea. Analyses were performed by B.F.M, following discussions with M.W., A.B., P.P.K., M.A.J., C.B., and D.B. A.B. performed additional literature research. B.F.M. and M.W. wrote the first manuscript draft, which was subsequently edited by B.F.M., M.W., A.B., P.P.K., M.A.J., C.B., and D.B.</p>", "<title>Peer review</title>", "<title>Peer review information</title>", "<p id=\"Par41\"><italic>Communications Medicine</italic> thanks Peter Klimek and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.</p>", "<title>Funding</title>", "<p>Open Access funding enabled and organized by Projekt DEAL.</p>", "<title>Data availability</title>", "<p>Analysis results produced in this study are given in the ##SUPPL##1##Supplementary Information## and on Zenodo (ref. 58). Source data for the figures are available as Supplementary Data ##SUPPL##2##1## and in the Zenodo repository. Data regarding the count of breakthrough infections and estimated vaccine efficacy during the study period may be found in ref. 11. Population sizes and contact numbers were chosen according to ref. 20, based on data from refs. 18–19.</p>", "<title>Code availability</title>", "<p>Code to reproduce the results of this study is available under github.com/benmaier/vaccontrib and on Zenodo^{##UREF##37##58##}.</p>", "<title>Competing interests</title>", "<p id=\"Par42\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Estimated contributions of infection pathways towards new cases within vaccinated and unvaccinated subpopulations.</title><p>Estimated contributions of infection pathways to in the (<bold>a</bold>) “high efficacy”, (<bold>b</bold>) “medium efficacy”, and (<bold>c</bold>) “low efficacy” scenarios as a graphical representation of Tabs. ##TAB##1##2##–##TAB##3##4##. The charts can be read as follows: Consider an infected population that caused a new generation of 100 new infecteds. Then for (<bold>a</bold>), 51 of those newly infected individuals will be unvaccinated people that have been infected by other unvaccinated people. Likewise, 25 newly infected individuals will be vaccinated people that have been infected by unvaccinated individuals. Hence, 76 new infections will have been caused by the unvaccinated. Along the same line, 15 newly infecteds will be unvaccinated people that have been infected by vaccinated individuals and 9 newly infecteds will be vaccinated people that have been infected by other vaccinated individuals, totaling 24 new infections that have been caused by vaccinated individuals.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Efficacy of potential interventions to achieve temporary epidemic control.</title><p><bold>a</bold> Required additional transmissibility reduction for the unvaccinated (horizontal axis) and vaccinated (vertical axis) population to lower to values below one, based on the assumption that the initial effective reproduction number is equal to . <bold>b</bold> The absolute contributions to of the unvaccinated (orange) and vaccinated population (green) as well as their sum (black) with decreasing mixing <italic>m</italic> between both groups, based on the “medium efficacy” scenario. The inset shows the respective relative contributions. Note that if heterogeneous mixing was already present during our observational period, the monotonically increasing contribution of the unvaccinated displayed in the inset implies that our results of Fig. ##FIG##0##1## are actually lower bounds of the true contribution. <bold>c</bold> Absolute contributions to for infections between and across groups of vaccinated and unvaccinated individuals at the vaccine uptake during the observational period (left bar) and a hypothetical vaccine uptake of 80% in the total population, i.e., 90% in the age groups that were, at the time, eligible for vaccination (right bar), based on the “medium efficacy” scenario. The latter would have sufficed to suppress sufficiently below one, assuming that other factors determining the base transmissibility remained on the same level.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Fraction of new cases caused by the unvaccinated and vaccinated population for varying age-independent vaccine efficacy <italic>s</italic>.</title><p>We consider an optimistic scenario with constant <italic>r</italic> = 0.1 and <italic>b</italic> = 3/2 (solid lines), and a pessimistic estimation in which <italic>r</italic> and <italic>b</italic> decrease according to <italic>r</italic> = <italic>s</italic>/10 and <italic>b</italic> = <italic>s</italic>/2 + 1 (dashed lines). As long as <italic>s</italic> remains larger than approximately 22% (optimistic, ) or 41% (pessimistic, ), the unvaccinated minority still causes the majority of infections, see also Supplementary Methods, Sec. ##SUPPL##0##1.3.1## and Sec. ##SUPPL##0##1.3.7##.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Share of breakthrough infections in the age groups eligible for vaccination according to official estimates by the Robert Koch Institute (RKI)^{##UREF##6##11##} and the model for “low efficacy”, “medium efficacy”, and “high efficacy” scenarios.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Age group</th><th>RKI report (symptomatic cases)</th><th>Model (“high eff.”)</th><th>Model (“medium eff.”)</th><th>Model (“low eff.”)</th></tr></thead><tbody><tr><td>adolescents</td><td>4.8%</td><td>5.1%</td><td>21.1%</td><td>25%</td></tr><tr><td>adults</td><td>41.6%</td><td>42.3%</td><td>51.2%</td><td>57%</td></tr><tr><td>elderly</td><td>61.9%</td><td>61.5%</td><td>74.1%</td><td>77.4%</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Contribution to from infections between vaccinated and unvaccinated populations for the upper parameter bounds.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th/><th>← (u)nvaccinated</th><th>← (v)accinated</th></tr></thead><tbody><tr><td>u ← </td><td>51.4%</td><td>15.0%</td></tr><tr><td>v ← </td><td>24.5%</td><td>9.1%</td></tr><tr><td>total</td><td>75.9%</td><td>24.1%</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Relative contributions to from infections between vaccinated and unvaccinated groups for the “medium efficacy” scenario.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th/><th>← (u)nvaccinated</th><th>← (v)accinated</th></tr></thead><tbody><tr><td>u ← </td><td>38.1%</td><td>17.4%</td></tr><tr><td>v ← </td><td>28.5%</td><td>16.0%</td></tr><tr><td>total</td><td>66.6%</td><td>33.4%</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Relative contributions to from infections between vaccinated and unvaccinated groups for the “low efficacy” scenario.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th/><th>← (u)nvaccinated</th><th>← (v)accinated</th></tr></thead><tbody><tr><td>u ← </td><td>31.6%</td><td>18.2%</td></tr><tr><td>v ← </td><td>29.5%</td><td>20.7%</td></tr><tr><td>total</td><td>61.1%</td><td>38.9%</td></tr></tbody></table></table-wrap>" ]

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id=\"M16\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq8\"><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M18\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq9\"><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M20\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${y}_{j}(g+1)=\\mathop{\\sum}\\limits_{i}{K}_{ji}{y}_{i}(g),\\qquad g=0,1,2,...$$\\end{document}</tex-math><mml:math id=\"M22\"><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>g</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"2.0em\"/><mml:mi>g</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq10\"><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{y}}_{i}={y}_{i}\\big/{\\sum }_{j}{y}_{j}$$\\end{document}</tex-math><mml:math id=\"M24\"><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mstyle mathsize=\"1.19em\"><mml:mfenced open=\"/\"><mml:mrow/></mml:mfenced></mml:mstyle><mml:msub><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{ji}={K}_{ji}{\\hat{y}}_{i}.$$\\end{document}</tex-math><mml:math id=\"M26\"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq11\"><alternatives><tex-math id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M28\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq12\"><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq13\"><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M32\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{u\\leftarrow u}=\\frac{{(1-v)}^{2}}{1-vs}{{{{{{{{\\mathcal{R}}}}}}}}}_{u}$$\\end{document}</tex-math><mml:math id=\"M34\"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi><mml:mo>←</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>v</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>v</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{u\\leftarrow v}=\\frac{v(1-v)(1-s)(1-r^{\\prime} )}{1-vs}{{{{{{{{\\mathcal{R}}}}}}}}}_{v}$$\\end{document}</tex-math><mml:math id=\"M36\"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi><mml:mo>←</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>v</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>v</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>v</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{v\\leftarrow u}=\\frac{v(1-v)(1-s)}{1-vs}{{{{{{{{\\mathcal{R}}}}}}}}}_{u}$$\\end{document}</tex-math><mml:math id=\"M38\"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mo>←</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>v</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>v</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>v</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ6\"><label>6</label><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{v\\leftarrow v}=\\frac{{v}^{2}{(1-s)}^{2}(1-r^{\\prime} )}{1-vs}{{{{{{{{\\mathcal{R}}}}}}}}}_{v},$$\\end{document}</tex-math><mml:math id=\"M40\"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mo>←</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>v</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:mfrac><mml:msub><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq14\"><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$r^{\\prime} =1-(1-r)/b$$\\end{document}</tex-math><mml:math id=\"M42\"><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mi>b</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq15\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\mathcal{R}}}}}}}}}_{u}$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:msub><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq16\"><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\mathcal{R}}}}}}}}}_{v}$$\\end{document}</tex-math><mml:math id=\"M46\"><mml:msub><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ7\"><label>7</label><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{array}{lll}{{{{{{{\\mathcal{R}}}}}}}}&amp;=&amp;{C}_{u\\leftarrow u}+{C}_{v\\leftarrow u}+{C}_{u\\leftarrow v}+{C}_{v\\leftarrow v}\\\\ &amp;=&amp;(1-v){{{{{{{{\\mathcal{R}}}}}}}}}_{u}+v(1-s)(1-r^{\\prime} ){{{{{{{{\\mathcal{R}}}}}}}}}_{v}.\\end{array}$$\\end{document}</tex-math><mml:math id=\"M48\"><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:mtd><mml:mtd columnalign=\"left\"><mml:mo>=</mml:mo></mml:mtd><mml:mtd columnalign=\"left\"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi><mml:mo>←</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mo>←</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi><mml:mo>←</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mo>←</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"/><mml:mtd columnalign=\"left\"><mml:mo>=</mml:mo></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>v</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>v</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq17\"><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$r^{\\prime} =1-(1-r)/b=40 \\%$$\\end{document}</tex-math><mml:math id=\"M50\"><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mn>40</mml:mn><mml:mi>%</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq18\"><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$r^{\\prime} =40 \\%$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>40</mml:mn><mml:mi>%</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq19\"><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{y}}_{i}$$\\end{document}</tex-math><mml:math id=\"M54\"><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq21\"><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M58\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq20\"><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\mathcal{R}}}}}}}}}_{u}={{{{{{{{\\mathcal{R}}}}}}}}}_{v}$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:msub><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq021\"><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{u\\leftarrow u}/{{{{{{{\\mathcal{R}}}}}}}}=50.1 \\%$$\\end{document}</tex-math><mml:math id=\"M64\"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi><mml:mo>←</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>50.1</mml:mn><mml:mi>%</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq022\"><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{v\\leftarrow u}/{{{{{{{\\mathcal{R}}}}}}}}=26.1 \\%$$\\end{document}</tex-math><mml:math id=\"M66\"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mo>←</mml:mo><mml:mi>u</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>26.1</mml:mn><mml:mi>%</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq023\"><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{u\\leftarrow v}/{{{{{{{\\mathcal{R}}}}}}}}=15.7 \\%$$\\end{document}</tex-math><mml:math id=\"M68\"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>u</mml:mi><mml:mo>←</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>15.7</mml:mn><mml:mi>%</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{v\\leftarrow v}/{{{{{{{\\mathcal{R}}}}}}}}=8.1 \\%$$\\end{document}</tex-math><mml:math id=\"M70\"><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mo>←</mml:mo><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>8.1</mml:mn><mml:mi>%</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}=1.2$$\\end{document}</tex-math><mml:math id=\"M72\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>1.2</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M73\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}} &lt; 1$$\\end{document}</tex-math><mml:math id=\"M74\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M75\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\sum }_{ji}{C}_{ji}={{{{{{{\\mathcal{R}}}}}}}}=1.2$$\\end{document}</tex-math><mml:math id=\"M76\"><mml:msub><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>1.2</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M77\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}=1$$\\end{document}</tex-math><mml:math id=\"M78\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M79\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}=1$$\\end{document}</tex-math><mml:math id=\"M80\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq30\"><alternatives><tex-math id=\"M81\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}=1$$\\end{document}</tex-math><mml:math id=\"M82\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq31\"><alternatives><tex-math id=\"M83\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\chi =v(1-s)(1-r^{\\prime} )/(1-v)$$\\end{document}</tex-math><mml:math id=\"M84\"><mml:mi>χ</mml:mi><mml:mo>=</mml:mo><mml:mi>v</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>s</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>v</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq32\"><alternatives><tex-math id=\"M85\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}=1$$\\end{document}</tex-math><mml:math id=\"M86\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq33\"><alternatives><tex-math id=\"M87\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$r^{\\prime} =40 \\%$$\\end{document}</tex-math><mml:math id=\"M88\"><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>40</mml:mn><mml:mi>%</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq38\"><alternatives><tex-math id=\"M89\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M90\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq39\"><alternatives><tex-math id=\"M91\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}=1.2$$\\end{document}</tex-math><mml:math id=\"M92\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>1.2</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq40\"><alternatives><tex-math id=\"M93\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M94\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq41\"><alternatives><tex-math id=\"M95\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M96\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq42\"><alternatives><tex-math id=\"M97\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M98\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq43\"><alternatives><tex-math id=\"M99\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$r^{\\prime} =40 \\% ^{\\prime}$$\\end{document}</tex-math><mml:math id=\"M100\"><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>40</mml:mn><mml:msup><mml:mrow><mml:mi>%</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq44\"><alternatives><tex-math id=\"M101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$r^{\\prime} =20 \\%$$\\end{document}</tex-math><mml:math id=\"M102\"><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>20</mml:mn><mml:mi>%</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq34\"><alternatives><tex-math id=\"M103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M104\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq35\"><alternatives><tex-math id=\"M105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M106\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq36\"><alternatives><tex-math id=\"M107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M108\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq37\"><alternatives><tex-math id=\"M109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}=0.86$$\\end{document}</tex-math><mml:math id=\"M110\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>0.86</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq038\"><alternatives><tex-math id=\"M111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}=1.2$$\\end{document}</tex-math><mml:math id=\"M112\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>=</mml:mo><mml:mn>1.2</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq039\"><alternatives><tex-math id=\"M113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M114\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq040\"><alternatives><tex-math id=\"M115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M116\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq041\"><alternatives><tex-math id=\"M117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}} &lt; 1$$\\end{document}</tex-math><mml:math id=\"M118\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq042\"><alternatives><tex-math id=\"M119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M120\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq043\"><alternatives><tex-math id=\"M121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M122\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq044\"><alternatives><tex-math id=\"M123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M124\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq45\"><alternatives><tex-math id=\"M125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M126\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq46\"><alternatives><tex-math id=\"M127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{R}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M128\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM4\"></supplementary-material>" ]

[ "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p><bold>Change history</bold></p><p>1/12/2024</p><p>A Correction to this paper has been published: 10.1038/s43856-024-00432-y</p></fn></fn-group>" ]

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[ "<media xlink:href=\"43856_2022_176_MOESM1_ESM.pdf\"><caption><p>Description of Additional Supplementary Files</p></caption></media>", "<media xlink:href=\"43856_2022_176_MOESM2_ESM.pdf\"><caption><p>Supplementary Information</p></caption></media>", "<media xlink:href=\"43856_2022_176_MOESM3_ESM.xlsx\"><caption><p>Supplementary Data 1</p></caption></media>", "<media xlink:href=\"43856_2022_176_MOESM4_ESM.pdf\"><caption><p>Reporting Summary</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>INTRODUCTION</title>", "<p>Since the discovery of oncogenic driver mutations, the therapeutic landscape of non-small-cell lung cancer (NSCLC) has evolved exponentially. Genomic profiling has set the stage for personalized cancer therapy. NSCLC is known to demonstrate somatic alterations in the <italic>BRCA1/2</italic> gene and homologous recombination repair (HRR) genes such as <italic>ATM</italic>, <italic>FANCA</italic>, and <italic>PALB2</italic> (##REF##26775620##Lord and Ashworth 2016##). Based on The Cancer Genome Atlas data set, the rates of <italic>BRCA1</italic> or <italic>BRCA2</italic> variants in lung adenocarcinoma were 3.0% and 4.8%, respectively. (##REF##25079552##Cancer Genome Atlas Research Network 2014##).</p>", "<p>Poly (ADP-ribose) polymerase (PARP) inhibitors trap PARP on DNA at sites of single strand breaks, preventing cellular DNA repair machinery from correcting the initial insult. HRR is necessary to resolve these PARP–DNA interactions via double-stranded break repair. Double-stranded break repair are defective in homologous recombination–deficient (HRD) cancer cells with <italic>BRCA1 or BRCA2</italic> pathogenic variants (PVs). PARP inhibitors confer synthetic lethality to these cells with accumulation of DNA damage causing cell death. PARP inhibitors are approved in malignancies associated with germline <italic>BRCA1/2</italic> PV such as breast, ovarian, prostate, and pancreatic cancer. The therapeutic relevance of olaparib remains less defined in NSCLC and other nongermline <italic>BRCA1</italic>/<italic>2</italic> related malignancies.</p>", "<p>A preclinical study (##REF##31753490##Ji et al. 2020##) depleted <italic>BRCA1</italic>/<italic>2</italic> in a NSCLC cell line and found that cells lacking homologous recombination (HR) proteins were hypersensitive to olaparib. The authors propose that olaparib induces apoptosis in HR-deficient NSCLC cells. Two phase II trials (##UREF##0##Fennell et al. 2020##; ##UREF##2##Postel-Vinay et al. 2021##) compared olaparib versus placebo monotherapy in patients with metastatic NSCLC. Both trials showed no progression-free survival (PFS) benefit or overall survival (OS) benefit with olaparib.</p>", "<p>With this report, we discuss the case of a 64-yr-old male patient with metastatic lung adenocarcinoma with a somatic <italic>BRCA2</italic> PV, whose treatment with olaparib led to a clinical and radiologic response of his disease.</p>" ]

[ "<title>METHODS</title>", "<p>The specimen obtained from the epigastric lump was sent for <italic>EGFR</italic> mutation analysis by direct Sanger sequencing. Genomic DNA was extracted from the tissue sample. The extracted DNA then undergoes polymerase chain reaction (PCR) amplification for exons 18, 19, 20, and 21, and the following mutations screened for in the assay are shown in the <ext-link xlink:href=\"http://www.molecularcasestudies.org/lookup/suppl/doi:10.1101/mcs.a006223/-/DC1\" ext-link-type=\"uri\">Supplemental Materials</ext-link>. For <italic>ALK</italic> and <italic>ROS1</italic> mutation analysis, this was determined via interphase FISH. <italic>ALK</italic> break-apart, <italic>ROS1</italic> break-apart, <italic>MET/CEP7</italic> enumeration, and <italic>RET</italic> break-apart probes were performed on paraffin-embedded tissue. A total of 100 nonoverlapped nuclei were scored manually by two independent observers, and interpretation of the results was based on prevailing guidelines.</p>", "<p>Peripheral blood was sent for <italic>EGFR</italic> T790M mutation. Isolation of cell-free DNA was performed using the Cobas cfDNA sample preparation kit. This was a real-time PCR assay for the semiquantification index (SQI) of mutations in exon 18 (G719X), deletion mutations in exon 19, T790M and S768I substitution mutations and insertion mutations in exon 20, and L858R and L861Q substitution mutations in exon 21 of the <italic>EGFR</italic> gene from serial collections of human plasma.</p>", "<p>The patient's lung biopsy was sent for <italic>EGFR</italic> mutation analysis by Roche Cobas EGFR mutation test V2, a commercial real-time allele-specific PCR test. DNA is extracted from the tissue sample. The extracted DNA undergoes PCR amplification for exon 18, 19, 20, 21. The list of mutations that are targeted for by the assay is described in the <ext-link xlink:href=\"http://www.molecularcasestudies.org/lookup/suppl/doi:10.1101/mcs.a006223/-/DC1\" ext-link-type=\"uri\">Supplemental Materials</ext-link>.</p>", "<p>The somatic tumor from the lung biopsy was sent for next generation sequencing via Oncomine Comprehensive Assay V3. Tumor tissue was obtained from archival formalin-fixed paraffin-embedded samples collected during the patient's lung biopsy. Genomic DNA and RNA from tumor were extracted from formalin-fixed paraffin-embedded or fresh-frozen tissue. A library was constructed using different panels, and the resulting amplicons were treated to partially digest, phosphorylate, and ligate to ion adapters with barcoding and purified. Quality and concentration of the libraries were determined using the Qubit 2.0 Fluorometer. Emulsion PCR and enrichment of template-positive Ion Sphere Particles, which contained clonally amplified DNA, were conducted using the Ion PGM OneTouch 2 system. Sequencing of this amplified DNA was subsequently performed on the Ion Torrent PGM sequencer. Data were analyzed primarily using Torrent Suite Variant Caller plugin, Ion Reporter software, and an in-house analysis pipeline v1.0.0 using Oncomine Reporter with reference genome hg19. The genomic analysis was developed to detect for somatic changes and was not designed or validated to interrogate for germline changes. Tumor mutation burden (TMB) was calculated based on the size of the panel and number of nonsynonymous mutations and mathematically approximated for 1 Mbp. The results of all molecular profiling were returned to the ordering clinician. The list of the genes analyzed from somatic tumor next-generation sequencing is found in the <ext-link xlink:href=\"http://www.molecularcasestudies.org/lookup/suppl/doi:10.1101/mcs.a006223/-/DC1\" ext-link-type=\"uri\">Supplemental Materials</ext-link>.</p>", "<p>For the germline sequencing, the patient's DNA was extracted from peripheral blood and sent to Invitae. Genomic DNA was obtained from the blood and enriched for targeted regions using a hybridization-based protocol and sequenced using Illumina technology. Reads were aligned to a reference sequence (GRCh37), and sequence changes were identified and interpreted. Enrichment and analysis focus on the coding sequence of the indicated transcripts, 20 bp of flanking intronic sequence, and other specific genomic regions demonstrated to be causative of disease at the time of assay design. Promoters, untranslated regions, and other noncoding regions are not otherwise interrogated. All clinically significant observations are confirmed by orthogonal technologies, except individually validated variants and variants previously confirmed in a first-degree relative. Confirmation technologies include any of the following: Sanger sequencing, Pacific Biosciences SMRT sequencing, MLPA, MLPA-seq, Array CGH. The list of the genes analyzed from germline sequencing is found in Supplemental Materials. The germline testing panel had included evaluation of 59 genes for variants that are associated with genetic disorders, shown in <ext-link xlink:href=\"http://www.molecularcasestudies.org/lookup/suppl/doi:10.1101/mcs.a006223/-/DC1\" ext-link-type=\"uri\">Supplemental Materials</ext-link>.</p>" ]

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[ "<title>DISCUSSION</title>", "<p>In this case report, we showed the clinical efficacy of olaparib in a patient with metastatic NSCLC harboring a somatic <italic>BRCA2</italic> PV and germline <italic>BRCA2</italic> wild type. The patient's PFS of 8 mo with olaparib was superior to pembrolizumab and pemetrexed-carboplatin. Although the partial response in the brain may be attributed to the effects of the WBRT, the disease control in the lymph nodes is likely due to the PARP inhibitor. These observations may support the role of extended genomic testing in this patient population. Multiple DNA repair genes in the HRR pathway may be associated with PARP inhibitor sensitivity. Not every patient with DNA repair defects will have clinical benefit with PARP inhibition, and there are ongoing efforts to identify biomarkers that will predict response to PARP inhibitors (##REF##26775620##Lord and Ashworth 2016##). To date, there are only case reports that describe the benefit of PARP inhibitors in metastatic NSCLC, but these reports were in patients harboring a germline <italic>BRCA1/2</italic> PV (##REF##31956609##Talreja et al. 2020##; ##REF##34616818##Zhang et al. 2021##; ##REF##34387603##Wu et al. 2022##).</p>", "<p>Given the tumor percentage of 70% and a variant allele frequency of 28.4%, it is presumed that the patient's <italic>BRCA2</italic> PV is a monoallelic <italic>BRCA2</italic> mutation. The NGS panel used did not provide information about the allelic status or the HRD status of the tumor. Biallelic alterations of HRR genes are significantly associated with genomic features of homologous recombination deficiency, whereas mono-allelic alterations are not (##REF##29021619##Riaz et al. 2017##). Apart from looking at mutations in genes involved in the HRR pathway, the HRD status may also be determined by evaluating the effect of genomic scarring. Genomic scars are aberrations that result in structural changes in the chromosomes. The most relevant genomic scars include the loss of heterozygosity (LOH), telomeric imbalance (TAI), and large-scale transitions (LSTs) (##REF##26775620##Lord and Ashworth 2016##). When measured together, this produces a genomic instability score that may be used as an indicator of HRD status and maximize the identification of samples with HRD (##REF##26015868##Marquad et al. 2015##). Germline testing of our patient did not identify any pathogenic variants of HRR genes that may have resulted in biallelic alterations. In our patient who had a somatic <italic>BRCA2</italic> pathogenic variant, LOH of the wild-type allele would have resulted in a biallelic alteration and explain the response to olaparib. Therefore, it would have been interesting to run a larger sequencing panel such as the Trusight Oncology 500 HRD panel, which could provide more information on the presence of genomic scars such as LOH.</p>", "<p>The Oncomine Comprehensive Assay v3 NGS panel, an amplicon-based assay, has its own limitations. The amplicon-based approach relies on primers that flank the interest of sequencing. This can lead to false negatives because of allele dropout or genomic deletions (##REF##35877256##Ionescu et al. 2022##). A study showed that amplicon-based assays had a lower rate of detecting gene fusions in NSCLC patients when compared to hybrid capture–based assays (##REF##33639937##Heydt et al. 2021##). Another study showed that commercially developed amplicon assays were limited in the detection of <italic>MET</italic> exon 14 skipping events in NSCLC patients (##REF##28779874##Poirot et al. 2017##).</p>", "<p>The response to olaparib runs contrary to the results from the two prior phase II studies that have looked at PARP inhibitors in platinum sensitive advanced NSCLC (##UREF##0##Fennell et al. 2020##; ##UREF##2##Postel-Vinay et al. 2021##). In both trials no genomic sequencing was performed to identify patients with <italic>BRCA1/2</italic> PVs. Only platinum sensitivity was hypothesized to enrich for patients who may have underlying HRR deficiency such as <italic>BRCA1/2</italic> PV and therefore be sensitive to olaparib. Yet, platinum sensitivity may not be sufficient to rule out clinical benefit from PARP inhibitors. Unlike PARP inhibitor sensitivity, which results from defective DNA single-stranded break repair, platinum sensitivity may result from defective nucleotide excision repair (##REF##27145721##Ceccaldi et al. 2016##). In addition, first-line chemotherapy for NSCLC is platinum-based. In the trial by Fennell et al. they concluded that design parameters justifying a phase III trial were met in the unadjusted PFS analysis, with a trend toward a longer PFS and OS in the olaparib arm of the study. This highlights the need for further translational studies to investigate the possibilities of olaparib use in patients with platinum sensitive NSCLC.</p>", "<p>In our case, we observed an initial response to PARP inhibition followed by progression of an aggressive tumor resistant to platinum-based chemotherapy. Overlapping mechanisms of resistance to platinum and PARP inhibitor treatment may be the reason for the patient's poorer response to pemetrexed-carboplatin. Restoration of the functional HRR pathway in a HRD tumor may occur via <italic>BRCA1/2</italic> or HR gene reversion mutation which functionally restores protein activity (##REF##26775620##Lord and Ashworth 2016##). These mechanisms of resistance are alluded to in the post-hoc analysis of the SOLO2 trial, in which patients with epithelial ovarian cancer who received maintenance olaparib had marked reduction in the efficacy of platinum-based chemotherapy as seen in the time to second progression (##UREF##1##Frenel et al. 2020##). A limitation of our research includes not performing a biopsy on the relapsed, progressive tumor to investigate the reasons for developing resistance to PARP inhibitors and platinum chemotherapy.</p>", "<p>According to the American College of Medical Genetics and Genomics (ACMG) guidelines, any patient with a known pathogenic or likely pathogenic variant in <italic>BRCA1/2</italic> in any tumor type should be further investigated with germline testing to look for germline mutations (##REF##27854360##Kalia et al. 2017##). ##REF##32259017##Vlessis et al. (2020)## found that 55% of patients with solid tumors who had a somatic <italic>BRCA1/2</italic> PV also carried the same germline <italic>BRCA1/2</italic> PV. Hence, it is important to refer clinically relevant genes found on somatic testing to a cancer genetics service. The guidelines by ACMG serve as a useful guide for practicing clinicians. Diagnosis of a pathogenic germline variant has implications in further management not just for patients, but for predictive testing and management of their family members who may be at elevated risk for developing cancers.</p>", "<p>Future clinical trials using <italic>BRCA1/2</italic> variant status or HRD status as an enrolment criterion may allow the indications of PARP inhibitors to be expanded beyond its current indications in <italic>BRCA1/2</italic>-associated cancer types. It is important for clinicians to ascertain somatic panel sequencing rather than relying solely on platinum sensitivity as a surrogate marker of HRD. The optimal choice and sequence of therapy in patients eligible for PARP inhibitor should also be studied. This will bring us closer to the goal of precision medicine with appropriate targeted therapies of low toxicity administered to genomically matched patients.</p>" ]

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[ "<p>Poly (ADP-ribose) polymerase (PARP) inhibitors have been approved in malignancies associated with germline <italic>BRCA1</italic> or <italic>BRCA2</italic> pathogenic variants, such as breast, ovarian, prostate, and pancreatic cancer. In malignancies not associated with germline <italic>BRCA1</italic> or <italic>BRCA2</italic> pathogenic variants, the therapeutic relevance of PARP inhibitors is less clear. Non-small-cell lung cancer (NSCLC) is known to demonstrate somatic alterations in <italic>BRCA1</italic> or <italic>BRCA2</italic> gene. The current report is on a gentleman with metastatic lung adenocarcinoma with a somatic <italic>BRCA2</italic> pathogenic variant, who was effectively treated with olaparib. Furthermore, we discuss the existing data for use of PARP inhibitors in NSCLC. This study highlights the utility of next-generation sequencing in identifying gene mutations and demonstrates how such information can be used to select targeted therapies in patients with actionable molecular alterations.</p>" ]

[ "<title>CLINICAL PRESENTATION</title>", "<p>The patient was a 64-yr-old Chinese gentleman with a smoking history of forty pack-years and hypertension who presented with a skin lump over the epigastrium in March 2019. Significant family history included his sister who was diagnosed with breast cancer in her 40s and had undergone surgery; further details were unavailable. Excision of the lump showed an adenocarcinoma strongly positive for CK7, TTF-1, and Napsin A. Computed tomography (CT) of the chest, abdomen, and pelvis showed a spiculated left upper lobe nodule with mediastinal lymphadenopathy and left adrenal metastases. Molecular testing on the lump revealed epidermal growth factor (<italic>EGFR</italic>) mutation c.2126A &gt; C(p.Glu709Ala) and c.2156_2157delins CA(p.Gly719Ala). No disruptions were seen in the <italic>ALK</italic> and <italic>ROS1</italic> genes, and the ratio of MET to CEP7 signals was 1.1. Programmed death ligand-1 (PD-L1) tumor proportion score (TPS) by immunohistochemistry was 40%.</p>", "<p>First-line treatment using afatinib 40 mg once daily commenced in April 2019 with partial response. A restaging CT scan 2 months later showed new lytic lesions in the left femur and left ulna. In view of pain at the left thigh and forearm and risk of fracture, the patient underwent a prophylactic left femur intramedullary nail insertion and left ulna plating in August 2019. Postoperatively, five fractions of radiotherapy were administered to both sites at 20 Gy each. Afatinib was restarted after radiotherapy and continued until progression in March 2020, with a disease control of 11 mo.</p>", "<p>In December 2019, a liquid biopsy was performed to look for T790M mutation, which was negative. In March 2020, a lung biopsy was performed on the lung primarily to look for T790M mutation, which was also negative. The lung biopsy showed a PD-L1 TPS score of 70%. Second-line pembrolizumab was started in April 2020. In May 2020, after two cycles of pembrolizumab, the patient had complained of left upper and lower limb weakness. A magnetic resonance imaging (MRI) brain scan showed multiple brain metastases, with the largest in the right frontal lobe (##FIG##0##Fig. 1##). The patient underwent five fractions of whole brain radiotherapy (WBRT) at 20 Gy.</p>", "<p>In May 2020, next-generation sequencing (NGS) (Oncomine Comprehensive Assay v3) was conducted on the lung biopsy, which revealed a somatic <italic>BRCA2</italic> c.1411G &gt; T (p.Glu471Ter) pathogenic variant. The patient was discussed at a molecular multidisciplinary tumor board in a tertiary cancer center and deemed to be suitable for PARP inhibitors (##REF##34250396##Seet et al. 2021##). A baseline CT of chest, abdomen, and pelvis in June 2020 showed mixed response, with increase in axillary lymphadenopathy, stable disease in the mediastinal and supraclavicular lymphadenopathy, and decrease in size of lung primary and skin metastases. Olaparib was then given from June 2020 at a dose of 300 mg twice a day. A CT scan in July 2020 showed stable disease in lymph nodes, bone, and skin. A partial response was then achieved, with an MRI of the brain in both September 2020 and December 2020 showing reduction in the size of the brain metastases (##FIG##0##Fig. 1##). A CT scan of the chest, abdomen, and pelvis in October 2020 showed reduction in the size of the supraclavicular and axillary lymphadenopathy (##FIG##0##Fig. 1##), and stable disease in the lung primary and hilar lymphadenopathy.</p>", "<p>A CT scan in January 2021 showed progression of the primary lung tumor and increasing lymphadenopathy, with a PFS of 8 mo while on olaparib. He was switched to pemetrexed-carboplatin. Two cycles later, his disease in the lung, lymph nodes, adrenals, and bone progressed. He was then switched to docetaxel. The patient was on best supportive care since May 2021 and died in June 2021.</p>", "<title>GENOMIC ANALYSIS</title>", "<p>The specimen obtained from the epigastric lump was sent for <italic>EGFR</italic> mutation analysis via direct Sanger sequencing and <italic>ALK, ROS1</italic>, <italic>RET</italic>, and <italic>MET/CEP7</italic> analyses by interphase fluorescence in situ hybridization (FISH); details are described in the Methods section. After progression on afatinib, peripheral blood and the biopsy of the lung primary was sent for <italic>EGFR</italic> mutation analysis to look for a <italic>EGFR</italic> T790M mutation, which would affect the choice of <italic>EGFR</italic> tyrosine kinase inhibitor.</p>", "<p>The specimen from the lung biopsy was sent for NGS panel to guide further treatment. The tumor percentage of the sequenced biopsy was 70%. Details of the sequencing are presented in ##TAB##0##Table 1##. Analysis of the tumor showed a somatic <italic>BRCA2</italic> c.1411G &gt; T(p.Glu471Ter) variant with variant frequency of 28.4%, an <italic>RB1</italic> c.1807G &gt; A(p.Ala603Thr) variant with variant frequency of 35.6%, and mean estimated tumor mutation burden of 14 Muts/Mb. The two <italic>EFGR</italic> mutations found on initial molecular testing were similarly demonstrated in the NGS panel. Three variants of unknown significance in the <italic>ATR</italic>, <italic>CREBBP</italic>, and <italic>SLX4</italic> genes were found; variant allele frequencies were not provided. No fusions, amplifications, nor copy number variations were detected.</p>", "<p>In the present case, the G &gt; T variant within <italic>BRCA2</italic> at genome coordinate [GRCh37] Chr 13:32907026 is predicted to result in an amino acid substitution of glutamic acid with a stop codon, causing a nonsense variant that leads to loss of normal protein function. Loss-of-function variants in <italic>BRCA2</italic> are known to be pathogenic (##REF##32398771##Mesman et al. 2020##). The absence of <italic>BRCA2</italic> leads to defective DNA double-strand break repair by homologous recombination, increasing the risk of tumorigenesis and susceptibility to PARP inhibitors. ##FIG##1##Figure 2## illustrates the lollipop plot of <italic>BRCA2</italic> mutations in an NSCLC cohort of 1144 patients from The Cancer Genome Atlas Pan-Lung Cancer data set. This has been edited to include our patient's variant, which was not observed in prior NSCLC cohorts (##REF##27158780##Campbell et al. 2016##).</p>", "<p>Germline testing performed on the patient's blood in March 2021 showed <italic>EGFR</italic> c.3353C &gt; T(p.Ala1118Val) variant of uncertain significance, but no germline <italic>BRCA2</italic> PV was found. Details of the germline testing are found in ##TAB##1##Table 2##. It did not find other genes associated with hereditary breast and ovarian cancer such as <italic>ATM, CHEK2, PALB2, RAD51C</italic>, and <italic>RAD51D</italic>. Based on the laboratory interpretation, algorithms developed to predict the effect of this missense change on protein structure and function (SIFT, PolyPhen-2, Align-GVGD) all suggest that the <italic>EGFR</italic> variant is likely to be tolerated, but these predictions had not been confirmed by functional studies and their clinical significance is uncertain.</p>", "<title>ADDITIONAL INFORMATION</title>", "<title>Data Deposition and Access</title>", "<p>The variants and their interpretations have been submitted to ClinVar (<uri xlink:href=\"https://www.ncbi.nlm.nih.gov/clinvar/\">https://www.ncbi.nlm.nih.gov/clinvar/</uri>) and can be found under accession number VCV000959846.3. Information can be made available upon reasonable request to the corresponding author.</p>", "<title>Ethics Statement</title>", "<p>The study was approved by the Singhealth Centralized Review Board (CIRB 2021/2593). The patient provided his consent to publish his information and radiological images for this work. All procedures followed were in accordance with the Declaration of Helsinki.</p>", "<title>Acknowledgments</title>", "<p>The authors thank our patient for sharing his presentation for this work.</p>", "<title>Author Contributions</title>", "<p>All authors participated in the conception and design of the project, collection and assembly of data, data analysis and interpretation, and manuscript writing and final approval of manuscript. J.C. provided study materials or patients and administrative support. All authors are accountable for all aspects of the work.</p>", "<title>Competing Interest Statement</title>", "<p>The authors have declared no competing interest.</p>", "<title>Referees</title>", "<p>Benjamin H. Lok\nAnonymous</p>", "<title>Supplementary Material</title>" ]

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[ "<fig position=\"float\" id=\"MCS006223SOOF1\"><label>Figure 1.</label><caption><p>Metastatic non-small-cell lung cancer (NSCLC) treatment using different regimens and results of next-generation sequencing (NGS). (BSC) Best supportive care, (PD) progression of disease, (PDL-1 TPS) programmed death-ligand 1 tumor proportion score, (WBRT) whole brain radiotherapy.</p></caption></fig>", "<fig position=\"float\" id=\"MCS006223SOOF2\"><label>Figure 2.</label><caption><p>Lollipop plot of all reported mutations in <italic>BRCA2</italic> among a total of 1144 sequenced non-small-cell lung cancers from The Cancer Genome Atlas Pan-Lung Cancer data set. (Green) missense mutation, (black) truncating mutation, (orange) splice mutation. These include the present case (lollipop with the black arrow) that contains a likely pathogenic <italic>BRCA2</italic> variant.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"MCS006223SOOTB1\"><label>Table 1.</label><caption><p>Genomic findings and their variant interpretation identified on somatic sequencing of lung primary</p></caption><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col align=\"left\" span=\"1\"/><col align=\"center\" span=\"1\"/><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" rowspan=\"1\" colspan=\"1\">Gene</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">Chromosome</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">HGVS DNA reference</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">HGVS protein reference</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">Variant type</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">Predicted effect</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">dbSNP/dbVar ID</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">Variant allele frequency (%)</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">\n<italic>BRCA2</italic>\n</td><td rowspan=\"1\" colspan=\"1\">Chr 13</td><td rowspan=\"1\" colspan=\"1\">c.1411G &gt; T (NM_000059.4)</td><td rowspan=\"1\" colspan=\"1\">p.Glu471Ter</td><td rowspan=\"1\" colspan=\"1\">Substitution</td><td rowspan=\"1\" colspan=\"1\">Nonsense</td><td rowspan=\"1\" colspan=\"1\">Rs80358428</td><td rowspan=\"1\" colspan=\"1\">28.8</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<italic>RB1</italic>\n</td><td rowspan=\"1\" colspan=\"1\">Chr 13</td><td rowspan=\"1\" colspan=\"1\">c.1807G &gt; A (NM_000321.3)</td><td rowspan=\"1\" colspan=\"1\">p.Ala603Thr</td><td rowspan=\"1\" colspan=\"1\">Substitution</td><td rowspan=\"1\" colspan=\"1\">Missense</td><td rowspan=\"1\" colspan=\"1\">Rs777791058</td><td rowspan=\"1\" colspan=\"1\">35.6</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<italic>EGFR</italic>\n</td><td rowspan=\"1\" colspan=\"1\">Chr 7</td><td rowspan=\"1\" colspan=\"1\">c.2126A &gt; C (NM_005228)</td><td rowspan=\"1\" colspan=\"1\">p.Glu709Ala</td><td rowspan=\"1\" colspan=\"1\">Substitution</td><td rowspan=\"1\" colspan=\"1\">Missense</td><td rowspan=\"1\" colspan=\"1\">Rs397517085</td><td rowspan=\"1\" colspan=\"1\">28.4</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<italic>EGFR</italic>\n</td><td rowspan=\"1\" colspan=\"1\">Chr 7</td><td rowspan=\"1\" colspan=\"1\">c. 2156G_2157delinsCA (NM_005228)</td><td rowspan=\"1\" colspan=\"1\">p.Gly719Ala</td><td rowspan=\"1\" colspan=\"1\">Deletion-insertion</td><td rowspan=\"1\" colspan=\"1\">Missense</td><td rowspan=\"1\" colspan=\"1\">N/A</td><td rowspan=\"1\" colspan=\"1\">27.8</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<italic>ATR</italic>\n</td><td rowspan=\"1\" colspan=\"1\">Chr 3</td><td rowspan=\"1\" colspan=\"1\">c.5386A &gt; C (NM_001184)</td><td rowspan=\"1\" colspan=\"1\">p.Lys1796Gln</td><td rowspan=\"1\" colspan=\"1\">Substitution</td><td rowspan=\"1\" colspan=\"1\">Missense</td><td rowspan=\"1\" colspan=\"1\">N/A</td><td rowspan=\"1\" colspan=\"1\">Not provided by lab</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<italic>CREBBP</italic>\n</td><td rowspan=\"1\" colspan=\"1\">Chr 16</td><td rowspan=\"1\" colspan=\"1\">c.2239A &gt; G (NM_004380)</td><td rowspan=\"1\" colspan=\"1\">p.Met747Val</td><td rowspan=\"1\" colspan=\"1\">Substitution</td><td rowspan=\"1\" colspan=\"1\">Missense</td><td rowspan=\"1\" colspan=\"1\">N/A</td><td rowspan=\"1\" colspan=\"1\">Not provided by lab</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\n<italic>SLX4</italic>\n</td><td rowspan=\"1\" colspan=\"1\">Chr 16</td><td rowspan=\"1\" colspan=\"1\">c.4765C &gt; T (NM_032444)</td><td rowspan=\"1\" colspan=\"1\">p.Arg1589Cys</td><td rowspan=\"1\" colspan=\"1\">Substitution</td><td rowspan=\"1\" colspan=\"1\">Missense</td><td rowspan=\"1\" colspan=\"1\">Rs181782315</td><td rowspan=\"1\" colspan=\"1\">Not provided by lab</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"MCS006223SOOTB2\"><label>Table 2.</label><caption><p>Genomic findings and their variant interpretation identified on germline testing of peripheral blood</p></caption><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col align=\"left\" span=\"1\"/><col align=\"char\" span=\"1\"/><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" rowspan=\"1\" colspan=\"1\">Gene</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">Chromosome</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">HGVS DNA reference</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">HGVS protein reference</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">Variant type</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">Predicted effect</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">dbSNP/dbVar ID</th><th align=\"center\" rowspan=\"1\" colspan=\"1\">Variant allele frequency (%)</th></tr></thead><tbody><tr><td rowspan=\"1\" colspan=\"1\">\n<italic>EGFR</italic>\n</td><td rowspan=\"1\" colspan=\"1\">Chr 7</td><td rowspan=\"1\" colspan=\"1\">c.3353C &gt; T (NM_005228)</td><td rowspan=\"1\" colspan=\"1\">p.Ala1118Val</td><td rowspan=\"1\" colspan=\"1\">Substitution</td><td rowspan=\"1\" colspan=\"1\">Missense</td><td rowspan=\"1\" colspan=\"1\">Rs773996588</td><td rowspan=\"1\" colspan=\"1\">Not provided by lab</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material id=\"PMC_1\" content-type=\"local-data\">\n<caption>\n<title>Supplemental Material</title>\n</caption>\n\n\n</supplementary-material>" ]

[ "<table-wrap-foot><fn><p>(HGVS) Human Genome Variation Society.</p></fn></table-wrap-foot>", "<table-wrap-foot><fn><p>(HGVS) Human Genome Variation Society.</p></fn></table-wrap-foot>", "<fn-group><fn fn-type=\"supplementary-material\"><p>[Supplemental material is available for this article.]</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

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[ "<title>Methodik</title>", "<p>Um die Entwicklung der Fahrradunfälle in Münster zu analysieren, wurden 4 Studien des UKM mit der offiziellen Verkehrsunfallstatistik der Polizei Münster verglichen. Alle Daten wurden mittels eines Patientenfragebogens und des Krankenhausinformationssystems erhoben. Die erste Studie wurde von Februar 2009 bis Januar 2010 in 6 Münsteraner Krankenhäusern durchgeführt und in der Zeitschrift <italic>Injury</italic> veröffentlicht [##REF##22105099##8##]. Um alle Studienergebnisse vergleichbar zu machen, wurden hier nur die Patienten berücksichtigt, die sich in diesem Zeitraum im UKM vorstellten. Die zweite Studie „International Bicycle Accident Study (IBAS)“ startete im Mai 2012 und endete im April 2013 und wurde vom Bundesamt für Straßenverkehr veröffentlicht [##UREF##3##4##]. 23 der 25 Kliniken aus dem TraumaNetzwerk NordWest nahmen daran teil. Aus Gründen der Vergleichbarkeit wurden auch hier nur die am UKM erhobenen Daten verwendet. Die dritte Studie umfasste einen Zeitraum von November 2017 bis Mai 2019 und wurde vom UKM durchgeführt. Die Daten aus dieser Studie sind bisher nicht veröffentlicht. In einer vierten Studie wurden vom Mai 2018 bis April 2019 gezielt Patienten angesprochen, welche angaben, mit einem E‑Bike verunglückt zu sein. Es stellten sich ausschließlich Pedelecfahrer vor. Diese Studie wurde vom Bundesamt für Straßenverkehr finanziell unterstützt und veröffentlicht [##UREF##19##24##], welches daher die Rechte an den erhobenen Daten hat und uns freundlicherweise die Daten der stationären Patienten für diese Arbeit zur Verfügung stellte. Da sich die Erhebungszeiträume der 3. und 4. Studie überschneiden, wurden Patientendaten, die sowohl in der 3. als auch in der 4. Studie auftauchten, aus dem Datensatz der 3. Studie entfernt, um Doppelzählungen zu vermeiden. Dies betraf 5 Fälle.</p>", "<p>Eine Übersicht über alle eingeschlossenen Studien, ihre Dauer und die einbezogenen Fahrradtypen ist in Tab. ##TAB##0##1## enthalten. Die Datenanalyse und Grafikerstellung wurden mittels Excel und R‑Studio (Posit PBC, Boston, Massachusetts, USA) durchgeführt.</p>", "<p>Alle Studien wurden von der Ethikkommission der Westfälischen Wilhelms-Universität Münster genehmigt<xref ref-type=\"fn\" rid=\"Fn1\">1</xref>.</p>", "<p>Um einen Vergleich zu den polizeilich erfassten Verkehrsunfällen ziehen zu können, wurden die online frei zugängliche Verkehrsunfallstatistik der Polizei in Münster aus den jeweiligen Jahren in die Auswertung einbezogen.</p>" ]

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[ "<p>Carl Neuerburg, München</p>", "<p>Ben Ockert, München</p>", "<p>Hans Polzer, München</p>", "<title>Hintergrund</title>", "<p>Fahrräder sind seit vielen Jahren ein beliebtes Verkehrsmittel. Gerade in Zeiten der verstärkten Klimadiskussion ist das Fahrrad als umweltfreundliches und kostengünstiges Verkehrsmittel weiter in den Fokus gerückt. Die Radwege und Straßen werden immer voller, und neue Verkehrsmittel wie Pedelecs oder E‑Scooter kommen auf.</p>", "<title>Methoden</title>", "<p>Es wurden insgesamt 4 Studien des Universitätsklinikum Münster zu Fahrradunfällen und die amtliche Unfallstatistik der Polizei Münster miteinander verglichen. Im Zeitraum von 2009 bis 2019 wurden 3 Studien durchgeführt, die alle Fahrradunfälle, und eine Studie, welche nur Pedelecfahrer gesondert berücksichtigt.</p>", "<title>Ergebnisse</title>", "<p>Die Altersverteilung sowie Hauptunfallursachen sind über die Jahre hinweg nahezu gleich geblieben. Die Anzahl an Pedelecunfällen hat zugenommen. Pedelecfahrer weisen ein höheres Durchschnittsalter und einen höheren Anteil an intensivstationären Aufenthalten auf. Jedoch weist gerade diese Kohorte auch eine hohe Quote an Helmträgern auf. Insgesamt scheint die Bereitschaft zum Tragen eines Helmes gestiegen zu sein.</p>", "<title>Schlussfolgerung</title>", "<p>Es ist zu bedenken, dass bei zunehmendem Radverkehr die Sicherheitsmaßnahmen entsprechend erhöht werden müssen. Dabei sollte sich die Unfallverhütung auf 3 große Bereiche konzentrieren: Technik, Erziehung, Durchsetzung.</p>", "<title>Background</title>", "<p>Bicycles have been a popular means of transport for many years. Especially in times of increased climate discussion, the bicycle has moved further into focus as an environmentally friendly and cost-effective means of transport. Bike lanes and roads are becoming more crowded and new means of transport such as pedelecs or e‑scooters are emerging.</p>", "<title>Methods</title>", "<p>A total of four studies by Münster University Hospital on bicycle accidents and the official accident statistics of the Münster police were compared. In the period from 2009 to 2019, three studies were conducted that considered all bicycle accidents and one study that only considered pedelec riders separately.</p>", "<title>Results</title>", "<p>The age distribution as well as main causes of accidents remained almost the same over the years. The number of pedelec accidents has increased. Pedelec riders have a higher average age and a higher proportion of intensive care stays; however, this cohort also has a high rate of helmet wearers. Overall, the willingness to wear a helmet seems to have increased.</p>", "<title>Conclusion</title>", "<p>It should be considered that with increasing bicycle traffic, safety measures must be increased accordingly. In this respect, accident prevention should focus on three major areas, engineering, education and enforcement.</p>", "<title>Schlüsselwörter</title>", "<title>Keywords</title>", "<p>Open Access funding enabled and organized by Projekt DEAL.</p>" ]

[ "<p>Fahrräder sind seit Jahren ein beliebtes Transportmittel. Sie bieten Vorteile für Umwelt und Gesundheit, dennoch sind Fahrradunfälle im Straßenverkehr häufig folgenschwer. Mit dem Wandel der Fahrradnutzung hin zur E‑Mobilität müssen sich die Bedingungen für Radfahrer im Sinne der Unfallverhütung weiterentwickeln. Viele Sicherheitsmaßnahmen beruhen jedoch auf den offiziellen Unfallstatistiken der Polizei, welche eine große Dunkelziffer aufweisen. In diesem Beitrag wird die Entwicklung der im Universitätsklinikum Münster (UKM) erfassten Radunfälle der vergangenen 10 Jahren betrachtet.</p>", "<p>Im Laufe der Jahre hat sich das Radfahren zu einer Art Kultur entwickelt, und auch das Freizeitradeln nimmt immer weiter zu [##UREF##4##5##]. Diese Tendenzen wurden durch die COVID-19-Pandemie verstärkt. In einer Studie zur Fahrradnutzung in Deutschland aus Juni 2020 gaben 25 % der Teilnehmer an, ihr Fahrrad häufiger zu benutzen; 31 % gaben an, das Fahrrad als Ersatz für andere Freizeitaktivitäten zu nutzen [##UREF##11##15##].</p>", "<p>Vor allem Pedelecs haben an Beliebtheit gewonnen. Pedelec ist die Abkürzung für „pedal electric cycle“. Als Untergruppe der E‑Bikes (alle elektrisch unterstützten Zweiräder) sind Pedelecs elektrisch unterstützte Zweiräder, die dann Unterstützung bieten, wenn der Fahrer aktiv in die Pedale tritt. <italic>Die motorisierte Unterstützung besteht bis zu einer maximalen Geschwindigkeit von 25</italic> <italic>km/h und wird progressiv verringert. Eine Anfahr- bzw. Schiebehilfe bis 6</italic> <italic>km/h ist zulässig. Ein Mindestalter oder eine Helmpflicht für die Nutzung von Pedelecs besteht nicht </italic>[##UREF##0##1##].</p>", "<p>Die Zahl der E‑Bikes in deutschen Haushalten ist in den letzten 6 Jahren von 1,6 Mio. (2013) auf 5,4 Mio. (2019) gestiegen [##UREF##18##23##].</p>", "<p>Die Vorteile des Radfahrens wie Umweltschonung und Gesundheitsförderung scheinen die Einwohner der nordrhein-westfälischen Stadt Münster schon lange überzeugt zu haben. Bereits im Jahr 2009 lag die Zahl der Fahrräder in der Stadt bei rund 400.000 und damit weit über der Einwohnerzahl (273.000) [##UREF##12##16##]. Im Jahr 2019 lebten dort über 310.000 Menschen, die mittlerweile über 500.000 Fahrräder besitzen [##UREF##7##11##]. Der Radverkehrsanteil in Münster liegt bei 39,1 %, damit ist das Fahrrad das führende Verkehrsmittel in dieser Stadt, gefolgt vom Pkw mit 29,0 % („modal split“) [##UREF##2##3##]. In dieser Arbeit wird die Stadt Münster beispielhaft für die Entwicklung von Fahrradunfällen in den letzten 10 Jahren genutzt. Sie konzentriert sich auf den Vergleich verschiedener Fahrradunfallstatistiken des UKM und der Polizei Münster von 2009 bis 2019.</p>", "<title>Ergebnisse</title>", "<p>Aufgrund strengerer Datenschutzbedingungen und personeller Veränderungen ist ein Rückgang der Erfassungszahlen am UKM zu verzeichnen. Dies bedeutet jedoch keinen Rückgang der Unfallzahlen in den Münsteraner Krankenhäusern; vielmehr ist davon auszugehen, dass die Unfallzahlen angesichts der steigenden Zahlen in der amtlichen Statistik ähnlich hoch oder sogar höher sein müssten (Tab. ##TAB##1##2##).</p>", "<p><italic>Die Altersverteilung aller Studien ist in etwa gleich, mit einem Höhepunkt im Alter von 20 bis 29 Jahren und einem Durchschnittsalter von 35 bis 40 Jahren. Der jüngste Patient war 4 Jahre und der Älteste 95 Jahre alt. Das Durchschnittsalter der Kohorte der Pedelecfahrer ist deutlich höher mit einem Mittelwert von 66 Jahren </italic>(Abb. ##FIG##0##1##).<italic> Der jüngste Pedelecfahrer war 49 Jahre und der Älteste 79 Jahre alt</italic>.</p>", "<p><italic>Bei den am UKM erfassten E‑Bike-Unfällen gab es im betrachteten Zeitraum einen leichten Anstieg </italic>(Tab. ##TAB##0##1##) <italic>(p-Wert: &lt;</italic> <italic>0,01</italic> <italic>%). Würde man die Fälle aus der reinen Pedelecstudie miteinberechnen, wäre dieser vermutlich deutlicher. In der amtlichen Unfallstatistik liegt der Anteil der E‑Bike-Unfälle im Jahr 2019 bei 8,36</italic> <italic>% und ist damit gegenüber den anfänglichen 3</italic> <italic>% aus dem Jahr 2015 gestiegen </italic>(Abb. ##FIG##1##2##).</p>", "<p>Über alle Studien hinweg ist die deutliche Hauptunfallursache der Alleinunfall, gefolgt von Kollisionen mit anderen Fahrrädern und Autos. In der offiziellen Unfallstatistik für 2019 sind die Hauptunfallursachen falsches Abbiegen, Missachtung des Abstands oder der Vorfahrt und Alkoholeinfluss. Im Jahr 2009 war die Hauptunfallursache erhöhter Alkoholeinfluss, gefolgt vom Benutzen der falschen Fahrspur und der Missachtung der Vorfahrt.</p>", "<p><italic>In den klinischen Studien wurde der Alkoholeinfluss auf freiwilliger Basis abgefragt. Im Jahre 2009 und 2018 lag der Anteil der Patienten, die Alkohol konsumierten, bei rund 3</italic> <italic>% (2009: 3,1</italic> <italic>%; 2018: 2,8</italic> <italic>%), und im Jahre 2012 zeigte sich ein deutlicher Peak mit 16,7</italic> <italic>% an alkoholisiert Verunfallten (p-Wert: &lt;</italic> <italic>0,01</italic> <italic>%). In der Pedelecstudie wurde der Alkoholkonsum nicht abgefragt. </italic>In der amtlichen Statistik ging die Zahl der Fahrradunfälle aufgrund von Trunkenheit leicht zurück, von 69 Unfällen unter Alkoholeinfluss im Jahr 2009 auf 60 im Jahr 2019.</p>", "<p>Positiv entwickelte sich die Helmnutzung in den Studien<italic>. So lag die Quote der Helmträger im Jahre 2009 bei 8,1</italic> <italic>% und stieg auf 24,1</italic> <italic>% im Jahre 2018. Bei den Pedelecfahrern zeigt sich die höchste Rate an Helmträgern mit 35,3</italic> <italic>% (p-Wert:&lt;</italic> <italic>0,01</italic> <italic>%)</italic> (Abb. ##FIG##2##3##). Leider gibt die offizielle Unfallstatistik der Polizei Münster keine Auskunft über die Anzahl der verunglückten Helmträger.</p>", "<p><italic>Bei den Verletzungsmustern fällt auf, dass die Zahl der Kopfverletzungen bei den Radfahrern rückläufig ist. In der zweiten Studie aus 2012 hatten 23,6</italic> <italic>% der Patienten eine Kopfverletzung erlitten, im Jahre 2018 waren es nur noch 12,0</italic> <italic>%. Von den Pedelecfahrern erlitten 17,7</italic> <italic>% eine Kopfverletzung </italic>(Abb. ##FIG##3##4##) <italic>(p-Wert: 0,03</italic> <italic>%).</italic></p>", "<p><italic>Die häufigsten Verletzungen in den 2012 und 2018 betrafen die obere Extremität </italic>(<italic>p</italic>-Wert: &lt; 0,01 %)<italic>. Die Pedelecfahrer hatten am häufigsten eine Verletzung an der unteren Extremität </italic>(Tab. ##TAB##2##3##) (<italic>p</italic>-Wert: &lt; 0,01 %).</p>", "<p>Aus Studie 1 lagen keine vergleichbaren Daten zu Verletzungsmustern vor. Die Bewertung der Verletzungsschwere erfolgte in allen weiteren Studien über den AIS 2005.</p>", "<p>Per Definition liegt ein Polytrauma bei einem ISS (Injury Severity Score) von 16 und mehr vor. <italic>Davon wurden 8 Patienten im Jahr 2012 und einer im Jahr 2018 erfasst. In der Kohorte der Pedelecfahrer wurden 2 polytraumatisierte Patienten aufgenommen.</italic> Der Patient mit dem höchsten ISS von 45 gehört ebenfalls zur Kohorte der Pedelecfahrer (Abb. ##FIG##4##5##).</p>", "<p>Laut der offiziellen Statistik der Polizei Münster liegt die Quote der schwer verletzten Radfahrer bei 17 %. Der Anteil der schwer verletzten Pedelecfahrer ist mit 38,9 % deutlich höher.</p>", "<p>Die polizeiliche Definition eines Schwerverletzten basiert jedoch nicht auf dem ISS, sondern darauf, ob ein Patient ins Krankenhaus eingeliefert wird und 24 h verbleibt oder nicht.</p>", "<p><italic>Die Zahl der stationären Einweisungen nach einem Unfall ist in 2018 (18,5</italic> <italic>%) im Vergleich zu den letzten beiden Studien (2009: 30,8</italic> <italic>%; 2012: 35,1</italic> <italic>%) deutlich rückläufig </italic>(Abb. ##FIG##5##6##) (<italic>p</italic>-Wert: &lt; 0,01 %).</p>", "<p>Wie in der Methodik beschrieben, konnten bei Pedelecfahrern nur stationäre Patienten berücksichtigt werden. Dennoch musste ein hoher Anteil der Patienten auf der Intensivstation behandelt werden. Von den 17 Patienten, die in die Studie aufgenommen wurden, mussten 6 mindestens eine Nacht auf der Intensivstation verbringen. Aus Studie 1 liegen keine Informationen über einen Aufenthalt auf der Intensivstation vor.</p>", "<title>Diskussion</title>", "<p>Erst seit 2015 gibt es einen eigenen Bericht des Statistischen Bundesamtes ausschließlich für Fahrrad- und E‑Bike-Unfälle, was zeigt, wie sehr das Interesse an Fahrrädern und deren Unfallbeteiligung gewachsen ist. Die Zahl der Fahrradunfälle mit Personenschaden ist von 2009 bis 2019 um 14,68 % gestiegen. Die Gesamtzahl der Verkehrsunfälle mit Personenschaden im gleichen Zeitraum sank um 3,68 % [##UREF##15##19##]. Dies könnte durch einen starken Anstieg der Fahrradnutzung bei nichtausreichendem Verkehrsraum bedingt sein.</p>", "<p>Die polizeiliche Verkehrsunfallstatistik 2019 der Stadt Münster zeigt, dass die Verkehrswege, insbesondere in der Innenstadt, oft stark überlastet und nicht für die derzeit über 300.000 Einwohner ausgelegt sind [##UREF##8##12##]. Besonders betroffen hiervon sind die Radfahrer, welche seit 2018 mehr als die Hälfte der Verunglückten auf Münsters Straßen ausmachen [##UREF##9##13##].</p>", "<p>Die Polizeistatistik aus dem Jahr 2019 zeigt, dass der Abstand zwischen Kraftfahrzeugen und Radfahrern eine häufige Unfallursache bei Kollisionen ist. Es bleibt abzuwarten, ob die neue Straßenverkehrsordnung vom 28.04.2020, die beim Überholen eines einspurigen Verkehrsteilnehmers innerorts einen Mindestabstand von 1,5 m und außerorts von 2 m vorschreibt, hier präventiv wirkt [##UREF##1##2##].</p>", "<p>Radfahren unter erhöhtem Alkoholeinfluss (&gt; 0,16 %) ist in Deutschland eine Straftat [##UREF##5##6##]. Bundesweit ist ein Rückgang der Fahrradunfälle unter Alkoholeinfluss zu beobachten. Waren es im Jahr 2009 noch 53 Radfahrer/1000 Beteiligte, die unter Alkoholeinfluss fuhren, sank diese Zahl auf 46/1000 Beteiligte im Jahr 2019 [##UREF##15##19##]. Zu bedenken sind hier die hohe Dunkelziffer und die Tatsache, dass es sich um ein Kontrolldelikt handelt. Die Studie des Bundesamts für Straßenverkehr zeigte, dass Radfahrer, die unter Alkoholeinfluss standen, eine höhere Wahrscheinlichkeit hatten, ins Krankenhaus eingeliefert oder intensivmedizinisch versorgt zu werden [##UREF##3##4##].</p>", "<p>Alleinunfälle tauchen in den von der Polizei erfassten Daten nur sehr selten auf, da die Polizei oft nicht zu Radunfällen gerufen wird, an denen keine anderen Personen beteiligt sind, und der Sachschaden an Fahrrädern zumeist weniger schwerwiegend ist als an Autos. Im Jahr 2009 zeigten Juhra et al. eine deutliche Diskrepanz zwischen der Zahl der von der Polizei erfassten Fahrradunfälle und der Zahl der verunglückten Radfahrer, die sich im Krankenhaus vorstellten. 67,9 % der Fahrradunfälle von Patienten, die sich selbst im Krankenhaus vorstellten, wurden von der Polizei nicht erfasst [##REF##22105099##8##]. Die daraus resultierende hohe Dunkelziffer von Fahrradunfällen in vielen Statistiken, die auf den Daten polizeilich erfasster Unfälle beruhen, wie z. B. die Statistiken des Bundesamtes für Verkehr, ist problematisch, da diese Statistiken oft die Grundlage für politische und infrastrukturelle Entscheidungen sind.</p>", "<p>Die Zahl der verunglückten Pedelecfahrer ist in den letzten Jahren angestiegen. Bundesweit lag der Anteil der verunglückten Pedelecfahrer im Jahr 2015 bei 3,76 % und hat sich in nur 4 Jahren auf 12,63 % (2019) erhöht [##UREF##13##17##, ##UREF##14##18##]. Dies könnte u. a. auf die zunehmende Beliebtheit von Pedelecs in der Bevölkerung zurückzuführen sein. Im Jahr 2009 wurden ca. 150.000 Elektrofahrräder/Jahr verkauft. Im Jahr 2019 waren es bereits 1,36 Mio. Elektrofahrräder, und dieser Trend hält an [##UREF##16##20##]. Im Jahre 2020 wurden 43,4 % mehr E‑Bikes verkauft als noch 2019 [##UREF##17##22##]. Da hiermit der Anteil von Elektrofahrrädern im Straßenverkehr weitersteigt, liegt ein Anstieg der Unfallzahlen nahe.</p>", "<p>Weiss et al. kamen 2013 zu dem Ergebnis, dass das Pedelec keinen Einfluss auf das Unfallrisiko und die Schwere der Verletzungen hat. Hier zeigte sich eine Tendenz, dass Pedelecfahrer über 65 Jahre häufiger ins Krankenhaus eingeliefert wurden als solche unter 65 Jahren. Bei den stationären Aufnahmen gab es jedoch keinen Unterschied zwischen den über 65-Jährigen, die ein herkömmliches Fahrrad fuhren, und den Pedelecfahrern. Es wurde daher vermutet, dass das höhere Durchschnittsalter und die erhöhte Anzahl an Komorbiditäten für die stationären Behandlungen verantwortlich sind [##REF##29166687##21##]. Lefarth et al. zeigten in ihrer Studie aus dem Jahr 2021, dass Pedelecfahrer signifikant mehr stationäre Aufnahmen (Fahrrad: 34 %; Pedelec: 53 %) und Intensivbehandlungen (Fahrrad: 1 %; Pedelec 7 %) sowie einen signifikant höheren ISS-Score (Fahrrad: 3,4 Punkte; Pedelec: 5,2 Punkte) in dieser Kohorte aufwiesen. Allerdings gab es auch hier bei den Pedelecfahrern ein höheres Durchschnittsalter und mehr Komorbiditäten, was für die stationären Aufnahmen und schwereren Verletzungen ausschlaggebend sein könnte [##REF##33665719##10##]. Auch die Ergebnisse der vorliegenden Studien scheinen intensivmedizinische Behandlungen in der Kohorte der Pedelecfahrer häufiger notwendig zu sein als in den anderen Kohorten. Allerdings ist zu beachten, dass die Zahl der Studienteilnehmer in der Pedelecstudie kleiner ist als in den anderen Studien und somit die Repräsentativität fraglich ist.</p>", "<p><italic>Eine forsa-Umfrage aus 2013 zeigt, dass v. a. Senioren (zwei Drittel der Befragten) einen Ratschlag ihres Arztes bezüglich der Fahreignung annehmen würden, jedoch wurden in der Umfrage gerade mal 4</italic> <italic>% von ihrem Hausarzt auf die Fahrtauglichkeit angesprochen </italic>[##UREF##10##14##]<italic>. Dies gilt nicht nur für Rad- und Pedelecfahrer, sondern auch für Pkw-Fahrer.</italic></p>", "<p>Der leichte Anstieg der Helmnutzung bei den verunglückten Radfahrern im Jahr 2018 lässt hoffen, dass die generelle Bereitschaft zum Tragen eines Helms bei den Münsteraner Radfahrern gestiegen ist. Möglicherweise haben die verstärkten Aufklärungsprogramme der Polizei und der Stadt Münster in den vergangenen Jahren dazu beigetragen, dass die Helmnutzung in den Fokus gerückt und die Akzeptanz des Helmtragens in der Münsteraner Bevölkerung gestiegen ist [##UREF##6##9##]. Erfreulich ist v. a., dass die Kohorte der Pedelecfahrer eine sehr hohe Helmtragequote aufweist, da gerade sie im Falle eines Sturzes ein erhöhtes Risiko für Hirnblutungen tragen [##REF##29549679##7##].</p>", "<title>Limitationen</title>", "<p>Um die Aussagekraft der Untersuchungsergebnisse zu erhöhen, wären u. a. folgende Änderungen notwendig:<list list-type=\"bullet\"><list-item><p>Betrachtung anderer Städte mit geringerem Fahrradanteil im Straßenverkehr,</p></list-item><list-item><p>Vergrößerung der Stichprobe v. a. hinsichtlich der Pedelecnutzer,</p></list-item><list-item><p>objektive Angaben zum Alkoholkonsum,</p></list-item><list-item><p><italic>Anpassung der Gruppenstärke bei hier vorliegenden großen Unterschieden in der Gruppengröße (Pedelec n</italic> <italic>=</italic> <italic>17; 2009 n</italic> <italic>=</italic> <italic>452).</italic></p></list-item></list></p>", "<title>Fazit für die Praxis</title>", "<p>\n<list list-type=\"bullet\"><list-item><p>Eine effektivere Vernetzung zwischen den Krankenhäusern und der Polizei führt zu einer Reduzierung der Dunkelziffer, lässt Ursachen und Zielgruppen genauer bestimmen und führt so zu einer Effektivitätssteigerung präventiver, aber auch repressiver Verkehrssicherheitsmaßnahmen.</p></list-item><list-item><p>Zur Reduzierung der Dunkelziffer und zur Objektivierung bei Fahrradverunfallten hinsichtlich des Alkoholkonsums wäre in zukünftigen Studien eine aktivere Ansprache der Thematik bei Krankenhausbehandlung und Testung mittels Blutanalyse wünschenswert, um aussagekräftigere Daten über Verletzungsschwere und -muster bei alkoholisierten Radfahrern zu erhalten.</p></list-item><list-item><p>Die Unfallverhütung bedarf einer ganzheitlichen, kooperativen Verkehrssicherheitsstrategie. Hierzu zählen die Analyse (Unfallforschung/-untersuchung), die Verkehrsraumgestaltung („engineering“), die Prävention („education“), die Verkehrsüberwachung („enforcement“) und eine zielgruppenadäquate Sicherheitskommunikation.</p></list-item><list-item><p>Eine Beratung der Patienten könnte durch den Hausarzt hinsichtlich der <italic>Fahrtauglichkeit </italic>aus medizinischer Sicht erfolgen, z. B. durch aktive Bewerbung der verschiedenen polizeilichen Verkehrssicherheitsübungen und anderen Partner (z. B. ADFC) zu den Besonderheiten von elektrisch unterstützten Fahrrädern, ggf. in den Pedelecverkaufsstellen.</p></list-item></list>\n</p>" ]

[ "<title>Funding</title>", "<p>Open Access funding enabled and organized by Projekt DEAL.</p>", "<title>Einhaltung ethischer Richtlinien</title>", "<title>Interessenkonflikt</title>", "<p>Die Autoren D. Schlautmann, M. Raschke, U. Weiss, B. Wieskötter, J. Ueberberg und C. Juhra erklären, dass sie keinen Interessenkonflikt haben. Die Studie 2012/2013 und die Pedelecstudie 2018/2019 wurden vom Kraftfahrt-Bundesamt gefördert. Die Studie aus dem Jahr 2009 wurde durch den Gesamtverband der Deutschen Versicherungswirtschaft/Unfallforschung gefördert.</p>", "<p>Für diesen Beitrag wurden von den Autor/-innen keine Studien an Menschen oder Tieren durchgeführt. Für die aufgeführten Studien gelten die jeweils dort angegebenen ethischen Richtlinien.</p>" ]

[ "<fig id=\"Fig1\"></fig>", "<fig id=\"Fig2\"></fig>", "<fig id=\"Fig3\"></fig>", "<fig id=\"Fig4\"></fig>", "<fig id=\"Fig5\"></fig>", "<fig id=\"Fig6\"></fig>" ]

[ "<table-wrap id=\"Tab1\"><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Studien</th><th>Studienzeitraum</th><th>Fahrradtypen</th><th><italic>n</italic></th><th>Pedelecs, absolut</th><th>Pedelecs, prozentual (%)</th></tr></thead><tbody><tr><td>Studie 1</td><td>01.02.2009–31.01.2010</td><td>Alle</td><td>452</td><td>1</td><td>0,2</td></tr><tr><td>Studie 2</td><td>01.05.2012–30.04.2013</td><td>Alle</td><td>329</td><td>13</td><td>4,0</td></tr><tr><td>Studie 3</td><td>01.11.2017–31.05.2019</td><td>Alle</td><td>108</td><td>5</td><td>4,6</td></tr><tr><td>Studie 4</td><td>01.05.2018–30.04.2019</td><td>Nur E‑Bikes</td><td>17</td><td>17</td><td>100,0</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Polizei Münster</th><th>2009</th><th>2012</th><th>2018</th><th>2019</th></tr></thead><tbody><tr><td>Insgesamt</td><td>650</td><td>669</td><td>863</td><td>873</td></tr><tr><td>Pedelecs, absolut</td><td>–</td><td>–</td><td>74</td><td>73</td></tr><tr><td>Pedelecs, prozentual</td><td>–</td><td>–</td><td>8,6 %</td><td>8,4 %</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Köperregion</th><th>2012 (in %)</th><th>2018 (in %)</th><th>Pedelec (in %)</th></tr></thead><tbody><tr><td>Kopf</td><td>23,6</td><td>12,0</td><td>17,7</td></tr><tr><td>Gesicht</td><td>24,0</td><td>16,7</td><td>17,7</td></tr><tr><td>Hals</td><td>1,2</td><td>0,0</td><td>0,0</td></tr><tr><td>Thorax</td><td>8,2</td><td>3,7</td><td>5,9</td></tr><tr><td>Abdomen</td><td>1,5</td><td>0,9</td><td>0,0</td></tr><tr><td>Wirbelsäule</td><td>6,3</td><td>1,9</td><td>11,8</td></tr><tr><td>Obere Extremität</td><td>46,6</td><td>28,7</td><td>47,1</td></tr><tr><td>Untere Extremität</td><td>34,6</td><td>22,2</td><td>70,5</td></tr><tr><td>Äußere Verletzungen</td><td>1,2</td><td>1,9</td><td>0,0</td></tr><tr><td>Becken</td><td>6,4</td><td>3,7</td><td>0,0</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group><fn id=\"Fn1\"><label>1</label><p>Ethikantragsnummern: Studie 1: Die Ethiksantragsnummer ist nach 10-jähriger Archivierung verfallen. Studie 2: 2012-171-f‑S Studie 3: Az.: 2017-281-b‑S; Studie 4: Az.: 2018-198-f‑S.</p></fn><fn><p>\n\n</p><p>QR-Code scannen &amp; Beitrag online lesen</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p id=\"Par2\">Since the beginning of the twentieth century, cardiovascular diseases (CVD) have remained the most common cause of death, taking significant hold, especially in industrialized countries in Western Europe and North America [##REF##23902484##1##]. CVD contributed to 17.5 million deaths annually, approximately 31% of global mortality [##UREF##0##2##]. Increasing prevalence of CVD conditions such as strokes and ischemic heart disease decreases Disability Life Adjusted Years (DALYs) due to high-blood pressure, diabetes, obesity, poor nutrition, and lack of exercise. This ever-growing concern demonstrates the need for additional therapies in treating CVD [##UREF##0##2##].</p>", "<p id=\"Par3\">Atherosclerosis is the primary cause and determinant of CVD. It is characterized by luminal occlusion of arteries formed by heavy extracellular fat and lipid deposition and thrombus precipitation onto intimal walls of major arteries and develop into atheromatous plaques [##UREF##1##3##]. This buildup of plaques narrows the arteries, restricting the volume and flow of blood, and leading to ischemia and hypoxia [##UREF##2##4##]. Coronary artery disease, myocardial infarction (MI), strokes, and peripheral artery disease are the major manifestations of atherosclerosis when left untreated.</p>", "<p id=\"Par4\">Atheromatous plaques in general are of two types: stable (non-vulnerable) and unstable (vulnerable) plaques [##UREF##3##5##]. The common factors mediating plaque vulnerability are size and depth of injury to the intima, levels of reactive oxidative species (ROS), number of macrophages and lymphocytes present, low-density lipoproteins (LDL) levels, artery size, and blood pressure in the lumen [##REF##23687352##6##]. Among these factors, endothelial cell (EC) dysfunction or injury during vascular intervention is an important risk factor mediating plaque formation. The injury along the intimal layer results in the release of damage-associated molecular patterns (DAMPs) which produce downstream activation of an inflammatory cascade of receptors such as Trigering Receptor Expressed on Myeloid Cells 1 (TREM-1) and Toll-like receptors (TLRs) playing a critical role in atherosclerosis and plaque vulnerability [##REF##23880853##7##]. The activation of these receptors induce increased secretion of inflammatory cytokines such as Tumor Necrosis Factor (TNF)-α, interleukins IL-1 and IL-6, and protease-like matrix metalloproteinases (MMPs), specifically MMP-9 [##REF##27017522##8##] (Fig. ##FIG##0##1##). The presence of these inflammatory markers makes the plaque more susceptible to damage [##REF##28746820##9##, ##REF##27148736##10##]. Although anti-inflammatory therapies are crucial in reducing atherosclerosis, acute inflammation is vital as it plays a key role in the regeneration of the arterial lining at the time of the earliest lesion. However, when acute inflammation sustains for a long time, chronic inflammation that is maladaptive to the lumen develops and causes the progression of stable plaque to unstable plaque [##REF##30973815##11##, ##REF##27331093##12##]. Along with inflammation, various other factors including hypoxia, oxidative stress, calcification, and neoangiogenesis play a critical role in plaque vulnerability [##UREF##4##13##]. This review focuses on correlating the possible role of sirtuins in plaque vulnerability.</p>", "<p id=\"Par5\">Additionally, increased expression of HIF-1α leads to activation of NF-κB and thus targeting sirtuins may attenuate oxidative stress and inflammation and enhance plaque stability. High Mobility Group Box 1 (HMGB-1), Heat Shock Protein (HSP), Nuclear factor kappa-light-chain-enhancer of activated B cells (NF-κB), NLR family pyrin domain containing 3 (NLRP3), Interleukin-1 (IL-1), Interleukin-6 (IL-6), Tumor Necrosis Factor-α (TNF-α), Hypoxia Inducible Factor (HIF)-α, Reactive Oxidative Species (ROS), Hydroxyl Radical Ion (OH⁻), Hydrogen Peroxide (H₂O₂), Forkhead Box O3 (FOXO3).</p>" ]

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[ "<title>Conclusion</title>", "<p id=\"Par27\">Regulation of vascular function is dependent on the balance between the adaptive inflammatory mechanisms in the body that are meant to produce fibrosis, wound healing, scarring, and maladaptive infiltration by pro-inflammatory cytokines. Factors such as diabetes, high cholesterol, high blood pressure, and other adverse cardiac and systemic manifestations can predispose individuals to maladaptive mechanisms. Infiltration by oxidized LDL which can recruit macrophage-derived foam cells, lymphocytes, cholesterol crystal development, and ECM fibrosis can subsequently lead to the generation of an atheromatous plaque. The development and vulnerability of a plaque to rupture are dependent on a combination of the oxidative stress, metabolism, and susceptibility to develop chronic inflammation. Studies have revealed and highlighted targets throughout this inflammatory pathway that can attenuate plaque vulnerability. Outside traditional therapeutic and pharmacological ways of targeting atherosclerosis, emerging studies show that targeting sirtuins through sirtuin activators and modulators can potentially yield significant attenuation of plaque formation and vulnerability. Sirtuins are expressed in highly inflammatory, hypoxic, and cell-cycle dysregulated states, all of which are prone to developing more atheromatous plaques. Regulated epigenetically, sirtuins can be targeted to regulate downstream inflammatory cascade and antioxidant scavengers. In view of the limited literature on the role of sirtuins in regulating vascular health, further investigations are warranted.</p>" ]

[ "<p id=\"Par1\">Atherosclerosis is characterized by the development of intimal plaque, thrombosis, and stenosis of the vessel lumen causing decreased blood flow and hypoxia precipitating angina. Chronic inflammation in the stable plaque renders it unstable and rupture of unstable plaques results in the formation of emboli leading to hypoxia/ischemia to the organs by occluding the terminal branches and precipitate myocardial infarction and stroke. Such delibitating events could be controlled by the strategies that prevent plaque development or plaque stabilization. Despite the use of statins to stabilize plaques, there is a need for novel targets due to continuously increasing cases of cardiovascular events. Sirtuins (SIRTs), a family of signaling proteins, are involved in sustaining genome integrity, DNA damage response and repair, modulating oxidative stress, aging, inflammation, and energy metabolism. SIRTs play a critical role in modulating inflammation and involves in the development and progression of atherosclerosis. The role of SIRTs in relation to atherosclerosis and plaque vulnerability is scarcely discussed in the literature. Since SIRTs regulate oxidative stress, inflammation, and aging, they may also regulate plaque progression and vulnerability as these molecular mechanisms underlie the pathogenesis of plaque development, progression, and vulnerability. This review critically discusses the role of SIRTs in plaque progression and vulnerability and the possibility of targeting SIRTs to attenuate plaque rupture, focusing on the highlights in genomics, molecular pathways, and cell types involved in the underlying pathophysiology.</p>", "<title>Keywords</title>", "<p>Open access funding provided by SCELC, Statewide California Electronic Library Consortium</p>" ]

[ "<title>Oxidative stress and sirtuins</title>", "<p id=\"Par6\">Intimal injury, the predisposing factor for plaque formation and progression, and vascular inflammation induce local upregulation of the renin-angiotensin system [##REF##26260307##14##, ##REF##22479397##15##]. Renin–angiotensin–aldosterone system (RAAS) is a highly studied hormonal mechanism that elicits change throughout the body primarily in response to decreased renal perfusion of filtrate and electrolytes. Secretion of renin elicits the release of Angiotensin II (AngII) which serves to increase blood pressure and fluid volume by systematic vasoconstriction [##REF##9095097##16##]. Along with this, Ang II also contributes to several other mechanisms such as in the presence of chronic inflammation, AngII significantly increases free radicals and ROS [##REF##20597104##17##]. Local activation of RAAS also enhances the activation of NF-<italic>κ</italic>B and release of inflammatory cytokines such as IL-6 through AngII, providing a positive feedback loop leading to higher levels of AngII, perpetuating vascular inflammation and plaque vulnerability [##REF##24804145##18##]. Lowering AngII levels may mediate important anti-inflammatory and regenerative vascular tissue growth to stabilize plaque formation.</p>", "<p id=\"Par7\">Oxidative stress is involved in plaque progression, which is partly regulated by the Sirtuin family proteins. Sirtuins have a wide range of roles such as energy metabolism, transcription regulation, DNA repair, circadian rhythm regulation, and more importantly inflammation through their nicotinamide adenine dinucleotide-dependent deacetylase activity (SIRT1-7) and function to suppress gene transcription epigenetically [##REF##27461006##19##]. The role of various sirtuins, their epigenetic mechanism, targets, and biological effects have been reviewed earlier [##REF##21537393##20##]. Among SIRT1 to SIRT7, nuclear sirtuins including SIRT 1, 2, 6, and 7 play a critical role in regulating inflammation; SIRT1 is known to suppress NF-κB, COX-2, and iNOS production, SIRT2 deacetylates p65 subunit of NF-κβ and RIP-1, and SIRT6 interacts with p65/RelA bound to the NF-κβ promoter region and represses transcription [##REF##29132743##21##]. Acute and chronic inflammation alter metabolism, bioenergy reprogramming, and homeostasis which ultimately lead to increased ROS production. The alteration in glycolysis and fatty acid metabolism is linked with NAD + dependent function of sirtuins [##REF##26904696##22##]. SIRT1, whose expression is regulated by various upstream activators and suppressors operating on the transcriptional and post-transcriptional levels, involves in modulating inflammation via its biological effect by deacetylating various proteins and post-translational modifications [##REF##35359990##23##]. In addition to playing a role in homeostasis and inflammation, sirtuins may play role in cellular senescence and aging by regulating insulin/IGF-1 signaling pathway, AMP-activated protein kinase, and forkhead box O (FOXO) [##REF##30526767##24##]. Sirtuins are known to be localized in different regions throughout the cell including cytosol, nucleus, and mitochondria with SIRT1 mainly in nucleus, SIRT2 in cytosol, and SIRT3, 4, and 5 in mitochondria. SIRT3 is also located in the cytosol and nucleus [##REF##21586315##25##]. SIRT3 is closely tied to lipid metabolism and oxidative stress. SIRT3, a stress‐responsive deacetylase, is involved in mitochondrial metabolism and homeostasis, and protects cells from genotoxic and oxidative stress‐mediated cell death [##REF##32724473##26##, ##REF##27686535##27##]. SIRT3 levels change during oxidative stress, genotoxic stress, metabolic stress, and stroke in order to maintain homeostasis to protect the cells [##REF##30450031##28##]. SIRT3 also plays a role in aging and is associated with longevity and it has recently been documented that exercise-induced SIRT3 decreases cellular stress and may contribute to longevity [##UREF##5##29##]. Sirtuins are also involved in regulating glucose-6-phosphate dehydrogenase (G6PD) activity (SIRT2), SOD2 activity (SIRT3), suppressing inflammation by suppressing AP-1 signaling (SIRT1, 3, and 6), regulating inflammatory signaling in dendritic cells, which play a critical role in plaque vulnerability, thereby regulating T cells and Tregs cells population (SIRT1), inhibiting NLRP3 inflammasome (SIRT1 and SIRT2), and promoting osteoblast differentiation and bone formation (SIRT1, 6, and 7) [##REF##27148736##10##, ##REF##36060706##30##, ##REF##36581622##31##]. With the important role of sirtuins in inflammation, cellular stress, and aging, it is important to delineate the role of sirtuins in plaque vulnerability as inflammation, cellular stress, and aging are the risk factors in plaque vulnerability and atherosclerosis.</p>", "<p id=\"Par8\">Sirtuins expression at the transcriptional level is regulated by transcription factors FOXOs and during nutritional stress SIRT1 expression is regulated by FOXO3a [##REF##15604409##32##]. However, during oxidative stress, sirtuins can regulate the expression of FOXO3a by deacetylation causing increased expression of downstream target genes involved in mitochondrial homeostasis, anti-apoptosis, and anti-oxidative stress [##REF##36060706##30##]. The role of FOXOs in atherosclerosis is also supported by the fact that FOXOs are involved in vessel development, growth, maintenance, and function. FOXOs are also involved in controlling tissue differentiation, growth and maintenance, cell cycle progression, ROS detoxification, programmed cell death, and glucose metabolism [##REF##22539757##33##]. FOXOs also play a role in aging and age-related metabolic disorder like diabetes mellitus type-2 (DMII) and both aging and DMII are risk factors for atherosclerosis [##REF##34727995##34##].</p>", "<p id=\"Par9\">Forkhead box transcription factors (FOXOs), which are regulated by the Sirtuin family and implicated to be an anti-oxidant mediator in many processes, have been implicated in decreasing age-related mortality through the breakdown of ROS and regulation of cell death [##REF##25832544##35##]. Release of ROS from inflammatory mediators is known to increase atheromatous plaque size and pressure by thinning out the fibrotic cap overlying vascular tissue, characterizing vulnerable plaque, suggesting a possible method of attenuating plaque vulnerability by targeting inflammation via the downregulation of oxidative stress imposed by ROS through SIRT3-downstream effect on FOXOs [##REF##23687352##6##] (Fig. ##FIG##0##1##). However, there is a lack of clear understanding of the underlying molecular mechanisms involving sirtuins and FOXOs and the relationship between these protective measures [##REF##25804908##36##]. Studies on the role of SIRT1 and SIRT2 in histone demethylation and deacetylation in response to immediate stress and cellular death have given reasons to implicate the role of SIRT3 in epigenetic regulation, but it remains controversial. Clear relationships in the expression levels of FOXO3 and SIRT 1–2 and SIRT 6–7 have also similarly implied the role of SIRT3 in specifical regulation of FOX03a. Since epigenetic factors may also affect plaque development, it is important to investigate and understand how sirtuins can affect plaque vulnerability as they have histone modification capability [##REF##34419168##37##, ##REF##34976029##38##]. The following sections are focussed on critically discussing the probable role of sirtuins in regulating plaque vulnerability and the possible therapeutic implications.</p>", "<title>Plaque development and plaque rupture</title>", "<p id=\"Par10\">With the advanced technology capable of classifying atheromatous plaques to be rupture-prone, major addendums have been provided through extensive research on the molecular mechanisms of these plaques. It is thought that micro-injuries to the intimal layer of the artery lead to breaches in the vascular barrier that protect endothelial cells [##REF##16631513##39##]. This breach can lead to deposition and infiltration of various extracellular matrices, collagen, and lipids such as triglycerides and cholesterol carried by various lipoproteins in plasma. LDL accumulation and developing ischemia can result in the recruitment of macrophages by the release of chemoattractants which attempt to prevent further accumulation of cholesterol. As the plaque develops, the intima becomes further leaky, promoting further LDL recruitment, suggesting that high LDL or cholesterol count is a major risk factor for developing atherosclerosis [##REF##34552965##40##] (Fig. ##FIG##1##2##). Furthermore, several studies showed how epigenetic factors such as both microRNA and long-noncoding RNA help in the regulation of cholesterol efflux, lipid metabolism, and control of inflammation.</p>", "<p id=\"Par11\">About 75% of all coronary events such as strokes and MI stem from plaque rupture and erosion following a period of arterial necrosis and ischemia. These rupture-prone plaques have been classified as vulnerable and unstable [##UREF##3##5##]. Plaque rupture depends on many internal and external factors of the arterial lining, but generally, it is thought to depend on plaque structural stress (PSS) crossing the threshold of maximal strength [##REF##28734911##41##]. Inflammatory cytokines can promote additional oxidation, necrosis of cellular tissue, and extracellular matrix degradation and attenuate collagen synthesis to break down the integrity of the fibrous cap [##REF##35328769##42##]. This loss of integrity of the fibrous tissue can lead to luminal compromise, which is healed but cause further narrowing [##REF##20554950##43##] (Fig. ##FIG##1##2##). This suggests that advanced plaques develop in a significantly different manner from early lesions as advanced plaques have higher remodeling rates than earlier plaques. Narrowing of the luminal lining forms regions of high blood-shear stress, mechanically damaging the intimal lining. With this reasoning, stable plaques can be characterized by small or no necrotic cores, intact fibrous caps, and few calcification nodules. Unstable or vulnerable plaques are characterized by thinning of the fibrous cap, presence of necrotic core, chronic inflammation, decreased vascular smooth muscle cells, decreased collagen and other extracelluar matrix, and neo-angiogenesis [##REF##27017522##8##].</p>", "<p id=\"Par12\">Clinical factors like high blood pressure, high cholesterol, diabetes, poor nutrition, and stress are known to drive the development of atherosclerosis by increasing the LDL composition, number, and size as well as arterial permeability to be infiltrated and susceptibility to being inflamed [##REF##11001066##44##]. Since atherosclerosis is a multifocal condition, vulnerable plaques at the risk of rupture and thrombosis in one region can be a marker of advanced cardiovascular disease elsewhere in the body: with regions at the highest risk being coronary arteries [##UREF##2##4##]. These findings reemphasize the importance of preventing the worsening of stable plaques at the earliest time point to decrease the risk of rupture and thrombosis and resulting coronary events. Since inflammation precedes and superimposes plaque rupture, understanding and targeting the inflammatory pathway to reduce the chance of cardio-ischemic events remain essential [##REF##10341836##45##].</p>", "<title>Loss of protective mechanisms in chronic inflammation and plaque vulnerability</title>", "<p id=\"Par13\">As lipid molecules deposit and are endocytosed into macrophages and dendritic cells, the vascular endothelium necrotizes and forms a fibrotic cap. Tissue hardening and vascular remodeling promote the release of pro-inflammatory molecules, leading to inflammation [##REF##28314799##46##]. Macrophages, after significant uptake of oxidized LDLs and other lipids, become foam cells. Although there are myriad mechanisms of modifications in LDLs and macrophages, nitric oxide (NO) is a key pro-oxidant that is produced by both endothelial cells and macrophages. The effect of NO can be both protective and atherogenic, depending on its source and dosage. However, through inducible NO synthase expressed in foam cells, overproduction of NO lead to dysfunction of the local vasodilatory effects meant to protect the intima from occlusion [##REF##12490960##47##]. Increased macrophage foam cell formation leads to increase oxidative stress and ROS production causing endothelial dysfunction and increased endothelium permeability allowing increased entry of low density lipoproteins (LDL) [##UREF##6##48##]. Increased LDL deposition enhances plaque vulnerability, thus attenuating oxidative stress by targeting sirtuins may attenuate oxidative stress and thereby reduced LDL deposition and plaque vulnerability [##REF##35771356##49##]. This notion is supported by the fact that SIRT2 significantly decreases plaque area, macrophage infiltration, expression of iNOS and increases the levels of ARG-1 (arginase-1) in the atheroma of LDLR knockout mice and enhance plaque stability by attenuating macrophage polarization towards M1 phenotype [##REF##29145149##50##].</p>", "<p id=\"Par14\">In periods of non-infectious inflammation, loss of intima prompts the release of DAMPs such as high-mobility group box 1 (HMGB1), heat-shock proteins (HSPs), fibronectin, and serum amyloid A (SAA). These DAMPs induce inflammation by activating downstream signaling involving the NF-κB pathway or directly stimulating macrophages by binding to Toll-like receptors (TLRs), particularly TLR4 [##REF##30181915##51##]. Chemoattractants released by macrophages at the site also recruit other components of the innate and adaptive immune systems [##REF##26648785##52##] (Fig. ##FIG##0##1##). Initially, macrophages were thought to be the primary key players in plaque development. However, later studies revealed the combined effect of CD4 + T cells, regulatory T cells, and myeloid cells that produce immature macrophages and mast cells. In normal vascular regenerative processes, mast cells and CD4 + T cells are not present in significant numbers, as the concentration of cytokines to recruit these cells is not sufficient. Mast cells in vulnerable plaques overexpress vascular endothelial growth factor (VEGF), leading to the overdevelopment of collagen and ECM layer, promoting the hardening of the necrotic tissue. Continued overexpression of TLR4 and its downstream signaling cascades create an abnormal positive feedback loop of intimal lesion and inflammation, rendering the plaque to be further infiltrated and more vulnerable [##REF##23892452##53##]. Therefore, the breakdown of the protective effects of acute inflammation, presence of persistent inflammation, inflammatory immune cell recruitment, and functional disorders in developing atherosclerotic plaque mediate the loss of protective mechanisms and plaque vulnerability [##REF##27017522##8##]. In the case of neuroinflammation in Alzheimer’s (AD), triggering DAMPs activate a cascade of NF-κB and its subsequent downstream inflammatory pathways and the formation of NLRP3 inflammasomes, both of which are critically involved in inflammation related to atherosclerosis [##UREF##7##54##] (Fig. ##FIG##0##1##). Several studies in the neuroinflammation of AD showed that Sirtuins, specifically SIRT1 and SIRT2, exhibit protective effects by targeting DAMPs and downregulating NF-κB, yielding sirtuins as a possible therapeutic target [##REF##33014276##55##]. DAMPs and other pattern recognition receptors (PRR) can further propagate hypoxia-mediated disorders as described in studies investigating TLR4 signaling in obstructive sleep apnea [##REF##29673358##56##]. Hypoxia is cited to be one of the important mediators in plaque vulnerability as lower oxygen potentials caused by stenotic arteries lead to higher expression of IL-1β and caspase-1 activation, along with altering lipid metabolism [##REF##27472406##57##]. Chronicity of inflammation due to dysregulated innate and adaptive immune response plays a critical role in plaque progression, thrombosis, and vessel stenosis [##REF##35937643##58##].</p>", "<p id=\"Par15\">Expression and activation of NF-κB is one of the central events of inflammation, and once stimulated by TREM-1, TLR4, and DAMPs, NF-κB translocates to the nucleus to regulate transcriptional activity to increase pro-inflammatory molecules. One such molecule that is activated by NF-κB is IL-1β, which proteolytically activates a multiportion assembly called inflammasome, in particular the NLRP3 inflammasome [##UREF##7##54##]. Inflammasomes, in general, are responsible for sensing cellular stressors such as cellular damage, electrolyte imbalance, and ROS levels. Components of NLRP3 inflammasomes are highly expressed in foam cells within atheromatous plaques, additionally enhanced through low oxygen tension and aggravating inflammation (Fig. ##FIG##1##2##). Along with indirectly increasing NLRP3 levels through IL-1β, NF-κB can proteolytically activate NLRP3 in foam cells [##REF##31184236##59##]. In addition to regulating other proteins, NF-κB can also be modulated by protein modifications such as phosphorylation, methylation, and acetylation; one such epigenetic regulator is the Sirtuin family.</p>", "<title>Hypoxia and plaque vulnerability</title>", "<p id=\"Par16\">ROS such as superoxide (O²⁻) and hydroxyl (OH·) radicals is a highly reactive molecule class that is derived from aerobic metabolism. Homeostasis of ROS is a key determinant factor of many organisms and affects their capacity to produce energy from coupling mechanisms [##REF##31428618##60##]. As cells regularly age, apoptosis and proper cleanup of ROS are handled in a very complex, but regulated manner. However, at the sites of necrosis and chronic inflammation, ROS can be released in the ECM, forming foam cells, and increasing the size of the fibrotic cap (Fig. ##FIG##1##2##). One such enzyme that has been a key player in the overproduction of ROS is NADPH oxidase, which also induces autophagy. Autophagy involves cells rapidly degrading old cellular components to generate a nutrient pool to remodel as well as remove damaged mitochondrial organelles. Autophagy has also been involved with lipid homeostasis in blood, further complicating LDL levels in the body as well as activating vascular disorders such as atherosclerosis [##REF##25866599##61##]. It has been shown that defective autophagy significantly stimulates inflammatory responses with the activation of TLR-9 through the release of mitochondrial DNA [##REF##23326634##62##]. Overly active autophagy in response to NADPH oxidase in pre-existing chronic inflammation can induce fibrous membrane formation, detrimentally reducing the nutrient pool, especially in collagen, leading to more vulnerable plaques [##UREF##8##63##]. Homeostasis of ROS can be key to controlling plaque size.</p>", "<p id=\"Par17\">Occlusion of intimal layers limits blood flow and exchange between blood in the vessels and cells, inducing hypoxia. Although the exact mechanisms of how hypoxia induces and is inducible by ROS are still debated, it seems like Complexes I, II, and III of the electron transport chain leads to the production of ROS which then stabilizes the production of Hypoxia-inducible Factor alpha (HIF1α) [##REF##26798421##64##] (Fig. ##FIG##1##2##). HIF1α is a major transcription factor involved in tissue regeneration, cellular adaption to low O₂, as well as regulating NF-κB and Sirtuins. Chronic periods of inflammation will result in prolonged periods of hypoxia throughout the body, spreading inflammation at various vascular sites. NF-κB is dependent on ROS and is expressed to transcript pro-inflammatory genes such as monocyte chemotactic protein (MCP)-1) and IL-6. In vascular smooth muscle cells, Ang II induces MCP-1 through NF-κB. In a similar manner to DAMPs, AngII-induced NF-κB DNA binding and activation causes apoptosis in the media of the blood vessels and mediates an increase in adhesion with MCP-1 [##REF##10864918##65##]. Growth of adhesion molecules, infiltration of inflammatory cells and the development of necrotic tissue due to cell death in this manner directly lead to plaque destabilization and further inflammation, showing that high levels of ROS and AngII can directly play a key role in the development of atherosclerotic plaques (Fig. ##FIG##1##2##).</p>", "<p id=\"Par18\">It has been found that SIRT1 increases during hypoxia. An increase in HIF1α is shown to increase gene expression of SIRT1 as well as a reduction of SIRT1 levels in HIF1α knockout mice with high sensitivity has been documented [##REF##21345792##66##] (Fig. ##FIG##0##1##). However, the role of SIRT1 on HIF1α levels is still unknown. Since AngII and ROS largely destabilize plaques through the downstream NF-κB pathway, understanding and investigating the epigenetic regulation of NF-κB through the ROS-SIRT1-FOXO3 axis may be beneficial, and targeting sirtuins and FOXO proteins can attenuate or slow thrombotic plaque pathogenesis [##REF##22539757##33##].</p>", "<title>Molecular mechanisms of FOXOs and sirtuins in plaque vulnerability</title>", "<p id=\"Par19\">Although Sirtuins have various roles in several locations, in general, they produce tissue-protective effects. Numerous cancer studies have shown significant sirtuin involvement in gene silencing and detoxification. For example, SIRT1-6 increase genomic stability by repairing single and double-stranded DNA breaks from oxidative stress from tumor growth [##REF##26383140##67##]. The most studied in the group, SIRT 1 in particular has been implicated with reducing inflammation in multiple cell types undergoing obesity-induced insulin resistance, which involves defects in insulin signaling, systemic inflammation, mitochondrial disruption, and cellular stress [##REF##30574122##68##]. SIRT1 is also cleaved in inflammatory conditions and low NAD + levels during oxidative stress, which then impair SIRT1 signaling [##REF##21305533##69##]. This suggests both the protective and antagonistic role of SIRT1 during inflammatory signaling. One such mechanism in how SIRT1 achieves this is by deacetylating and thus inhibiting the expression of NF-κB [##REF##23770291##70##]. Since inflammation, oxidative stress, and mitochondrial dysfunction play a critical role in the pathogenesis of plaque formation, progression, and atherosclerosis, sirtuins may play a role in plaque vulnerability, however, this remains obscure [##UREF##9##71##]. What remains unclear about the role of SIRT1 in atherosclerosis is during the periods of hypoxia. There is no yet clear relationship between SIRT1 expression and HIF-1α levels, exhibiting increasing effects in some tissues and decreasing in others [##REF##22479397##15##, ##REF##20620956##72##]. However, it has been demonstrated that SIRT1 inhibits NFκB by deacetylating the p65 subunit of the NF-κB complex as well as activating AMPK, PPARα, and PGC-1α, which then inhibit NF-κB signaling and suppress inflammation [##REF##23770291##70##]. By reducing NF-κB levels, SIRT1 could stabilize vulnerable plaques before excessive intimal lining is inflamed and there is a foam cell overgrowth, making the plaque rupture-prone [##REF##30374331##73##] (Fig. ##FIG##0##1##).</p>", "<p id=\"Par20\">Although SIRT3 is less widely studied and understood than its counterparts in the Sirtuin family, its location of expression and effects in the mitochondria provide promising hopes for new therapy and alternatives in the field. SIRT3 has similar roles to SIRT1 in being negatively associated with ROS and protecting cells from oxidant-induced cell death [##REF##21094524##74##]. Studies about metabolism and caloric reduction highlight SIRT3 as an essential player in the mitochondrial glutathione antioxidant defense system. A plethora of work that has been done around sirtuins revolves around cancer research, and it has been found that SIRT3 knockout cells have, in fact, higher levels superoxide radicals and are prone to being genetically vulnerable and more likely to develop tumors [##REF##20129246##75##]. More directly, mtDNA induced by higher superoxide levels can predict higher-risk plaques [##REF##23841983##76##].</p>", "<p id=\"Par21\">SIRT3 has also been further identified as a tumor suppressor by decreasing levels of ROS, minimizing HIF1α [##REF##21397863##77##]. There is still a lack of information on whether this relationship extends to arterial narrowing-induced hypoxia. However, by reducing HIF1α levels, SIRT3 could yield protective mechanisms in response to oxidative stress posed by atheromatous plaques as HIF1α levels are directly correlated to plaque pathogenesis [##REF##28942242##78##].</p>", "<p id=\"Par22\">Through cancer research, it has been found that SIRT3 reduces ROS through the activation of transcription factors, particularly FOXO3a in adipocytes by expressing ROS-scavenging enzymes [##REF##25979314##79##]. Significant decreases in ROS following overexpression of SIRT levels have been found in multiple cell types such as age-related auditory issues, cardiomyocytes that have been oxidatively damaged due to doxorubicin, and most particularly endothelial cells undergoing hypoxic stress [##REF##24505357##80##–##REF##25162939##82##]. It has been implied that SIRT3 stabilizes FOXO3a by deacetylation which enhances the mitochondrial antioxidant defense system, and FOX03a levels are reduced in periods of SIRT3 deficiency through HIF1α playing an intermediary role [##REF##21397863##77##].</p>", "<title>Targeting sirtuins to attenuate plaque vulnerability</title>", "<p id=\"Par23\">It has been shown that myeloid deletion of SIRT3 accelerates adipocyte inflammation in cells overexpressing AngII [##REF##33327751##83##]. This could be due to SIRT3 mediating deacetylation of pyruvate dehydrogenase E1 alpha (PDHA1). PDHA1 deficiency correlates with mitochondrial dysfunction and promotes glycolysis, which further creates radical species [##REF##29700317##84##]. AngII levels probably aggravate and accelerate infiltration of foam cells and secretion of inflammatory interleukins in SIRT3 knockout mice, due to the loss of protective mechanisms set in place by the stabilization of FOX03a. As previously discussed, AngII elicits chronic inflammation, exacerbating plaque vulnerability and leading to the growth of necrotic tissue at the vascular lesion. Vulnerable plaques have a high likelihood to rupture, and in most worsened conditions can lead to ischemia-induced events such as MI and strokes. Targeting and utilizing sirtuins, in particular SIRT3, remains a potential therapeutic target to reduce such occurrences. Many such cancer studies have shown potential targets for therapy using Sirtuin Activators (STAC) to target the anti-inflammatory cascade potentiated by sirtuins.</p>", "<p id=\"Par24\">There is growing evidence that cellular aging or cellular senescence can promote atherosclerotic plaques [##UREF##10##85##]. Cells undergoing senescence cause pro-inflammatory secretion which can lead to loss of tissue-repair capacity. Senescent endothelial cells have an increased likelihood of recruiting monocytes and releasing more tissue factors, leading to coagulation and thrombotic plaques. Originally discovered in aging studies, pharmacological modifiers of sirtuins such as STACs can inhibit senescent cell-mediated secretion of chemokines, particularly by overexpression of SIRT1 [##UREF##10##85##]. SIRT1 inhibits p53/p21 signaling along with maintaining telomere health, both of which are considered anti-senescent effects [##REF##20203304##86##].</p>", "<p id=\"Par25\">The concurrent prevalence of both diabetes and CVD points to insulin regulation as an important mediator of atherosclerotic plaques [##UREF##11##87##]. In addition to promoting LDL dysregulation, there is increased medial and intimal calcification, which can lead to loss of fibrotic cap. In a study measuring SIRT6 expression in diabetic patients, SIRT6 is very much involved with the inflammatory pathways as it is expressed significantly higher in non-diabetics [##REF##25325735##88##]. This study highlights the loss of sirtuin-led downregulation of calcification, inflammation, and infiltration in diabetes, which leads to a higher likelihood of plaques. Furthermore, SIRT6 overexpression decreased the proliferation of atherogenic genes such as tumor necrosis factor superfamily member 4 (TNFSF4), which may impair the adhesion of monocytes to endothelial cells, directly impairing the formation of the plaques [##REF##27249230##89##]. Targeting SIRTs to attenuate plaque vulnerability and progression of atherosclerosis is also supported by the fact that SIRTs attenuate inflammation by inhibiting NF-κB (SIRT1 and SIRT6), decrease apoptosis by inhibiting p53 by deacetylation (SIRT2), decrease oxidative stress (SIRT3), and regulate LDL by inhibiting PCSK9 (SIRT2) [##REF##33614337##90##]. The results of the studies discussed above and in Table ##TAB##0##1## suggest that SIRTs might be important players and therapeutic target in attenuating plaque vulnerability and atherosclerosis progression (Fig. ##FIG##2##3##) [##UREF##12##91##].</p>", "<p id=\"Par26\">The epidemic of COVID-19 has also led to increased number of cases of incident venous thromboembolism, heart failure, and overall cardiovascular deaths [##UREF##13##92##]. It is suspected that SARS-CoV-2 has potential to accelerate the progression of atheromatous plaques by destabilizing old plaques and inducing further endothelial damage [##UREF##14##93##]. Host response to the virus includes innate inflammation in the lungs that is largely mediated by alveolar macrophages and can cause onset of acute respiratory distress. Inflammatory responses remain high and consistent for longer times than expected. Interaction between SARS-CoV-2 and ACE2 receptors in endothelium is well noted to reduce levels of ACE2 preventing the degradation of atherosclerotic AngII. The role of AngII in atherosclerosis has been been discussed above, and the effect of Sirtuins in downregulating AngII and anti-oxidant activity can promote the downregulation of pro-inflammatory cytokines to being transported to atheromatous plaques as seen highly in COVID-19 patients.\n</p>" ]

[ "<title>Acknowledgements</title>", "<p>VR is thankful to WesternU for the startup funds and DKA thanks the National Institutes of Health, USA for funding.</p>", "<title>Author contribution</title>", "<p>VP: literature search; design; critical review and interpretation of the published reports; preparation of figures and table; manuscript editing. VR: concept and design; analysis of the published information; preparation of figures and table; manuscript preparation; manuscript editing. DKA: conceptualization and design; manuscript preparation; manuscript editing; resources; funding.</p>", "<title>Funding</title>", "<p>Open access funding provided by SCELC, Statewide California Electronic Library Consortium. The research work of DKA is supported by the R01 HL144125 and R01 HL147662 grants from the National Institutes of Health, USA. The content of this critical review is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.</p>", "<title>Data availability</title>", "<p>Not applicable since the information is gathered from published articles.</p>", "<title>Declarations</title>", "<title>Competing interests</title>", "<p id=\"Par28\">The authors declare no competing interests.</p>", "<title>Conflict of interest</title>", "<p id=\"Par29\">The authors declare no competing interests. All the authors have read the manuscript and declare no conflict of interest. No writing assistance was utilized in the production of this manuscript.</p>", "<title>Consent fot publication</title>", "<p id=\"Par30\">All the authors have read the manuscript and consented for publication.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Inflammatory pathway leading to cascade resulting in plaque vulnerability and the effect of Sirtuins attenuating pro-inflammatory mediators<bold>.</bold> Intimal injury during vascular intervention or during atherogenesis induce release of DAMPs which stimulate inflammatory signaling by activating Toll-like receptors (TLRs) and receptor for advanced glycation end-products (RAGE) leading to immune response involving increased recruitment of innate and adaptive immune cells, secretion of pro-inflammatory cytokines, and NLRP3 inflammasome activation. Altogether, they cause chronic inflammation in the plaque and induce vulnerability. Intimal injury also causes disruption of vasa-vasorum and leads to hypoxia and oxidative stress which in turn activate sirtuins. Increased sirtuins act as oxidative stress scavenger and reduce oxygen radicals and decrease oxidative stress which may lead to attenuated plaque vulnerability</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>Pathological sequence of Ischemic events in the cardiovascular system resulting from vascular injuries and acute inflammation is gone awry. Oxidized Low-density Lipoprotein (oxLDL), NLR family pyrin domain containing 3 (NLRP3), Interleukin-1 (IL-1), Interleukin-6 (IL-6), Interleukin-8 (IL-8), Tumor Necrosis Factor-α (TNF-α), Interferon-γ (IFNγ)</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>Pharmacological modulators to be used in sequential steps of development of atheromatous plaques. Angiotensin II (AngII), Oxidized Low-density Lipoprotein (oxLDL), NLR family pyrin domain containing 3 (NLRP3), Platelet-derived growth factor (PDGF), phosphoinositide-3-kinase–protein kinase B/Akt (PI3K-PKB/Akt), Monocyte chemoattractant protein-1 (MCP-1/CCL2), SRT (SIRT activator), 1,4 Dihydropyridine (DHP)</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Summary of the studies investigating the role of sirtuins (SIRTs) in atherosclerosis</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Aim of the study</th><th align=\"left\">Experimental model</th><th align=\"left\">Strategy</th><th align=\"left\">Study outcome</th></tr></thead><tbody><tr><td align=\"left\">To evaluate the role of SIRT6 in decreasing atheroprogression</td><td align=\"left\">ApoE^−/− mice fed a high-fat diet</td><td align=\"left\">High fat-fed mice were evaluated for plaque and atherosclerosis in SIRT6 KO mice</td><td align=\"left\"><p>SIRT6 decreased the TNFSF4 gene by deacetylating H3K9 [##REF##27249230##89##]</p><p>SIRT6 attenuated monocyte adhesion to ECs</p></td></tr><tr><td align=\"left\">To evaluate and measure SIRT1 levels in CAD and explain risk factors</td><td align=\"left\">Human adults</td><td align=\"left\">A case–control study with peripheral venous blood samples</td><td align=\"left\">SIRT1 is negatively correlated to cholesterol content and risk of CVD [##UREF##15##94##]</td></tr><tr><td align=\"left\">To investigate the role of SIRT6 in smooth muscle cells</td><td align=\"left\">SIRT6 Knockout Mice</td><td align=\"left\">SMC-specific SIRT6-deficient (SIRT6KO) mice were infused with Ang II to induce oxidative stress</td><td align=\"left\">SIRT6KO mice developed aortitis, aortic hemorrhage, and aneurysms in response to Ang II [##UREF##16##95##]</td></tr><tr><td align=\"left\">To investigate the role of SIRT6 in atherosclerotic lesion development</td><td align=\"left\">ApoE^−/− mice</td><td align=\"left\">SIRT6 was knocked down in ApoE^(−/−) mice using small hairpin RNAs (shRNAs) lentivirus injection and then evaluated for plaque size and vulnerability</td><td align=\"left\">Intracellular adhesion molecule-1 expression was significantly upregulated in aortic endothelium without SIRT6 expression leading to higher monocyte adhesion [##REF##26924042##96##]</td></tr><tr><td align=\"left\">To measure the risk of coronary plaques in asymptomatic patients using SIRT1 levels</td><td align=\"left\">Human adults</td><td align=\"left\">A case–control study between high-risk and non-high-risk groups considering age, total cholesterol, gender</td><td align=\"left\">SIRT1 may play a predictive role in plaque screening before coronary angiography [##REF##31577623##97##]</td></tr><tr><td align=\"left\">Role of mi-RNA and other epigenetic modulators in atherosclerosis</td><td align=\"left\">Human adults</td><td align=\"left\">A case–control study between blood samples obtained from healthy patients and individuals with atherosclerosis</td><td align=\"left\">SIRT1 is a target of miR-217-mediated downregulation of macrophage apoptosis and subsequent inflammatory response</td></tr><tr><td align=\"left\">Role of SIRT6 expression in diabetic patients</td><td align=\"left\">Human adults</td><td align=\"left\">A case–control study between diabetics and nondiabetic patients undergoing carotid endarterectomy</td><td align=\"left\">Higher presence of SIRT1 and downregulation of NF-kB in non-diabetics [##REF##25325735##88##]</td></tr><tr><td align=\"left\">Investigate genetic polymorphisms associated with Sirtuins with CAD</td><td align=\"left\">Human adults</td><td align=\"left\">A case–control study between blood samples obtained from healthy patients and individuals with CAD using genomic DNA</td><td align=\"left\">Genetic polymorphisms in SIRT3 and SIRT6 were found in higher correlation with patients with CAD [##REF##35224092##98##]</td></tr><tr><td align=\"left\">Role of SIRT3 in vascular inflammation</td><td align=\"left\">ApoE^−/− mice</td><td align=\"left\">Evaluated if endothelial-selective SIRT3 deletion accelerates vascular inflammation and oxidative stress and the protective effect of NAD + to alleviate these changes in endothelial cells using mice model</td><td align=\"left\">Upregulation of vascular inflammatory markers and oxidative stress with plaque embolization was found significantly with SIRT3 knockout [##REF##35453391##99##]</td></tr><tr><td align=\"left\" rowspan=\"2\">Role of SIRT7 in mitochondria stability</td><td align=\"left\" rowspan=\"2\"><italic>SIRT7</italic>^−/− mice</td><td align=\"left\">Generation of <italic>SIRT7</italic>^{<italic>−/−</italic>} mice gene targeting strategy in 129 Sv embryonic stem cells</td><td align=\"left\" rowspan=\"2\">SIRT7 is a dynamic nuclear regulator of mitochondrial function through its impact on GABPβ1 function [##REF##25200183##100##]</td></tr><tr><td align=\"left\">In vitro acetylation and deacetylation assays using GABPβ1 protein</td></tr><tr><td align=\"left\" rowspan=\"2\">Role of SIRT3 in cardiac hypertrophy and interstitial fibrosis</td><td align=\"left\" rowspan=\"2\">SIRT3-deficient and SIRT3-overexpressing transgenic mice</td><td align=\"left\">SIRT3-deficient and SIRT3-expressing transgenic mice were exposed to hypertrophic stimuli</td><td align=\"left\" rowspan=\"2\">SIRT3 suppresses levels of ROS, thus endogenously negatively regulating cardiac hypertrophy [##REF##19652361##101##]</td></tr><tr><td align=\"left\">Cardiomyocytes were used to study the role of FOXO3a in cardiac protection and oxidative stress</td></tr><tr><td align=\"left\" rowspan=\"2\">Role of SIRT1 in cellular aging</td><td align=\"left\" rowspan=\"2\">ApoE^−/− mice</td><td align=\"left\">Downregulation of SIRT1 in VSMCs isolated from human plaques</td><td align=\"left\" rowspan=\"2\">SIRT1 regulates DNA damage repair and survival in vascular smooth muscle cells [##REF##23224247##102##]</td></tr><tr><td align=\"left\">SM22-SIRT1^+/+, SIRT1^+/Δex4, and SM22-SIRT1^Δex4/Δex4 littermate mice and ApoE^−/− were fed normal chow or high-fat diet</td></tr><tr><td align=\"left\" rowspan=\"2\">Role of SIRT2 in pathological cardiac hypertrophy</td><td align=\"left\" rowspan=\"2\">LDLr^−/− mice</td><td align=\"left\">Female LDLr^−/− mice were treated with saline, empty lentivirus, lentivirus-SIRT2, or shRNA-SIRT2 for 4 weeks</td><td align=\"left\" rowspan=\"2\">By inhibiting macrophage polarization, SIRT2 inhibited atherosclerotic plaque progression, but the degree of effect is small and not significant [##REF##29145149##50##]</td></tr><tr><td align=\"left\">Atherosclerotic plaques were assessed in the aortic sinus and macrophage polarization was evaluated</td></tr><tr><td align=\"left\">SIRT6 role in Atherosclerosis</td><td align=\"left\">Human VSMC and ApoE^−/− mice</td><td align=\"left\">SIRT6 expression was measured in human VSMC derived from plaques as well as ApoE^−/− high-fat diet</td><td align=\"left\">SIRT6 expression was reduced in human and mouse plaques and was shown to maintain telomeres while inhibiting atherogenesis [##REF##33353368##103##]</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p>Abbreviations: Ang, Angiotensin; CAD, coronary artery disease; CVD, cardiovascular disease; ECs, endothelial cells; KO, knockout; LDLr, low density lipoprotein receptor; TNFSF4, TNF Superfamily Member 4</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par13\">Health-related stigma refers to a social process by which people devalue or exclude others on the basis of their perceived health condition or features associated with that health condition [##REF##17130065##1##]. Hepatitis B virus (HBV) is a blood-borne virus, with two main transmission routes: vertical transmission occurring perinatally (mother-to-child at birth), and horizontal transmission occurring after birth through exposure to blood or sexual fluids where the virus is present [##REF##15602165##2##]. In the context of HBV, stigma may be related to negative attitudes towards potential transmission routes (e.g., unprotected sex, shared use of equipment for injecting drugs) [##REF##32214859##3##], as well as fears around the infectiousness of the virus [##REF##32214859##3##, ##REF##28101498##4##]. Smith-Palmer et al. [##REF##32214859##3##] recently conducted a literature review on stigma among people living with HBV, and found social stigma was common as was the internalisation or acceptance of negative judgements towards the virus by people living with HBV (i.e., internalised stigma). Stigma can also be enacted in the form of discrimination, such as the denial of health care, workplace discrimination, and exclusion by family, friends, or intimate partners [##REF##32214859##3##]. There is evidence to indicate that stigma is a barrier to screening for HBV [##REF##22993729##5##, ##REF##24933144##6##] and access to monitoring and treatment (where needed) [##REF##32214859##3##, ##REF##25000917##7##–##REF##23679261##10##].</p>", "<p id=\"Par14\">HBV affects approximately 257 million people or 3.5% of the world’s population [##UREF##0##11##]. It is a primary risk factor for liver cancer [##REF##29489473##12##], and one of the leading causes of mortality related to cancer [##UREF##1##13##]. In Australia, an estimated 1% of the population or 233,947 people are living with chronic HBV (when an infection persists for 6 months or longer) [##REF##28712127##14##]; however, it is estimated that only around 68% of people living with chronic HBV in Australia have been diagnosed [##UREF##2##15##]. Treatment for chronic HBV can reduce the risk of cancer by up to 75% [##REF##20483498##16##] and is a cost-effective public health strategy in at-risk populations [##REF##19303657##17##], although not all people will require treatment. Ongoing monitoring of HBV (every six months) is recommended within clinical guidelines; however, data suggest that only 11.9% of people living with chronic HBV were receiving the optimal annual monitoring [##UREF##3##18##]. Aboriginal and Torres Strait Islander people represent 3.2% of the Australian population [##UREF##4##19##], yet estimates suggest that 7% of the population of people living with chronic HBV are Aboriginal and Torres Strait Islander people [##UREF##2##15##, ##UREF##5##20##]. Rates of HBV overall are declining in Australia due to universal vaccination [##UREF##6##21##], but despite this HBV notifications were more than one and a half times higher among Aboriginal and Torres Strait Islander compared to non-Aboriginal or Torres Strait Islander people in Australia in 2020 [##UREF##5##20##], and some researchers suggest that transmissions may be occurring both vertically and horizontally among Aboriginal and Torres Strait Islander children [##REF##36575515##22##].</p>", "<p id=\"Par15\">The National Aboriginal and Torres Strait Islander BBV and STI and the National Hepatitis B strategies have a reported goal to eliminate the negative impacts of HBV and other BBV and STI-related stigma and discrimination on people’s health, and recognise that Aboriginal and Torres Strait Islander communities face multiple layers of stigma [##UREF##7##23##, ##UREF##8##24##]. However, there is limited research on HBV-related stigma generally, and particularly among Aboriginal and Torres Strait Islander communities [##REF##30483598##9##]. Most of the evidence that exists on HBV and stigma relates to migrant communities. For instance, research suggests that migrant communities contend with a range of structural and systemic barriers that impede access to HBV testing, monitoring, and treatment. These may include racism, a lack of culturally and linguistically diverse resources, a lack of culturally competent care, and a health system that does not recognise differences in cultural beliefs about health care and treatment [##REF##32214859##3##, ##REF##22151101##8##, ##REF##25338513##25##]. Additionally, health workers may have concerns about treating clients with HBV, including that they, as health workers, do not have the skills or knowledge to treat these clients, and that clients may not understand the need for ongoing monitoring and care [##UREF##9##26##].</p>", "<p id=\"Par16\">Research has shown that structural and systematic barriers, such as prohibitive cost of care, inappropriate location of services, lack of transportation, and lack of culturally appropriate and sensitive care, impede access to primary care more broadly among Aboriginal and Torres Strait Islander people [##REF##27716235##27##]. Although there is limited research in the context of HBV specifically, there is evidence to suggest, for example, that health workers who provide care to Aboriginal and Torres Strait Islander communities may not have the appropriate knowledge about HBV and available treatment [##UREF##10##28##]. There is very little research documenting HBV-related stigma among Aboriginal and Torres Strait Islander people. One study indicated that Aboriginal and Torres Strait Islander people describe HBV using negative terminology, suggesting that stigma surrounding HBV persists [##REF##25430502##29##]. Taken together, these barriers can impede access to testing, monitoring, and treatment (where needed), and result in low health literacy of HBV among Aboriginal and Torres Strait Islander people and health workers, mistrust in mainstream health service providers, and may result in further stigmatisation around HBV.</p>", "<p id=\"Par17\">Noting the above impediments to client participation in health care, the Deadly Liver Mob program (DLM) has set out to overcome these barriers for Aboriginal and Torres Strait Islander communities in New South Wales (NSW), Australia. DLM is an incentivised health promotion program offered in three sites in Sydney metropolitan area and six in rural and regional NSW. The program aims to increase access to testing and treatment for blood-borne viruses (BBVs) and sexually transmissible infections (STIs) among Aboriginal and Torres Strait Islander communities across the state. The program offers incentives for Aboriginal and Torres Strait Islander clients to be educated on viral hepatitis, recruit and educate peers, and receive screening and treatment for blood-borne viruses (BBVs) and sexually transmissible infections (STIs), and vaccination for hepatitis A and HBV. This program was developed and implemented in partnership with Aboriginal and Torres Strait Islander people, for Aboriginal and Torres Strait Islander people, as a culturally appropriate and sensitive and safe way of improving access to screening and treatment of BBVs and STIs. Findings from an early evaluation of two pilot sites running DLM found that attendance of Aboriginal and Torres Strait Islander clients at the two health services significantly increased following the introduction of the program, and acceptability of the program was high among both staff and clients [##REF##29391019##30##]. One of the key strengths of the program cited by workers and clients alike is the involvement of Aboriginal and Torres Strait Islander workers, who deliver the DLM program in a culturally sensitive and non-judgemental environment, overcoming some of the barriers to health care that exist for Aboriginal and Torres Strait Islander people, including stigma and discrimination [##REF##29391019##30##]. Thus, this program and the health workers who deliver it can provide unique insights into HBV and HBV stigma among Aboriginal and Torres Strait Islander people, including identifying barriers to screening and treatment, as well as identifying ways that mainstream health care could better engage with Aboriginal and Torres Strait Islander clients. The aim of this paper is therefore to explore Aboriginal and Torres Strait Islander and non-Aboriginal or Torres Strait Islander health workers’ perspectives on HBV and HBV-related stigma, using data drawn from an evaluation of the DLM program.</p>" ]

[ "<title>Methods</title>", "<p id=\"Par18\">This paper reports on qualitative data, drawn from a broader mixed-methods evaluation of the DLM program, conducted by the Centre for Social Research in Health at UNSW Sydney [##UREF##11##31##]. The broader evaluation aimed to evaluate the effectiveness of the DLM program in increasing access to BBV and STI testing among Aboriginal and Torres Strait Islander people, to examine client and provider attitudes and barriers towards acceptability of the program, and to develop a scale-up plan and implementation toolkit to guide future sites. While hepatitis C (HCV) is the core focus of the program, the DLM program seeks to improve access to testing, treatment, and vaccination for other BBVs and STIs, including HBV.</p>", "<p id=\"Par19\">An email invitation was sent by the research team to the HIV and Related Programs (HARP) Managers within the Local Health District (LHD) in NSW where DLM sites were located (seven LHDs in total covering nine DLM sites). HARP Managers then forwarded the invitation to the relevant staff involved in the development, implementation, and/or management of the DLM program within the participating services. The invitation email contained the Participant Information Statement providing additional information about the research and a consent form, and prospective participants were instructed to contact a member of the research team if they were interested in taking part.</p>", "<p id=\"Par20\">Semi-structured telephone interviews were conducted with five Aboriginal and Torres Strait Islander and six non-Aboriginal or Torres Strait Islander health workers and key informants from five LHDs (total <italic>n</italic> = 11). In keeping with Braun and Clarke’s [##UREF##12##32##] approach to sample size selection and the issue of data saturation, sample size was largely a pragmatic decision, based on the exploratory nature of the research, the timeframe, and that these interviews were a subset of other interviews conducted as part of the research, with these interviews focussing on HBV. The interviews were conducted by one Aboriginal and Torres Strait Islander (MB) and one non-Aboriginal or Torres Strait Islander (MH) member of the research team between August and October 2020. Interviews explored the perspectives of health workers on HBV infection among Aboriginal and Torres Strait Islander clients and in the context of the DLM program, HBV-related stigma, and its impacts on both clients and staff within the program, and the potential for the program to improve health literacy and address the stigma surrounding HBV. Interviews were between 20 and 40 min duration. All interviews were audio-recorded and transcribed, with any identifying information about the participant removed from the transcript. Ethics approval for the evaluation of the DLM program was obtained from South Eastern Sydney LHD and the Aboriginal Health and Medical Research Council Ethics Committees. Site-specific approvals were also obtained from each LHD involved in the study.</p>", "<p id=\"Par21\">Thematic analysis of the transcripts was conducted by experienced qualitative researchers (MH, EC). In this paper, we are interested in individual and structural factors which impact on HBV stigma. This is important as stigma research has thus far largely focused on individual or interpersonal factors [##REF##30770756##33##]. However, structural factors must be considered in exploration of health conditions of groups experiencing significant disadvantage and racism. We used health literacy as a means to explore workers’ perspectives of community understandings of HBV. However, like Christie [##UREF##13##34##] we take the position that ‘effective health literacy is largely to do with effective communication’ (p. 40). Thus, when we speak about clients’ HBV health literacy, it must be understood in the context of whether there is effective communication from health systems to communities, rather than focusing on an individual’s lack of knowledge about complex health problems.</p>", "<p id=\"Par22\">We complemented our analytic framework with analysis of intersecting stigma. Intersecting stigma and discrimination acts as a significant barrier to accessing mainstream health care broadly, with racism and particularly institutional racism being one of the key concerns [##REF##30764816##50##]. Institutional racism refers to the ways in which our health systems are both intentionally and unintentionally steeped in racist beliefs and values [##UREF##14##35##], including through policies and processes that serve to discriminate against racial minority groups and maintain and reproduce health inequalities. For First Nations people globally, institutional racism manifests in various ways in health systems, through cultural misunderstandings that undermine the quality of care provided, poorer health outcomes including shorter life expectancy of Aboriginal and Torres Strait Islander people compared to non-Aboriginal or Torres Strait Islander people, longer waiting times for medical care, and reduced likelihood of receiving health treatments [##REF##30458120##36##–##REF##17725009##38##].</p>", "<p id=\"Par23\">Finally, as research should avoid furthering the discourse of ‘deficiency’ or ‘pathology’ of Aboriginal and Torres Strait Islander people and communities [##REF##20477739##39##], our analysis focused on aspects of strength and resilience. Thus, we focus on the strategies that workers have used to resist and address HBV health literacy and stigma in their work.</p>", "<p id=\"Par24\">Two authors (MH, EC) closely read each transcript to produce data summaries informed by conceptual tools outlined above guided by Braun and Clarke’s approach to thematic analysis [##UREF##15##40##]. These summaries explored issues within the data aligned with our approaches to health literacy and stigma (particularly structural and intersectional stigma). These data summaries were used in a number of capacity building workshops attended by Aboriginal and Torres Strait Islander authors (MB, KB) and non-Aboriginal or Torres Strait Islander authors (EC, CT) to review interpretation and presentation of the findings. All authors reviewed the findings. Due to the small sample size and the close-knit community of people involved in the DLM development, implementation, and management, demographic characteristics of the sample were not collected to protect the confidentiality of participants.</p>" ]

[ "<title>Results and discussion</title>", "<p id=\"Par25\">The sample consisted of five Aboriginal and Torres Strait Islander and six non-Aboriginal or Torres Strait Islander health workers and key informants involved in the DLM program. Health workers were involved in varying aspects of the DLM program, from the initial development, planning, implementation, and delivery (including frontline Aboriginal and Torres Strait Islander workers and clinical staff). This section presents the findings from interviews with the 11 health workers as well as our interpretation of how these findings relate to the broader evaluation of DLM as well as existing literature. We refer to ‘viral hepatitis’ more broadly due to the challenges in disentangling stigma related to HBV and HCV, as well as the intersecting nature of stigma.</p>", "<title>Intersectional stigma surrounding viral hepatitis</title>", "<p id=\"Par26\">Among Aboriginal and Torres Strait Islander clients of the DLM program, stigma is complex and intersectional. Aboriginal and Torres Strait Islander people often contend with multiple layers of stigma in their lives [##UREF##16##41##], so HBV-related stigma, reportedly, did not stand out as a specific problem. Instead, it was described by health workers as just one dimension within a cascade of intersecting stigma and discrimination, particularly racism.</p>", "<p id=\"Par28\">In a meta-narrative systematic review of Aboriginal and Torres Strait Islander peoples’ experiences of health care, Jones, Heslop, and Harrison [##REF##33317556##42##] found that client experiences were in part informed by the level of trust that Aboriginal and Torres Strait Islander people have in health systems. Trust was informed by past and current experiences within health care settings, but also historical traumas, including past experiences with non-health-related institutions or through government policies that adversely impact Aboriginal and Torres Strait Islander peoples. The researchers also found that mistrust in health systems manifests through scepticism around health information, mistrust in the transfer of private health information between services, and reluctance to share information with their health providers [##REF##33317556##42##]. Similarly, in this research, as a result of intersecting stigmas and institutional racism, Aboriginal and Torres Strait Islander clients were reported as often not trusting ‘the system’ or the doctors who work in it, with many having experienced stigma and discrimination from health care services, such as hospitals. It was reported by health workers in this study that doctors and other health workers represent authority, which is challenging for some Aboriginal and Torres Strait Islander clients to trust given the devastating impacts of colonisation in Australian history, including ongoing inequality and racism. Treloar et al. [##UREF##16##41##] elsewhere described Aboriginal and Torres Strait Islander people describing ‘automatic’ expectations that their communities would have stigmatised health conditions such as hepatitis C, with these expectations potentially imposed by health workers but which can become internalised as well. As one client in their research said, ‘I think I would have to say that the automatic racist attitude that, you know, you'll come up against, you know, like, I expect you to have these things, you are black’. (p. 24). Thus, institutional racism represents an ongoing structural and systemic barrier for Aboriginal and Torres Strait Islander communities to access primary health care.</p>", "<p id=\"Par29\">Intersecting stigma also referred to the stigma that surrounds viral hepatitis, including both hepatitis B and C. This was perceived to be driven by shame about specific social practices, and by fear of unknown transmission risks and health outcomes. For instance, participants perceived the stigma to surround HBV due to the broader stigma that surrounds viral hepatitis more broadly. Each type of viral hepatitis was perceived to be stigmatised due to their association with one another.</p>", "<p id=\"Par32\">Such stigma was also linked to negative attitudes around practices that may place people at risk for viral hepatitis transmission, such as injecting drug use and sexual practices. HBV is most commonly transmitted perinatally or vertically (mother-to-child at birth) [##REF##15602165##2##], with most historical transmissions of HBV in Aboriginal and Torres Strait Islander communities occurring perinatally or in early childhood [##REF##12526729##43##]. Despite this, the association between HBV and HCV meant that the focus among Aboriginal and Torres Strait Islander people was on transmission through injecting drug use and sexual intercourse. Health workers reported that Aboriginal and Torres Strait Islander communities often feel shame about injecting drug use and sexual practices, and thus HBV stigma is fuelled by its association with these practices. The stigma that surrounds HCV is largely fuelled by negative attitudes towards injecting drug use [##REF##17630381##44##, ##REF##15385226##45##], and thus it was no surprise that participants spoke of the shame among Aboriginal and Torres Strait Islander communities that surrounds injecting drug use. Most of the clients who enter the DLM program do so because they have a history of injecting drugs or are otherwise at risk of HCV, and thus it is difficult to disentangle the stigma that surrounds HBV and HCV among this client group. As the second health worker highlights below, they were unsure of whether HBV was seen as worse than HCV due to potential transmission via sexual intercourse or injecting drugs.</p>", "<title>The connection between viral hepatitis health literacy and stigmatisation</title>", "<p id=\"Par35\">Health workers perceived that there were significant knowledge gaps in relation to viral hepatitis, which begin with health workers and extend to the general community. This is supported by prior research, which reports low levels of HBV knowledge among both health workers and Aboriginal and Torres Strait Islander people in Australia [##UREF##10##28##, ##REF##23550250##46##]. Workers discussed that there is a general lack of understanding of the differences between different types of viral hepatitis, the availability of a vaccine for HBV, and the varying treatment options. This is supported by previous research in a community sample which found that HCV knowledge may be better than HBV knowledge, with some important gaps existing in relation to vaccines and treatment [##UREF##17##47##]. Although DLM program staff were well informed about viral hepatitis in general, participants noted the complexities in understanding HBV and acknowledged that it had taken them quite some time to understand the differences between hepatitis A, B, and C.</p>", "<p id=\"Par38\">As we have noted, HBV health literacy must be understood within the context of the presence or absence of effective communication between health systems and community [##UREF##13##34##]. Health systems must provide effective, culturally appropriate, and culturally safe education and resources to Aboriginal and Torres Strait Islander communities in order to disentangle complex medical information. Thus, it is no surprise that as a result of the broader knowledge gaps around HBV among health workers, viral hepatitis health literacy was said to be poor among Aboriginal and Torres Strait Islander people upon entry to the DLM program. Participants perceived that the lack of HBV knowledge was linked to a focus on HCV within the broader health sector, resulting in HBV not being seen as a priority for Aboriginal and Torres Strait Islander clients:</p>", "<p id=\"Par40\">This supports prior academic literature which highlights significant knowledge gaps around protective behaviours, transmission, monitoring, and treatment of the infection [##REF##25430502##29##, ##REF##23550250##46##].</p>", "<p id=\"Par41\">Although relatively few clients tested positive for HBV, many clients were found to require HBV vaccinations (which are offered as part of the DLM program) despite being in the age groups in which they were expected to have received vaccination as part of Australia’s national immunisation program. Therefore, the lack of knowledge around the different types of viral hepatitis was concerning for health workers, who perceived that clients were unaware how serious this preventable infection can be. As the following quote shows, clients conflated HBV and HCV, and thus there was confusion about there being no vaccine available for HCV.</p>", "<p id=\"Par43\">Participants reported that there was confusion among clients about the transmission routes for HBV infection. For instance, some clients associated HBV infection with injecting drug use, and thus HBV infection was not widely understood to be sexually transmissible:</p>", "<p id=\"Par45\">Some of the health workers perceived that there was little to no HBV-related stigma among clients, in part due to a lack of available information about the infection. This was also because some health workers perceived that HCV was more highly stigmatised due to its association with injecting drug use. A lack of awareness about HBV can be a double-edged sword, as on the one hand people might be less inclined to stigmatise a condition they know nothing about, yet on the other hand, low viral hepatitis health literacy can facilitate processes of stigmatisation by exacerbating misguided fears and confusion about risks of contagion. This is supported by previous research which has found that stigma surrounding HBV is linked to fears of the contagiousness of the infection [##REF##32214859##3##, ##REF##28101498##4##].</p>", "<p id=\"Par46\">\n\n</p>", "<p id=\"Par49\">However, others reported that HBV was stigmatised largely due to its association and conflation with HCV. This had important implications for clients, with workers describing that HBV-positive clients had concerns about disclosing their positive status to potential sexual partners, and often attended the DLM program by themselves without a support person. This was understood by one health worker as an indication of stigma and the liminal status of HBV infection within Aboriginal and Torres Strait Islander communities: it is barely perceived as distinct from HCV infection, and people living with HBV infection fear they will be similarly stigmatised.</p>", "<title>Strategies to reduce viral hepatitis-related stigma</title>", "<p id=\"Par51\">In addition to proposing effective communication as core to health literacy, Christie [##UREF##18##48##] also suggests that culturally appropriate and culturally safe approaches that work towards a shared understanding of issues are more effective than imposing Western biomedical understandings of health. A culturally appropriate or safe approach would be one that develops respectful relationships between health systems and Aboriginal and Torres Strait Islander people, and which ‘is mindful of language, worldview, existing knowledge and beliefs’ [##REF##25430502##29##] (p. 2). As others have noted, employing Aboriginal and Torres Strait Islander staff, ensuring community ownership, and engaging with community are critical to improving health care access and accessibility among Aboriginal and Torres Strait Islander communities [##REF##27716235##27##, ##REF##25091076##49##]. When participants spoke about strategies to reduce viral hepatitis-related stigma, they often referred to DLM as uniquely able to counter stigma and bridge the gap between Aboriginal and Torres Strait Islander people and mainstream health services, particularly due to the presence of a frontline Aboriginal and Torres Strait Islander DLM worker. The presence of an Aboriginal and Torres Strait Islander worker has elsewhere been described as a core strength of the DLM program in facilitating connections between clients and mainstream sexual health services [##REF##29391019##30##]. In this context, health workers described that Aboriginal and Torres Strait Islander health workers could deliver health promotion messages around HBV screening, vaccination, monitoring, and treatment (where required) in culturally appropriate and culturally safe ways. Research suggests that yarning style interactions are positively received by Aboriginal and Torres Strait Islander clients because they are informal and relaxed, allowing clients to communicate as they would within their communities [##REF##33317556##42##]. Within DLM, having honest conversations with clients about viral hepatitis through yarning, coupled with resources from community organisations, reportedly reduced stigma through clearing up misinformation:</p>", "<p id=\"Par53\">One health worker described the importance of the Aboriginal and Torres Strait Islander worker in establishing relationships with Aboriginal and Torres Strait Islander clients, as well as fostering constructive and respectful relationships with non-Aboriginal or Torres Strait Islander health staff and addressing health literacy concerns. In this way, DLM can act to build trust between Aboriginal and Torres Strait Islander communities and mainstream health services. Trust is critical in improving Aboriginal and Torres Strait Islander people’s access to and experiences within health care [##REF##33317556##42##]. Trust and rapport take time to build, and they were described in this research as preconditions for addressing stigma in whatever shape or form it takes. Aboriginal and Torres Strait Islander DLM workers needed to build rapport and trust with Aboriginal and Torres Strait Islander clients before the clients felt comfortable enough to attend the DLM program or before they would consider visiting a liver clinic accompanied by an Aboriginal and Torres Strait Islander peer worker. This is particularly because beyond the frontline Aboriginal and Torres Strait Islander DLM worker, much of the staff within Australian health systems are non-Aboriginal or Torres Strait Islander. As the quote below illustrates, the Aboriginal and Torres Strait Islander workers have to date assisted in facilitating clients’ access to mainstream services, by accompanying clients to various appointments at a nearby hospital:</p>", "<p id=\"Par55\">Participants believed that education was critical to reducing the stigma that surrounds viral hepatitis. As the following health workers described, it is important to have open conversations about HBV both to raise awareness about the infection and treatment options, but also to address stigma, similarly to mental health:</p>", "<p id=\"Par58\">Because the core focus of the DLM program continues to be HCV, HBV infection generally remains a ‘hidden issue’ (Non-Aboriginal or Torres Strait Islander health worker). To date, efforts to reduce HBV-related stigma within the DLM program have mostly been opportunistic, such as when a client returns a positive test result and is given information to address their concerns around transmission to family and friends. Due to the intersecting nature of viral hepatitis stigma, health workers stressed the importance of educating clients on hepatitis A, B and C in future:</p>", "<p id=\"Par60\">It was suggested that HBV-related stigma might be addressed via a program to increase HBV testing, monitoring, and treatment (where required), and by focusing upon the prevention of liver cancer from uptake of the HBV vaccine. Given that stigma was believed to be fuelled by negative attitudes towards social practices that place people at risk of viral hepatitis, the following health worker suggested that focusing on the risk of cancer might help to alleviate viral hepatitis-related stigma:</p>", "<p id=\"Par62\">Participants reported that the DLM program has improved the lives of many clients and that it is important to use this model to build rapport, to reduce stigma and to open discussions about difficult topics, so that DLM appointments become an opportunity for clients to have a conversation rather than solely a medical consultation. This approach has been useful for addressing HCV-related stigma within Aboriginal and Torres Strait Islander communities. A similar reduction in stigma might be expected if the profile of HBV infection was carefully and sensitively increased within the DLM program. Although HCV is the focus of the program, ultimately, the DLM program was described as being a culturally appropriate and safe mechanism through which to address viral hepatitis health literacy more broadly, improve access to testing and treatment, and address other health concerns:</p>" ]

[ "<title>Results and discussion</title>", "<p id=\"Par25\">The sample consisted of five Aboriginal and Torres Strait Islander and six non-Aboriginal or Torres Strait Islander health workers and key informants involved in the DLM program. Health workers were involved in varying aspects of the DLM program, from the initial development, planning, implementation, and delivery (including frontline Aboriginal and Torres Strait Islander workers and clinical staff). This section presents the findings from interviews with the 11 health workers as well as our interpretation of how these findings relate to the broader evaluation of DLM as well as existing literature. We refer to ‘viral hepatitis’ more broadly due to the challenges in disentangling stigma related to HBV and HCV, as well as the intersecting nature of stigma.</p>", "<title>Intersectional stigma surrounding viral hepatitis</title>", "<p id=\"Par26\">Among Aboriginal and Torres Strait Islander clients of the DLM program, stigma is complex and intersectional. Aboriginal and Torres Strait Islander people often contend with multiple layers of stigma in their lives [##UREF##16##41##], so HBV-related stigma, reportedly, did not stand out as a specific problem. Instead, it was described by health workers as just one dimension within a cascade of intersecting stigma and discrimination, particularly racism.</p>", "<p id=\"Par28\">In a meta-narrative systematic review of Aboriginal and Torres Strait Islander peoples’ experiences of health care, Jones, Heslop, and Harrison [##REF##33317556##42##] found that client experiences were in part informed by the level of trust that Aboriginal and Torres Strait Islander people have in health systems. Trust was informed by past and current experiences within health care settings, but also historical traumas, including past experiences with non-health-related institutions or through government policies that adversely impact Aboriginal and Torres Strait Islander peoples. The researchers also found that mistrust in health systems manifests through scepticism around health information, mistrust in the transfer of private health information between services, and reluctance to share information with their health providers [##REF##33317556##42##]. Similarly, in this research, as a result of intersecting stigmas and institutional racism, Aboriginal and Torres Strait Islander clients were reported as often not trusting ‘the system’ or the doctors who work in it, with many having experienced stigma and discrimination from health care services, such as hospitals. It was reported by health workers in this study that doctors and other health workers represent authority, which is challenging for some Aboriginal and Torres Strait Islander clients to trust given the devastating impacts of colonisation in Australian history, including ongoing inequality and racism. Treloar et al. [##UREF##16##41##] elsewhere described Aboriginal and Torres Strait Islander people describing ‘automatic’ expectations that their communities would have stigmatised health conditions such as hepatitis C, with these expectations potentially imposed by health workers but which can become internalised as well. As one client in their research said, ‘I think I would have to say that the automatic racist attitude that, you know, you'll come up against, you know, like, I expect you to have these things, you are black’. (p. 24). Thus, institutional racism represents an ongoing structural and systemic barrier for Aboriginal and Torres Strait Islander communities to access primary health care.</p>", "<p id=\"Par29\">Intersecting stigma also referred to the stigma that surrounds viral hepatitis, including both hepatitis B and C. This was perceived to be driven by shame about specific social practices, and by fear of unknown transmission risks and health outcomes. For instance, participants perceived the stigma to surround HBV due to the broader stigma that surrounds viral hepatitis more broadly. Each type of viral hepatitis was perceived to be stigmatised due to their association with one another.</p>", "<p id=\"Par32\">Such stigma was also linked to negative attitudes around practices that may place people at risk for viral hepatitis transmission, such as injecting drug use and sexual practices. HBV is most commonly transmitted perinatally or vertically (mother-to-child at birth) [##REF##15602165##2##], with most historical transmissions of HBV in Aboriginal and Torres Strait Islander communities occurring perinatally or in early childhood [##REF##12526729##43##]. Despite this, the association between HBV and HCV meant that the focus among Aboriginal and Torres Strait Islander people was on transmission through injecting drug use and sexual intercourse. Health workers reported that Aboriginal and Torres Strait Islander communities often feel shame about injecting drug use and sexual practices, and thus HBV stigma is fuelled by its association with these practices. The stigma that surrounds HCV is largely fuelled by negative attitudes towards injecting drug use [##REF##17630381##44##, ##REF##15385226##45##], and thus it was no surprise that participants spoke of the shame among Aboriginal and Torres Strait Islander communities that surrounds injecting drug use. Most of the clients who enter the DLM program do so because they have a history of injecting drugs or are otherwise at risk of HCV, and thus it is difficult to disentangle the stigma that surrounds HBV and HCV among this client group. As the second health worker highlights below, they were unsure of whether HBV was seen as worse than HCV due to potential transmission via sexual intercourse or injecting drugs.</p>", "<title>The connection between viral hepatitis health literacy and stigmatisation</title>", "<p id=\"Par35\">Health workers perceived that there were significant knowledge gaps in relation to viral hepatitis, which begin with health workers and extend to the general community. This is supported by prior research, which reports low levels of HBV knowledge among both health workers and Aboriginal and Torres Strait Islander people in Australia [##UREF##10##28##, ##REF##23550250##46##]. Workers discussed that there is a general lack of understanding of the differences between different types of viral hepatitis, the availability of a vaccine for HBV, and the varying treatment options. This is supported by previous research in a community sample which found that HCV knowledge may be better than HBV knowledge, with some important gaps existing in relation to vaccines and treatment [##UREF##17##47##]. Although DLM program staff were well informed about viral hepatitis in general, participants noted the complexities in understanding HBV and acknowledged that it had taken them quite some time to understand the differences between hepatitis A, B, and C.</p>", "<p id=\"Par38\">As we have noted, HBV health literacy must be understood within the context of the presence or absence of effective communication between health systems and community [##UREF##13##34##]. Health systems must provide effective, culturally appropriate, and culturally safe education and resources to Aboriginal and Torres Strait Islander communities in order to disentangle complex medical information. Thus, it is no surprise that as a result of the broader knowledge gaps around HBV among health workers, viral hepatitis health literacy was said to be poor among Aboriginal and Torres Strait Islander people upon entry to the DLM program. Participants perceived that the lack of HBV knowledge was linked to a focus on HCV within the broader health sector, resulting in HBV not being seen as a priority for Aboriginal and Torres Strait Islander clients:</p>", "<p id=\"Par40\">This supports prior academic literature which highlights significant knowledge gaps around protective behaviours, transmission, monitoring, and treatment of the infection [##REF##25430502##29##, ##REF##23550250##46##].</p>", "<p id=\"Par41\">Although relatively few clients tested positive for HBV, many clients were found to require HBV vaccinations (which are offered as part of the DLM program) despite being in the age groups in which they were expected to have received vaccination as part of Australia’s national immunisation program. Therefore, the lack of knowledge around the different types of viral hepatitis was concerning for health workers, who perceived that clients were unaware how serious this preventable infection can be. As the following quote shows, clients conflated HBV and HCV, and thus there was confusion about there being no vaccine available for HCV.</p>", "<p id=\"Par43\">Participants reported that there was confusion among clients about the transmission routes for HBV infection. For instance, some clients associated HBV infection with injecting drug use, and thus HBV infection was not widely understood to be sexually transmissible:</p>", "<p id=\"Par45\">Some of the health workers perceived that there was little to no HBV-related stigma among clients, in part due to a lack of available information about the infection. This was also because some health workers perceived that HCV was more highly stigmatised due to its association with injecting drug use. A lack of awareness about HBV can be a double-edged sword, as on the one hand people might be less inclined to stigmatise a condition they know nothing about, yet on the other hand, low viral hepatitis health literacy can facilitate processes of stigmatisation by exacerbating misguided fears and confusion about risks of contagion. This is supported by previous research which has found that stigma surrounding HBV is linked to fears of the contagiousness of the infection [##REF##32214859##3##, ##REF##28101498##4##].</p>", "<p id=\"Par46\">\n\n</p>", "<p id=\"Par49\">However, others reported that HBV was stigmatised largely due to its association and conflation with HCV. This had important implications for clients, with workers describing that HBV-positive clients had concerns about disclosing their positive status to potential sexual partners, and often attended the DLM program by themselves without a support person. This was understood by one health worker as an indication of stigma and the liminal status of HBV infection within Aboriginal and Torres Strait Islander communities: it is barely perceived as distinct from HCV infection, and people living with HBV infection fear they will be similarly stigmatised.</p>", "<title>Strategies to reduce viral hepatitis-related stigma</title>", "<p id=\"Par51\">In addition to proposing effective communication as core to health literacy, Christie [##UREF##18##48##] also suggests that culturally appropriate and culturally safe approaches that work towards a shared understanding of issues are more effective than imposing Western biomedical understandings of health. A culturally appropriate or safe approach would be one that develops respectful relationships between health systems and Aboriginal and Torres Strait Islander people, and which ‘is mindful of language, worldview, existing knowledge and beliefs’ [##REF##25430502##29##] (p. 2). As others have noted, employing Aboriginal and Torres Strait Islander staff, ensuring community ownership, and engaging with community are critical to improving health care access and accessibility among Aboriginal and Torres Strait Islander communities [##REF##27716235##27##, ##REF##25091076##49##]. When participants spoke about strategies to reduce viral hepatitis-related stigma, they often referred to DLM as uniquely able to counter stigma and bridge the gap between Aboriginal and Torres Strait Islander people and mainstream health services, particularly due to the presence of a frontline Aboriginal and Torres Strait Islander DLM worker. The presence of an Aboriginal and Torres Strait Islander worker has elsewhere been described as a core strength of the DLM program in facilitating connections between clients and mainstream sexual health services [##REF##29391019##30##]. In this context, health workers described that Aboriginal and Torres Strait Islander health workers could deliver health promotion messages around HBV screening, vaccination, monitoring, and treatment (where required) in culturally appropriate and culturally safe ways. Research suggests that yarning style interactions are positively received by Aboriginal and Torres Strait Islander clients because they are informal and relaxed, allowing clients to communicate as they would within their communities [##REF##33317556##42##]. Within DLM, having honest conversations with clients about viral hepatitis through yarning, coupled with resources from community organisations, reportedly reduced stigma through clearing up misinformation:</p>", "<p id=\"Par53\">One health worker described the importance of the Aboriginal and Torres Strait Islander worker in establishing relationships with Aboriginal and Torres Strait Islander clients, as well as fostering constructive and respectful relationships with non-Aboriginal or Torres Strait Islander health staff and addressing health literacy concerns. In this way, DLM can act to build trust between Aboriginal and Torres Strait Islander communities and mainstream health services. Trust is critical in improving Aboriginal and Torres Strait Islander people’s access to and experiences within health care [##REF##33317556##42##]. Trust and rapport take time to build, and they were described in this research as preconditions for addressing stigma in whatever shape or form it takes. Aboriginal and Torres Strait Islander DLM workers needed to build rapport and trust with Aboriginal and Torres Strait Islander clients before the clients felt comfortable enough to attend the DLM program or before they would consider visiting a liver clinic accompanied by an Aboriginal and Torres Strait Islander peer worker. This is particularly because beyond the frontline Aboriginal and Torres Strait Islander DLM worker, much of the staff within Australian health systems are non-Aboriginal or Torres Strait Islander. As the quote below illustrates, the Aboriginal and Torres Strait Islander workers have to date assisted in facilitating clients’ access to mainstream services, by accompanying clients to various appointments at a nearby hospital:</p>", "<p id=\"Par55\">Participants believed that education was critical to reducing the stigma that surrounds viral hepatitis. As the following health workers described, it is important to have open conversations about HBV both to raise awareness about the infection and treatment options, but also to address stigma, similarly to mental health:</p>", "<p id=\"Par58\">Because the core focus of the DLM program continues to be HCV, HBV infection generally remains a ‘hidden issue’ (Non-Aboriginal or Torres Strait Islander health worker). To date, efforts to reduce HBV-related stigma within the DLM program have mostly been opportunistic, such as when a client returns a positive test result and is given information to address their concerns around transmission to family and friends. Due to the intersecting nature of viral hepatitis stigma, health workers stressed the importance of educating clients on hepatitis A, B and C in future:</p>", "<p id=\"Par60\">It was suggested that HBV-related stigma might be addressed via a program to increase HBV testing, monitoring, and treatment (where required), and by focusing upon the prevention of liver cancer from uptake of the HBV vaccine. Given that stigma was believed to be fuelled by negative attitudes towards social practices that place people at risk of viral hepatitis, the following health worker suggested that focusing on the risk of cancer might help to alleviate viral hepatitis-related stigma:</p>", "<p id=\"Par62\">Participants reported that the DLM program has improved the lives of many clients and that it is important to use this model to build rapport, to reduce stigma and to open discussions about difficult topics, so that DLM appointments become an opportunity for clients to have a conversation rather than solely a medical consultation. This approach has been useful for addressing HCV-related stigma within Aboriginal and Torres Strait Islander communities. A similar reduction in stigma might be expected if the profile of HBV infection was carefully and sensitively increased within the DLM program. Although HCV is the focus of the program, ultimately, the DLM program was described as being a culturally appropriate and safe mechanism through which to address viral hepatitis health literacy more broadly, improve access to testing and treatment, and address other health concerns:</p>" ]

[ "<title>Conclusions</title>", "<p id=\"Par64\">Health workers in this research perceived that HBV was not a priority in Aboriginal and Torres Strait Islander communities, and this was partly due to the focus of health promotion programs on screening and treatment for HCV and the conflation among health workers and the community of the different types of viral hepatitis. Stigma is complex and may be intersectional or layered according to multiple identities or social practices [##REF##30764816##50##]. Aboriginal and Torres Strait Islander communities face significant racism and discrimination more broadly [##REF##17725009##38##, ##REF##21835522##51##], and it can be difficult to pinpoint stigma that exists specifically in relation to viral hepatitis. Both racism and stigma surrounding HBV act as a significant barrier to Aboriginal and Torres Strait Islander communities accessing mainstream health services, thus further research is needed to attempt to disentangle the complex nature and impacts of viral hepatitis stigma within these communities.</p>", "<p id=\"Par65\">The DLM program is uniquely situated in being able to reduce hepatitis-related stigma in Aboriginal and Torres Strait Islander communities through the one-on-one support provided to clients (including Aboriginal and Torres Strait Islander workers accompanying clients to appointments at hospitals and liver clinics), through yarning (including the education sessions delivered by Aboriginal and Torres Strait Islander health workers about viral hepatitis), establishing trust in mainstream health services (by Aboriginal and Torres Strait Islander staff introducing clients to any non-Aboriginal or Torres Strait Islander clinical staff), and by health workers engaging clients in HBV vaccination, assessment, monitoring, and treatment (where required). Elsewhere, we have demonstrated the success of the early pilot DLM programs in encouraging Aboriginal and Torres Strait Islander communities to go through the DLM ‘cascade of care’ (education, screening, results, additional treatment if required) [##REF##29391019##30##]. Hla et al. [##REF##32381081##52##] documented the positive impacts of a ‘one stop liver shop’ program (that includes Aboriginal and Torres Strait Islander health workers) in remote Aboriginal and Torres Strait Islander communities. Thus, the DLM model of care could be further developed to focus on HBV education and screening in Aboriginal and Torres Strait Islander communities, and thereby acting as a ‘one stop shop’ across all BBVs and STIs. While clients are screened for HBV and offered vaccination if applicable, the focus of DLM and the education provided to clients is currently on HCV. An expanded focus of the education component to include HBV might address the conflation between the different types of viral hepatitis that was noted in this study. Scholars have recently co-designed and evaluated culturally appropriate and safe HBV-related education for the Aboriginal and Torres Strait Islander health workforce, which could help guide such an expanded focus for the DLM program [##UREF##19##53##].</p>", "<p id=\"Par66\">There are several study limitations of this research that should be noted. Firstly, the sample was recruited through the DLM program; a health promotion program that specifically aims to educate Aboriginal and Torres Strait Islander communities about BBVs and STIs and improve access to screening and treatment. The sample is not generalisable to all DLM workers within NSW, nor is it generalisable to health workers beyond the program. These are workers who have greater education and awareness of the various types of viral hepatitis, with interview participants noting that other health workers may lack such awareness and knowledge. Previous research has found that some health workers who provide care to Aboriginal and Torres Strait Islander communities lack knowledge about available treatment for HBV [##UREF##10##28##]. More generally, research has also found that health workers may be concerned that they lack the skills and confidence to treatment clients with HBV [##UREF##9##26##]. The DLM program is opportunistic in the sense that people are recruited to the program in relation to their potential risk of having HCV and are screened and treated for other BBVs and STIs in the process. Future research among health workers should diversify the sampling approach to include other health workers who may have contact with or treat clients living with HBV, and who may be unaware of the need to screen for the full suite of BBVs and STIs. This study was also limited in that we only interviewed Aboriginal and Torres Strait Islander and non-Aboriginal or Torres Strait Islander health workers about their perspectives of HBV and HBV-related stigma. These perspectives are valuable as a first step and to begin to map the field; however, future research should also seek perspectives of Aboriginal and Torres Strait Islander people to explore perceptions of HBV and HBV-related stigma within these communities. Finally, Aboriginal and Torres Strait Islander communities come from diverse language and cultural groups across Australia, and therefore may have different perceptions of HBV and stigma. Unfortunately, we were unable to explore such differences, and future research could seek to explore such variations across communities in Australia.</p>", "<p id=\"Par67\">The data presented in this paper have important implications for health promotion interventions among Aboriginal and Torres Strait Islander communities. Although the DLM program more broadly aims to improve health literacy related to viral hepatitis among Aboriginal and Torres Strait Islander communities, it is apparent that the focus is on HCV. This results in confusion among both staff and clients about the different types of viral hepatitis. Given that the infrastructure is already in place, including the good will of health workers in wanting to improve knowledge and health care access in relation to HBV, it would be useful for the DLM to have a stronger focus on HBV education moving forward. Health promotion programs like DLM have the potential to reduce stigma by acting as a ‘one stop shop’ for BBVs and STIs, through employment of Aboriginal and Torres Strait Islander peers to increase trust in health services, one-on-one support, yarning, and promotion of screening, monitoring, treatment (where required), and vaccination.</p>" ]

[ "<title>Background</title>", "<p id=\"Par1\">Experiences of stigma and discrimination can act as a significant barrier to testing, monitoring, and treatment for hepatitis B virus (HBV). Aboriginal and Torres Strait Islander Australians are a population disproportionately impacted by HBV and yet limited research has explored HBV-related stigma in these communities. To begin preliminary explorations of HBV-related stigma among Aboriginal and Torres Strait Islander people, we interviewed health workers about their perceptions regarding HBV infection and HBV-related stigma.</p>", "<title>Methods</title>", "<p id=\"Par2\">Participants were recruited from staff involved in the Deadly Liver Mob (DLM) program which is a health promotion program that offers incentives for Aboriginal and Torres Strait Islander clients to be educated on viral hepatitis, recruit and educate peers, and receive screening and treatment for blood-borne viruses (BBVs) and sexually transmissible infections (STIs), and vaccination. Semi-structured interviews were conducted with 11 Aboriginal and Torres Strait Islander and non-Aboriginal or Torres Strait Islander health workers who have been involved in the development, implementation, and/or management of the DLM program within participating services in New South Wales, Australia.</p>", "<title>Results</title>", "<p id=\"Par3\">Findings suggest that stigma is a barrier to accessing mainstream health care among Aboriginal and Torres Strait Islander clients, with stigma being complex and multi-layered. Aboriginal and Torres Strait Islander people contend with multiple and intersecting layers of stigma and discrimination in their lives, and thus HBV is just one dimension of those experiences. Health workers perceived that stigma is fuelled by multiple factors, including poor HBV health literacy within the health workforce broadly and among Aboriginal and Torres Strait Islander clients, shame about social practices associated with viral hepatitis, and fear of unknown transmission risks and health outcomes. The DLM program was viewed as helping to resist and reject stigma, improve health literacy among both health workers and clients, and build trust and confidence in mainstream health services.</p>", "<title>Conclusions</title>", "<p id=\"Par4\">Health promotion programs have the potential to reduce stigma by acting as a ‘one stop shop’ for BBVs and STIs through one-on-one support, yarning, and promotion of the HBV vaccine, monitoring for chronic HBV, and treatment (where required).</p>", "<title>Keywords</title>" ]

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[ "<title>Acknowledgements</title>", "<p>This research was funded by a National Health and Medical Research Council Partnership Grant and by the NSW Ministry of Health. Funding and in-kind contributions were also provided by participating sites. The authors wish to acknowledge the contributions of everyone who was involved in DLM over the years, including but not limited to (listed alphabetically by surname): Clayton Anderson, Jade Christian, Margaret Crowley, John de Wit, Basil Donovan, Brian Doyle, Peter Emeana, Gary Gahan, Kim Grant, Jen Heslop, Max Hopwood, Kendall Jackman, Aunty Clair Jackson, Franklin John-Leader, Stevie Kemp, Catherine Kostovski, Jade Lane, Jo Lenton, Wendy Machin, Lou Maher, Julie Page, Kate Pearce, Syl Phillips-Ayre, Ann Ryan, Felicity Sheaves, Edmund Silins, Trevor Slattery, Kerri-Anne Smith, Larissa Smyth, Annabelle Stevens, Victor Tawil, Donna Tilley, Bev Tyson, James Vernon, Aunty Estelle Wade, Andrew Walden, Melinda Walker, James Ward, and David Webb. Thanks to Kerryn Drysdale and Morgan Richards for their work in developing the DLM toolkit. Thanks also to Ron Horwood (original artwork), Aunty Clair Jackson (logo image), Emma Hicks (logo image and digitisation) for development of the DLM logo.</p>", "<title>Author contributions</title>", "<p>CT obtained the grant funding and conceived of the original evaluation. MH and EC conducted initial analysis of the data and EC wrote the first draft. CT, EC, KB, and MB all participated in workshops for analyses and interpretation. All authors contributed to analysis, interpretation, and final manuscript.</p>", "<title>Funding</title>", "<p>This research was funded by a National Health and Medical Research Council Partnership Grant and by the NSW Ministry of Health, with in-kind and program costs provided by DLM sites.</p>", "<title>Availability of data and materials</title>", "<p>The datasets generated and/or analysed during the current study are not publicly available due to the possibility of individual privacy being compromised.</p>", "<title>Declarations</title>", "<title>Ethics approval and consent to participate</title>", "<p id=\"Par68\">This project has ethics approval from South Eastern Sydney LHD Ethics Committee and the Aboriginal Health and Medical Research Council Ethics Committee. Site-specific approvals were also obtained from the LHDs governing each of the DLM sites.</p>", "<title>Consent for publication</title>", "<p id=\"Par69\">Not applicable.</p>", "<title>Competing interests</title>", "<p id=\"Par70\">The authors declare that they have no competing interests.</p>" ]

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[ "<disp-quote><p id=\"Par27\">I mean there are so many forms of stigma that conflate into and comes down upon an individual’s head and on a community shoulders, that being able to unpick … Being treated shabbily by health care workers or community members or family members and peers, because of an individual kind of health concern … there’s so much intersection and potential to elicit a stigmatised reaction from a whole range of different concerns, I don’t reckon people can have a particular need to identify it coming from one particular virus. People just are treated like shit all over the place. (Non-Aboriginal or Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par30\">Hep [Hepatitis] B, the stigma around hep B … I mean, it’s just the hep word. (Aboriginal and Torres Strait Islander health worker)</p><p id=\"Par31\">I know there is stigma, there’s stigma with any hepatitis for anybody and it’s real and I hate to say that people don’t … people especially even health care workers can have stigma. I’ve seen it firsthand or treat people differently with hepatitis whether it be B or C… (Non-Aboriginal or Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par33\">I think you know it’s also a bit shame in some Aboriginal communities around injecting drug use, so in terms of transmission and how the transmission occurred, was it sexually transmitted or was it from sharing injecting equipment or you know any IV [intravenous] use, so I think, yeah, there is a lot of shame around that. (Non-Aboriginal or Torres Strait Islander health worker)</p><p id=\"Par34\">You know the fact that it’s sexually transmitted, rather than just injecting related, you know like to some people it might make it a worse disease or a more dirty disease to have, whereas to other people, it’s like ‘oh no, I didn’t get it from needles, I just had unprotected sex’ so I think there’s a lot of value judgement going on there. (Non-Aboriginal or Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par36\">Not a lot of them know about A, B, and C, I mean, it’s a big thing for even us to [understand] you know the differences between A, B, and C even as health workers trying to work out in the brain what’s the difference. (Aboriginal and Torres Strait Islander health worker)</p><p id=\"Par37\">I think that’s tricky [laughs]. I think that’s really tricky and I don’t know that that’s limited to Aboriginal community, because I think generally people get very confused and it’s true to say that even at the very beginning with our frontline project workers, we took a long time to really drill down with them what the differences were. (Non-Aboriginal or Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par39\">I don’t think [HBV is] a major issue for [Aboriginal and Torres Strait Islander clients]. I don’t think it’s a priority. Most of the conversations and most of the health messaging that we provide, a lot of focus is on building clients health literacy around blood-borne viruses, with a really sharp focus on hepatitis C, because the majority of people that we work with have a history of injecting drug use and just the relative priority that is placed on hep C a whole range of programs, both clinical, needle and syringe, the community sector. (Non-Aboriginal or Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par42\">I think there’s still an enormous amount of confusion between B and C and A and even clients we have treated for hepatitis C will come in and say ‘I had my hep C vaccination’ or something like that, they don’t quite understand. I don’t think any of the clients realise how serious hepatitis B can be, because they think it’s just like hep C sort of that, but it’s not quite, and I have been very surprised on doing blood tests on the Deadly Liver Mob clients how many of the people are not immune to hepatitis B when they are in the age group where they should be. (Non-Aboriginal or Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par44\">… it’s like you know when people are in fear of sexually transmitted infections, hepatitis B isn’t the top one that springs to your mind is it? Chlamydia, gonorrhoea and syphilis or something, you know top three, you are not going to name hep B. (Non-Aboriginal or Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par47\">Hep B, I don't think there is much stigma around that, because I don't think a lot of people understand that it’s very different from Hep C. See hep C has that injecting drug users stigma and I really don't believe that hep B is being talked about enough in the community, so the understanding is different and there is a vaccination and it just didn’t seem even to the elders that hepatitis B was in anyway stigmatised as hep C was. (Aboriginal and Torres Strait Islander health worker)</p><p id=\"Par48\">… [The DLM clients] don’t know much about [HBV infection], so and yeah … there’s not much stigma, because there’s not much information about it going around. Maybe there is, but a lot of Aboriginal people aren’t getting it, so … (Aboriginal and Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par50\">They know that they’ve got hep B and they’re concerned about sexual partners and future and whether that will affect their relationships… That’s the concerns especially for the person that knows that you know if they’re going to have a relationship they then need to cross that bridge and how they’re going to feel about … how that person is going to treat them and whether the relationship will go further or not you know. (Non-Aboriginal or Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par52\">I think that peer Aboriginal workers can probably deliver the message better than non-Aboriginal health workers, potentially they have got more of a connection with the clients, they can maybe relate it back to country and their mob better than you know hearing it from a non-Aboriginal person who may not understand the cultural significance of a vaccination for somebody and be able to explain it in a way that makes them feel culturally safe about making that decision to vaccinate themselves. (Non-Aboriginal or Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par54\">You know some people don't like going up to the hospital, they come to us and ask us to take them up to the hospital. (Aboriginal and Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par56\">I think it’s talking about it … it’s like anything else, it’s like I suppose mental health or anything else if people don't want to talk about it, they think it’s going to upset somebody, but you know anything about health you should talk about it openly… I think once you gain their confidence to say, you know, let’s encourage people to talk about things if they have an issue, because two brains are better than one if you work together, at the end of the day, you will get something that satisfies everyone you know… the word gets around, like you know, if you are going to educate one person to educate somebody else and it will go around, might take a while but it will get there I think sooner or later. Might take a while but I think it’s talking and education around stuff. (Aboriginal and Torres Strait Islander health worker)</p><p id=\"Par57\">I just think going out into the community, having our Aboriginal peer worker and visiting places where Aboriginal people meet and talking about it is perhaps the best way and talking about what we do. (Non-Aboriginal or Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par59\">I know DLM is mainly about hepatitis C, but I believe that to understand hepatitis C, you need to educate on A and B as well so that they can be defined and separated from each other in the education. Yeah, that C is a very different strain I mean, when we started the project, there wasn’t a cure for hep C then, there wasn’t a vaccination and you know to let them know that there is a vaccination for hep A and for hepatitis B but if you got hepatitis B like chronically, there was no cure for that either. So I used to say to them, ‘you know, you are going to be screened and it will be hep B too and if they say that you need the vaccination, we will pay you to do that because once you have had the vaccination you are covered and you don’t want to get hep B because it’s lifelong’. (Aboriginal and Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par61\">It’s okay to talk about cancer but it’s not okay to talk about this virus in your blood. So, you know, that’s just a messaging thing, ‘Get screened and get tested and help prevent cancer’. And immunisation for hep B you know, ‘It’s an anticancer vaccination’ is basically what you are asking people to do. That’s pretty impressive if you talk about that, so that’s a messaging thing, whether that really changes people’s feeling of stigma, I don’t know … (Non-Aboriginal or Torres Strait Islander health worker)</p></disp-quote>", "<disp-quote><p id=\"Par63\">…understanding that the health and social needs and community needs of Aboriginal people need to be addressed as a whole, so working with individuals and communities and identifying and fostering constructive useful, respectful and authentic ways of working with Aboriginal people and then the mechanism by which we get to prioritise or we get to raise, you know discussing health literacy concerns around people with intravenous drug use, people at risk of viral hepatitis, mapping out … identifying what they are going to want to do with the information that we are providing and then setting a really clear and you know easy way to engage with the health service and working through the various health needs that are identified… DLM has been such an awesome mechanism for us, … so while hep C is basically the entry point, we are really keen to use that mechanism, that way of working, because it’s authentic, it works and it provides people with incentives, you know it’s wholly focused around the crucial role of the relationship between the Aboriginal health worker and the Aboriginal client and supporting that and then enabling whatever comes through that assessment process to be managed effectively. (Non-Aboriginal or Torres Strait Islander health worker)</p></disp-quote>" ]

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[ "<fn-group><fn><p>The original online version of this article was revised: reference number 31 has been added \n</p></fn><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p><bold>Change history</bold></p><p>1/13/2024</p><p>A Correction to this paper has been published: 10.1186/s12954-024-00925-y</p></fn></fn-group>" ]

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{ "acronym": [ "BBV", "DLM", "HARP", "HBV", "HCV", "LHD", "NSW", "STI" ], "definition": [ "Blood-borne virus", "Deadly Liver Mob", "HIV and Related Programs", "Hepatitis B", "Hepatitis C", "Local Health District", "New South Wales", "Sexually transmissible infection" ] }

[ "<title>Introduction</title>", "<p>The impact of the digital turn on our culture is undeniable. It has changed the way we consume and create content, breaking down geographical barriers, and empowering individuals to participate in cultural expression on a global scale. With the rise of social media, digital media has also transformed how we communicate and interact with each other, changing the way we form communities and share ideas. One of the major questions that are systematically asked in the debates on the digital is the scope and stakes of this impact. Is the digital turn a real break with previous cultural frameworks? Or, on the contrary, it is better to think the digital as a remediation of the older media? In this article, I aim to examine the impact of the digital turn on ‘collective memory’\n<italic toggle=\"yes\">.</italic> Following Schwartz, collective memory can be defined as “the distribution throughout society of what individuals know, believe, and feel about the past, how they judge the past morally, how closely they identify with it, and how much they are inspired by it as a model for their conduct and identity” (\n##UREF##48##Schwartz, 2016##, 10)</p>", "<p>Since what has been called the ‘memory boom’ in the 80s, collective memory has been an object of continuous academic inquiry as well as a constant presence in public debate. Fueled by various factors —including the exceptional scope of violence and wars of the 20th century, changes in media technology, the end of the Cold War, and developments within academia (\n##UREF##18##Erll, 2011a##, 4–5)— the memory boom led to a detailed investigation of the role of memory in the formation of collective identities, its political function, its stability and variability over time and space, its fidelity or distortion of historical events, its ‘insufficiency’ or ‘excess’ (\n##UREF##10##Connerton, 1989##;\n##UREF##13##De Cesari &amp; Rigney, 2014##;\n##UREF##35##Levy &amp; Sznaider, 2006##;\n##UREF##41##Olick\net al., 2011a##;\n##UREF##43##Radstone &amp; Schwarz, 2010##;\n##UREF##46##Rothberg, 2009##;\n##UREF##54##Tota &amp; Hagen, 2016##).</p>", "<p>In the last 30 years, memory studies not only became institutionalized, as evidenced by the existence of specialized journals and editorial collections, but they also, in some way, became standardized —or at least a part of them— around an approach to the representation of the past that ended up being repetitive and sterile. Assessing the current state of the field and the future of memory studies, Astrid Erll stated,</p>", "<p>“With the methodology at hand, memory studies will easily be able to keep generations of scholars busy, charting the mnemonic practices of all ages and places. However, the question arises whether ‘memory’ is thus turning into a mere ‘stencil’, and memory studies into an additive project: we add yet another site of memory, we address yet another historical injustice. While such memory work is for many historical, political and ethical reasons an important activity, memory research finds itself faced with the decisive question of how it envisages its future” (\n##UREF##19##Erll, 2011b##, 4)</p>", "<p>Erll's critique of the need to update the conceptual and methodological framework of memory studies is even more evident in the contemporary scenario. Today, it is impossible to understand the shaping and transmission of collective memory without addressing the changes brought about by the digital turn. However, although there is consensus on acknowledging the impact of electronic media —something that would be futile to deny— the extent of these changes is a subject of debate. Is it an evolution or a revolution? Are we witnessing a radical break from pre-digital forms or a cultural continuity where mnemonic practices do not undergo substantial modification?</p>", "<p>In the recent debate, the prevailing idea is that digital memory represents a radical change, giving rise to a new form of memory referred to as “algorithmic memory,” “connective memory,” “new memory,” or “the memory of the multitude” (\n##UREF##3##Blom, 2017##;\n##UREF##23##Garde-Hansen\net al., 2009##;\n##UREF##37##Makhortykh, 2021##;\n##UREF##27##Hoskins, 2009##;\n##UREF##28##Hoskins, 2011##;\n##UREF##30##Hoskins, 2018b##). According to these perspectives, digital memory not only implies a different ontology but, in more radical proposals, even entails the “end of collective memory” (\n##UREF##29##Hoskins, 2018a##).</p>", "<p>Andrew Hoskins, one of the leading figures in the field of digital memory studies, undoubtedly best represents this position. Hoskins argues that the impact of digitization implies an ontological shift that renders the conceptual vocabulary previously used in memory studies obsolete, especially its central concept, collective memory. In his formulation, digital memory is based on hyperconnectivity, where agency is both technological and human. This marks a key difference from the previous era. Technological agency —the role of algorithms, the dynamism of archives, the quantity and scope of available information— leads to fragmentation, dispersion, unpredictability, and a loss of control over memory:</p>", "<p>“temporary technological arrangements […] open up a new kind of social articulation of digital memory: immediate, hyperconnective, remote, but whose parameters (size and duration) are unpredictable as they have a feature that the traditional idea of the social does necessarily possess, that they are archival. And it is through shadow archives that the individual and the social are inescapably entwined and also suspended, in a kind of extratemporal and extraspatial existence, and made contingent on the potential of their reinvigoration or rediscovery (emergence)” (\n##UREF##29##Hoskins, 2018a##, 93).</p>", "<p>In Hoskins’ pessimistic view, the digital realm radically destabilizes the temporal and spatial boundaries of collective memory. A central factor of destabilization is the inherently archival nature of digital media:</p>", "<p>“The notion of the archive as static is replaced by the much more fluid temporalities and dynamics of ‘permanent data transfer’ (\n##UREF##21##Ernst, 2018##) defining the new memory ecology. The archives […] are rendered fluid through their hyperconnectivity” (\n##UREF##30##Hoskins, 2018a##, 87).</p>", "<p>While archives in the previous era ensured the stability of collective memory, anchored in both time and space, the archive in the digital era possesses its own temporality that we do not control. As Hoskins claims,</p>", "<p>“Today, the connective turn has rendered most communications, everyday and exceptional, as potentially part of shadow archives, in which sharing without sharing has no limits. Once media became potentially infinite (in time and space), then duration of our multiple presences-in-the-world became uncertain and uncontrollable. Collective memory has its limits. Memory of the multitude does not” (\n##UREF##29##Hoskins, 2018a##, 106).</p>", "<p>And the same happens with temporal anchoring, as Hoskins asserts, “the archive today is the inverted archive, which challenges the significance of place and location. Clearly, place and location are still important, but in many ways they are transcended, if not displaced, by connectivity, time and algorithmic search” (\n##UREF##31##Hoskins &amp; Halstead, 2021##, 678).</p>", "<p>The second factor that destabilizes collective memory lies in the digital practices of individuals —what we\n<italic toggle=\"yes\">do</italic> in the digital environment. Hoskins describes these practices as “compulsive”, once again highlighting the lack of control, in which individuals seem to be engaged in frenetic activity, “posting, linking, liking, recording, swiping, scrolling, forwarding, etc., digital media content” […] constituting a new coercive multitude that does not debate but rather digitally emotes (as in via emoticons) (\n##UREF##30##Hoskins, 2018b##, 2).</p>", "<p>While the participatory nature of digital media has often been interpreted as a democratization of collective memory, in this case, digital practices lack democratizing potential and instead result in a radical dispersion and fragmentation of collective memory. Hoskins then proposes the term “the memory of the multitude” to replace the concept of collective memory. Unlike the stable or organic communities of the pre-digital era, the multitude is characterized by its diffusion, dispersion, provisional nature, and perpetual becoming. Inspired by the work of Hard and Negri, and Virno, “the multitude is an internally different, multiple social subject whose constitution and action are based not on identity or unity (or, much less, indifference) but on what it has in common”. The individual, placed at the center of the media ecology, connects with other individuals and forms “aggregates” or “masses”. In Merrin’s words, “these masses are fluid, flexible, variable, ephemeral, constructed by our voluntary participation and continually evolving, and we are involved in many of these at once. What networked computing realizes, therefore, is ‘the multitude’” (93).</p>", "<p>While Hoskins asserts that the multitude as a collective subject is an organic community of a different kind, “not characterized by stability and homogeneity, but rather by diffusion, dispersal, and emergence”, his final assessment accentuates the vulnerability of this social formation. Hoskins even questions whether the multitude might give rise to a society without memory.</p>", "<p>“What prospects are there, then, for the translation of the digital present into some kind of usable past? Is it that the multitude is the basis of a society without memory? As the social and cultural ‘frameworks’ of remembering have moved inside the machine and inside us, the multitude is increasingly vulnerable to the vagaries of the network […] In sum, the multitude is the impossible crowd: attracted by the immediacy, continuity and sociability of its hyperconnective relations, and yet, with diminishing means to ever escape its digital shadow” (\n##UREF##29##Hoskins, 2018a##, 105–6).</p>", "<p>These words ultimately condemn the multitude as a collective subject of meaningful memory. Digital memory does not represent a new phase in the development of our relationship with the past but the “end of collective memory”. Are we indeed facing the end of collective memory? Should we abandon the conceptual framework and insights gained along the way, closing the debates that have occupied us by deeming them irrelevant in the contemporary setting? I would claim the opposite. Digital memory is not a new form of memory, and far from representing the end of collective memory, it materializes and puts into practice the characteristics with which we have defined collective memory since the inception of the field. In fact, we could argue that what we have always advocated about collective memory finds its true realization in the digital era.</p>", "<p>My argument is that Hoskins’ ‘pessimistic’ view is derived from a conception of collective memory that emphasizes its permanence and stability. In the traditional conception, the past provides coherence and certainty. The title of the introduction to the influential volume\n<italic toggle=\"yes\">Digital Memory Studies</italic> is precisely “The restless past”, a past that has been (fatally) deprived of its stability, and, therefore, its ability to provide coherence to the present —and the future. In Hoskins’s words:</p>", "<p>“Memory has been lost to the hyperconnective illusion of an open access world of the availability, accessibility, and reproduceability of the past. I say ‘illusion’ as our submergence in post-scarcity culture has also elided what is really at stake here: the loss of the security of vision that the past once afforded (a clear sense of the why of the difference) and a slippage of the grasp of what effect all our current entanglements with media will have on remembering and forgetting. The past has been stripped of its once retrospective coherence and stability, entangled in today’s melee of uncertainty” (\n##UREF##30##Hoskins, 2018b##, 5–6).</p>", "<p>The emphasis on coherence and stability as central features of collective memory is evident in the factors that Hoskins analyzes as indicators of the change between the previous era and the present: the ontology of the archive —which is no longer static but dynamic; technological agency —now dominant; the inversion of the dialectic between memory and forgetting —previously forgetting was the norm, while now it is the exception; the transformation from an organic community into ‘the multitude’. From these changes arises the obsolescence of the concept of collective memory, which essentially designates a coherent, stable, and bounded memory and, therefore, would no longer be useful for describing digital memory.</p>", "<p>Overall, I agree with the changes that Hoskins highlights as central to understanding the current state of collective memory under the impact of the digital. Factors like the new ontology of the archive are indeed central to the debate on digital memory and have been extensively theorized in fundamental works by scholars like Wolfgang Ernst (\n##UREF##20##Ernst, 2013##), Røssaak (\n##UREF##44##Røssaak, 2010##), Blom (\n##UREF##4##Blom\net al., 2017##) or Brügger (\n##UREF##8##Brügger, 2018##). However, I argue that these changes do not represent a new ontology or the end of collective memory but rather that this new phase constitutes an update and materialization of the characteristics by which we have always defined it.</p>", "<p>My position is that the primary feature of collective memory does not lie in its stability but in its dynamism. This does not imply denying continuity or coherence —always provisional but effective— of collective representations of the past but rather highlighting that dynamism or variability is what underlies reflections on collective memory. Maurice Halbwachs’ theory of the “social frameworks of memory” is, in fact, a response to the variability that Henri Bergson observed regarding individual memory. Halbwachs discusses Bergson —as I will develop in the following section— with a theory that explains this dynamism based on changing social frameworks. At its core, change and not permanence is the defining feature of memory. Hence my argument that collective memory —understood as essentially dynamic— does not come to an end with the ‘digital turn’ but, on the contrary, is enhanced.</p>", "<p>Starting from the essential dynamism of collective memory, in what follows I will examine digital memory in relation to three topics.</p>", "<p>First, I will focus on the definition of collective memory to demonstrate how the digital realm allows us to rethink the social nature of memory through a different concept of the social.</p>", "<p>Maurice Halbwachs laid the foundation in the field of memory studies by theorizing the existence of a collective memory —shared representations of a common past— that guides our understanding of the present. However, despite the consensus that Halbwachs occupies a foundational place, the concept of collective memory inherited from Halbwachs has been criticized for its imprecision, stemming from its disembodied, homogeneous, and abstract conceptualization of the social. The digital era represents an exceptional opportunity to rethink the conception of the social inherited from Halbwachs because it reveals that an organic, homogeneous, and stable community associated with a specific territory and defined in generational terms does not adequately describe ‘the social’ as it manifests today. Does this mean that we need to abandon the term collective memory? No. The term collective memory defines the social, cultural and political dimension of shared memory that goes beyond the individual. I do not see any reason why collective memory should be a concept limited to the era of mass media, rather than referring broadly to this shared dimension. What is necessary is to review whether the way Halbwachs defined the social is still valid. It is not just a matter of stating that this concept of the social was valid when Halbwachs theorized and is no longer valid today, but of exploring what alternative conceptualizations of the social existed at that time and which Halbwachs did not choose. We know that Halbwachs’ theory is indebted to Emile Durkheim in his view of the social. However, alongside Durkheim, other sociologists proposed a different way of conceiving the social. Among them, Gabriel Tarde stands out, who has been rediscovered in recent years, and who Bruno Latour has called a precursor of digital theory. By contrasting Halbwachs’ notion of the social, which forms the basis of memory studies, with the alternative proposal of Gabriel Tarde, I argue that the latter enables us to refine the concept of the ‘collective’ that we have inherited from the founding figure of memory studies.</p>", "<p>The second topic that I address is the new ontology of the digital archive. Following the proposed hypothesis, my argument is that the new dynamism of the archive does not threaten the stability or coherence of collective memory. On the contrary, the new ontology of ‘the archive in motion’ (Røssaak) represents the utopia of an archive that remains and changes simultaneously. An archive that, instead of remaining static —outside of time— inscribes the drift of time on its very surface. In this sense, rather than representing a threat, the digital archive is the realization of the desire to possess a device capable of recording the variability of memory and evolving alongside it.</p>", "<p>Finally, I address the dialectic between memory and forgetting. One of the key points of debate in the field of digital memory is the new balance —or lack thereof— in the relationship between memory and forgetting. Theses regarding the preeminence of one over the other are contradictory, and in many cases, apocalyptic. On one hand, it is argued that in the digital age, forgetting does not exist, and we are constantly threatened by the possibility of the involuntary reemergence of that which we would prefer to remain in the past. Regulations such as the ‘Right to be forgotten’ (referred to in the third section) strive to guarantee that right, attempting to correct the ‘excess’ of memory fostered by digital media. On the other hand, it is pointed out that digital memory, far from being the utopia of perfect memory, is fragile, vulnerable, inconsistent, ephemeral. In this sense, the digital represents a radical threat to the very possibility of remembering. If the supports that house traces of the past disappear, if all that is solid melts into air—or rather dissolves in the broken links of web pages and obsolete devices—how can memory be ensured? I argue that, far from apocalyptic visions, beyond the truth value they hold, digital memory invites us to rethink new, dynamic forms of memory and forgetting.</p>", "<p>Just as memory has traversed different phases following technological changes, ranging from orality, the emergence of writing, the era of mass media, to the digital age, accompanied by the various metaphors that define it, forgetting can also be thought of as variable and dependent on the evolution of media. In this sense, the suggestion by Hoskins to reconsider forgetting is undoubtedly indispensable. How can we conceive a notion of memory and a notion of forgetting suitable for the digital era? In this section, I draw on the reflections of Esposito and Bouchardon, who suggest particular forms of memory and forgetting in the digital age.</p>", "<p>When I state that digital memory is not new, I do not mean to say that memory has not undergone changes under the impact of the digital realm—which would be absurd to claim. On the contrary, the changes are profound and they affect both the actors involved, as well as the objects and practices. What I propose is that these changes, instead of constituting a new memory, actually materialize or implement the essential characteristics that define collective memory. What digital memory offers us is a memory that can now align more perfectly with its own definition, bridging the gaps between theory and practice. It also allows us to reconsider problematic aspects in theory under a new light.</p>" ]

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[ "<title>Conclusion</title>", "<p>The digital revolution has had a significant impact on collective memory, resulting in notable changes. Archives, which were once static repositories, have undergone a transformation and now function as dynamic entities. This shift challenges traditional storage methods and promotes new archiving practices that aim to make information more accessible to everyone.</p>", "<p>The digital era has prompted a reversal in the relationship between memory and forgetting. What used to be a demanding endeavor requiring significant resources, as memory was a scarce commodity, now appears as an activity we do not even have to worry about. Our visits, purchases, the number of pages we have read in a book, the photos we take on vacation —everything is automatically recorded and preserved. At the same time, the networked operation of our devices can potentially make this data available beyond our intentions and into an unpredictable future.</p>", "<p>Now, forgetting is what appears to be a valuable asset to preserve, something that attempts to be regulated through laws like the ‘Right to be Forgotten’. As\n##UREF##37##Makhortykh (2021)## suggests, it is interesting to consider whether the ‘Right to be Forgotten’ could also apply collectively and what the consequences would be. What happens if a perpetrator or a victim requests the deletion of information related to their involvement in atrocities? What would occur if a state implicated in crimes against humanity claimed this right?</p>", "<p>The flip side of this process is that, despite the apparent persistence of data on the web, the internet is not an ideal mnemonic medium. On the contrary, the fragility of data, caused by factors such as the rapid obsolescence of storage devices, led early critics of the internet in the mid-90s to warn that we might be living in a ‘dark age’ where records will not be preserved. Various responses have been proposed to address the difficulty of preserving data in the digital age, or more specifically, digitally-born content, where content is not an object but fundamentally a process that can only exists through actualization, with interactivity being an essential part of it.</p>", "<p>Among these responses, the most interesting one is the intertwining of preservation with reinvention. The metaphor of the ‘Sappho syndrome’\n^{\n<xref rid=\"FN5\" ref-type=\"other\">5</xref>\n} is valid in envisioning a memory appropriate to the digital era. If works disappear to the extent that all that remains are fragments and descriptions made by others, comments on comments, all that is left is to reinvent them. But isn't that what collective memory has always done? Reinventing the past based on its remnants and the narratives others have constructed about it.</p>", "<p>Digital media has introduced powerful tools that allow us to observe interactions between actors —both human and non-human— with unprecedented precision and abundance of data. What are the consequences of this for the way we conceive of the social? How can a mode of conceiving the social enable us to capture social memory more faithfully? Tarde's proposal, which considers the articulation between the individual and the collective, the micro and the macro, without favoring one at the expense of the other, can provide us with an appropriate conception of the social for thinking about it in the digital age, as well as refining the shortcomings inherited from Halbwachs.</p>", "<p>In sum, the extent of the changes introduced by the digital turn is undeniable. However, does it make sense to categorize digital memory as a completely new and radically different form of memory? Does the digital render the concept of collective memory obsolete? As I hope to have showed, collective memory, conceived as a process that is mediated and remediated by multiple media, with the participation of dynamic communities that perform rather than represent the past, is still valid. Furthermore, not only is it valid, but the digital realm materializes its processual nature —which is inscribed in the very archive— brings to the forefront the role of technological mediation and make visible the associations that produce ‘the collective’ without representing collective memory as a disembodied entity that transcend the individuals who comprise it.</p>" ]

[ "<p>No competing interests were disclosed.</p>", "<p>This article explores the configuration of collective memory under the impact of the digital turn. In recent debates, there has been a marked tendency to interpret ‘digital memory’ as a new type of memory, which is radically different from the traditional conceptualization. Even leading authors in the field claim that the digital revolution implies the end of collective memory. However, I argue that despite the transformations that memory undergoes in the digital age, these changes do not imply a new ontology of memory but rather a materialization of the theoretical claims made by Memory Studies since the field's inception.</p>", "<p>To support this hypothesis, I analyze digital memory in relation to three topics: first, I focus on the problematic definition of collective memory to demonstrate how the digital realm allows us to rethink the social nature of memory through a different concept of the social. By contrasting Halbwachs' notion of the social, which forms the basis of memory studies, with the alternative proposal of Gabriel Tarde, I argue that the latter enables us to refine the concept of the ‘collective’ that we have inherited from the founding figure of memory studies. Second, I delve into the new ontology of the digital archive showing how it materializes one of the defining features of collective memory: its mobile, dynamic, and procedural nature. Lastly, I address the inversion of the dialectic between memory and forgetting to highlight the specificity of these practices in the digital environment. I demonstrate how these changes effectively implement, surpassing older technologies, the concept of collective memory as a distributed and dynamic technological process that shapes our shared representations of the past.</p>", "<title>Plain language summary</title>", "<p>In everyday language, memory is conceived of as an individual faculty; however, remembering is an eminently social act. Societies elaborate representations of the past that are fundamental to the constitution of their identity. Following this, the field of memory studies has explored the way collective memory, the relationship that a social group establishes with 'its' past, is socially constructed and circulates, as well as its multiple effects on our experience of the present. One of the fundamental insights of memory studies is that memory is always mediated. Different media shape memory in different ways, providing memory with specific affordances and constraints. If the advent of writing in early civilisation radically altered the constitutions of memory cultures, and the invention of print marked a new shift putting memory in circulation in an unprecedented way, the onset and spread of digital media signals the more recent revolution in collective memory and mnemonic communities. This article explores the configuration of collective memory under the impact of the digital turn. Recent debates suggest that ‘digital memory’ is a completely new form of memory that is fundamentally different from traditional memory. I argue that while memory has certainly transformed in the digital age, these changes don’t create a new type of memory but materialize the theoretical claims made by memory studies from its inception. To support this argument, I examine digital memory in three key aspects. First, I discuss the definition of collective memory and how the digital realm allows us to reconsider the social nature of memory. Second, I explore the new nature of digital archives and how they embody the essential characteristics of collective memory: mobility, dynamism, and procedural nature. Lastly, I address the inversion of the dialectic between memory and forgetting to highlight the specificity of these practices in the digital environment.</p>", "<title>Amendments from Version 1</title>", "<p>Following the reviewers’ suggestions, I have made the following changes: I have rewritten the first section, presenting the main points of Andrew Hoskins' argumentation, who posits that digital memory implies the obsolescence of the concept of collective memory. In this section, I outline the central points of his theory and develop my argument about why digital memory, far from implying the end of collective memory, materializes and puts into practice the characteristics with which we have defined collective memory since the inception of the field.   I have also reworked the 'Rethinking the Social' section. To make this section clearer, especially for a reader not familiar with Tarde, I have rewritten the section into two subsections. In the first, I outline the different notions of Durkheim (the foundational basis of Halbwachs) and Tarde on the social. In the second, I specify why Tarde's conception is useful for rethinking “the collective” in the digital age and in particular, its importance for rethinking collective memory.</p>" ]

[ "<title>Rethinking the social</title>", "<p>In 1925, Maurice Halbwachs published\n<italic toggle=\"yes\">Social Frameworks of Memory</italic> (\n<italic toggle=\"yes\">Les cadres sociaux de la mémoire</italic>), a book in which he advanced the concept of ‘collective memory’. While Halbwachs was not the only one to address the social nature of memory at that time —Aby Warburg is often cited as representing an alternative genealogy in the field— the current concept of collective memory derives from his reflections\n^{\n<xref rid=\"FN1\" ref-type=\"other\">1</xref>\n}.</p>", "<p>Halbwachs's exploration of memory integrated perspectives from two influential figures in late nineteenth-century France, philosopher Henri Bergson and sociologist Emile Durkheim (\n##UREF##41##Olick\net al., 2011a##, 30–36). Bergson conducted a radical philosophical analysis of the experience of time, emphasizing the central role of memory within it. He challenged the notion of memory as a passive storage mechanism and instead characterized remembering as an active process. Bergson's exploration of memory brought to Halbwachs's attention the distinction between objective (often transcendental) and subjective perceptions of the past. Against the uniform time of the clock, individual memory was essentially variable and dynamic, vividly capturing short periods while leaving longer periods only vaguely outlined.</p>", "<p>Durkheim agreed with Bergson in rejecting transcendentalist explanations of time but, unlike Bergson, Durkheim attributed the variability of perceptual categories not to subjective experiences but to differences among various forms of social organization. According to Durkheim, groups share ‘collective representations’ —narratives, symbols, meanings— that constitute the group but not need to be shared by all members of the group. This is the reason Durkheimian approaches has been often criticized “for being radically anti-individualist, conceptualizing society in disembodied terms, as an entity existing in and of itself, over and above the individuals who comprise it” (\n##UREF##41##Olick\net al., 2011a##, 30). Another important characteristic, as Olick, Vinitzky-Seroussi, and Levy highlight, is “the tendency to assume, without justification, that these societies constituted by ‘collective representations’ are homogenous entities. Consequently, a Durkheimian approach to collective memory can lead us to attribute a single collective memory or a set of memories to entire, well-defined societies” (33). As J. Olick, Vinitzky-Seroussi, and Levy summarize, “by connecting cognitive order (time perception) with social order (division of labor), Durkheim thus provided for Halbwachs a sociological framework for studying the variability of memory raised by Bergson” (\n##UREF##41##Olick\net al., 2011a##, 31).</p>", "<p>For Halbwachs, society is composed of different groups —the family, social classes, religious groups. Each of these groups has its own distinctive memories that give the group continuity, coherence, and stability over time. The group’s own memory comprises a shared body of concerns and ideas and is sufficiently general and even impersonal to keep its meaning beyond the variations that may arise from the perspectives of each member.</p>", "<p>Halbwachs says:</p>", "<p>“Each man is immersed successively or simultaneously in several groups. Moreover, each group is confined in space and time. Each has its own original collective memory, keeping alive for a time important remembrances; the smaller the group, the greater the interest members have in these events […] In such milieus all persons think and remember in common. Each has his own perspective, but each is connected so closely to everyone else that, if his remembrances become distorted, he need only place himself in the viewpoint of others to rectify them” (\n##UREF##25##Halbwachs, 1980##, 78)</p>", "<p>Collective memory then appears as an exclusive property of the group —exclusive in the sense that individual differences never seem to threaten the homogeneity of the ‘common to all’.</p>", "<p>It is this conception of collective memory as an abstract, disembodied, and homogeneous entity that has been criticized for not providing an adequate framework to capture the social nature of remembrance. In the current digital environment, this inadequacy becomes even more apparent.</p>", "<p> Following a Durkheimian approach, the term ‘collective’, in the classical theory of Halbwachs, refers to social groups with a dynamic but stable identity. Even if an individual can be part of several groups, they appear as clear-cut formations, unified by a shared identity, a set of shared values, and attached to a territory. Today, on the contrary, the social —as in social media— is the product of a social engineering driven by the entanglement of algorithmic forces, shared digital values, and participatory practices. The meaning of ‘social’ hence encompasses both (human) connectedness and (automated) connectivity, since social media are inevitably automated systems that engineer and manipulate connections, by coding relationships between people, things, and ideas into algorithms (\n##UREF##55##Van Dijck, 2013##, 12). As\n##UREF##34##Latour (2005)## puts it, the social is not a thing or domain, not an explicative category but precisely what needs explaining. This becomes even more acute in the digital era, in which the social is not a fact but a doing, performed through digital connections (\n##UREF##6##Bucher, 2015##). As Blom wonders about the suitability of the concept of collective memory in the digital age,</p>", "<p>“But what if the material frameworks of memory seem to lack the type of stability and durability that confer identity on things? What is a society’s self-image if this image may be the object of instantaneous erasure, dispersal through multiple relays or information overflow, or transmutation through dynamic feedback circuits? What is society if its memory images are perhaps not even representations?” (\n##UREF##3##Blom, 2017##, 14).</p>", "<p>The question is: what conception of the social would allow us to understand the collective nature of memory in the digital era? Gabriel Tarde's theory may provide us with an answer.\n^{\n<xref rid=\"FN2\" ref-type=\"other\">2</xref>\n} It can constitute an alternative for rethinking the social ontology inherited from Durkheim, which forms the basis of Halbwachs’ conception.</p>", "<title>Gabriel Tarde versus Emile Durkheim</title>", "<p>The nineteenth-century sociologist Gabriel Tarde (1843–1904) has experienced a sort of revival in recent years. His theories have been revealed as inspiring or precursors to the theories of Foucault, Deleuze, or Latour, who has described Tarde as “the forgotten grandfather of Actor-Network Theory” (\n##UREF##9##Candea, 2010##, 1).</p>", "<p>During the 1890s, debates surrounding the object, methods, and the very definition of sociology multiplied, and increasingly, these sociological debates came to gravitate around the crucial contrast between Tarde’s and Durkheim’s answers to these questions. The disputes between Tarde and Durkheim —which can be traced in their respective publications, where one comments on and refutes the other – culminate in 1903 in an epic debate between the two thinkers at the École des hautes études sociales (\n##UREF##9##Candea, 2010##, 5). As we know, despite the recognition Tarde enjoyed in his time, it was Durkheim who ultimately took center stage. But, Candea wonders</p>", "<p>“What if Durkheimian sociology had had, from the very beginning, a thoughtful and vocal opponent; one who queried the ‘thingness’ of the social and the holistic, bounded nature of societies and human groups […] one who foregrounded imitations, oppositions and inventions where Durkheim saw conformism to a rule as the key component of the social; one who had already found a way to dissolve the linked contrasts between individual and society, micro and macro, agency and structure, freedom and constraint —Durkheim’s main (and for many, troublesome) legacy to twentieth-century social science?” (\n##UREF##9##Candea, 2010##, 1)</p>", "<p>Durkheim and Tarde differ in the role they assign to individuals in relation to society. While for Durkheim, social facts or collective tendencies impose themselves on individuals, Tarde places the individual at the center, and it is through events of association —and the laws of imitation —that the social is constituted.</p>", "<p>A central point of disagreement between Durkheim and Tarde lies in the definition of the social fact and in the force with which they impose themselves on individuals. According to Durkheim, ‘collective tendencies’ have their own existence and depend on forces external to individuals. Durkheim compares them to cosmic forces, and like them, their existence is demonstrated by the uniformity of their effects.</p>", "<p>For Durkheim, a social fact is identifiable by the power of external coercion it exerts or is capable of exerting on individuals. In fact, a mode of behavior that exists outside the consciousness of individuals only becomes generalized by exerting pressure on them. Therefore, two central features characterize a social fact for Durkheim: the first is that it exists independently of its individual expressions; the second, no less important, is its coercive nature.</p>", "<p>If, for Durkheim, social phenomena are explained through coercion, for Tarde, on the other hand, they find their foundation in imitation. For Tarde, all social facts are the product of individual inventions propagated by imitation. Any belief and any practice would have at its origin an original idea born from an individual brain. Whether they are considered useful or because their author is invested with authority, other members of society adopt them, and once generalized, the invention ceases to be an individual phenomenon and becomes a collective phenomenon. Imitation is, therefore, the key. Emerging from the individual and through the laws of propagation, the fact becomes social.</p>", "<p>Durkheim criticizes the idea of imitation, arguing that a movement repeated by all individuals does not constitute a social fact for this reason. “What constitutes a social fact”, —he answers— “is a belief, tendency, or practice of the group taken collectively, which is something else entirely than the form it may assume when it is refracted through individuals” (\n##UREF##16##Durkheim, 1894##, 54).</p>", "<p>Then, there is a fundamental disagreement in the relation between the whole and the parts. For Durkheim, the whole is always more and essentially different from the form it acquires in the refractions of individuals. On the contrary, for Tarde, “how could it be refracted before existing, and how could it exist, let us speak intelligibly, outside of all individuals?” It is Tarde himself who presents his theory in the following passage, and it is why it is convenient to quote him extensively:</p>", "<p>“The truth is that a social thing, whatever it might be [...] devolves and passes on, not from the social group collectively to the individual, but rather from one individual [...] to another individual, and that, in the passage of one mind into another mind, it is refracted. The sum of these refractions, from the initial impulse of an inventor, a discoverer, an innovator or modifier, whoever it might be, unknown or illustrious, is the entire reality of a social thing at a given moment; a reality which is constantly changing, just like any other reality, through imperceptible nuances; this does not prevent a collectivity from emerging out of these individual varieties, an almost unchanging [\n<italic toggle=\"yes\">constante</italic>] collectivity, which immediately strikes the eye and gives rise to Mr Durkheim’s ontological illusion” (\n##UREF##50##Tarde, 1895##, 66–67).</p>", "<p>In summary, Tarde does not deny the existence of a collectivity, but he does not place that collectivity above and beyond individual expressions. The collective emerges from a generalization of individual varieties, and while it may acquire consistency and stability, this stability is provisional —even if it can produce the ontological illusion that Tarde accuses Durkheim of. Tarde’s view is, unlike Durkheim’s, essentially dynamic. He describes the social world as the result of events of association in which contingency plays a significant role. Thus, while Durkheim is interested in studying ‘social facts as things’, as entities that exist beyond individuals, Tarde chooses to focus on ‘things as social facts’, which implies assigning a social character to all entities. “This relocation”, assert Harvey and Venkatesan “has radical potential in relation to established, positivist social science, for it foregrounds the processual and captures the importance of relational dynamics of becoming, the open-endedness of all things, the potential of all things to transform through their inherent situated relationality” (\n##UREF##26##Harvey &amp; Venkatesan, 2010##, 129–30).</p>", "<p>Tarde’s view is a conception of the social that is not limited to human relationships. Not only are human beings social, but social behavior is inherent in all phenomena in the universe, from atoms, nations, cells, and bacteria to plants and animals. As Tarde remarks, “every thing is a society and every phenomenon a social fact” (\n##UREF##50##Tarde, 1999##, 58). Society is thus not an exclusive trait of humans but extends far beyond to encompass all kind of phenomena.</p>", "<p>Why does Tarde’s conception of the social become relevant in the digital age? And what are the implications of this conception for collective memory?</p>", "<p>Tarde’s conception of the social becomes relevant in the digital age at least for three reasons:</p>", "<p>First, Tarde’s idea that the social is constituted through relationality resonates in the digital culture, where connectivity is a defining feature. In the digital era, online interactions, social networks, and constant connectivity among individuals reflect the notion of social entities that Tarde proposed.</p>", "<p>Second, Tarde’s vision, which does not limit the social to human relationships, is particularly relevant in the digital era, where technology plays a central role. Relationality extends to algorithmic connections and the interaction between humans and technology. This provides a broader and more comprehensive perspective on sociability in the digital context.</p>", "<p>Third, Tarde’s thesis of imitation serves as a model for understanding the spread of ideas, behaviors, and emotions in digital networks. In an environment where information spreads rapidly through online platforms, Tarde’s notion of imitation provides a lens for analyzing how trends disseminate in the digital space. A book like\n<italic toggle=\"yes\">Virality: Contagion Theory in the Age of Networks\n##UREF##47##(2012)##\n</italic> by Tony Sampson, which draws on the social epidemiology developed by Tarde, serves as an example.</p>", "<p>The rediscovery of Gabriel Tarde in recent decades is framed within the preeminence that several concepts have gained in the social sciences and humanities. These concepts emphasize that social phenomena are complex, procedural, indeterminate, relational, and constantly open to the effects of contiguous processes. This epistemological shift shares a common ontology that erases the differences between the social and the natural, the mind and the body, the cognitive and the affective. It is grounded in concepts such as assemblage, flow, emergence, becoming, relationality, the machinic, the inventive, the event, the virtual, and the informational (\n##UREF##2##Blackman &amp; Venn, 2010##).</p>", "<p>Precisely in the digital environment, the distinction between the human component and the technological component dissolves, and what emerges, instead, is a “human-data assemblage” (\n##UREF##36##Lupton, 2016##). By not restricting the social to human actors, Tarde’s theory allows us to explore collective memory as a process encompassing both (human) connectedness and (automated) connectivity. The role of algorithms in encoding the relationships between people, things, and ideas is essential for understanding how shared representations of the past are formed and transmitted in the contemporary landscape. Algorithms play a crucial role in shaping the way information is curated, filtered, and presented to individuals within digital platforms and networks. They influence what content is prioritized, recommended, and made visible to users, thereby shaping their understanding and perception of historical events, cultural narratives, and collective memory. Understanding the role of algorithms in the encoding of relationships and the formation of shared representations of the past is crucial for critically analyzing and navigating the contemporary digital landscape. It prompts us to consider the potential biases, limitations, and power dynamics inherent in algorithmic systems and their impact on collective memory. Based on a conception of collective memory indebted to Durkheim’s paradigm, this exploration is simply impossible.</p>", "<p>There is a fundamental difference between Durkheim and Tarde that revolves around attention to detail, as well as the significance given to contingency. As Harvey and Venkatesan argue, for Durkheim, “the details of specific interactions were necessarily subordinated to the ‘bigger’ picture. Contingency, error, uncertainty were simply not relevant in his view of the social, where the whole was necessarily more than the sum of its parts and thus beyond the detail. But for Tarde, as for many contemporary ethnographers, the detail is not approached as less than the whole, for it is through attention to the detail that we can find different kinds of collectivity in formation” (\n##UREF##26##Harvey &amp; Venkatesan, 2010##, 130). In the writings of Halbwachs, groups appeared as organic entities in which the individual was subsumed. Even though an individual could be part of various groups and move between them, this mobility did not imply any contradiction or friction. Upon ‘entering’ a group, the individual adopted the collective point of view, and any individual variation became subsumed in the whole. What we call detail, therefore, is nothing less than the specificity of individual variations in the interactions that shape a community.</p>", "<p>This is precisely what digital technologies design and make visible: the associations that produce “the collective” without subsuming them under a ‘structure’ a ‘social fact’, or a ‘collective representation’. Following Tarde’s proposal makes it possible to overcome the abstract and disembodied nature that the notion of collective memory had in Halbwachs. Viewed from a Tardean framework, the collective nature of memory is not a representation of the group that imposes itself from the outside on individuals. Instead, it is the result —always provisional and contingent, in other words, dynamic—of individual narratives that connect and integrate within a broader context that gives them meaning.</p>", "<p>If, as Tarde proposes, the central mechanism that underlies sociality is imitation, it is through the propagation of ideas, behaviors, and images that the social is constituted. It is then —in this generalization— that we can speak of collectivity. The social, or collective, character of memory does not necessitate the consistency, stability, and homogeneity proclaimed by Halbwachs. On the contrary, contingency and variability are essential traits. Contingency and variability are essential to explain the dynamic nature of collective memory. The dynamic character is not supported, as Halbwachs argued, in the different social frameworks, in the variety of the groups we are part of, or in the evolution of the groups we are part of. The dynamic character is based on continuous processes of imitation, on the propagation of representations of the past that give rise to a provisional aggregate —a collective— that then dissolves again, fades, decreases in intensity, or stops. It is the conception of the social that harbors in Tarde a provisional, contingent, malleable character —not a fact, a structure, a law, but a becoming. A conception of the social that, shaped by human and non-human actors, a ‘human-data assemblage’, allows us to rethink the collective nature of memory in the digital age.</p>", "<title>The archive in motion</title>", "<p>In this section I explore the emerging ontology of the digital archive. In line with the proposed hypothesis, I contend that the newfound dynamism within the archive does not pose a threat to the stability or coherence of collective memory. On the contrary, the novel ontology of the archive embodies the ideal of an archive that both endures and transforms concurrently.</p>", "<p>In theorizations of memory, the archive has arguably been the most powerful and enduring metaphor. Well-known analogies of memory, such as Plato’s wax seal or table, the loci of classical art or Freud’s mystic writing-pad, rely on the “imprints-on-a-substrate paradigm”, an imagery that conceives memory as a content preserved in a specific place —whether it's the brain, the library, or the screen (\n##UREF##7##Burton, 2008##, 322). The trait that defines the archive in these metaphors is stability. The archive extracts something from the flow of time and safeguard it from decay, preserving it 'intact' for the future.</p>", "<p>Clear examples of this can be found in well-known theories on collective memory, such as the opposition between ‘canon’ and ‘archive’ proposed by Aleida Assmann. For Assmann, “the institution of the archive is part of cultural memory in the passive dimension of preservation” (\n##UREF##0##Assmann, 2008##, 103). The knowledge saved in the archive is inert, “it is stored and potentially available, but it is not interpreted” (103). While the archive refers to the passive dimension of cultural memory, the canon “stand for the active working of memory of a society that defines and supports the cultural identity of a group” (\n##UREF##0##Assmann, 2008##, 106). In a similar vein, Diana Taylor opposes two epistemological systems, coded in the dichotomy ‘archive/repertoire’. ‘Archival’ memory exists as documents, maps, literary texts, letters, archaeological remains, bones, videos, films, and all those items supposedly resistant to change. The repertoire, on the other hand, enacts embodied memory and encompasses mnemonic practices such as performances, gestures, orality, movement, dance, or singing (\n##UREF##52##Taylor, 2003##).</p>", "<p>In the digital age the static archive is replaced by an “archive in motion” (\n##UREF##44##Røssaak, 2010##). The archive, conceived as the guarantor of the stability of collective memory, undergoes an ontological change characterized by dynamism and mobility. This dynamism is evident in different domains and levels, involving diverse topics such as techno-mathematical operations governing data transfer, access to previously unavailable materials, and the emergence of new archival practices carried out by amateur communities.</p>", "<p>The complexity of the debate surrounding the relationship between the archive and digital technologies lies in the fact that, as Taylor argues, archive simultaneously refers to a place, an object, and a practice.</p>", "<p>“An archive is simultaneously an authorized place (the physical or digital site housing collections), a thing/object (or collection of things —the historical records and unique or representative objects marked for inclusion), and a practice (the logic of selection, organization, access, and preservation over time that deems certain objects “archivable”). Place, thing, and practice function in a mutually sustaining way” (\n##UREF##53##Taylor, 2012##).</p>", "<p>The changes experienced by the archive in the digital era are observed in the three domains.</p>", "<p>Firstly, the archive as thing/object. What is stored in an archive can be any object, ranging from the text of a law, a literary work, a painting, the recording of a musical score, to an Aztec’s mask. All these tangible and material things are transformed into the logic of zeroes and ones. Although digital technologies may speak the language of storage and containment, as Blom argues, in digital media “nothing is stored but code: the mere potential for generating an image of a certain material composite again and again by means of numerical constellations (\n##UREF##3##Blom, 2017##, 12). Blom emphasizes mobility, asserting that “once the archive is based on networked data circulation, its emphatic form dissolves into the coding and protocol layer, into electronic circuits or data flow” (13).</p>", "<p>Objects in the archive have, of course, always circulated. In fact, the purpose of an archive is not only to preserve documents but also to make them available in a potential future. The difference is that preservation now relies on inherent dynamism, as the conservation of the object in the logic of zeros and bytes involves its translation into protocols and codes that are then updated again when we request the image or text to read it through the interface. The dynamism of digital objects lies in these algorithmic operations formed by constant regenerations, updates and affordances. Archives are “dynamic living systems, constantly transformed and updated” (\n##UREF##45##Røssaak, 2017##, 202–4). Or as Blom puts it: “The conflation of memory with storage is, in other words, undermined by a technical emphasis on dynamic processes of memorizing. To the extent that computer memory exists, it is essentially activity; virtual as well as actual, and its images are electronic events (\n##UREF##3##Blom, 2017##, 12).</p>", "<p>Secondly, the archive as place. The spatial arrangement of an archive within a public building has been replaced by the ‘placelessness’ of documents distributed along multiple sites. Digital memories are often perceived as 'placeless', meaning that one notable change brought about by digital media is the diminished connection to a specific location. Under this perspective, memories in the pre-digital era were strongly tied to physical spaces, which played a crucial role in their interpretation. However, in the present era, this connection fades away, leading to memories that are fluid, detached, fragmented, and omnipresent, constantly shifting without precise spatial references. The loss of a meaningful bond with place connotes, in this imaginary, a loss of meaning, given the centrality that the category of place plays in the shaping, preservation, and transmission of collective memory. The idea of placelessness is linked to the changing traits of the archive and the database as the privileged cultural forms of the digital era. The notion of the traditional archive as static, selective, organised, and restricted to a particular place, has been replaced by a dynamic, non-selective, and multimedia online archive, whose logic is fluidity and ubiquity, and which is always ‘in becoming’ (\n##UREF##38##Mandolessi, 2021##)</p>", "<p>However, this imaginary has been questioned by highlighting how digital memory practices actively engage in place-making, rather than succumbing to the logic of hyper-connectivity and all the attributes upon which it is predicated (\n##UREF##38##Mandolessi, 2021##). Rather than deterritorizaling memories, digital archives become key tools for new engagements with place (for different approaches on the issue of digital memory and place see the Special Issue “Locating “Placeless” Memories: The Role of Place in Digital Constructions of Memory and Identity”, edited by Huw Halstead, Memory Studies 14.3 (2021).</p>", "<p>Thirdly, the archive as a practice. The practice of archiving involves the selection and organization of objects that will be stored in the archive. The authority of experts is crucial, as the decision to archive certain documents and discard others grants them a legitimization that they do not possess prior to their entry into the archive. In this sense, archivization is performative, its operations of selection and organization, as Derrida claims “produces as much as it records the event” (\n##UREF##15##Derrida, 1996##, 17). The importance of selection is emphasized by Ketelaar, who coined the term ‘archivalization’ to denote a phase prior to the archiving process, meaning “the conscious or unconscious choice (determined by social and cultural factors) to consider something worth archiving. Archivalization precedes archiving” (\n##UREF##33##Ketelaar, 2001##, 133)</p>", "<p>The digital has radically altered the phase of ‘archivalization’ in at least two senses. Digital media are inherently archival, meaning that even if we do not consciously decide to archive specific content, it will be automatically stored. For example, our tweets or posts on social media platforms like Twitter or Instagram, the locations we visit through geo-tagging, or our selections on online shopping platforms like Amazon. There are no experts behind the scenes determining which content is worthy of archiving; instead, algorithms are designed to collect and process data. This automation is closely tied to the idea of the Internet as a ‘perfect memory machine’ capable of recording and preserving everything. The persistence of data challenges our right or desire to forget, to leave certain things in the past (which I will address in the next section). The lack of selection in automated storage seems to complicate matters rather than offering a solution to the scarcity of memory.</p>", "<p>However, changes in archival practices not only involve algorithms recording information with or without our consent but also encompass alternative selection criteria that go beyond traditional notions of expertise. In the digital era, archives can be constructed by anyone who believes that certain materials are worth archiving.</p>", "<p>In her book\n<italic toggle=\"yes\">Rogue Archives: Digital Cultural Memory and Media Fandom</italic> (\n##UREF##14##2016##), De Kosnik extensively explores the work of technovolunteer archivists. These ‘rogue archives’ are established and maintained by fans with the aim of preserving and sharing cultural artifacts that are often neglected, banned, or considered outside the mainstream cultural canon. The efforts of these technovolunteers play a crucial role in curating and safeguarding the cultural memory of marginalized groups, rather than relying solely on the assumed archival nature of the internet. These archives possess a democratizing potential that challenges the criteria upheld by official institutions regarding what should be preserved or not. This expansion of value in determining preservation challenges the established norms and opens up possibilities for a more inclusive approach to archiving. Moreover, rogue archives not only enable the preservation of materials that would otherwise be lost but also fundamentally transform the mnemonic function of the archive. According to De Kosnik, the archive, which was traditionally seen as a record of cultural production, now becomes a source of cultural production itself. She introduces the term “rogue memory” to signify this shift from preservation to creativity.</p>", "<p>Returning to the question posed in the introduction: In what way does the new ontology of the archive materialize the concept of collective memory?</p>", "<p>Collective memory is defined as a practice, an active process that interprets the past in light of the present. However, the metaphor of the archive pointed to a static content, those objects, traces, and remnants of the past that could potentially be activated but also remain inert. This potentiality gave the archive a liminal or ambiguous status. If we define memory as an active process, in what sense can we speak of the archive as a ‘passive dimension of memory’? The new ontology of the digital archive, an archive in motion, brings together potentiality and actualization. In the digital era, practices are inscribed in the archive, which no longer functions as a mere passive container or collection of traces of the past but as a dynamic entity. The archive becomes the site in which the traces of the past are continuously interpreted, updated, and collectively reappropriated in an ongoing process.</p>", "<title>The inversion of the dialectics between memory and forgetting</title>", "<p>In principle, collective memory refers to any shared representation of the past by a social group. However, the meaning of memory that dominates the field of memory studies has a more restricted character. It is a memory of violence, atrocity, and human right abuses (\n##UREF##32##Huyssen, 2011##;\n##UREF##35##Levy &amp; Sznaider, 2006##). This focus on memory centered on violence is deeply linked to the emergence and globalization of a powerful discourse on human rights, which is materialized in the adoption of the Universal Declaration of Human Rights in 1948. In this foundational text, memory holds a central place as an instrument to promote the defense of human rights, a role summarized in the ‘duty to remember.’ There are two underlying assumptions regarding the ethical and moral responsibility to remember that form the core of the relationship between memory and human rights: The first assumption is that acknowledging human rights abuses and honoring victims through memory represents the ethically correct and necessary response to violence. The second assumption further intertwines memory with human rights: the remembrance of past violence is considered one of the most effective measures to prevent future violence (\n##UREF##49##Sodaro, 2018##). The link between memory and human rights explains the positive role assigned to the act of remembering and the corresponding stigmatization of forgetting. This also helps to explain why, in comparison to the extensive research on the various ways in which societies remember, there is limited focus on how societies forget. In general terms, forgetting has been viewed as something to be combated rather than as an object that requires scrutiny. One notable exception is Paul Connerton's\n<italic toggle=\"yes\">Seven Types of Forgetting</italic> (\n##UREF##11##2008##) and\n<italic toggle=\"yes\">How Societies Forget</italic> (\n##UREF##12##2009##). Connerton does not stigmatize forgetting. On the contrary, he demonstrates the variety and complexity of the mechanisms involved in the act of forgetting and how forgetting can be necessary, for various reasons and in different contexts, for political and social life.</p>", "<p>This lack of interest in forgetting as an object of study —its forms, its actors, its means, and its value— is currently being reversed, becoming central in the debate on digital memory. The value attributed to forgetting is also being inverted. Beyond memory associated with human rights, where the ‘duty to remember’ prevails, the question about the function and benefits of forgetting is being brought back into the spotlight. In the digital age, the dialectic between memory and forgetting has been reversed As Mayer-Schönberger claims “Quite obviously, remembering has become the norm, and forgetting the exception”\n^{\n<xref rid=\"FN3\" ref-type=\"other\">3</xref>\n} (\n##UREF##39##Mayer-Schönberger, 2009##, 52). We need to forget. To forget what? To forget how? And why? The debate can be structured around these questions.</p>", "<p>In the debate about memory in the digital era, the essential function of forgetting as an individual and social mechanism is affirmed. The importance of forgetting has been even ratified by a legal framework known as ‘The Right to be Forgotten’\n^{\n<xref rid=\"FN4\" ref-type=\"other\">4</xref>\n}. ‘The Right to be Forgotten’ is a legal concept that refers to an individual's right to request the removal or deletion of personal information from online platforms and search engine results. It recognizes the importance of privacy and allows individuals to have control over their personal data and its availability on the internet.</p>", "<p>Despite the existence of a legal framework that guarantees the right to be forgotten, the question of whether it is possible to forget —or be forgotten— in a digital environment remains. Is it true that forgetting becomes impossible in the digital era? Or, on the contrary, is it true that collective memory comes to an end? Perhaps, these bold claims about the impossibility to forget or the impossibility to remember in a digital era stem from a conception of memory and forgetting that remains confined to the framework of an analog or pre-digital culture. It would be more appropriate to examine the specific ways in which we forget and remember in the current media ecology. Following this line of thought, I would like to briefly discuss two examples of alternative forms of (digital) memory and forgetting explored by Elena Esposito and Serge Bouchardon.</p>", "<title>Re-inventing Forgetting</title>", "<p>In ‘Algorithmic Memory and The Right to be Forgotten’ (\n##UREF##22##2022##), Elena Esposito discusses a judgment that the European Court of Justice issued in 2014 in favor of the plaintiff on case C-131/12 about the ‘right to be forgotten’. The European Court held Google accountable for this ‘excess of memory.’ On the other hand, Google claimed that it could not be held responsible, arguing that the processing of data is carried out by the search engine, and the company does not exercise control over that data. However, according to the European Court, although Google may not be directly responsible for data processing, the search engine’s activity makes that data accessible to users. Esposito remarks that this raises the question of what conception of social memory and forgetting is implied in the ruling:</p>", "<p>“Is memory the ability to store information in an archive, even if it is inaccessible? Or does it depend on the ability to find the information when you need it? Is computer memory storage or remembering? Ascribing to Google the management of the right to oblivion implies a clear choice: data are considered forgotten if they are made difficult to find, while social memory should be preserved by the storage of data in the pages of newspapers and in other archives” (\n##UREF##22##Esposito, 2022##, 67–68).</p>", "<p>Esposito acknowledges the difficulty of requesting certain information to be erased, among other reasons, because it draws attention to the deleted content, achieving the opposite effect of what is desired. How, then, to deal with forgetting on the web? Esposito suggests that forgetting should follow the logic of algorithmic memory by implementing a procedure that involves multiplying the information instead of erasing it. Therefore, in order to manage forgetting on the web in a way that aligns with algorithmic memory, one could consider implementing a procedure that goes against the conventional practice of deleting or making content unavailable. This involves employing strategies of obfuscation, which generate misleading, false, or ambiguous data alongside each transaction on the web. In practice, this multiplication of information production aims to impede the meaningful contextualization of data (\n##UREF##22##Esposito, 2022##, 73).</p>", "<p>This proliferation of information renders each individual piece of data more marginal, getting lost within the vast volume. Rather than being erased, information becomes invisible due to the obfuscation caused by the excess of data.</p>", "<p>As Esposito suggests in her conclusion, the issues arising from the digital ecology need to be addressed from a digital perspective, which entails a shift in the reference frame:</p>", "<p>“Algorithms participating in communication can implement, for the first time, the classical insight that it might be possible to reinforce forgetting—not by erasing memories but by multiplying them. This requires a radical change in perspective. It does not solve all the problems of digital memory and of the difficulty in controlling the continuous production of an excess of data, but moves these problems to a different and much more effective level: from the reference frame of individuals to that of communication” (76–77).</p>", "<title>Re-inventing memory</title>", "<p>In the article ‘Preservation of Digital Literature: From Stored Memory to Reinvented Memory’ (\n##UREF##5##2013##) Bouchardon and Bachimont address the issue of preservation in the digital age. This is the opposite problem to the one described above. On the one hand, the persistence and availability of digital data raise the issue of the difficulty of forgetting, exemplified by ‘The Right to be Forgotten’. On the other hand, concerning preservation, the digital era is likely the most fragile and complex context in human history. Compared to a medium like books, which have remained virtually unchanged, allowing us to read works produced hundreds or thousands of years ago, the lifespan of digital works is short. They quickly become obsolete. This is explained by the fact that in the digital medium, content is situated at two different levels.</p>", "<p>Bachimont makes a distinction between the \"inscription form\" and the \"restitution form\" (\n##UREF##1##Bachimont, 2007##). In the context of printed material, both forms are the same, represented by the printed text. However, in the case of digital media, these two forms are separate due to the intervention of computational processes that mediate between them.</p>", "<p>The existence of two forms raises the question of how we define content. Is the content what is stored on the hard drive (the resource), or is it the content that appears on the screen (the rendering)? If we preserve the resource but not the representation, can we still speak of preservation? This problem is evident in the field of digital literature, the topic that Bouchardon and Bachimon address in their article, since a digital literary work is not an object —like a printed book— but fundamentally a process that can only exist through actualization. Bouchardon and Bachimon distinguish four possible strategies for the preservation of digital literature: museology, migration, emulation and description.</p>", "<p>The museological approach consists in “preserving contents as they are as well as the tools permitting playability. This way, it is not only the information which is preserved, but the technological environment characteristic of a certain time and content” (\n##UREF##5##Bouchardon &amp; Bachimont, 2013##, 187). Migration involves “updating the technical format of the contents so that they should remain compatible with and adapted to the reading tools available in the current technological environment” (188). In emulation “the contents are not made to evolve. Instead, the reading tools of the old formats are simulated on current environments” (188). Lastly, description, an approach that is “counter-intuitive but the most potent on a theoretical level” consists in “discarding recorded contents in as far as they are incomplete, partial or ill-defined. Therefore, it is better to preserve a description of the content which permits us to reproduce it. The description may concern the reproduction of key elements, of the authors’ intention, and the variable media approach, of the graphic appearance, etc;” (\n##UREF##5##Bouchardon &amp; Bachimont, 2013##, 189–90).</p>", "<p>The authors compare the description-based preservation method with the preservation model of classical music. How do we know how to play Baroque music today when we do not have recordings of how it was performed in its time? Thanks to the combination of three elements: score, instrument, and instrumental practice. The score serves as a set of instructions for producing music on an instrument. Organology ensures the preservation of instruments and the techniques involved in their creation. Furthermore, through the practice of music, which involves reading scores and playing instruments, knowledge is continuously taught and passed down. Preservation goes hand in hand with constant usage. In this sense, the concept of music serves as a model for preserving a content that cannot be directly recorded. Instead, it relies on saving resources (the score), a player (the instrument), and a practice (music education). This unique musical solution allows for the preservation of a non-variable description of the performance, even in the absence of the original content.</p>", "<p>As Bouchardon and Bachimon point out, this preservation model —which does not preserve the content intact but rather the elements necessary for its reconstruction— is essentially an interpretative act. In this model, preservation is synonymous with (re)invention: “Preserving is keeping intact the interpretability of the work to be able to reinvent it. In other words, preserving is saving the knowledge of its re-invention” (\n##UREF##5##Bouchardon &amp; Bachimont, 2013##, 191).</p>", "<p>This model, which the authors argue is \"from an anthropological point of view\" \"more valuable and more authentic than the model of printed media which is a memory of storage\" (200) effectively embodies one of the central characteristics of collective memory. Like the ontological dynamism of archives, the dialectic between memory and forgetting in the digital age brings us closer to the functioning of collective memory understood as the continuous evolution, change, and adaptation of representations of the past in the present. What collective memory preserves is not an ‘intact’ past. It does not faithfully preserve the facts, characters, or significant events of that past for a community —which does not imply that the representations of the past it preserves are false. Collective memory re-appropriates, adapts, and re-actualizes the facts of the past, along with their affects and values. Re-invention makes room for the creation of new relationships with the past, that is ultimately, the goal of collective memory.</p>" ]

[ "<title>Data availability</title>", "<p>No data associated with this article.</p>" ]

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[ "<fn-group content-type=\"pub-status\"><fn><p>[version 2; peer review: 1 approved, 2 approved with reservations]</p></fn></fn-group>", "<fn-group><fn><p id=\"FN1\">\n¹ In fact, reflection on memory is as old as humanity itself. It not only has a long history predating the 'memory boom,' but often, by crowning Halbwachs as the founder of memory studies, important contemporary figures of Halbwachs are obscured. For a detailed account of the history of the reflections on memory, see Olick, Vinitzkt-Seroussi and Levy's introduction to\n<italic toggle=\"yes\">The Collective Memory Reader</italic> (\n##UREF##42##2011b##). </p></fn><fn><p id=\"FN2\">\n²\n##UREF##3##Blom (2017)## and\n##UREF##45##Røssaak (2017)## have also advocated for the relevance of Tarde’s social theory to examine digital memory. Although not focusing on memory, Sampson’s book\n<italic toggle=\"yes\">Virality: Contagion Theory in the Age of Networks</italic> (\n##UREF##47##2012##) provides an analysis of the spreading of emotions and affects on digital networks using a Tarde inspired imitation thesis.</p></fn><fn><p id=\"FN3\">\n³ According to Mayer-Schönberger, four main technological drivers have facilitated this shift: digitization, cheap storage, easy retrieval, and global reach” (\n##UREF##39##Mayer-Schönberger, 2009##, 52). See also\n<italic toggle=\"yes\">The End of Forgetting: Growing up with Social Media</italic> by\n##UREF##17##Kate Eichhorn (2019)##.</p></fn><fn><p id=\"FN4\">\n⁴ In 2012, the European Commission published a ‘Proposal for a Regulation on the Protection of Individuals with Regard to Processing of Personal Data and on the Free Movement of Such Data’ (COM (2012), which includes the ‘right to be forgotten’ (RtbF), that is, a person’s right to have their personal data deleted when these data were voluntarily (or not) made available on the Internet. The RTBF has been put into practice not only in the European Union but in several jurisdictions, including Argentina and the Philippines. For a discussion about the role and the implications of the RTBF from different perspectives see Ghezzi, Alessia, Ângela Guimarães Pereira, and Lucia Vesnić-Alujević, eds. \n<italic toggle=\"yes\">The Ethics of Memory in a Digital Age: Interrogating the Right to Be Forgotten (\n##UREF##24##2014##)</italic>​.</p></fn><fn><p id=\"FN5\">\n⁵ In\n<italic toggle=\"yes\">Traversals: The Use of Preservation for Early Electronic Writing</italic> (2017) Stuart Moulthrop and Dene Grigar uses the methapor of the “Sappho syndrome” to refer to the difficulty of preserving electronic literature: “We are haunted by a condition we call the Sappho Syndrome: the disappearance of literary works to the extent that all that remains are fragments and references to them by others” (\n##UREF##40##Moulthrop &amp; Grigar, 2017##, 230)</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p id=\"Par2\">The <italic>Domb numbers</italic>\n, defined byfor non-negative integers <italic>n</italic>, first appeared in an extensive study by  Domb [##UREF##3##4##] on interacting particles on crystal lattices. In particular, Domb showed that counts the number of 2<italic>n</italic>-step polygons on the diamond lattice.</p>", "<p id=\"Par3\">The Domb numbers also appear in a variety of other settings, such as in the coefficients in several known series for . For example, from [##UREF##1##2##, Equation (1.3)] we know thatIn [##UREF##9##10##, Theorem 3.1],  Rogers showed the following generating function for the Domb numbers by applying a rather intricate method:where |<italic>u</italic>| is sufficiently small. Mu and Sun [##UREF##8##9##, Equation (1.11)] proved a congruence involving the Domb numbers by applying the telescoping method: For any prime , we have the supercongruencewhere denotes the Fermat quotient .</p>", "<p id=\"Par4\">In [##UREF##4##5##], Liu proved a couple of conjectures of Sun and Sun. In particular he confirmed [##UREF##4##5##, Theorem 1.3] that for any positive integer <italic>n</italic> the two sumsare also positive integers.</p>", "<p id=\"Par5\">Sun [##UREF##20##21##, Conjecture 4.1] conjectured the following congruence for the Domb numbers: Let be a prime. Thenwhere are the Bernoulli numbers given byThis conjecture was confirmed by the first author and  Wang [##UREF##7##8##]. For more research on Domb numbers, we kindly refer the readers to [##UREF##4##5##, ##UREF##6##7##, ##UREF##13##14##, ##UREF##18##19##, ##UREF##21##22##] (and the references therein).</p>", "<p id=\"Par6\">The main result of this paper is Theorem <xref ref-type=\"sec\" rid=\"FPar1\">1.1</xref> which contains two supercongruences that were originally conjectured by Sun in [##UREF##12##13##, Conjecture 3.5, Conjecture 3.6]. What makes them interesting is that their formulations involve the binary quadratic form for primes <italic>p</italic> that are congruent to 1 modulo 3. (It is well-known that any prime can be expressed as for some integers <italic>x</italic> and <italic>y</italic>, an assertion first made by Fermat and subsequently proved by Euler, see [##UREF##2##3##]. In his paper [##UREF##12##13##], Sun stated further conjectures of similar type, involving different moduli, and other binary quadratic forms.) First, Sun definedThe two supercongruences which we will confirm are as follows.</p>", "<title>Theorem 1.1</title>", "<p id=\"Par7\">Let be a prime. Then</p>", "<p>Our preparations for the proof of this theorem consist of seven lemmas that we give in Sect. <xref rid=\"Sec2\" ref-type=\"sec\">2</xref>. These are used in Sect. <xref rid=\"Sec3\" ref-type=\"sec\">3</xref>, devoted to the actual proof of Theorem <xref ref-type=\"sec\" rid=\"FPar1\">1.1</xref>. As tools for establishing the results in Sects. <xref rid=\"Sec2\" ref-type=\"sec\">2</xref> and <xref rid=\"Sec3\" ref-type=\"sec\">3</xref> we utilize some congruences from [##UREF##5##6##, ##UREF##6##7##] and several combinatorial identities that can be found and proved by the package Sigma [##UREF##10##11##] via the software Mathematica.</p>" ]

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[ "<p id=\"Par1\">In this paper, we prove two recently conjectured supercongruences (modulo , where <italic>p</italic> is any prime greater than 3) of Zhi-Hong Sun on truncated sums involving the Domb numbers. Our proofs involve a number of ingredients such as congruences involving specialized Bernoulli polynomials, harmonic numbers, binomial coefficients, and hypergeometric summations and transformations.</p>", "<title>Keywords</title>", "<title>Mathematics Subject Classification</title>", "<p>Open access funding provided by Austrian Science Fund (FWF).</p>" ]

[ "<title>Preliminary lemmas</title>", "<p id=\"Par9\">Recall that the Bernoulli polynomials are given bywhere, as before, are the Bernoulli numbers. We will also use the classical Legendre symbol (for integer <italic>a</italic> and odd prime <italic>q</italic>). The following lemma involving the (generalized) harmonic numbers can be easily deduced from [##UREF##15##16##, Theorem 5.2 (c)], [##UREF##16##17##, Theorem 3.9 (ii), (iii), (iv)], [##UREF##16##17##, third equation on p. 302], and the simple identity</p>", "<title>Lemma 2.1</title>", "<p id=\"Par10\">Let be a prime. Then</p>", "<p>The above lemma can be compared to results established in [##UREF##17##18##], which contains similar congruences involving the Bernoulli polynomials but, rather than for the harmonic numbers , for the special numbers , which in [##UREF##17##18##] were recursively defined by</p>", "<title>Lemma 2.2</title>", "<p id=\"Par12\">([##UREF##6##7##, Lemma 2.2]) Let be a prime. If , then we have</p>", "<title>Lemma 2.3</title>", "<p id=\"Par13\">([##UREF##6##7##, Lemma 2.3]) Let be a prime. For any <italic>p</italic>-adic integer <italic>t</italic>, we have</p>", "<title>Lemma 2.4</title>", "<p id=\"Par14\">Let be a prime. If , then</p>", "<title>Proof</title>", "<p id=\"Par15\">By using Sigma, we establish the following identity:(In terms of classical identities for hypergeometric series, this evaluation is equivalent to the case of the Pfaff–Saalschütz summation [##UREF##11##12##, Appendix III, Equation (III.2)].) So modulo , we havewhere we used the standard notation for the shifted factorial (cf. [##UREF##11##12##, Section 1.1.1]). It is easy to check thatThese identities, together with [##UREF##5##6##, pp. 14], yieldAgain, by [##UREF##5##6##, pp. 14–15], we haveIt is easy to check that the right-side of the above congruence is congruent to modulo . Therefore we immediately get the desired result stated in Lemma <xref ref-type=\"sec\" rid=\"FPar5\">2.4</xref>. </p>", "<title>Lemma 2.5</title>", "<p id=\"Par16\">Let be a prime with and let . Thenand</p>", "<title>Proof</title>", "<p id=\"Par17\">It is easy to see thatBy Lemma <xref ref-type=\"sec\" rid=\"FPar3\">2.2</xref> we haveThus,Together with (##FORMU##45##2.1##) and (cf. [##UREF##22##23##]), this yieldswhich completes the proof of the first congruence in Lemma <xref ref-type=\"sec\" rid=\"FPar7\">2.5</xref>. The proof of the second congruence is similar and therefore omitted. </p>", "<title>Lemma 2.6</title>", "<p id=\"Par18\">Let be a prime. If , then we haveIf , then</p>", "<title>Proof</title>", "<p id=\"Par19\">If , we can get the result from [##UREF##14##15##, pp. 24–25]. If , thenwhich completes the proof of Lemma <xref ref-type=\"sec\" rid=\"FPar9\">2.6</xref>. </p>", "<title>Lemma 2.7</title>", "<p id=\"Par20\">Let be a prime with . Then</p>", "<title>Proof</title>", "<p id=\"Par21\">By using Sigma, we establish the following identity:Substituting into the above identity, then modulo we haveIn view of [##UREF##6##7##, pp. 9] and [##UREF##5##6##, pp. 14–15], we haveHence,The proof of Lemma <xref ref-type=\"sec\" rid=\"FPar11\">2.7</xref> is complete. </p>", "<title>Proof of Theorem <xref ref-type=\"sec\" rid=\"FPar1\">1.1</xref></title>", "<p id=\"Par22\">Our proof of Theorem <xref ref-type=\"sec\" rid=\"FPar1\">1.1</xref> heavily relies on the following two transformation formulas due to  Chan and Zudilin [##UREF##0##1##] and  Sun [##UREF##18##19##] respectively,</p>", "<title>Proof of Theorem 1.1</title>", "<p id=\"Par23\">We first consider the first congruence in Theorem <xref ref-type=\"sec\" rid=\"FPar1\">1.1</xref> in the case . By (##FORMU##82##3.2##), we haveBy using Sigma, we establish the following identity:whereThus,LetIn view of Lemma <xref ref-type=\"sec\" rid=\"FPar3\">2.2</xref>, we have for the supercongruencewhere is defined by the following expression with ,In view of [##UREF##19##20##, Equation (12)] and [##UREF##6##7##, Equation (2.10)], we haveAnd in view of [##UREF##6##7##, pp. 13–14], we haveHence by (##FORMU##96##3.5##), Lemma <xref ref-type=\"sec\" rid=\"FPar7\">2.5</xref> and [##UREF##5##6##, Theorem 1.2], we haveThus we immediately obtain the desired resultNow we are ready to prove the case with . As before,In view of [##UREF##6##7##, pp. 9–10], we haveand in view of Lemma <xref ref-type=\"sec\" rid=\"FPar5\">2.4</xref>, we haveThus,In view of [##UREF##6##7##, Equation (4.2)], we haveandHenceNow we consider the other congruences in Theorem <xref ref-type=\"sec\" rid=\"FPar1\">1.1</xref>. Similar to above, by (##FORMU##81##3.1##), we haveBy using Sigma, we establish the following identity:whereLetThus, if , then by Lemma <xref ref-type=\"sec\" rid=\"FPar9\">2.6</xref>, we have, modulo ,where is defined by the following expression with ,Hence, as above, by (##FORMU##94##3.3##), (##FORMU##95##3.4##), (##FORMU##96##3.5##), Lemmas <xref ref-type=\"sec\" rid=\"FPar7\">2.5</xref> and <xref ref-type=\"sec\" rid=\"FPar11\">2.7</xref>, we haveIt is easy to see thatIn view of [##UREF##6##7##, pp. 18], we haveThese yieldIf with (the case can be checked directly), then modulo , we haveHence, similar to above, we haveThis, together with (##FORMU##98##3.6##), (##FORMU##107##3.9##) and (##FORMU##121##3.10##), completes the proof of Theorem <xref ref-type=\"sec\" rid=\"FPar1\">1.1</xref>. </p>" ]

[ "<title>Acknowledgements</title>", "<p>The authors would like to thank the anonymous referees for helpful comments.</p>", "<title>Funding</title>", "<p>Open access funding provided by Austrian Science Fund (FWF).</p>", "<title>Data availability</title>", "<p>Data sharing is not applicable to this article as no new data were created or analyzed in this study.</p>" ]

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[ "<inline-formula id=\"IEq1\"><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p^3$$\\end{document}</tex-math><mml:math id=\"M2\"><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq2\"><alternatives><tex-math id=\"M3\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{D_n\\}$$\\end{document}</tex-math><mml:math 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open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq3\"><alternatives><tex-math id=\"M7\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$D_n$$\\end{document}</tex-math><mml:math id=\"M8\"><mml:msub><mml:mi>D</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq4\"><alternatives><tex-math id=\"M9\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$1/\\pi $$\\end{document}</tex-math><mml:math id=\"M10\"><mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ13\"><alternatives><tex-math id=\"M11\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{n=0}^\\infty \\frac{5n+1}{64^n}D_n=\\frac{8}{\\sqrt{3}\\pi }. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M12\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:munderover><mml:mfrac><mml:mrow><mml:mn>5</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:msup><mml:mn>64</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>8</mml:mn><mml:mrow><mml:msqrt><mml:mn>3</mml:mn></mml:msqrt><mml:mi>π</mml:mi></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ14\"><alternatives><tex-math id=\"M13\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{n=0}^\\infty D_nu^n= \\frac{1}{1-4u}\\sum _{k=0}^\\infty \\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2 \\left( {\\begin{array}{c}3k\\\\ k\\end{array}}\\right) \\left(\\frac{u^2}{(1-4u)^3}\\right)^k, \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M14\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:munderover><mml:msub><mml:mi>D</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:msup><mml:mi>u</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:mi>u</mml:mi></mml:mrow></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:munderover><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:msup><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:mfrac></mml:mfenced><mml:mi>k</mml:mi></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq5\"><alternatives><tex-math id=\"M15\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p&gt;3$$\\end{document}</tex-math><mml:math id=\"M16\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ15\"><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{p-1}\\frac{3k^2+k}{16^k}D_k\\equiv -4p^4q_p(2)\\quad \\ (\\mathrm{{mod}}\\ p^5), \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M18\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq6\"><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q_p(a)$$\\end{document}</tex-math><mml:math id=\"M20\"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq7\"><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(a^{p-1}-1)/p$$\\end{document}</tex-math><mml:math id=\"M22\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>a</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ16\"><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{1}{n}\\sum _{k=0}^{n-1}(2k+1)D_k8^{n-1-k}\\ \\ \\ \\text{ and }\\ \\ \\ \\frac{1}{n}\\sum _{k=0}^{n-1}(2k+1)D_k(-8)^{n-1-k} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M24\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>8</mml:mn><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mspace width=\"4pt\"/><mml:mspace width=\"4pt\"/><mml:mspace width=\"4pt\"/><mml:mspace width=\"0.333333em\"/><mml:mtext>and</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"4pt\"/><mml:mspace width=\"4pt\"/><mml:mspace width=\"4pt\"/><mml:mfrac><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>8</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq8\"><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p&gt;3$$\\end{document}</tex-math><mml:math id=\"M26\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ17\"><alternatives><tex-math id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} D_{p-1}\\equiv 64^{p-1}-\\frac{p^3}{6}B_{p-3}\\quad \\ (\\mathrm{{mod}}\\ p^4), \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M28\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msup><mml:mn>64</mml:mn><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mn>6</mml:mn></mml:mfrac><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq9\"><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{B_n\\}$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ18\"><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} B_0=1,\\ \\ \\ \\sum _{k=0}^{n-1}\\left( {\\begin{array}{c}n\\\\ k\\end{array}}\\right) B_{k}=0\\ \\ (n\\ge 2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M32\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"4pt\"/><mml:mspace width=\"4pt\"/><mml:mspace width=\"4pt\"/><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mi>n</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msub><mml:mi>B</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"4pt\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq10\"><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x^2+3y^2$$\\end{document}</tex-math><mml:math id=\"M34\"><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq11\"><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p\\equiv 1\\ (\\mathrm{{mod}}\\ 3)$$\\end{document}</tex-math><mml:math id=\"M36\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mspace width=\"4pt\"/><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq12\"><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p=x^2+3y^2$$\\end{document}</tex-math><mml:math id=\"M38\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ19\"><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} R_3(p)=\\left( 1+2p+\\frac{4}{3}(2^{p-1}-1) -\\frac{3}{2}(3^{p-1}-1)\\right) \\left( {\\begin{array}{c}\\frac{p-1}{2}\\\\ \\lfloor p/6\\rfloor \\end{array}}\\right) ^2. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M40\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mn>3</mml:mn><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfenced><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mo>⌊</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>6</mml:mn><mml:mo>⌋</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq13\"><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p&gt;3$$\\end{document}</tex-math><mml:math id=\"M42\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ20\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;\\sum _{k=0}^{p-1}k^3\\frac{D_k}{4^k}\\\\&amp;\\quad \\equiv {\\left\\{ \\begin{array}{ll}-\\frac{64}{45}x^2+\\frac{32}{45}p+\\frac{43p^2}{90x^2}\\;\\ (\\mathrm{{mod}}\\ p^3) &amp;{} \\quad \\text {if} \\,\\,\\ \\textit{p}=\\textit{x}^2+3\\textit{y}^2\\equiv 1\\ (\\mathrm{{mod}}\\ 3), \\\\ \\frac{28}{9}R_3(p)\\;\\ (\\mathrm{{mod}}\\ p^2) &amp;{} \\quad \\text {if} \\,\\,\\ \\textit{p}\\equiv 2\\ (\\mathrm{{mod}}\\ 3)\\ \\text{ and }\\ \\textit{p}\\ne 5,\\end{array}\\right. }\\\\&amp;\\sum _{k=0}^{p-1}k^3\\frac{D_k}{16^k}\\\\&amp;\\equiv {\\left\\{ \\begin{array}{ll}\\frac{4}{45}x^2-\\frac{2}{45}p+\\frac{p^2}{45x^2}\\;\\ (\\mathrm{{mod}}\\ p^3) &amp;{}\\text {if} \\,\\,\\ \\textit{p}=\\textit{x}^2+3\\textit{y}^2\\equiv 1\\ (\\mathrm{{mod}}\\ 3), \\\\ -\\frac{4}{9}R_3(p)\\;\\ (\\mathrm{{mod}}\\ p^2) &amp;{}\\text {if} \\,\\, \\ \\textit{p}\\equiv 2\\ (\\mathrm{{mod}}\\ 3).\\end{array}\\right. } \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M44\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>4</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mo>≡</mml:mo><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>64</mml:mn><mml:mn>45</mml:mn></mml:mfrac><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:mn>32</mml:mn><mml:mn>45</mml:mn></mml:mfrac><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>43</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>90</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"1em\"/><mml:mtext>if</mml:mtext><mml:mspace width=\"0.166667em\"/><mml:mspace width=\"0.166667em\"/><mml:mspace width=\"4pt\"/><mml:mi mathvariant=\"italic\">p</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi mathvariant=\"italic\">x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi mathvariant=\"italic\">y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mfrac><mml:mn>28</mml:mn><mml:mn>9</mml:mn></mml:mfrac><mml:msub><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"1em\"/><mml:mtext>if</mml:mtext><mml:mspace width=\"0.166667em\"/><mml:mspace width=\"0.166667em\"/><mml:mspace width=\"4pt\"/><mml:mi mathvariant=\"italic\">p</mml:mi><mml:mo>≡</mml:mo><mml:mn>2</mml:mn><mml:mspace width=\"4pt\"/><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mspace width=\"4pt\"/><mml:mspace width=\"0.333333em\"/><mml:mtext>and</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"4pt\"/><mml:mi mathvariant=\"italic\">p</mml:mi><mml:mo>≠</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mfrac><mml:mn>4</mml:mn><mml:mn>45</mml:mn></mml:mfrac><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>45</mml:mn></mml:mfrac><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mn>45</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mtext>if</mml:mtext><mml:mspace width=\"0.166667em\"/><mml:mspace width=\"0.166667em\"/><mml:mspace width=\"4pt\"/><mml:mi mathvariant=\"italic\">p</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi mathvariant=\"italic\">x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi mathvariant=\"italic\">y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mo>-</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mn>9</mml:mn></mml:mfrac><mml:msub><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mtext>if</mml:mtext><mml:mspace width=\"0.166667em\"/><mml:mspace width=\"0.166667em\"/><mml:mspace width=\"4pt\"/><mml:mi mathvariant=\"italic\">p</mml:mi><mml:mo>≡</mml:mo><mml:mn>2</mml:mn><mml:mspace width=\"4pt\"/><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq14\"><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{B_n(x)\\}$$\\end{document}</tex-math><mml:math id=\"M46\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ21\"><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} B_n(x)=\\sum _{k=0}^n\\left( {\\begin{array}{c}n\\\\ k\\end{array}}\\right) B_kx^{n-k}\\ \\ (n=0,1,2,\\ldots ), \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M48\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mi>n</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msub><mml:mi>B</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mspace width=\"4pt\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq15\"><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{B_n\\}$$\\end{document}</tex-math><mml:math id=\"M50\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq16\"><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\big (\\frac{a}{q}\\big )$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mfrac><mml:mi>a</mml:mi><mml:mi>q</mml:mi></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ22\"><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{1\\le k&lt;\\frac{2p}{3}}\\frac{1}{k}= \\sum _{1\\le k&lt;\\frac{2p}{3}}\\frac{(-1)^{k-1}}{k} +\\sum _{1\\le k&lt;\\frac{p}{3}}\\frac{1}{k}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M54\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>k</mml:mi><mml:mo>&lt;</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:munder><mml:mfrac><mml:mn>1</mml:mn><mml:mi>k</mml:mi></mml:mfrac><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>k</mml:mi><mml:mo>&lt;</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:munder><mml:mfrac><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>k</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>k</mml:mi><mml:mo>&lt;</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:munder><mml:mfrac><mml:mn>1</mml:mn><mml:mi>k</mml:mi></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq17\"><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p&gt;5$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ23\"><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} H_{\\frac{p-1}{2}}&amp;\\equiv -2q_p(2)\\;\\ (\\mathrm{{mod}}\\ p),\\\\ H_{\\lfloor \\frac{p}{6}\\rfloor }&amp;\\equiv -2q_p(2)-\\frac{3}{2}q_p(3)\\;\\ (\\mathrm{{mod}}\\ p),\\\\ H_{\\lfloor \\frac{p}{3}\\rfloor }^{(2)}&amp;\\equiv \\frac{1}{2}\\left( \\frac{p}{3}\\right) B_{p-2}\\Big (\\frac{1}{3}\\Big )\\;\\ (\\mathrm{{mod}}\\ p),\\\\ H_{\\lfloor \\frac{p}{3}\\rfloor }&amp;\\equiv -\\frac{3}{2}q_p(3)+\\frac{3p}{4}q^2_p(3)-\\frac{p}{6}\\left( \\frac{p}{3}\\right) B_{p-2} \\Big (\\frac{1}{3}\\Big )\\;\\ (\\mathrm{{mod}}\\ p^2),\\\\ H_{\\lfloor \\frac{2p}{3}\\rfloor }&amp;\\equiv -\\frac{3}{2}q_p(3)+\\frac{3p}{4}q^2_p(3)+\\frac{p}{3}\\left( \\frac{p}{3}\\right) B_{p-2} \\Big (\\frac{1}{3}\\Big )\\;\\ (\\mathrm{{mod}}\\ p^2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M58\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:msub><mml:mi>H</mml:mi><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:msub></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mo>⌊</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mo>⌋</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mo>⌊</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>⌋</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac></mml:mfenced><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mo>⌊</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>⌋</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:mfrac><mml:msubsup><mml:mi>q</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>6</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac></mml:mfenced><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mo>⌊</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:mo>⌋</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:mfrac><mml:msubsup><mml:mi>q</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac></mml:mfenced><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq18\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{H_n\\}$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq19\"><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{U_n\\}$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ24\"><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} U_0=1,\\ \\ \\ U_n=-2\\sum _{k=1}^{\\lfloor n/2\\rfloor }\\left( {\\begin{array}{c}n\\\\ 2k\\end{array}}\\right) U_{n-2k}\\ \\ \\ (n\\ge 1). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M64\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace width=\"4pt\"/><mml:mspace width=\"4pt\"/><mml:mspace width=\"4pt\"/><mml:msub><mml:mi>U</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mo>⌊</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mo>⌋</mml:mo></mml:mrow></mml:munderover><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mi>n</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mspace width=\"4pt\"/><mml:mspace width=\"4pt\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq20\"><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p&gt;5$$\\end{document}</tex-math><mml:math id=\"M66\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq21\"><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$0\\le j\\le (p-1)/2$$\\end{document}</tex-math><mml:math id=\"M68\"><mml:mrow><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ25\"><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\left( {\\begin{array}{c}3j\\\\ j\\end{array}}\\right) \\left( {\\begin{array}{c}p+j\\\\ 3j+1\\end{array}}\\right) \\equiv \\frac{p}{3j+1}(1-pH_{2j}+pH_j)\\;\\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M70\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>≡</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p&gt;3$$\\end{document}</tex-math><mml:math id=\"M72\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ26\"><alternatives><tex-math id=\"M73\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\left( {\\begin{array}{c}\\frac{2p-2}{3}+pt\\\\ \\frac{p-1}{2}\\end{array}}\\right) \\equiv \\left( {\\begin{array}{c}\\frac{2p-2}{3}\\\\ \\frac{p-1}{2}\\end{array}}\\right) \\left( 1+pt(H_{\\frac{2p-2}{3}}-H_{\\frac{p-1}{6}})\\right) \\;\\ (\\mathrm{{mod}}\\ p^2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M74\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>≡</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>6</mml:mn></mml:mfrac></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mfenced><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M75\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p&gt;3$$\\end{document}</tex-math><mml:math id=\"M76\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M77\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p=x^2+3y^2\\equiv 1\\ (\\mathrm{{mod}}\\ 3)$$\\end{document}</tex-math><mml:math id=\"M78\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ27\"><alternatives><tex-math id=\"M79\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} p\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{(3k+4)16^k} \\equiv \\frac{4}{25}\\left( 4x^2-2p-\\frac{p^2}{4x^2}\\right) \\;\\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M80\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>p</mml:mi><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mn>25</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mfenced><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ28\"><alternatives><tex-math id=\"M81\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^n\\frac{\\left( {\\begin{array}{c}n\\\\ k\\end{array}}\\right) \\left( {\\begin{array}{c}n+k\\\\ k\\end{array}}\\right) (-1)^k}{3k+4}=-\\frac{1}{(3n-1)(3n+1)(3n+4)}\\prod _{k=1}^n\\frac{3k-1}{3k-2}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M82\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mfrac><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mi>n</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M83\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(a,b,c)\\mapsto (n+1,4/3,1)$$\\end{document}</tex-math><mml:math id=\"M84\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>↦</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M85\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p^3$$\\end{document}</tex-math><mml:math id=\"M86\"><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ29\"><alternatives><tex-math id=\"M87\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;p\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{(3k+4)16^k}\\\\&amp;\\equiv p\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}\\frac{p-1}{2}\\\\ k\\end{array}}\\right) \\left( {\\begin{array}{c}\\frac{p-1}{2}+k\\\\ k\\end{array}}\\right) (-1)^k}{3k+4}+\\left( {\\begin{array}{c}-\\frac{1}{2}\\\\ \\frac{p-4}{3}\\end{array}}\\right) ^2+\\left( {\\begin{array}{c}\\frac{p-1}{2}\\\\ \\frac{p-4}{3}\\end{array}}\\right) \\left( {\\begin{array}{c}\\frac{p-1}{2}+\\frac{p-4}{3}\\\\ \\frac{p-4}{3}\\end{array}}\\right) \\\\&amp;=\\frac{4}{25-9p^2}\\frac{(\\frac{2}{3})_{\\frac{p-1}{2}}}{(\\frac{1}{3})_{\\frac{p-1}{3}}(\\frac{p}{3}+1)_{\\frac{p-1}{6}}}+\\left( {\\begin{array}{c}-\\frac{1}{2}\\\\ \\frac{p-4}{3}\\end{array}}\\right) ^2+\\left( {\\begin{array}{c}\\frac{p-1}{2}\\\\ \\frac{p-4}{3}\\end{array}}\\right) \\left( {\\begin{array}{c}\\frac{p-1}{2}+\\frac{p-4}{3}\\\\ \\frac{p-4}{3}\\end{array}}\\right) , \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M88\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mi>p</mml:mi><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mi>p</mml:mi><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mrow><mml:mn>25</mml:mn><mml:mo>-</mml:mo><mml:mn>9</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mfrac><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:msub><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>6</mml:mn></mml:mfrac></mml:msub></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M89\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(a)_n=\\prod _{j=0}^{n-1}(a+j)$$\\end{document}</tex-math><mml:math id=\"M90\"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∏</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mo>+</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><label>2.1</label><alternatives><tex-math id=\"M91\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\left( {\\begin{array}{c}-\\frac{1}{2}\\\\ \\frac{p-4}{3}\\end{array}}\\right) ^2&amp;=\\frac{4(p-1)^2}{(2p-5)^2}\\left( {\\begin{array}{c}-\\frac{1}{2}\\\\ \\frac{p-1}{3}\\end{array}}\\right) ^2,\\nonumber \\\\ \\left( {\\begin{array}{c}\\frac{p-1}{2}\\\\ \\frac{p-4}{3}\\end{array}}\\right) \\left( {\\begin{array}{c}\\frac{p-1}{2}+\\frac{p-4}{3}\\\\ \\frac{p-4}{3}\\end{array}}\\right)&amp;=\\frac{4(p-1)}{5(p+5)}\\left( {\\begin{array}{c}\\frac{p-1}{2}\\\\ \\frac{p-1}{3}\\end{array}}\\right) \\left( {\\begin{array}{c}\\frac{p-1}{2}+\\frac{p-1}{3}\\\\ \\frac{p-1}{3}\\end{array}}\\right) . \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M92\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>5</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ30\"><alternatives><tex-math id=\"M93\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;p\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{(3k+4)16^k}\\\\&amp;\\equiv \\frac{4}{25-9p^2}\\frac{(\\frac{2}{3})_{\\frac{p-1}{2}}}{(\\frac{1}{3})_{\\frac{p-1}{3}}(\\frac{p}{3}+1)_{\\frac{p-1}{6}}}+\\frac{4(p-1)^2}{(2p-5)^2} \\left( {\\begin{array}{c}-\\frac{1}{2}\\\\ \\frac{p-1}{3}\\end{array}}\\right) ^2\\\\&amp;\\quad \\;+\\frac{4(p-1)}{5(p+5)}\\left( {\\begin{array}{c}\\frac{p-1}{2}\\\\ \\frac{p-1}{3}\\end{array}}\\right) \\left( {\\begin{array}{c}\\frac{p-1}{2}+\\frac{p-1}{3}\\\\ \\frac{p-1}{3}\\end{array}}\\right) \\\\&amp;\\equiv \\frac{4}{25}\\left( 1+\\frac{9p^2}{25}\\right) (-1)^{\\frac{p-1}{6}}\\left( {\\begin{array}{c}\\frac{p-1}{2}\\\\ \\frac{p-1}{3}\\end{array}}\\right) \\left( {\\begin{array}{c}\\frac{2p-2}{3}\\\\ \\frac{p-1}{2}\\end{array}}\\right) \\\\&amp;\\quad \\;\\times \\left( 1-\\frac{2p}{3}q_p(2)+\\frac{5p^2}{9}q^2_p(2) +\\frac{5p^2}{12}\\left( \\frac{p}{3}\\right) B_{p-2}\\Big (\\frac{1}{3}\\Big )\\right) \\\\&amp;\\quad \\;+\\frac{4}{25}\\left( 1-\\frac{6p}{5}-\\frac{3p^2}{25}\\right) \\left( {\\begin{array}{c}\\frac{p-1}{2}\\\\ \\frac{p-1}{3}\\end{array}}\\right) ^2\\\\&amp;\\quad \\;\\;\\times \\left( 1-\\frac{3p}{2}q_p(3)+\\frac{15p^2}{8}q^2_p(3)+\\frac{5p^2}{24}\\left( \\frac{p}{3}\\right) B_{p-2}\\Big (\\frac{1}{3}\\Big )\\right) \\\\&amp;\\quad \\;-\\frac{4}{25}\\left( 1-\\frac{6p}{5}+\\frac{6p^2}{25}\\right) \\left( {\\begin{array}{c}\\frac{p-1}{2}\\\\ \\frac{p-1}{3}\\end{array}}\\right) \\left( {\\begin{array}{c}\\frac{5p-5}{6}\\\\ \\frac{p-1}{3}\\end{array}}\\right) \\quad \\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M94\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mi>p</mml:mi><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mrow><mml:mn>25</mml:mn><mml:mo>-</mml:mo><mml:mn>9</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mfrac><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:msub><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>6</mml:mn></mml:mfrac></mml:msub></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>5</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mn>25</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>9</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>25</mml:mn></mml:mfrac></mml:mfenced><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>6</mml:mn></mml:mfrac></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>×</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>5</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>9</mml:mn></mml:mfrac><mml:msubsup><mml:mi>q</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>5</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>12</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac></mml:mfenced><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">)</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>+</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mn>25</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>6</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>25</mml:mn></mml:mfrac></mml:mfenced><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>×</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>15</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>8</mml:mn></mml:mfrac><mml:msubsup><mml:mi>q</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>5</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>24</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac></mml:mfenced><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">)</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>-</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mn>25</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>6</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>6</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>25</mml:mn></mml:mfrac></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfrac><mml:mrow><mml:mn>5</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>5</mml:mn></mml:mrow><mml:mn>6</mml:mn></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ31\"><alternatives><tex-math id=\"M95\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;p\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{(3k+4)16^k} \\\\ {}&amp;\\equiv \\frac{4}{25}\\left( 1+\\frac{9p^2}{25}\\right) \\left( 4x^2-2p-\\frac{p^2}{4x^2}\\right) \\\\&amp;\\quad \\;\\times \\left( 1+\\frac{2p}{3}q_p(2)-\\frac{p^2}{9}q^2_p(2) +\\frac{5p^2}{24}\\left( \\frac{p}{3}\\right) B_{p-2}\\Big (\\frac{1}{3}\\Big )\\right) \\\\&amp;\\quad \\;\\times \\left( 1-\\frac{2p}{3}q_p(2)+\\frac{5p^2}{9}q^2_p(2) +\\frac{5p^2}{12}\\left( \\frac{p}{3}\\right) B_{p-2}\\Big (\\frac{1}{3}\\Big )\\right) \\\\&amp;\\quad \\;+\\frac{4}{25}\\left( 1-\\frac{6p}{5}-\\frac{3p^2}{25}\\right) \\left( 4x^2-2p-\\frac{p^2}{4x^2}\\right) \\\\&amp;\\quad \\;\\;\\times \\bigg (1-\\frac{3p}{2}q_p(3) +\\frac{15p^2}{8}q^2_p(3)+\\frac{5p^2}{24}\\left( \\frac{p}{3}\\right) B_{p-2}\\Big (\\frac{1}{3}\\Big )\\bigg )\\\\&amp;\\quad \\;\\;\\times \\bigg (1-\\frac{4p}{3}q_p(2)+\\frac{3p}{2}q_p(3) +\\frac{14p^2}{9}q^2_p(2)-2p^2q_p(2)q_p(3)\\\\&amp;\\quad +\\frac{3p^2}{8}q^2_p(3) +\\frac{p^2}{8}\\left( \\frac{p}{3}\\right) B_{p-2} \\Big (\\frac{1}{3}\\Big )\\bigg )\\\\&amp;\\quad \\;-\\frac{4}{25}\\left( 1-\\frac{6p}{5}+\\frac{6p^2}{25}\\right) \\left( 4x^2-2p-\\frac{p^2}{4x^2}\\right) \\\\&amp;\\quad \\;\\;\\times \\left( 1-\\frac{4p}{3}q_p(2)+\\frac{14p^2}{9}q^2_p(2) +\\frac{23p^2}{24}\\left( \\frac{p}{3}\\right) B_{p-2} \\Big (\\frac{1}{3}\\Big )\\right) \\;\\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M96\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mi>p</mml:mi><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mn>25</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>9</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>25</mml:mn></mml:mfrac></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>×</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mn>9</mml:mn></mml:mfrac><mml:msubsup><mml:mi>q</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>5</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>24</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac></mml:mfenced><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">)</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>×</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>5</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>9</mml:mn></mml:mfrac><mml:msubsup><mml:mi>q</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo 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open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>6</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>25</mml:mn></mml:mfrac></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>×</mml:mo><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>15</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>8</mml:mn></mml:mfrac><mml:msubsup><mml:mi>q</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>5</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>24</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac></mml:mfenced><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\" stretchy=\"true\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>×</mml:mo><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>14</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>9</mml:mn></mml:mfrac><mml:msubsup><mml:mi>q</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>8</mml:mn></mml:mfrac><mml:msubsup><mml:mi>q</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mn>8</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac></mml:mfenced><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\" stretchy=\"true\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>-</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mn>25</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>6</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>6</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>25</mml:mn></mml:mfrac></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>×</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:msub><mml:mi>q</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>14</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>9</mml:mn></mml:mfrac><mml:msubsup><mml:mi>q</mml:mi><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>23</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>24</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac></mml:mfenced><mml:msub><mml:mi>B</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">)</mml:mo></mml:mrow></mml:mfenced><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M97\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\frac{4}{25}\\left( 4x^2-2p-\\frac{p^2}{4x^2}\\right) $$\\end{document}</tex-math><mml:math id=\"M98\"><mml:mrow><mml:mfrac><mml:mn>4</mml:mn><mml:mn>25</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M99\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p^3$$\\end{document}</tex-math><mml:math id=\"M100\"><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq30\"><alternatives><tex-math id=\"M101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M102\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq31\"><alternatives><tex-math id=\"M103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p&gt;3$$\\end{document}</tex-math><mml:math id=\"M104\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq32\"><alternatives><tex-math id=\"M105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p=x^2+3y^2\\equiv 1\\ (\\mathrm{{mod}}\\ 3)$$\\end{document}</tex-math><mml:math id=\"M106\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq33\"><alternatives><tex-math id=\"M107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k=(p-4)/3$$\\end{document}</tex-math><mml:math id=\"M108\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ32\"><alternatives><tex-math id=\"M109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;\\big (k(k+1)(k+3)+(2-k^2)(3k+1)p-(k+2)(3k+1)(3k+2)p^2\\big )\\\\&amp;\\qquad \\times \\frac{\\left( {\\begin{array}{c}-\\frac{1}{2}\\\\ k\\end{array}}\\right) ^2}{(k+1)(3k+2)} \\left( \\frac{\\left( {\\begin{array}{c}3k\\\\ k\\end{array}}\\right) \\left( {\\begin{array}{c}p+k\\\\ 3k+1\\end{array}}\\right) }{3k+4} -\\frac{1-pH_{2k}+pH_k}{3k+1}\\right) \\\\&amp;\\qquad \\quad \\equiv -\\frac{184p^2x^2}{125}\\;\\ (\\mathrm{{mod}}\\ p^3) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M110\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mi>k</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"2em\"/><mml:mo>×</mml:mo><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"2em\"/><mml:mspace width=\"1em\"/><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>184</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>125</mml:mn></mml:mfrac><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ33\"><alternatives><tex-math id=\"M111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;\\big (k(1+2k)+2p(k+1)(3k+1)-2p^2(3k+1)(3k+2)\\big )\\\\&amp;\\qquad \\times \\frac{\\left( {\\begin{array}{c}-\\frac{1}{2}\\\\ k\\end{array}}\\right) ^2}{3k+2} \\left( \\frac{\\left( {\\begin{array}{c}3k\\\\ k\\end{array}}\\right) \\left( {\\begin{array}{c}p+2k\\\\ 3k+1\\end{array}}\\right) }{3k+4} +\\frac{1+pH_{2k}-pH_k}{3k+1}\\right) \\\\&amp;\\quad \\qquad \\equiv -\\frac{184p^2x^2}{125}\\;\\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M112\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mi>k</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"2em\"/><mml:mo>×</mml:mo><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"2em\"/><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>184</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>125</mml:mn></mml:mfrac><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ34\"><alternatives><tex-math id=\"M113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;\\frac{\\left( {\\begin{array}{c}3k\\\\ k\\end{array}}\\right) \\left( {\\begin{array}{c}p+k\\\\ 3k+1\\end{array}}\\right) }{3k+4}-\\frac{1-pH_{2k}+pH_k}{3k+1}\\\\&amp;\\quad =\\left( {\\begin{array}{c}3k+3\\\\ k+1\\end{array}}\\right) \\left( {\\begin{array}{c}p+k+1\\\\ 3k+4\\end{array}}\\right) \\frac{2(2p-5)(p-1)^2}{(4p-1)(p-3)(p+2)(p+5)}\\\\&amp;\\qquad \\;-\\frac{1-pH_{2k+2}+pH_{k+1}+\\frac{p}{2k+2}+\\frac{p}{2k+1}-\\frac{p}{k+1}}{3k+1}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M114\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mfrac><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mo>=</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"2em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ35\"><alternatives><tex-math id=\"M115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\left( {\\begin{array}{c}3k+3\\\\ k+1\\end{array}}\\right) \\left( {\\begin{array}{c}p+k+1\\\\ 3k+4\\end{array}}\\right) \\equiv 1-pH_{2k+2}+pH_{k+1}\\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M116\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ36\"><alternatives><tex-math id=\"M117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{\\left( {\\begin{array}{c}3k\\\\ k\\end{array}}\\right) \\left( {\\begin{array}{c}p+k\\\\ 3k+1\\end{array}}\\right) }{3k+4}-\\frac{1-pH_{2k} +pH_k}{3k+1}\\equiv -\\frac{207p^2}{100}\\;\\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M118\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfrac><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>207</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>100</mml:mn></mml:mfrac><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq34\"><alternatives><tex-math id=\"M119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\left( {\\begin{array}{c}(p-1)/2\\\\ (p-1)/3\\end{array}}\\right) \\equiv 2x\\ (\\mathrm{{mod}}\\ p)$$\\end{document}</tex-math><mml:math id=\"M120\"><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>≡</mml:mo><mml:mn>2</mml:mn><mml:mi>x</mml:mi><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ37\"><alternatives><tex-math id=\"M121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;\\big (k(k+1)(k+3)+(2-k^2)(3k+1)p-(k+2)(3k+1)(3k+2)p^2\\big )\\\\&amp;\\quad \\times \\frac{\\left( {\\begin{array}{c}-\\frac{1}{2}\\\\ k\\end{array}}\\right) ^2}{(k+1)(3k+2)} \\left( \\frac{\\left( {\\begin{array}{c}3k\\\\ k\\end{array}}\\right) \\left( {\\begin{array}{c}p+k\\\\ 3k+1\\end{array}}\\right) }{3k+4} -\\frac{1-pH_{2k}+pH_k}{3k+1}\\right) \\\\&amp;\\qquad \\equiv -\\frac{184p^2x^2}{125}\\quad \\ (\\mathrm{{mod}}\\ p^3), \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M122\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mi>k</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mo>×</mml:mo><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"2em\"/><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>184</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>125</mml:mn></mml:mfrac><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq35\"><alternatives><tex-math id=\"M123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M124\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq36\"><alternatives><tex-math id=\"M125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p&gt;2$$\\end{document}</tex-math><mml:math id=\"M126\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq37\"><alternatives><tex-math id=\"M127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$0\\le j\\le (p-1)/2$$\\end{document}</tex-math><mml:math id=\"M128\"><mml:mrow><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ38\"><alternatives><tex-math id=\"M129\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\left( {\\begin{array}{c}3j\\\\ j\\end{array}}\\right) \\left( {\\begin{array}{c}p+2j\\\\ 3j+1\\end{array}}\\right) \\equiv \\frac{p(-1)^j}{3j+1}(1+pH_{2j}-pH_j)\\;\\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M130\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>j</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq38\"><alternatives><tex-math id=\"M131\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(p+1)/2\\le j\\le p-1$$\\end{document}</tex-math><mml:math id=\"M132\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ39\"><alternatives><tex-math id=\"M133\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\left( {\\begin{array}{c}3j\\\\ j\\end{array}}\\right) \\left( {\\begin{array}{c}p+2j\\\\ 3j+1\\end{array}}\\right) \\equiv \\frac{2p(-1)^j}{3j+1}\\;\\ (\\mathrm{{mod}}\\ p^2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M134\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>j</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq39\"><alternatives><tex-math id=\"M135\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$0\\le j\\le (p-1)/2$$\\end{document}</tex-math><mml:math id=\"M136\"><mml:mrow><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq40\"><alternatives><tex-math id=\"M137\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(p+1)/2\\le j\\le p-1$$\\end{document}</tex-math><mml:math id=\"M138\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ40\"><alternatives><tex-math id=\"M139\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\left( {\\begin{array}{c}3j\\\\ j\\end{array}}\\right) \\left( {\\begin{array}{c}p+2j\\\\ 3j+1\\end{array}}\\right)&amp;=\\frac{(p+2j)\\cdots (2p+1)(2p)(2p-1)\\cdots (p+1)p(p-1)\\cdots (p-j)}{(3j+1)j!(2j)!}\\\\&amp;\\equiv \\frac{2p^2(2j)\\cdots (p+1)(p-1)!(-1)^{j}(j)!}{(3j+1)j!(2j)!}\\equiv \\frac{2p(-1)^j}{3j+1}\\;\\ (\\mathrm{{mod}}\\ p^2), \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M140\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⋯</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⋯</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⋯</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>j</mml:mi><mml:mo>!</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>⋯</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>!</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>j</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>!</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>j</mml:mi><mml:mo>!</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>j</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq41\"><alternatives><tex-math id=\"M141\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M142\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq42\"><alternatives><tex-math id=\"M143\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p&gt;3$$\\end{document}</tex-math><mml:math id=\"M144\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq43\"><alternatives><tex-math id=\"M145\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p=x^2+3y^2\\equiv 1\\ (\\mathrm{{mod}}\\ 3)$$\\end{document}</tex-math><mml:math id=\"M146\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ41\"><alternatives><tex-math id=\"M147\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} p\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2(H_{2j}-H_j)}{(3j+4)16^j}\\equiv -\\frac{18}{125}(4x^2-2p)\\;\\ (\\mathrm{{mod}}\\ p^2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M148\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>p</mml:mi><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>j</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mn>18</mml:mn><mml:mn>125</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ42\"><alternatives><tex-math id=\"M149\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;\\sum _{j=0}^n\\frac{\\left( {\\begin{array}{c}n\\\\ j\\end{array}}\\right) \\left( {\\begin{array}{c}n+j\\\\ j\\end{array}}\\right) (-1)^j(H_{2j}-H_j)}{3j+4} =-\\frac{9(2n+1)}{10(3n-1)(3n+4)}\\\\&amp;\\quad +\\frac{(\\frac{2}{3})_n}{(3n-1)(3n+1)(3n+4)(\\frac{1}{3})_n}\\left( \\frac{9}{10} +\\sum _{k=1}^n\\frac{(\\frac{1}{3})_k}{k(\\frac{2}{3})_k}\\right) . \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M150\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mfrac><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mi>n</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>j</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>9</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>10</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mo>+</mml:mo><mml:mfrac><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mn>9</mml:mn><mml:mn>10</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mfrac><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mi>k</mml:mi><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq44\"><alternatives><tex-math id=\"M151\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n=(p-1)/2$$\\end{document}</tex-math><mml:math id=\"M152\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq45\"><alternatives><tex-math id=\"M153\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p^2$$\\end{document}</tex-math><mml:math id=\"M154\"><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ43\"><alternatives><tex-math id=\"M155\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} p\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2(H_{2j}-H_j)}{(3j+4)16^j}\\equiv \\frac{p(\\frac{2}{3})_{\\frac{p-1}{2}}}{\\frac{3p-5}{2}\\frac{3p-1}{2}\\frac{3p+5}{2}(\\frac{1}{3})_{\\frac{p-1}{2}}}\\left( \\frac{9}{10}+\\sum _{k=1}^{\\frac{p-1}{2}}\\frac{(\\frac{1}{3})_k}{k(\\frac{2}{3})_k}\\right) . \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M156\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>p</mml:mi><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>j</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:msub></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>5</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>5</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:msub></mml:mrow></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mn>9</mml:mn><mml:mn>10</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mi>k</mml:mi><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ44\"><alternatives><tex-math id=\"M157\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{(\\frac{2}{3})_{\\frac{p-1}{2}}}{(\\frac{1}{3})_{\\frac{p-1}{3}}(\\frac{p}{3}+1)_{\\frac{p-1}{6}}}&amp;\\equiv 4x^2-2p\\quad \\ (\\mathrm{{mod}}\\ p^2),\\\\ \\frac{(\\frac{2}{3})_{\\frac{p-1}{2}}}{(\\frac{1}{3})_{\\frac{p-1}{2}}} \\sum _{k=1}^{\\frac{p-1}{2}}\\frac{(\\frac{1}{3})_k}{k(\\frac{2}{3})_k}&amp;\\equiv 0\\quad \\ (\\mathrm{{mod}}\\ p). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M158\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mfrac><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:msub><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>6</mml:mn></mml:mfrac></mml:msub></mml:mrow></mml:mfrac></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:mfrac><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:msub><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:msub></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mi>k</mml:mi><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ45\"><alternatives><tex-math id=\"M159\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} p\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2(H_{2j}-H_j)}{(3j+4)16^j}&amp;\\equiv \\frac{9}{10}\\frac{p(\\frac{2}{3})_{\\frac{p-1}{2}}}{\\frac{3p-5}{2}\\frac{3p-1}{2}\\frac{3p+5}{2}(\\frac{1}{3})_{\\frac{p-1}{2}}}\\\\&amp;\\equiv -\\frac{9}{10}\\frac{4}{25}(4x^2-2p)=-\\frac{18}{125}(4x^2-2p)\\quad \\ (\\mathrm{{mod}}\\ p^2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M160\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>p</mml:mi><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>j</mml:mi></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mn>9</mml:mn><mml:mn>10</mml:mn></mml:mfrac><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:msub></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>5</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mfrac><mml:mrow><mml:mn>3</mml:mn><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>5</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mn>9</mml:mn><mml:mn>10</mml:mn></mml:mfrac><mml:mfrac><mml:mn>4</mml:mn><mml:mn>25</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mn>18</mml:mn><mml:mn>125</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq46\"><alternatives><tex-math id=\"M161\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M162\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ2\"><label>3.1</label><alternatives><tex-math id=\"M163\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^n\\left( {\\begin{array}{c}n\\\\ k\\end{array}}\\right) ^2\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) \\left( {\\begin{array}{c}2n-2k\\\\ n-k\\end{array}}\\right) =\\sum _{k=0}^{n}(-1)^k\\left( {\\begin{array}{c}n+2k\\\\ 3k\\end{array}}\\right) \\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2\\left( {\\begin{array}{c}3k\\\\ k\\end{array}}\\right) 16^{n-k}, \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M164\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mi>n</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msup><mml:mn>16</mml:mn><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ3\"><label>3.2</label><alternatives><tex-math id=\"M165\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^n\\left( {\\begin{array}{c}n\\\\ k\\end{array}}\\right) ^2\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) \\left( {\\begin{array}{c}2n-2k\\\\ n-k\\end{array}}\\right) =\\sum _{k=0}^{\\lfloor n/2\\rfloor }\\left( {\\begin{array}{c}n+k\\\\ 3k\\end{array}}\\right) \\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2\\left( {\\begin{array}{c}3k\\\\ k\\end{array}}\\right) 4^{n-2k}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M166\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mi>n</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>⌊</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mo>⌋</mml:mo></mml:mrow></mml:munderover><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msup><mml:mn>4</mml:mn><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq47\"><alternatives><tex-math id=\"M167\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p=x^2+3y^2\\equiv 1\\ (\\mathrm{{mod}}\\ 3)$$\\end{document}</tex-math><mml:math id=\"M168\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ46\"><alternatives><tex-math id=\"M169\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{p-1}k^3\\frac{D_k}{4^k}&amp;=\\sum _{k=0}^{p-1}\\frac{k^3}{4^k}\\sum _{j=0}^{\\lfloor k/2\\rfloor } \\left( {\\begin{array}{c}k+j\\\\ 3j\\end{array}}\\right) \\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2\\left( {\\begin{array}{c}3j\\\\ j\\end{array}}\\right) 4^{k-2j}\\\\&amp;=\\sum _{j=0}^{(p-1)/2}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2\\left( {\\begin{array}{c}3j\\\\ j\\end{array}}\\right) }{16^j}\\sum _{k=2j}^{p-1}k^3\\left( {\\begin{array}{c}k+j\\\\ 3j\\end{array}}\\right) . \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M170\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>4</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:msup><mml:mn>4</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo>⌊</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mo>⌋</mml:mo></mml:mrow></mml:munderover><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msup><mml:mn>4</mml:mn><mml:mrow><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>j</mml:mi></mml:msup></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ47\"><alternatives><tex-math id=\"M171\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=2j}^{n-1}k^3\\left( {\\begin{array}{c}k+j\\\\ 3j\\end{array}}\\right) =\\frac{\\sigma _1}{(j+1)(3j+2)(3j+4)}\\left( {\\begin{array}{c}n+j\\\\ 3j+1\\end{array}}\\right) , \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M172\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>σ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ48\"><alternatives><tex-math id=\"M173\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sigma _1&amp;=j(j+1)(j+3)+n(2-j^2)(3j+1)-n^2(j+2)(3j+1)(3j+2)\\\\&amp;\\quad \\;+n^3(j+1)(3j+1)(3j+2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M174\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:msub><mml:mi>σ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mi>j</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>n</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi>j</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>+</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ49\"><alternatives><tex-math id=\"M175\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{p-1}k^3\\frac{D_k}{4^k} =\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2\\left( {\\begin{array}{c}3j\\\\ j\\end{array}}\\right) \\left( {\\begin{array}{c}p+j\\\\ 3j+1\\end{array}}\\right) }{16^k}\\frac{\\sigma _1}{(j+1)(3j+2)(3j+4)}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M176\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>4</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mfrac><mml:msub><mml:mi>σ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ50\"><alternatives><tex-math id=\"M177\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sigma _2=k(k+1)(k+3)+p(2-k^2)(3k+1)-p^2(k+2)(3k+1)(3k+2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M178\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>k</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq48\"><alternatives><tex-math id=\"M179\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p\\equiv 1\\ (\\mathrm{{mod}}\\ 3)$$\\end{document}</tex-math><mml:math id=\"M180\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mspace width=\"4pt\"/><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ51\"><alternatives><tex-math id=\"M181\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{p-1}k^3\\frac{D_k}{4^k}\\nonumber&amp;\\equiv \\sum _{k=0}^{\\frac{p-1}{2}} \\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2\\left( {\\begin{array}{c}3k\\\\ k\\end{array}}\\right) \\left( {\\begin{array}{c}p+k\\\\ 3k+1\\end{array}}\\right) }{16^k}\\frac{\\sigma _2}{(k+1)(3k+2)(3k+4)}\\\\&amp;\\equiv \\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{16^k}\\frac{p(1-pH_{2k} +pH_k)\\sigma _2}{(k+1)(3k+1)(3k+2)(3k+4)}+S_1\\quad \\ (\\mathrm{{mod}}\\ p^3), \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M182\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>4</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mfrac><mml:msub><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq49\"><alternatives><tex-math id=\"M183\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$S_1$$\\end{document}</tex-math><mml:math id=\"M184\"><mml:msub><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq50\"><alternatives><tex-math id=\"M185\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k=(p-4)/3$$\\end{document}</tex-math><mml:math id=\"M186\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ52\"><alternatives><tex-math id=\"M187\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} S_1&amp;=\\big (k(k+1)(k+3)+(2-k^2)(3k+1)p-(k+2)(3k+1)(3k+2)p^2\\big )\\\\&amp;\\quad \\;\\times \\frac{\\left( {\\begin{array}{c}-\\frac{1}{2}\\\\ k\\end{array}}\\right) ^2}{(k+1)(3k+2)}\\left( \\frac{\\left( {\\begin{array}{c}3k\\\\ k\\end{array}}\\right) \\left( {\\begin{array}{c}p+k\\\\ 3k+1\\end{array}}\\right) }{3k+4} -\\frac{1-pH_{2k}+pH_k}{3k+1}\\right) . \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M188\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:msub><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mi>k</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>×</mml:mo><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ4\"><label>3.3</label><alternatives><tex-math id=\"M189\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{(k+1)16^k}\\equiv 0\\ (\\mathrm{{mod}}\\ p^2),\\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M190\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ5\"><label>3.4</label><alternatives><tex-math id=\"M191\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2(H_{2k}-H_k)}{(3k+1)16^k} \\equiv 0\\ (\\mathrm{{mod}}\\ p). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M192\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ6\"><label>3.5</label><alternatives><tex-math id=\"M193\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{2p^2}{3}\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2(H_{2k}-H_k)}{(3k+2)16^k} \\equiv \\sum _{k=0}^{\\frac{p-1}{2}}\\frac{p\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{(3k+2)16^k}\\equiv -\\frac{p^2}{x^2}\\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M194\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ53\"><alternatives><tex-math id=\"M195\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{p-1}k^3\\frac{D_k}{4^k}+\\frac{184p^2x^2}{125}&amp;\\equiv \\frac{p}{27} \\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{16^k}\\left( \\frac{-8}{3k+1} +\\frac{21}{3k+2}-\\frac{10}{3k+4}\\right) \\\\&amp;\\quad \\;+\\frac{p^2}{3}\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{16^k} \\left( \\frac{-3}{k+1}+\\frac{7}{3k+2}+\\frac{1}{3k+4}\\right) \\\\&amp;\\quad \\;-p^3\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{16^k} \\left( \\frac{1}{k+1}-\\frac{2}{3k+4}\\right) \\\\&amp;\\quad \\;-\\frac{p^2}{27}\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2(H_{2k}-H_k)}{16^k}\\left( \\frac{-8}{3k+1}+\\frac{21}{3k+2}-\\frac{10}{3k+4}\\right) \\\\&amp;\\quad \\;-\\frac{p^3}{3}\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2 (H_{2k}-H_k)}{16^k}\\left( \\frac{-3}{k+1}+\\frac{7}{3k+2}+\\frac{1}{3k+4}\\right) \\\\&amp;\\equiv \\left( -\\frac{8}{27}{-}\\frac{10}{27}\\frac{4}{25}\\right) \\left( 4x^2{-}2p{-}\\frac{p^2}{4x^2}\\right) {-}\\frac{21}{27}\\frac{p^2}{x^2}{+}\\frac{p}{3}\\frac{4}{25}(4x^2-2p)\\\\&amp;\\quad \\;+2p^2\\frac{16x^2}{25}-\\frac{10p}{27}\\frac{18}{125} (4x^2-2p)+\\frac{21}{27}\\frac{3}{2}\\frac{p^2}{x^2}+\\frac{p^2}{3}\\frac{18}{125}4x^2\\\\&amp;\\equiv -\\frac{64x^2}{45}+\\frac{32p}{45}+\\frac{43p^2}{90x^2}+\\frac{184p^2x^2}{125}\\quad \\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M196\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>4</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>184</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>125</mml:mn></mml:mfrac></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>27</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>8</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>21</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mn>10</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>+</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mn>3</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>7</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>-</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mn>27</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>8</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>21</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mn>10</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mn>3</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>7</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mo>-</mml:mo><mml:mfrac><mml:mn>8</mml:mn><mml:mn>27</mml:mn></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mn>10</mml:mn><mml:mn>27</mml:mn></mml:mfrac><mml:mfrac><mml:mn>4</mml:mn><mml:mn>25</mml:mn></mml:mfrac></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>-</mml:mo><mml:mfrac><mml:mn>21</mml:mn><mml:mn>27</mml:mn></mml:mfrac><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mfrac><mml:mn>4</mml:mn><mml:mn>25</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mfrac><mml:mrow><mml:mn>16</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>25</mml:mn></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>10</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>27</mml:mn></mml:mfrac><mml:mfrac><mml:mn>18</mml:mn><mml:mn>125</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mn>21</mml:mn><mml:mn>27</mml:mn></mml:mfrac><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mn>3</mml:mn></mml:mfrac><mml:mfrac><mml:mn>18</mml:mn><mml:mn>125</mml:mn></mml:mfrac><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>64</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>45</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>32</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>45</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>43</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>90</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>184</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>125</mml:mn></mml:mfrac><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ7\"><label>3.6</label><alternatives><tex-math id=\"M197\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{p-1}k^3\\frac{D_k}{4^k}\\equiv -\\frac{64x^2}{45}+ \\frac{32p}{45}+\\frac{43p^2}{90x^2}\\quad \\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M198\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>4</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>64</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>45</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>32</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>45</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>43</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>90</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq51\"><alternatives><tex-math id=\"M199\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p\\equiv 2\\ (\\mathrm{{mod}}\\ 3)$$\\end{document}</tex-math><mml:math id=\"M200\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>≡</mml:mo><mml:mn>2</mml:mn><mml:mspace width=\"4pt\"/><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq52\"><alternatives><tex-math id=\"M201\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p&gt;5$$\\end{document}</tex-math><mml:math id=\"M202\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ54\"><alternatives><tex-math id=\"M203\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{p-1}k^3\\frac{D_k}{4^k}\\nonumber&amp;\\equiv \\sum _{k=0}^{\\frac{p-1}{2}} \\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2\\left( {\\begin{array}{c}3k\\\\ k\\end{array}}\\right) \\left( {\\begin{array}{c}p+k\\\\ 3k+1\\end{array}}\\right) }{16^k}\\frac{k(k+1)(k+3) +p(2-k^2)(3k+1)}{(k+1)(3k+2)(3k+4)}\\\\&amp;\\equiv \\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{16^k} \\frac{p(1-pH_{2k}+pH_k)(k(k+1)(k+3)+p(2-k^2)(3k+1))}{(k+1)(3k+1)(3k+2)(3k+4)}\\\\&amp;\\equiv \\frac{p}{27}\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{16^k} \\left( \\frac{-8}{3k+1}+\\frac{21}{3k+2}-\\frac{10}{3k+4}\\right) \\\\&amp;\\quad \\;+\\frac{p^2}{3}\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{16^k} \\left( \\frac{-3}{k+1}+\\frac{7}{3k+2}+\\frac{1}{3k+4}\\right) \\\\&amp;\\quad \\;-\\frac{p^2}{27}\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2(H_{2k}-H_k)}{16^k}\\left( \\frac{-8}{3k+1}+\\frac{21}{3k+2}-\\frac{10}{3k+4}\\right) \\;\\ (\\mathrm{{mod}}\\ p^2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M204\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>4</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mfrac><mml:mrow><mml:mi>k</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>27</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>8</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>21</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mn>10</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>+</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mn>3</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>7</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mn>27</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>8</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>21</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mn>10</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mfenced><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ55\"><alternatives><tex-math id=\"M205\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{(3k+1)16^k}\\equiv 0\\quad \\ (\\mathrm{{mod}}\\ p), \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M206\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ56\"><alternatives><tex-math id=\"M207\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{(3k+4)16^k}\\equiv 0\\quad \\ (\\mathrm{{mod}}\\ p). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M208\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:mn>0</mml:mn><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ57\"><alternatives><tex-math id=\"M209\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{p-1}k^3\\frac{D_k}{4^k}&amp;\\equiv \\frac{7p}{9}\\sum _{k=0}^{\\frac{p-1}{2}} \\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{(3k+2)16^k}+\\frac{7p^2}{3} \\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{(3k+2)16^k}\\\\&amp;\\quad \\;-\\frac{7p^2}{9}\\sum _{k=0}^{\\frac{p-1}{2}} \\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2(H_{2k}-H_k)}{(3k+2)16^k}\\quad \\ (\\mathrm{{mod}}\\ p^2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M210\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>4</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mn>7</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>9</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>7</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>3</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>7</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>9</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ8\"><label>3.7</label><alternatives><tex-math id=\"M211\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2(H_{2k}-H_k)}{16^k(3k+2)} \\equiv 3\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{16^k(3k+2)}\\quad \\ (\\mathrm{{mod}}\\ p), \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M212\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:mn>3</mml:mn><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ9\"><label>3.8</label><alternatives><tex-math id=\"M213\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} p\\sum _{k=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) ^2}{(3k+2)16^k} \\equiv 4R_3(p)\\quad \\ (\\mathrm{{mod}}\\ p^2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M214\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>p</mml:mi><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:mn>4</mml:mn><mml:msub><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ10\"><label>3.9</label><alternatives><tex-math id=\"M215\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{p-1}k^3\\frac{D_k}{4^k}\\equiv \\frac{28}{9}R_3(p)\\quad \\ (\\mathrm{{mod}}\\ p^2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M216\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>4</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mo>≡</mml:mo><mml:mfrac><mml:mn>28</mml:mn><mml:mn>9</mml:mn></mml:mfrac><mml:msub><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ58\"><alternatives><tex-math id=\"M217\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{p-1}k^3\\frac{D_k}{16^k}&amp;=\\sum _{k=0}^{p-1}\\frac{k^3}{16^k} \\sum _{j=0}^{k}(-1)^j\\left( {\\begin{array}{c}k+2j\\\\ 3j\\end{array}}\\right) \\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2\\left( {\\begin{array}{c}3j\\\\ j\\end{array}}\\right) 16^{k-j}\\\\&amp;=\\sum _{j=0}^{p-1}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2\\left( {\\begin{array}{c}3j\\\\ j\\end{array}}\\right) }{(-16)^j}\\sum _{k=j}^{p-1}k^3\\left( {\\begin{array}{c}k+2j\\\\ 3j\\end{array}}\\right) . \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M218\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>k</mml:mi></mml:munderover><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>j</mml:mi></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msup><mml:mn>16</mml:mn><mml:mrow><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>16</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>j</mml:mi></mml:msup></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ59\"><alternatives><tex-math id=\"M219\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=j}^{n-1}k^3\\left( {\\begin{array}{c}k+2j\\\\ 3j\\end{array}}\\right) =\\frac{\\sigma _3}{(3j+2)(3j+4)}\\left( {\\begin{array}{c}n+2j\\\\ 3j+1\\end{array}}\\right) , \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M220\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>σ</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ60\"><alternatives><tex-math id=\"M221\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sigma _3=j(1+2j)+2n(j+1)(3j+1)-2n^2(3j+1)(3j+2)+n^3(3j+1)(3j+2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M222\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>j</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ61\"><alternatives><tex-math id=\"M223\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sigma _4=j(1+2j)+2p(j+1)(3j+1)-2p^2(3j+1)(3j+2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M224\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>j</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq53\"><alternatives><tex-math id=\"M225\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p\\equiv 1\\ (\\mathrm{{mod}}\\ 3)$$\\end{document}</tex-math><mml:math id=\"M226\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>≡</mml:mo><mml:mn>1</mml:mn><mml:mspace width=\"4pt\"/><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq54\"><alternatives><tex-math id=\"M227\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p^3$$\\end{document}</tex-math><mml:math id=\"M228\"><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ62\"><alternatives><tex-math id=\"M229\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;\\sum _{k=0}^{p-1}k^3\\frac{D_k}{16^k}-\\frac{1}{18p(p+1)}\\left( {\\begin{array}{c}-1/2\\\\ \\frac{2p-2}{3}\\end{array}}\\right) ^2 \\left( {\\begin{array}{c}2p-2\\\\ \\frac{2p-2}{3}\\end{array}}\\right) \\left( {\\begin{array}{c}p+\\frac{4p-4}{3}\\\\ 2p-1\\end{array}}\\right) \\\\&amp;\\quad \\equiv \\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2\\left( {\\begin{array}{c}3j\\\\ j\\end{array}}\\right) \\left( {\\begin{array}{c}p+2j\\\\ 3j+1\\end{array}}\\right) }{(-16)^j}\\frac{\\sigma _4}{(3j+2)(3j+4)}\\\\&amp;\\quad \\equiv \\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2p(-1)^j (1+pH_{2j}-pH_j)}{(-16)^j}\\frac{\\sigma _4}{(3j+1)(3j+2)(3j+4)}+S_2, \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M230\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>18</mml:mn><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mo>≡</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>16</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>j</mml:mi></mml:msup></mml:mfrac><mml:mfrac><mml:msub><mml:mi>σ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mo>≡</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mi>p</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>j</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mn>16</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>j</mml:mi></mml:msup></mml:mfrac><mml:mfrac><mml:msub><mml:mi>σ</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq55\"><alternatives><tex-math id=\"M231\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$S_2$$\\end{document}</tex-math><mml:math id=\"M232\"><mml:msub><mml:mi>S</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq56\"><alternatives><tex-math id=\"M233\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k=(p-4)/3$$\\end{document}</tex-math><mml:math id=\"M234\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ63\"><alternatives><tex-math id=\"M235\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;S_2=-\\frac{\\left( {\\begin{array}{c}-\\frac{1}{2}\\\\ k\\end{array}}\\right) ^2(k(1+2k)+2p(k+1)(3k+1)-2p^2(3k+1)(3k+2))}{3k+2}\\\\&amp;\\quad \\;\\times \\left( \\frac{\\left( {\\begin{array}{c}3k\\\\ k\\end{array}}\\right) \\left( {\\begin{array}{c}p+2k\\\\ 3k+1\\end{array}}\\right) }{3k+4}+\\frac{1+pH_{2k}-pH_k}{3k+1}\\right) . \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M236\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>×</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ64\"><alternatives><tex-math id=\"M237\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;\\sum _{k=0}^{p-1}k^3\\frac{D_k}{16^k}-\\frac{1}{18p(p+1)}\\left( {\\begin{array}{c}-1/2\\\\ \\frac{2p-2}{3}\\end{array}}\\right) ^2 \\left( {\\begin{array}{c}2p-2\\\\ \\frac{2p-2}{3}\\end{array}}\\right) \\left( {\\begin{array}{c}p+\\frac{4p-4}{3}\\\\ 2p-1\\end{array}}\\right) \\\\&amp;\\equiv \\frac{p}{27}\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2}{16^j} \\left( \\frac{-1}{3j+1}+\\frac{-3}{3j+2}+\\frac{10}{3j+4}\\right) \\\\&amp;\\quad \\;+\\frac{p^2}{3}\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2}{16^j} \\left( \\frac{1}{3j+2}+\\frac{1}{3j+4}\\right) -2p^3\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2}{(3j+4)16^j}\\\\&amp;\\quad \\;+\\frac{p^2}{27}\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2}{16^j} \\left( \\frac{-1}{3j+1}+\\frac{-3}{3j+2}+\\frac{10}{3j+4}\\right) (H_{2j}-H_j)\\\\&amp;\\quad \\;+\\frac{p^3}{3}\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2}{16^j} \\left( \\frac{1}{3j+2}+\\frac{1}{3j+4}\\right) (H_{2j}-H_j)+\\frac{184p^2x^2}{125}\\\\&amp;\\equiv \\left( -\\frac{1}{27}+\\frac{10}{27}\\frac{4}{25}\\right) \\left( 4x^2-2p- \\frac{p^2}{4x^2}\\right) +\\frac{1}{9}\\frac{p^2}{x^2}+\\frac{p}{3}\\frac{4}{25}(4x^2-2p)\\\\&amp;\\quad \\;-2p^2\\frac{16x^2}{25}-\\frac{10p}{27}\\frac{18}{125}(4x^2-2p)+ \\frac{1}{9}\\frac{3}{2}\\frac{p^2}{x^2}-\\frac{p^2}{3}\\frac{18}{125}4x^2+\\frac{184p^2x^2}{125}\\\\&amp;\\equiv \\frac{4x^2}{45}-\\frac{2p}{45}+\\frac{49p^2}{180x^2}\\quad \\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M238\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>18</mml:mn><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>27</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>j</mml:mi></mml:msup></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>10</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>+</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mn>3</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>j</mml:mi></mml:msup></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>j</mml:mi></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>+</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mn>27</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>j</mml:mi></mml:msup></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>10</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mfenced><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>+</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mn>3</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>j</mml:mi></mml:msup></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mfenced><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>184</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>125</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>27</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>10</mml:mn><mml:mn>27</mml:mn></mml:mfrac><mml:mfrac><mml:mn>4</mml:mn><mml:mn>25</mml:mn></mml:mfrac></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>9</mml:mn></mml:mfrac><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:mfrac><mml:mfrac><mml:mn>4</mml:mn><mml:mn>25</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mfrac><mml:mrow><mml:mn>16</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>25</mml:mn></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>10</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>27</mml:mn></mml:mfrac><mml:mfrac><mml:mn>18</mml:mn><mml:mn>125</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>9</mml:mn></mml:mfrac><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mn>3</mml:mn></mml:mfrac><mml:mfrac><mml:mn>18</mml:mn><mml:mn>125</mml:mn></mml:mfrac><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>184</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>125</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>45</mml:mn></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>45</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>49</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>180</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ65\"><alternatives><tex-math id=\"M239\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\left( {\\begin{array}{c}2p-2\\\\ \\frac{2p-2}{3}\\end{array}}\\right) \\left( {\\begin{array}{c}p+\\frac{4p-4}{3}\\\\ 2p-1\\end{array}}\\right) \\equiv -2p\\quad \\ (\\mathrm{{mod}}\\ p^2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M240\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ66\"><alternatives><tex-math id=\"M241\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\left( {\\begin{array}{c}-1/2\\\\ \\frac{2p-2}{3}\\end{array}}\\right) ^2\\equiv \\frac{9p^2}{4x^2}\\quad \\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M242\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mn>3</mml:mn></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mn>9</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ11\"><label>3.10</label><alternatives><tex-math id=\"M243\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{p-1}k^3\\frac{D_k}{16^k}&amp;\\equiv \\frac{4x^2}{45}-\\frac{2p}{45} +\\frac{49p^2}{180x^2}-\\frac{p^2}{4x^2}\\nonumber \\\\&amp;=\\frac{4x^2}{45}-\\frac{2p}{45}+\\frac{p^2}{45x^2}\\quad \\ (\\mathrm{{mod}}\\ p^3). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M244\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>45</mml:mn></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>45</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>49</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>180</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>45</mml:mn></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>p</mml:mi></mml:mrow><mml:mn>45</mml:mn></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mn>45</mml:mn><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq57\"><alternatives><tex-math id=\"M245\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p\\equiv 2\\ (\\mathrm{{mod}}\\ 3)$$\\end{document}</tex-math><mml:math id=\"M246\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>≡</mml:mo><mml:mn>2</mml:mn><mml:mspace width=\"4pt\"/><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq58\"><alternatives><tex-math id=\"M247\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p&gt;5$$\\end{document}</tex-math><mml:math id=\"M248\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq59\"><alternatives><tex-math id=\"M249\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p=5$$\\end{document}</tex-math><mml:math id=\"M250\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq60\"><alternatives><tex-math id=\"M251\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p^2$$\\end{document}</tex-math><mml:math id=\"M252\"><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ67\"><alternatives><tex-math id=\"M253\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;\\sum _{k=0}^{p-1}k^3\\frac{D_k}{16^k}\\equiv \\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2}{16^j}\\frac{pj(1+2j)+ 2p^2(j+1)(3j+1)+p^2j(2j+1)(H_{2j}-H_j)}{(3j+1)(3j+2)(3j+4)}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M254\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac><mml:mo>≡</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>j</mml:mi></mml:msup></mml:mfrac><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mi>j</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>j</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ68\"><alternatives><tex-math id=\"M255\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{k=0}^{p-1}k^3\\frac{D_k}{16^k}&amp;\\equiv \\frac{p}{27}\\sum _{j=0}^{\\frac{p-1}{2}} \\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2}{16^j}\\left( \\frac{-1}{3j+1}+\\frac{-3}{3j+2}+\\frac{10}{3j+4}\\right) \\\\&amp;\\quad \\;+\\frac{p^2}{3}\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2}{16^j} \\left( \\frac{1}{3j+2}+\\frac{1}{3j+4}\\right) \\\\&amp;\\quad \\;+\\frac{p^2}{27}\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2}{16^j} \\left( \\frac{-1}{3j+1}+\\frac{-3}{3j+2}+\\frac{10}{3j+4}\\right) (H_{2j}-H_j)\\\\&amp;\\equiv -\\frac{p}{9}\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2}{(3j+2)16^j} +\\frac{p^2}{3}\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2}{(3j+2)16^j}\\\\&amp;\\quad \\;-\\frac{p^2}{9}\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2}{(3j+2)16^j}(H_{2j}-H_j)\\\\&amp;\\equiv -\\frac{p}{9}\\sum _{j=0}^{\\frac{p-1}{2}}\\frac{\\left( {\\begin{array}{c}2j\\\\ j\\end{array}}\\right) ^2}{(3j+2)16^j}\\equiv -\\frac{4}{9}R_3(p)\\quad \\ (\\mathrm{{mod}}\\ p^2). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M256\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:msup><mml:mi>k</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:msup><mml:mn>16</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mfrac></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mi>p</mml:mi><mml:mn>27</mml:mn></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:munderover><mml:mfrac><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mn>16</mml:mn><mml:mi>j</mml:mi></mml:msup></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mo>-</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mn>10</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd/><mml:mtd 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open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mn>2</mml:mn><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>j</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mn>16</mml:mn><mml:mi>j</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>≡</mml:mo><mml:mo>-</mml:mo><mml:mfrac><mml:mn>4</mml:mn><mml:mn>9</mml:mn></mml:mfrac><mml:msub><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"4pt\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">mod</mml:mi><mml:mspace width=\"4pt\"/><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq61\"><alternatives><tex-math id=\"M257\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M258\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>" ]

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[ "<fn-group><fn><p>Guo-Shuai Mao was supported by the National Natural Science Foundation of China (grant nos. 12001288 and 12071208) and China Scholarship Council (grant no. 202008320187). Michael J. Schlosser was partially supported by FWF Austrian Science Fund grant P 32305.</p></fn><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p id=\"Par5\">Neutropenia is a risk factor for certain infections that may appear as a frequent and severe complication in oncological patients; its only typical symptom is fever [##REF##16925575##1##].</p>", "<p id=\"Par6\">This entity has classically been treated through hospitalized management and intravenous (i.v.) administration of wide-spectrum antibiotics in all patients. Currently, only a fraction of patients with febrile neutropenia present serious complications, and there is evidence of a diverse prognostic spectrum [##REF##23358983##2##–##REF##8490912##10##]. In light of this fact, the main scientific societies at a national and international level, including the American Society for Clinical Oncology (ASCO) [##REF##29461916##11##] and the Spanish Society for Medical Oncology (SEOM) [##REF##30470991##12##], make a distinction between patients with febrile neutropenia who are at high or low risk, depending on defined criteria set out in globally validated scales like the ones from the Talcott group [##REF##21931024##3##], the Multinational Association of Supportive Care in Cancer (MASCC) [##REF##10944139##13##], and the clinical index of stable febrile neutropenia (CISNE) [##REF##25559804##14##]. Better understanding of these different risk profiles, plus the emergence of new care models like home hospital care, have raised the possibility of less aggressive treatment options, for instance simplified oral monotherapy and immediate or early discharge following an observation period.</p>", "<p id=\"Par7\">This qualitative systematic review aimed to assess ambulatory care strategies in adult patients with febrile neutropenia through a triple aim, examining efficacy, cost-effectiveness, and quality of life. These results can inform the definition of the most adequate setting for managing these patients (ambulatory care, home hospitalization, or inpatient care); contribute to updating the action protocols in emergency services; and identify areas where there is uncertainty in the available evidence in order to trace new lines of research.</p>" ]

[ "<title>Methods</title>", "<title>Design and search strategy</title>", "<p id=\"Par8\">Using the MEDLINE (PubMed), Embase, and Cochrane Library databases, a search was undertaken of all meta-analyses, clinical trials, and observational studies published up to February 2021. We also performed backward reference searching on the bibliographies of the most recent ASCO [##REF##29461916##11##] and SEOM [##REF##30470991##12##] clinical guidelines to look for additional studies that met our inclusion criteria.</p>", "<p id=\"Par9\">The search strategy used the following terms:</p>", "<title>Study selection</title>", "<p id=\"Par11\">Eligible studies were those comparing in-hospital versus ambulatory management with immediate or early discharge, those assessing simple or combined treatment regimens, and those aiming to define the most appropriate administration route. We also included studies that analyzed the costs of different interventions or evaluated patient-reported quality of life according to the applied treatment strategy.</p>", "<p id=\"Par12\">Studies in pediatric or adolescent populations (&lt; 18 years of age) were excluded, as oncological pathologies have peculiarities in this group that are not generalizable to adults.</p>" ]

[ "<title>Results</title>", "<p id=\"Par13\">The results of the search and the study flow chart are presented in Fig. ##FIG##0##1##.</p>", "<title>Description of included studies</title>", "<p id=\"Par14\">The criteria for defining the concepts of fever and neutropenia in different studies are presented in annex tables ##SUPPL##0##2## and ##SUPPL##0##3##. Included studies used heterogeneous criteria to define risk levels in their samples (Table ##TAB##0##1##).\n</p>", "<p id=\"Par15\">At present, there are three available scales for estimating the risk of medical complications in patients with febrile neutropenia: the rules developed by Talcott et al. [##REF##21931024##3##] and the MASCC [##REF##10944139##13##] (from 0 to 26 points, a score ≥ 21 is predictive of low risk) and CISNE [##REF##25559804##14##] (from 0 to 8 points, with three levels of risk: I or low ≥ 2; II or intermediate ≥ 4; and III or high &gt; 4). These validated scoring systems assume neutropenia and fever status for certain patients without regard to the severity or duration of the neutropenia as predictors of medical complications, which could require or prolong hospitalization.</p>", "<p id=\"Par16\">Validated criteria have been used to identify patients with febrile neutropenia at low risk in clinical trials (since 2008) and observational studies (since 2010). Except for the studies by Talcott et al. [##REF##21931024##3##], Mizuno et al. [##REF##16941132##15##], and Rolston et al. [##REF##16628654##16##], which use the Talcott criteria [##REF##21931024##3##], all other clinical trials and observational studies have used the MASCC score, generally with a cutoff of 21 points to define low risk (Kern et al. [##REF##16925575##1##] used a cutoff of 20 points). No included studies used the CISNE scale. However, some studies have employed clinical, biochemical, and sometimes psychosocial criteria to establish risk levels, including the studies by Rubenstein et al. [##REF##8490912##10##] and Hidalgo et al. [##REF##9921995##6##]. Likewise, Weycker et al. [##REF##23824496##21##] also did not use a specific scale in their retrospective cohort study, instead choosing treatment arms based on the diagnostic setting.</p>", "<p id=\"Par17\">In all included clinical trials, the ratio of patients with hematological neoplasms (leukemia or lymphoma) to solid tumors was approximately 1:4 to 1:3. All but two observational studies assessing a low-risk intervention protocol included only patients with solid tumors: the ratio in the study by Poprawski et al. [##REF##30517995##20##] was 1:3, while the distribution between tumor types in the study by Goodman et al. [##REF##28282274##22##] was roughly equal.</p>", "<p id=\"Par18\">Antibiotic treatments used, administration routes, and treatment settings are presented in annex tables ##SUPPL##0##2 ##and ##SUPPL##0##3##. Moreover, the samples of almost all clinical trials and observational studies included 25 to 38% of people being treated with colony-stimulating factors (CSFs), except for Innes et al. [##REF##12838298##5##], who excluded them.</p>", "<p id=\"Par19\">Between 6 and 20% of patients treated on an ambulatory basis were readmitted.</p>", "<title>Analysis of efficacy</title>", "<p id=\"Par20\">As for the results of the efficacy analysis, Rubenstein et al. [##REF##7872337##9##], Minotti et al. [##REF##9921995##6##], and Sebban et al. [##REF##18197434##4##] did not find differences between oral versus intravenous administration in low-risk patients with neutropenia. The latter two studies evaluated a monotherapy antibiotic regimen. Malik et al. [##REF##7872337##9##] found that oral fluoroquinolone monotherapy was equally effective whether administered in hospitals or other settings. Rapoport et al. [##REF##10567777##8##] likewise obtained similar results in patients receiving classical inpatient care or early discharge with ambulatory therapeutic management. In line with these results, Hidalgo et al. [##REF##9921995##6##] did not find differences in treatment response between an inpatient i.v. double regimen versus oral monotherapy, administered in ambulatory care following good response to the first dose in hospital and early discharge. These findings were also reproduced by Innes et al. in 2003 [##REF##12838298##5##], who compared double i.v. therapy in-hospital versus double oral treatment in ambulatory care. For their part, Talcott et al. [##REF##21931024##3##] did not observe a higher rate of complications using an ambulatory strategy. Finally, Kern et al. [##REF##23358983##2##] reported similar results between double versus monotherapy in the ambulatory setting.</p>", "<p id=\"Par21\">In the observational studies, Mizuno et al. [##REF##16941132##15##] suggested that telephone follow-up could be an effective strategy to support ambulatory management. Cooksley et al. [##REF##29675545##23##] drew similar conclusions, speculating that ambulatory treatment could be more feasible if clinical oncological support were available; this group provided such support in the form of an oncologist on call through a 24 h hotline. Rolston et al. [##REF##16628654##16##] evaluated an early discharge strategy with gatifloxacin monotherapy 400 mg versus moxifloxacin 400 mg, suggesting that quinolone monotherapy could be a safe and effective intervention in low-risk patients selected using the Talcott criterio [##REF##21931024##3##] and the MASCC score, respectively. In addition, Hocking et al. [##REF##23668271##19##] implemented an ambulatory management strategy aligned with Australian practice guidelines in 2012, concluding that this could be a valid strategy for selected patients with solid tumors and low-risk febrile neutropenia. In their retrospective analysis, Weycker et al. [##REF##23824496##21##] obtained similar results in a sample of patients with mostly solid tumors, a small proportion of whom were treated exclusively as outpatients. Goodman et al. in 2017 [##REF##28282274##22##] and Poprawski in 2018 [##REF##30517995##20##] also reached the same conclusion based on their studies, with ratios of 1:1 and 1:3, respectively, of hematological to solid tumors. Goodman et al. [##REF##28282274##22##] noted that applying social and non-medical criteria in addition to the MASCC score complicated ambulatory treatment approaches in low-risk patients with febrile neutropenia. In the same line as Hocking et al. [##REF##23668271##19##], Lingaratnam et al. [##REF##23809725##17##] analyzed the successful implementation of the Australian protocol in a center in Victoria. Even though there was a sizable proportion of patients who could not initially be classified as low risk, their evolution permitted a change to the oral treatment group, and none presented treatment failure.</p>", "<title>Cost-effectiveness analysis</title>", "<p id=\"Par22\">Our search strategy yielded a total of 8 articles with a focus on cost-effectiveness: 1 clinical trial, 2 cohort studies, and 5 other observational studies. The characteristics of the studies (research group, year of publication, country, definition of febrile neutropenia, mean age, design, perspective, source of economic data, length of hospital stay, direct and indirect costs, and summary of results) are shown in annex table ##SUPPL##0##4##.</p>", "<p id=\"Par23\">This analysis did not use data from other included studies that mentioned cost savings without providing a detailed explanation of the methods used to calculate them, as the 8 studies that are included in this analysis did.</p>", "<p id=\"Par24\">Three studies analyzed the direct costs derived from inpatient management of febrile neutropenia. Kuderer et al. [##REF##16575919##24##] reported that the health care expenditure from each episode of febrile neutropenia arose mainly from the costs of the hospital stay along with associated infections and comorbidities. Mayordomo et al. [##REF##19722781##25##] went into more detail, itemizing the costs for each classical intervention and showing that hospitalization accounted for 79% of the costs, followed by antibiotic treatment (10%), CSF (5%), complementary tests (4%), and transfusions (1%). Larger costs were incurred (in-hospital stay, interventions, and complementary tests) in patients with lymphoma compared to breast or lung cancers. Finally, O’Brien et al. [##REF##24472035##26##] added that health care costs in patients aged 65 years or older were significantly higher.</p>", "<p id=\"Par25\">Five included studies assessed the economic impact of an ambulatory-based treatment strategy. In 2000, Elting et al. [##REF##11054443##27##] found that an ambulatory strategy incurred lower costs than i.v. treatment during admission. In their 2008 cohort [##REF##18235119##28##], the same group found no differences in the efficacy of hospital versus ambulatory treatment, and even though just 21% of the ambulatory group were readmitted, the cost in that arm was double. In 2011, Hendricks et al. [##REF##21931037##29##] reported savings of USD 5354 using an ambulatory strategy with early discharge, considering the costs derived from the initial emergency department consultation, complementary studies, home interventions, pharmacological treatment, and daily nurse home visits. In 2017, Teh et al. [##REF##29018966##30##] reported 2.65 times lower costs after applying an ambulatory strategy in low-risk patients compared to a comparable historic cohort in the same hospital. Finally, Borget et al. [##REF##25373692##31##] aimed to identify where the largest economic burden came from, comparing the standard hospital strategy versus outpatient treatment in hospital and ambulatory treatment outside of it, reporting successive decreases in costs in each setting.</p>", "<title>Analysis of patient-reported quality of life</title>", "<p id=\"Par26\">Patients’ perceived health was only specifically analyzed in relation to ambulatory treatment strategies by Teuffel et al. in 2011 [##REF##22350594##32##], who asked cancer patients to hypothetically consider what their preferred treatment strategy would be if they presented febrile neutropenia during the course of their disease. Authors observed that patients generally preferred early discharge plus ambulatory treatment with oral rather than i.v. administration. Patients were willing to renounce up to 10 weeks of their life and pay between USD 255 and USD 327 more if they could avoid hospitalization. However, in individual interviews, some perceptions changed substantially. Details of this research are presented in annex table ##SUPPL##0##5##.</p>", "<p id=\"Par27\">In addition, two of the included clinical trials—one by the Talcott group in 2011 and another by Sebban et al. in 2008 [##REF##18197434##4##]—performed a simple quality of life analysis using two questionnaires. The first applied the European Organization for Research and Treatment of Cancer quality of life questionnaire (EORTC QLQ-C30), a validated, self-administered instrument, finding a significant reduction in the perception of pain in patients from the ambulatory treatment arm. The second applied a quality of life questionnaire 24 h after finalizing treatment, finding no differences between intervention groups.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par28\">Since at least the early 1990s, with Rubenstein et al. [##REF##8490912##10##], and possibly even before then, there has been recognition that the risk of complications is heterogeneous in populations with febrile neutropenia. The few studies available on this topic have been characterized by the diversity of their aims, approaches, and patient selection criteria. Factors contributing to this heterogeneity include evolving knowledge on the entity; the emergence of new oncological treatments (chemotherapies, CSFs, antibiotics); the growing tendency to simplify treatments to favor adherence and tolerability and to reduce adverse effects; and the decentralization of health care away from hospitals due to associated comorbidity, impaired quality of life, and increased cost.</p>", "<p id=\"Par29\">Indeed, even the definition of fever itself and the best place to measure it have changed several times since Carl Wunderlich’s early work on it in 1868. This branching evolution is evident in the current definitions of the Infectious Diseases Society of America and other societies in South America, Europe, and Asia, and in the different ways that our included studies identify it; however, it is not necessarily reflected in the publication year of our included studies.</p>", "<p id=\"Par30\">These changes are also reflected in the fact that Minotti et al. [##REF##10335931##7##] excluded patients with neutrophil counts under 300/μL because at that time, severe neutropenia, especially with neutrophil counts of less than 100/μL, was considered a marker of high risk. Later studies do not reflect the application of this criterion. Indeed, the interpretation of our results must consider the variations in the neutrophil count of included and excluded patients. Kern et al. [##REF##16925575##1##], Sebban et al. [##REF##18197434##4##], Innes et al. [##REF##12838298##5##], Hidalgo et al. [##REF##9921995##6##], and Malik et al. [##REF##7872337##9##] all included patients with moderate neutropenia (500–1000/μL) with the potential to become severe neutropenia (&lt; 500/μL). However, only Malik et al. [##REF##7872337##9##] clarified that the patients whose neutrophil counts did not drop below the 500/μL threshold were excluded. However, given the inclusion of patients with critical levels of neutropenia in other studies, this distinction may be irrelevant.</p>", "<p id=\"Par31\">Another fact suggestive of the diversity of selection criteria is that despite criticism of Talcott’s group and the MASCC and CISNE scales for not considering the neutrophil count as a criterion for risk assessment, the risk associated with this aspect does not seem to determine the appropriateness of managing neutropenia on an ambulatory basis. Indeed, good outcomes were achieved with simplified therapies in the ambulatory setting regardless of the criteria for selecting candidates. In the studies that did report the proportion of patients with neutrophil counts that dipped below 100/μL (Kern et al. [##REF##23358983##2##], Talcott et al. [##REF##21931024##3##], Hidalgo et al. [##REF##9921995##6##], Minotti et al. [##REF##10335931##7##], Malik et al. [##REF##7872337##9##], and Rubenstein et al. [##REF##8490912##10##]), these proportions were described as similar between groups in all studies but Rubenstein et al.’s, who did not specify.</p>", "<p id=\"Par32\">Regarding the risk assessment scales, the ASCO 2018 [##REF##29461916##11##] guidelines recommend the use of MASCC followed by CISNE, while the SEOM 2018 [##REF##30470991##12##] guidelines prefer only the CISNE. However, the included studies did not evaluate the latter scale as a method to identify candidates for ambulatory care. Moreover, CISNE has not been validated in hematological patients, restricting the subgroups eligible for less aggressive interventions. Furthermore, Ahn et al. [##REF##29168032##33##] demonstrated that the MASCC had greater discriminatory power than the CISNE for detecting low-risk patients, with greater sensitivity and negative predictive value, resulting in a smaller likelihood of including patients at true high risk of complications in ambulatory regimes. Thus, in 2013, Ahn et al. [##REF##23519568##34##] already proposed procalcitonin values as an additional parameter for the MASCC to refine the probability of presenting bacteremia or septic shock, in patients at both low and high risk. Even within the latter group, the authors recognized heterogeneous prognoses, calling into question the need for indiscriminately applying aggressive management.</p>", "<p id=\"Par33\">The evidence on antibiotic treatment approaches, both for double therapy with amoxycillin/clavulanic acid (or clindamycin in patients with allergies) plus quinolone and for monotherapy with fluoroquinolone, shows good efficacy outcomes in and outside of the hospital setting (Kern et al. [##REF##23358983##2##], Sebban et al. [##REF##18197434##4##], Hidalgo et al. [##REF##9921995##6##], Minotti et al. [##REF##10335931##7##], Malik et al. [##REF##7872337##9##]). Despite these results, the ASCO guidelines [##REF##29461916##11##] continue advising against fluoroquinolone monotherapy for ambulatory management of low-risk patients, while SEOM affirms that the only alternative in penicillin-allergic patients is hospital admission and i.v. treatment.</p>", "<p id=\"Par34\">In 20 to 30% of febrile neutropenia cases, the germs responsible are identified, predominantly gram-positive rather than gram-negative bacteria, followed by polymicrobial and fungal infections (Malik et al. [##REF##7872337##9##], Rapoport et al. [##REF##10567777##8##], Hidalgo et al. [##REF##9921995##6##], Minotti et al. [##REF##10335931##7##], Sebban et al. [##REF##18197434##4##]). These results are consistent with other available studies in the literature.</p>", "<p id=\"Par35\">With respect to the use of CSFs, it has been known for decades, as reported by studies like Mather et al. in 1994 [##REF##7520676##35##], that these interventions are capable of reducing the mean duration of the neutropenia by 1 day as well as the total time needed for resolution, although not the total days of fever. No study has explained how authors decided to administer this treatment, describing the criteria, type of neoplasm, level of neutrophils, or timing of administration (from the last session of chemotherapy or emergency presentation). Similarly, the literature has not explored how CSFs may have influenced the outcome (patient recovery or the success of the treatment strategy). There are also gaps in research investigating whether prescribing CSFs would result in savings given the implicit pharmacological costs associated with hospitalizing these patients in a hospital or their home, and whether it would be possible to reduce the duration of antibiotic therapy in patients with a fever of unknown origin and negative culture.</p>", "<p id=\"Par36\">Several strategies for ambulatory follow-up have been evaluated, including home visits from health professionals, hotlines, scheduled telephone calls, and outpatient hospital visits. However, no studies have compared these interventions with each other to assess treatment response, adverse effects, cost-efficiency, or quality of life. These aspects could be the focus of future lines of research.</p>", "<p id=\"Par37\">Although teenagers were not included as population in our systematic review, because we excluded studies with population under 19 years old in our criteria research, in many countries, they are treated around 16 years old in emergency departments as adults. Authors, such as Klastersky et al. [##REF##16943529##36##], carried on an early discharge strategy after 24 h, til 48 h in many of the episodes of febrile neutropenia in people that includeed 16 years old teenagers [##REF##16943529##36##].</p>", "<p id=\"Par38\">Their results show that we could safely proceed in low-risk teenagers according to MASK score, in the same way as adults with an oral combination therapy of peniciline and quinolones, if not allergic, with a proposed followed up by phone, temperature control each/6 h, and blood control test each 48 h until accomplishing criteria of recovering. We should highlight in this investigation team, they first give a definition more than absence of hospital readmission as a criteria of succes of oral empirical therapy as 5 days without fever, no clinic symptoms, and signs of infection and pathogen erradication.</p>", "<p id=\"Par39\">Within the sphere of ambulatory management, four studies proposed that patients receive hemograms every 2 to 3 days after discharge, at least until achieving neutrophil counts of 1000/μL, indicating mild neutropenia (Goodman et al. [##REF##28282274##22##], Kern et al. [##REF##23358983##2##], Sebban et al. [##REF##18197434##4##], Hocking et al. [##REF##23668271##19##]). Although several studies have evaluated the cost of ambulatory follow-up, there has been no calculation of the specific cost of such controls, which could generate substantial amounts of research information. It may be appropriate to restrict these control visits to cases with poor evolution or to limit them to one visit at the end of treatment.</p>", "<p id=\"Par40\">Another strategy that could be analyzed for its efficacy and cost-efficiency would be the performance of a preliminary assessment of oncological patients’ household situation (including family support) by home hospital team, prior to initiating chemotherapy. This assessment could help facilitate decision-making in emergency departments around managing these patients.</p>", "<p id=\"Par41\">Many studies have limited ambulatory approaches to patients who live at some distance from the reference center or hospital (Kern et al. [##REF##23358983##2##], Hidalgo et al. [##REF##9921995##6##], Talcott et al. [##REF##21931024##3##], Minotti et al. [##REF##10335931##7##], Rubenstein et al. [##REF##8490912##10##], Rolston et al. [##REF##19387695##18##], Lingaratnam et al. [##REF##23809725##17##]). To reduce the population of low-risk patients excluded from this strategy, future research could also assess the efficacy, patient satisfaction, and cost-efficiency associated with strategies in coordination with primary health care teams, for patients who live far from the hospital and whose only limitation for inclusion in an ambulatory regime is distance.</p>", "<p id=\"Par42\">Available studies suggest than an ambulatory strategy for low-risk patients with febrile neutropenia is cost-efficient as well as effective. This is true for interventions assessed both in clinical trials (Hendricks et al. [##REF##21931037##29##], Elting et al. [##REF##11054443##27##]) and in prospective studies, for example, in those evaluating the implementation of new protocols compared to data from historical cohorts fulfilling the same criteria (Teh et al. [##REF##29018966##30##], Mayordomo et al. [##REF##19722781##25##]). Economic studies should be considered in light of potential differences in the source of economic data and their perspective. Establishing the most appropriate design to allow comparison with other studies can be complicated because health systems are not managed the same across different countries, nor is access to different therapeutic options the same in privately managed systems. Thus, in future studies, it will be necessary to establish quality criteria in order to favor reliability and comparability of the economic data obtained.</p>", "<p id=\"Par43\">With regard to quality of life, Teuffel et al. [##REF##21468048##37##] suggested that patients prefer ambulatory strategies, although data did not show differences in the short quality of life evaluations performed in two clinical trials in people with febrile neutropenia. In any case, more structured trials that specifically aim to assess this domain are needed before drawing any firm conclusions.</p>", "<p id=\"Par44\">Despite the positive evidence, fewer patients benefit from ambulatory strategies than expected (for clinical, socioeconomic, and geographic reasons). Health services still seem reluctant to implement the action protocols in low-risk patients that are recommended by national and international clinical practice guidelines, even though these tightly restrict the application of ambulatory models.</p>", "<p id=\"Par45\">Available evidence on efficacy, quality of life, and cost-efficiency seems to support ambulatory approaches for low-risk patients with febrile neutropenia. The main societies and services involved in managing these patients should establish more standardized priorities and criteria to guide future research and reach more definitive conclusions. Studies are needed to analyze the facillitators and barriers for the instigation and maintenance of an ambulatory febrile neutropenia management programmes.</p>" ]

[]

[ "<title>Purpose</title>", "<p id=\"Par1\">Recent clinical practice guidelines have recommended ambulatory management of febrile neutropenia in patients with low risk of complications. Although some centers have begun developing management protocols for these patients, there appears to be a certain reluctance to implement them in clinical practice. Our aim is to evaluate the strengths and weaknesses of this strategy according to available evidence and to propose new lines of research.</p>", "<title>Methods</title>", "<p id=\"Par2\">Systematic review using a triple aim approach (efficacy, cost-effectiveness, and quality of life), drawing from literature in MEDLINE (PubMed), Embase, and Cochrane Library databases. The review includes studies that assess ambulatory management for efficacy, cost-efficiency, and quality of life.</p>", "<title>Results</title>", "<p id=\"Par3\">The search yielded 27 articles that met our inclusion criteria.</p>", "<title>Conclusion</title>", "<p id=\"Par4\">In conclusion, based on current evidence, ambulatory management of febrile neutropenia is safe, more cost-effective than inpatient care, and capable of improving quality of life in oncological patients with this complication. Ambulatory care seems to be an effective alternative to hospitalization in these patients.</p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s00520-023-08065-y.</p>", "<title>Keywords</title>" ]

[ "<title>Supplementary information</title>", "<p>\n</p>" ]

[ "<title>Author contribution</title>", "<p>All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by EF-Q. The first draft of the manuscript was written by EF-Q, DO-B, and CL-Q. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.</p>", "<p>EF-Q had the idea for the article; EF-Q, DO-B, and CL-Q carried out the bibliographic search and data analysis; and EF-Q, DO-B, and CL-Q wrote and/or critically reviewed the article.</p>", "<title>Funding</title>", "<p>Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.</p>", "<title>Data availability</title>", "<p>N/A.</p>", "<title>Code availability</title>", "<p>N/A.</p>", "<title>Declarations</title>", "<title>Ethics approval</title>", "<p id=\"Par46\">Approval was obtained from the ethics committee of University Miguel Hernandez de Elche. The procedures used in this study adhere to the tenets of the Declaration of Helsinki.</p>", "<title>Consent to participate</title>", "<p id=\"Par47\">Personal information is unidentifiable in this present study; and patient consent cannot be obtained.</p>", "<title>Consent for publication</title>", "<p id=\"Par48\">N/A.</p>", "<title>Conflict of interest</title>", "<p id=\"Par49\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Study flow chart</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1.</label><caption><p>Exclusion criteria for low-risk treatment group, by organ, organ system, or medical specialty</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Criteria</th><th>Studies</th></tr></thead><tbody><tr><td><italic>Cardiovascular system</italic></td><td/></tr><tr><td>  Heart failure</td><td>Hidalgo et al. 1999 [##REF##9921995##6##]</td></tr><tr><td>  Hypotension</td><td>Talcot et al. 2011 [##REF##21931024##3##], Hidalgo et al. 1999 [##REF##12838298##5##], Minoti et al. 1995 [##REF##10335931##7##], Rubenstein et al. 1993 [##REF##7872337##9##] (&lt; 90 mmHg), and Mizuno et al. 2006 [##REF##16941132##15##]</td></tr><tr><td>  Taking drugs causing QT prolongation</td><td>Sotalol and dofetilide in Rolston et al. 2003 [##REF##16628654##16##]</td></tr><tr><td><italic>Respiratory system</italic></td><td/></tr><tr><td>  Tachypnea/respiratory insufficiency</td><td>Talcott et al. 2011 [##REF##21931024##3##], Minotti et al. 1999 [##REF##10335931##7##] and Rubenstein et al. 1993 [##REF##8490912##10##] (&gt; 30 bpm), and Mizuno et al. 2006 [##REF##16941132##15##]</td></tr><tr><td><italic>Nervous system/psychiatry</italic></td><td/></tr><tr><td>  Impaired consciousness</td><td>Talcot et al. 2011 [##REF##21931024##3##], Hidalgo et al. 1999 [##REF##9921995##6##]</td></tr><tr><td>  Unfavorable psychological status</td><td>Sebban et al. 2008 [##REF##18197434##4##], Rapoport et al. 1999 [##REF##10567777##8##], Linaratnam et al. 2013 [##REF##23809725##17##]</td></tr><tr><td><italic>Kidney</italic></td><td/></tr><tr><td><p>  Acute kidney failure</p><p>  Oliguria/anuria</p></td><td><p>Talcott et al. 2011 [##REF##21931024##3##], Rapoport et al. 1999 [##REF##10567777##8##], Malik et al. 1995 [##REF##7872337##9##] (dialysis); Minotti et al. 1999 [##REF##10335931##7##] (eGFR &lt; 30 ml/min); Rolston et al. 2010 [##REF##19387695##18##], Rolston et al. 2003 [##REF##16628654##16##], and Rubenstein et al. 1993 [##REF##8490912##10##] (eGFR &lt; 50 ml min or creatinine &gt; 2.5 mg/dl)</p><p>Hidalgo et al. 1999 [##REF##9921995##6##]</p></td></tr><tr><td><italic>Liver</italic></td><td/></tr><tr><td>  Liver failure</td><td>Rolston et al. 2010 [##REF##19387695##18##], Rolston et al. 2003 [##REF##16628654##16##], Malik et al. 1995 [##REF##7872337##9##], Rubenstein et al. 1993 [##REF##8490912##10##], and Hocking et al. 2012 [##REF##23668271##19##] (AST, ALT &gt; 4× normal), Sebban et al. 2008 [##REF##21931024##3##] (AST, ALT &gt; 5× normal)</td></tr><tr><td><italic>Digestive system</italic></td><td/></tr><tr><td>  Nausea, vomiting, diarrhea</td><td>Mizuno et al. 2006 [##REF##16941132##15##]</td></tr><tr><td>  Severe mucositis</td><td>Hocking et al. 2012 [##REF##23668271##19##] and Mizuno et al. 2006 [##REF##16941132##15##]</td></tr><tr><td>  Dysphagia</td><td>Talcott et al. 2011 [##REF##21931024##3##], Hidalgo et al. 1999 [##REF##9921995##6##], Malik et al. 1995 [##REF##7872337##9##], and Poprawski et al. 2018 [##REF##30517995##20##]</td></tr><tr><td><italic>Endocrine system</italic></td><td/></tr><tr><td>  Malnutrition</td><td>Lingaratnam et al. 2013 [##REF##23809725##17##] and Mizuno et al. 2006 [##REF##16941132##15##]</td></tr><tr><td><italic>Metabolism</italic></td><td/></tr><tr><td>  Acidosis</td><td>Hidalgo et al. 1999 [##REF##9921995##6##]</td></tr><tr><td>  Hyponatremia</td><td>Rubenstein et al. 1993 [##REF##8490912##10##] (&lt; 128 mg/dl)</td></tr><tr><td>  Uncontrolled hypercalcemia</td><td>Rubenstein et al. 1993 [##REF##8490912##10##]</td></tr><tr><td><italic>Hematology</italic></td><td/></tr><tr><td>  Altered coagulation</td><td>Hidalgo et al. 1999 [##REF##9921995##6##]</td></tr><tr><td>  Prolonged neutropenia</td><td>Malik et al. 1995 [##REF##7872337##9##] (&gt; 7 d), Lingaratnam et al. 2013 [##REF##23809725##17##] (&gt; 7 d)</td></tr><tr><td>  Use of granulocyte colony-stimulating factors</td><td>Minotti et al. 1999 [##REF##10335931##7##]</td></tr><tr><td>  Transplant patients with hematopoietic parents</td><td>Rapoport et al. 1999 [##REF##10335931##7##], Minotti et al. 1999 [##REF##10335931##7##], and Weyckert et al. 2014 [##REF##23824496##21##]</td></tr><tr><td>  Absolute neutrophil count ≤ 0.5 × 10⁹/L 24 h</td><td>Rapoport et al. 1999 [##REF##10567777##8##]</td></tr><tr><td><italic>Infectious diseases</italic></td><td/></tr><tr><td>  HIV</td><td>Rapoport et al. 1999 [##REF##10567777##8##]</td></tr><tr><td>  Confirmed/severe site infection</td><td>Poprawski et al. 2018 [##REF##30517995##20##], Lingaratnam et al. 2013 [##REF##23809725##17##], Rolston et al. 2010 [##REF##19387695##18##], and Hidalgo et al. 1999 [##REF##9921995##6##]</td></tr><tr><td>  Previous use of antibiotics</td><td>14 days prior to recruitment, Rapoport et al. 1999 [##REF##10567777##8##]; 5 days prior, Minotti et al. 1999 [##REF##10335931##7##]; 4 days prior, Sebban et al. 2008 [##REF##18197434##4##] and Kern et al. 2013 [##REF##23358983##2##] (except 1 oral dose or IV in the previous 8 h); 96 h days prior, Hidalgo et al. 1999 [##REF##9921995##6##] and Malik et al. 1995 [##REF##7872337##9##]; and prophylactic quinolones, Goodman et al. 2017 [##REF##28282274##22##]</td></tr><tr><td><italic>Gynecology</italic></td><td/></tr><tr><td>  Pregnant or breastfeeding</td><td>All clinical trials and observational studies</td></tr><tr><td><italic>Other</italic></td><td/></tr><tr><td>  No decrease in fever</td><td>Rapoport et al. 1999 [##REF##10567777##8##] and Poprawski et al. 2018 [##REF##30517995##20##]</td></tr><tr><td>  Last cycle of chemotherapy &lt; 12 days prior</td><td>Weycker et al. 2014 [##REF##23824496##21##]</td></tr><tr><td>  Allergy to the antibiotic used</td><td>All clinical trials and observational studies</td></tr><tr><td>  Recurrent fever of unknown origin</td><td>Malik et al. 1995 [##REF##7872337##9##]</td></tr><tr><td>  More than one primary tumor</td><td>Weycker et al. 2014 [##REF##23824496##21##]</td></tr><tr><td>  Poor treatment adherence</td><td>Innes et al. 2003 [##REF##12838298##5##]</td></tr><tr><td>  Admitted for other reason</td><td>Hocking et al. 2012 [##REF##23668271##19##]</td></tr><tr><td>  Lack of written informed consent</td><td>All clinical trials and observational studies</td></tr><tr><td>  Absence of caregiver</td><td>Talcott et al. 2011 (24 h) [##REF##21931024##3##], Sebban et al. 2008 [##REF##18197434##4##], Innes et al. 2003 [##REF##12838298##5##], Hidalgo et al. 1999 [##REF##9921995##6##] (24 h)</td></tr><tr><td>  Patient refusal</td><td>Sebban et al. 2008 [##REF##18197434##4##]</td></tr><tr><td>  Distance between residence and reference center</td><td>Kern et al. 2013 (&gt;1 h) [##REF##23358983##2##]; Hidalgo et al. 1999 [##REF##9921995##6##] and Talcott et al. 2011 (&gt;2 h) [##REF##21931024##3##]; Minotti et al. 1999 [##REF##10335931##7##] (&gt; 20 miles [32 km] from hospital), Rubenstein et al. 1993 [##REF##8490912##10##] and Rolston et al. 2003 [##REF##16628654##16##] (&gt; 30 miles [48 km] from hospital), Lingaratnam et al. 2013 [##REF##23809725##17##] (&gt; 40 km radius)</td></tr><tr><td>  Decline following observation period</td><td>Kern et al. 2013 [##REF##23358983##2##], Sebban et al. 2008 [##REF##21931024##3##], Innes et al. 2003 [##REF##12838298##5##], Rapoport et al. 1999 [##REF##10567777##8##], Rubenstein et al. 1993 [##REF##8490912##10##]</td></tr><tr><td>  Progression of underlying illness</td><td>Hidalgo et al. 1999 [##REF##9921995##6##] and no remission in leukemia in Talcott et al. 2011 [##REF##21931024##3##]</td></tr><tr><td>  Hematological neoplasms</td><td>Lingaratnam et al. 2013 [##REF##23809725##17##]</td></tr><tr><td>  Prior episode of febrile neutropenia included in the study</td><td>Rapoport et al. 1999 [##REF##10567777##8##]</td></tr><tr><td>  Ease of obtaining educational requirements</td><td>Poprawski et al. 2018 [##REF##30517995##20##]</td></tr></tbody></table></table-wrap>" ]

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[ "<disp-quote><p id=\"Par10\">“Febrile neutropenia,” “low risk febrile neutropenia,” and “Low risk Neutropenic fever” in three search strings, each combined with the following terms using the Boolean operator AND: “Hospital admission,” “inpatient management,” “outpatient management,” “Home admission,” “Ambulatory management,” “Early discharge,” “immediate discharge,” “delay discharge,” “Observation before discharge,” and “outpatient treatment.”</p></disp-quote>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p><italic>AST</italic> aspartate transaminase, <italic>ALT</italic> alanine aminotransferase, <italic>eGFR</italic> estimated glomerular filtration rate</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher’s Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p><bold>Change history</bold></p><p>1/15/2024</p><p>Missing Open Access funding information has been added in the Funding Note.\n</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"520_2023_8065_Fig1_HTML\" id=\"MO1\"/>" ]

[ "<media xlink:href=\"520_2023_8065_MOESM1_ESM.pdf\"><label>ESM 1</label><caption><p>(PDF 344 kb)</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction and main results</title>", "<p id=\"Par2\">In many cases, completeness properties of various objects of General Topology or Topological Algebra can be characterized externally as closedness in ambient objects. For example, a metric space <italic>X</italic> is complete if and only if <italic>X</italic> is closed in any metric space containing <italic>X</italic> as a subspace. A uniform space <italic>X</italic> is complete if and only if <italic>X</italic> is closed in any uniform space containing <italic>X</italic> as a uniform subspace. A topological group <italic>G</italic> is Raĭkov complete if and only if it is closed in any topological group containing <italic>G</italic> as a subgroup.</p>", "<p id=\"Par3\">On the other hand, for topological semigroups there are no reasonable notions of (inner) completeness. Nonetheless we can define many completeness properties of semigroups via their closedness in ambient topological semigroups.</p>", "<p id=\"Par4\">A <italic>topological semigroup</italic> is a topological space <italic>X</italic> endowed with a continuous associative binary operation , .</p>", "<title>Definition 1.1</title>", "<p id=\"Par5\">Let be a class of topological semigroups. A topological semigroup <italic>X</italic> is called<list list-type=\"bullet\"><list-item><p id=\"Par6\">-<italic>closed</italic> if for any isomorphic topological embedding to a topological semigroup the image <italic>h</italic>[<italic>X</italic>] is closed in <italic>Y</italic>;</p></list-item><list-item><p id=\"Par7\"><italic>injectively</italic>\n<italic>-closed</italic> if for any injective continuous homomorphism to a topological semigroup the image <italic>h</italic>[<italic>X</italic>] is closed in <italic>Y</italic>;</p></list-item><list-item><p id=\"Par8\"><italic>absolutely</italic>\n<italic>-closed</italic> if for any continuous homomorphism to a topological semigroup the image <italic>h</italic>[<italic>X</italic>] is closed in <italic>Y</italic>.</p></list-item></list></p>", "<p>For any topological semigroup we have the implications:</p>", "<title>Definition 1.2</title>", "<p id=\"Par10\">A semigroup <italic>X</italic> is defined to be (<italic>injectively, absolutely</italic>) -<italic>closed</italic> if <italic>X</italic> endowed with the discrete topology has the corresponding closedness property.</p>", "<p>We will be interested in the (absolute, injective) -closedness for the classes:<list list-type=\"bullet\"><list-item><p id=\"Par12\"> of topological semigroups satisfying the separation axiom ;</p></list-item><list-item><p id=\"Par13\"> of Hausdorff topological semigroups;</p></list-item><list-item><p id=\"Par14\"> of Tychonoff zero-dimensional topological semigroups.</p></list-item></list>A topological space satisfies the separation axiom if all its finite subsets are closed. A topological space is <italic>zero-dimensional</italic> if it has a base of the topology consisting of <italic>clopen</italic> (= closed-and-open) sets. It is well-known (and easy to see) that every zero-dimensional topological space is Tychonoff.</p>", "<p>Since , for every semigroup we have the implications:Many examples distinguishing various categorical closedness properties can be found in [##UREF##8##9##].</p>", "<title>Historical Remark 1.3</title>", "<p id=\"Par16\">Similar complete objects in different classes have been investigated under different names. For instance:<list list-type=\"bullet\"><list-item><p id=\"Par17\">H-closed spaces (the class of Hausdorff topological spaces, by Velichko [##UREF##69##70##]);</p></list-item><list-item><p id=\"Par18\">complete topological groups (the class of Hausdorff topological groups, by Raikov [##UREF##60##61##]);</p></list-item><list-item><p id=\"Par19\">absolutely maximal topological semigroups (the class of Hausdorff topological semigroups, by Stepp [##UREF##62##63##]);</p></list-item><list-item><p id=\"Par20\">absolutely H-closed semilattices (the class of Hausdorff topological semilattices, by Gutik and Repovš [##UREF##42##43##]);</p></list-item><list-item><p id=\"Par21\">h-complete topological groups (the class of Hausdorff topological groups, by Dikranjan and Tonolo [##UREF##30##31##]);</p></list-item><list-item><p id=\"Par22\">categorically compact topological groups (the class of Hausdorff topological groups, by Dikranjan and Uspenskij [##UREF##31##32##]);</p></list-item><list-item><p id=\"Par23\">sealed topological groups (the class of Hausdorff topological groups, by Bader and Leibtag [##UREF##1##2##]).</p></list-item></list></p>", "<p>-Closed topological groups for various classes were investigated in [##UREF##0##1##–##UREF##2##3##, ##UREF##14##15##, ##UREF##31##32##, ##UREF##39##40##, ##UREF##44##45##, ##UREF##50##51##]. In particular, the closedness of commutative topological groups in the class of Hausdorff topological semigroups was investigated in [##UREF##46##47##, ##UREF##72##73##]; -closed topological semilattices were investigated in [##UREF##5##6##, ##UREF##6##7##, ##UREF##41##42##, ##UREF##42##43##, ##UREF##62##63##]. Completeness in Category Theory was investigated in [##UREF##23##24##, ##UREF##24##25##, ##UREF##33##34##, ##UREF##34##35##, ##UREF##36##37##, ##UREF##37##38##, ##UREF##49##50##]. In particular, closure operators in different categories were studied in [##UREF##18##19##, ##UREF##20##21##–##UREF##22##23##, ##UREF##27##28##, ##UREF##28##29##, ##UREF##38##39##, ##UREF##67##68##, ##UREF##70##71##]. This paper is a continuation of the papers [##UREF##7##8##–##UREF##9##10##, ##UREF##12##13##] providing inner characterizations of various closedness properties of (discrete topological) semigroups. In order to formulate such inner characterizations, let us recall some properties of semigroups.</p>", "<p>A semigroup <italic>X</italic> is called<list list-type=\"bullet\"><list-item><p id=\"Par26\"><italic>chain-finite</italic> if any infinite set contains elements such that ;</p></list-item><list-item><p id=\"Par27\"><italic>singular</italic> if there exists an infinite set such that <italic>AA</italic> is a singleton;</p></list-item><list-item><p id=\"Par28\"><italic>periodic</italic> if for every there exists such that is an idempotent;</p></list-item><list-item><p id=\"Par29\"><italic>bounded</italic> if there exists such that for every the <italic>n</italic>-th power is an idempotent;</p></list-item><list-item><p id=\"Par30\"><italic>group-finite</italic> if every subgroup of <italic>X</italic> is finite;</p></list-item><list-item><p id=\"Par31\"><italic>group-bounded</italic> if every subgroup of <italic>X</italic> is bounded;</p></list-item><list-item><p id=\"Par32\"><italic>group-commutative</italic> if every subgroup of <italic>X</italic> is commutative.</p></list-item></list>The following theorem (proved in [##UREF##7##8##]) characterizes -closed commutative semigroups.</p>", "<title>Theorem 1.4</title>", "<p id=\"Par33\">(Banakh–Bardyla) Let be a class of topological semigroups such that . A commutative semigroup <italic>X</italic> is -closed if and only if <italic>X</italic> is chain-finite, nonsingular, periodic, and group-bounded.</p>", "<p>A subset <italic>I</italic> of a semigroup <italic>X</italic> is called an <italic>ideal</italic> in <italic>X</italic> if . Every ideal determines the congruence on <italic>X</italic>. The quotient semigroup of <italic>X</italic> by this congruence is denoted by <italic>X</italic>/<italic>I</italic> and called the <italic>quotient semigroup</italic> of <italic>X</italic> by the ideal <italic>I</italic>. If , then the quotient semigroup can be identified with the semigroup <italic>X</italic>.</p>", "<p>Theorem <xref ref-type=\"sec\" rid=\"FPar4\">1.4</xref> implies that each subsemigroup of a -closed commutative semigroup is -closed. On the other hand, quotient semigroups of -closed commutative semigroups are not necessarily -closed, see Example 1.8 in [##UREF##7##8##]. This motivates the following notions.</p>", "<title>Definition 1.5</title>", "<p id=\"Par36\">A semigroup <italic>X</italic> is called<list list-type=\"bullet\"><list-item><p id=\"Par37\"><italic>projectively</italic>\n<italic>-closed</italic> if for any congruence on <italic>X</italic> the quotient semigroup is -closed;</p></list-item><list-item><p id=\"Par38\"><italic>ideally</italic>\n<italic>-closed</italic> if for any ideal the quotient semigroup <italic>X</italic>/<italic>I</italic> is -closed.</p></list-item></list></p>", "<p>It is easy to see that for every semigroup the following implications hold:Observe that a semigroup <italic>X</italic> is absolutely -closed if and only if for any congruence on <italic>X</italic> the semigroup is injectively -closed.</p>", "<p>For a semigroup <italic>X</italic>, letbe the set of idempotents of <italic>X</italic>.</p>", "<p>For an idempotent <italic>e</italic> of a semigroup <italic>X</italic>, let be the maximal subgroup of <italic>X</italic> that contains <italic>e</italic>. The union of all subgroups of <italic>X</italic> is called the <italic>Clifford part</italic> of <italic>S</italic>. A semigroup <italic>X</italic> is called<list list-type=\"bullet\"><list-item><p id=\"Par42\"><italic>Clifford</italic> if ;</p></list-item><list-item><p id=\"Par43\"><italic>Clifford + finite</italic> if is finite.</p></list-item></list>Ideally and projectively -closed commutative semigroups were characterized in [##UREF##7##8##] as follows.</p>", "<title>Theorem 1.6</title>", "<p id=\"Par44\">(Banakh–Bardyla) Let be a class of topological semigroups such that . For a commutative semigroup <italic>X</italic> the following conditions are equivalent: <list list-type=\"order\"><list-item><p id=\"Par45\"><italic>X</italic> is projectively -closed;</p></list-item><list-item><p id=\"Par46\"><italic>X</italic> is ideally -closed;</p></list-item><list-item><p id=\"Par47\">the semigroup <italic>X</italic> is chain-finite, group-bounded and Clifford + finite.</p></list-item></list></p>", "<p>In [##UREF##8##9##] it is shown that the injective (and absolute) -closedness is tightly related to the (projective) -discreteness.</p>", "<title>Definition 1.7</title>", "<p id=\"Par49\">Let be a class of topological semigroups. A semigroup <italic>X</italic> is called<list list-type=\"bullet\"><list-item><p id=\"Par50\">-<italic>discrete</italic> (or else -<italic>nontopologizable</italic>) if for any injective homomorphism to a topological semigroup the image <italic>h</italic>[<italic>X</italic>] is a discrete subspace of <italic>Y</italic>;</p></list-item><list-item><p id=\"Par51\">-<italic>topologizable</italic> if <italic>X</italic> is not -discrete;</p></list-item><list-item><p id=\"Par52\"><italic>projectively </italic><italic>-discrete</italic> if for every homomorphism to a topological semigroup the image <italic>h</italic>[<italic>X</italic>] is a discrete subspace of <italic>Y</italic>.</p></list-item></list></p>", "<p>The study of topologizable and nontopologizable semigroups is a classical topic in Topological Algebra that traces its history back to Markov’s problem [##UREF##51##52##] of topologizability of infinite groups, which was resolved in [##UREF##43##44##, ##UREF##61##62##] and [##UREF##55##56##] by constructing examples of nontopologizable infinite groups. An interplay between ultrafilters and topologizability of groups was expounded in the monograph [##UREF##71##72##]. For some other results on topologizability of semigroups, see [##UREF##4##5##, ##UREF##10##11##, ##UREF##25##26##, ##UREF##26##27##, ##UREF##29##30##, ##UREF##32##33##, ##UREF##35##36##, ##UREF##47##48##, ##UREF##48##49##, ##UREF##63##64##, ##UREF##68##69##].</p>", "<p>In Propositions 3.2 and 3.3 of [##UREF##8##9##] the following two characterizations are proved.</p>", "<title>Theorem 1.8</title>", "<p id=\"Par55\">(Banakh–Bardyla) A semigroup <italic>X</italic> is <list list-type=\"order\"><list-item><p id=\"Par56\">injectively -closed if and only if <italic>X</italic> is -closed and -discrete;</p></list-item><list-item><p id=\"Par57\">absolutely -closed if and only if <italic>X</italic> is projectively -closed and projectively -discrete.</p></list-item></list></p>", "<p>The following two theorems characterizing absolutely -closed commutative semigroups are the main results of this paper. In contrast to Theorems <xref ref-type=\"sec\" rid=\"FPar4\">1.4</xref> and <xref ref-type=\"sec\" rid=\"FPar6\">1.6</xref>, the characterizations of absolutely -closed semigroups essentially depend on the class , where we distinguish two cases: and .</p>", "<title>Theorem 1.9</title>", "<p id=\"Par59\">For a commutative semigroup <italic>X</italic> the following conditions are equivalent: <list list-type=\"order\"><list-item><p id=\"Par60\"><italic>X</italic> is absolutely -closed;</p></list-item><list-item><p id=\"Par61\"><italic>X</italic> is projectively -closed and projectively -discrete;</p></list-item><list-item><p id=\"Par62\"><italic>X</italic> is projectively -closed and projectively -discrete;</p></list-item><list-item><p id=\"Par63\"><italic>X</italic> is finite.</p></list-item></list></p>", "<title>Theorem 1.10</title>", "<p id=\"Par64\">Let be a class of topological semigroups such that . For a commutative semigroup <italic>X</italic> the following conditions are equivalent: <list list-type=\"order\"><list-item><p id=\"Par65\"><italic>X</italic> is absolutely -closed;</p></list-item><list-item><p id=\"Par66\"><italic>X</italic> is ideally -closed, injectively -closed and bounded;</p></list-item><list-item><p id=\"Par67\"><italic>X</italic> is ideally -closed, group-finite and bounded;</p></list-item><list-item><p id=\"Par68\"><italic>X</italic> is chain-finite, bounded, group-finite and Clifford + finite.</p></list-item></list></p>", "<p>Theorems <xref ref-type=\"sec\" rid=\"FPar9\">1.9</xref> and <xref ref-type=\"sec\" rid=\"FPar10\">1.10</xref> imply that the absolute -closedness of commutative semigroups is inherited by subsemigroups:</p>", "<title>Corollary 1.11</title>", "<p id=\"Par70\">Let be a class of topological semigroups such that either or . Every subsemigroup of an absolutely -closed commutative semigroup is absolutely -closed.</p>", "<title>Historical Remark 1.12</title>", "<p id=\"Par71\">Corollary <xref ref-type=\"sec\" rid=\"FPar11\">1.11</xref> does not generalize to noncommutative groups: by Theorem 1.17 in [##UREF##8##9##], every countable bounded group <italic>G</italic> without elements of order 2 is a subgroup of an absolutely -closed countable simple bounded group <italic>X</italic>. If the group <italic>G</italic> has infinite center, then <italic>G</italic> is not injectively -closed by Theorem <xref ref-type=\"sec\" rid=\"FPar13\">1.13</xref>(2) below. On the other hand, <italic>G</italic> is a subgroup of the absolutely -closed group <italic>X</italic>. This example also shows that the equivalences in Theorems <xref ref-type=\"sec\" rid=\"FPar9\">1.9</xref> and <xref ref-type=\"sec\" rid=\"FPar10\">1.10</xref> do not hold for non-commutative groups.</p>", "<p>For a semigroup <italic>X</italic> letbe the <italic>center</italic> of <italic>X</italic>, andbe the <italic>ideal center</italic> of <italic>X</italic>. Every commutative semigroup <italic>X</italic> has .</p>", "<p>The following theorem proved in [##UREF##7##8##, Sect. 5] and [##UREF##9##10##] describes some properties of the center of a semigroup possessing various closedness properties.</p>", "<title>Theorem 1.13</title>", "<p id=\"Par74\">(Banakh–Bardyla) Let <italic>X</italic> be a semigroup. <list list-type=\"order\"><list-item><p id=\"Par75\">If <italic>X</italic> is -closed, then the center <italic>Z</italic>(<italic>X</italic>) is chain-finite, periodic and nonsingular.</p></list-item><list-item><p id=\"Par76\">If <italic>X</italic> is -discrete or injectively -closed, then <italic>Z</italic>(<italic>X</italic>) is group-finite.</p></list-item><list-item><p id=\"Par77\">If <italic>X</italic> is ideally -closed, then <italic>Z</italic>(<italic>X</italic>) is group-bounded.</p></list-item></list></p>", "<p>In [##UREF##9##10##] it was proved that the (ideal) -closedness is inherited by the ideal center:</p>", "<title>Theorem 1.14</title>", "<p id=\"Par79\">(Banakh–Bardyla) Let be a class of topological semigroups such that . For any (ideally) -closed semigroup <italic>X</italic>, its ideal center is (ideally) -closed.</p>", "<p>Theorem <xref ref-type=\"sec\" rid=\"FPar14\">1.14</xref> suggests the following problem.</p>", "<title>Problem 1.15</title>", "<p id=\"Par81\">Let be a class of topological semigroups. Is the (ideal) center of any absolutely -closed semigroup <italic>X</italic> absolutely -closed?</p>", "<p>The “ideal” version of Problem <xref ref-type=\"sec\" rid=\"FPar15\">1.15</xref> has an affirmative answer.</p>", "<title>Theorem 1.16</title>", "<p id=\"Par83\">Let be a class of topological semigroups such that either or . Every absolutely -closed semigroup <italic>X</italic> has absolutely -closed ideal center .</p>", "<p>The “non-ideal” version of Problem <xref ref-type=\"sec\" rid=\"FPar15\">1.15</xref> has an affirmative answer for group-commutative <italic>Z</italic>-viable semigroups.</p>", "<p>Following Putcha and Weissglass [##UREF##59##60##] we call semigroup <italic>X</italic>\n<italic>viable</italic> if for any with we have . This notion can be localized using the notion of a viable idempotent.</p>", "<p>An idempotent <italic>e</italic> in a semigroup <italic>X</italic> is defined to be <italic>viable</italic> if the setis a <italic>coideal</italic> in <italic>X</italic> in the sense that is an ideal in <italic>X</italic>. By we denote the set of viable idempotents of a semigroup <italic>X</italic>.</p>", "<p>By Theorem 3.2 of [##UREF##3##4##], a semigroup <italic>X</italic> is viable if and only if each idempotent of <italic>X</italic> is viable if and only if for every with we have and . This characterization implies that every semigroup <italic>X</italic> with is viable. In particular, every commutative semigroup is viable.</p>", "<p>For ideally (absolutely) -closed semigroups we have the following description of the structure of maximal subgroups of viable idempotents, see [##UREF##9##10##, Theorem 1.7].</p>", "<title>Theorem 1.17</title>", "<p id=\"Par89\">(Banakh–Bardyla) Let <italic>e</italic> be a viable idempotent of a semigroup <italic>X</italic> and be the maximal subgroup of <italic>e</italic> in <italic>X</italic>. <list list-type=\"order\"><list-item><p id=\"Par90\">If <italic>X</italic> is ideally -closed, then the group is bounded.</p></list-item><list-item><p id=\"Par91\">If <italic>X</italic> is absolutely -closed, then the group is finite.</p></list-item></list></p>", "<p>A semigroup <italic>X</italic> is called <italic>Z</italic>-<italic>viable</italic> if , i.e., if each central idempotent of <italic>X</italic> is viable. It is clear that each viable semigroup is <italic>Z</italic>-viable. On the other hand, there exist semigroups which are not <italic>Z</italic>-viable, see Remark <xref ref-type=\"sec\" rid=\"FPar30\">2.6</xref>.</p>", "<p>For a subset <italic>A</italic> of semigroup <italic>X</italic> letA subset <italic>B</italic> of a semigroup <italic>X</italic> is called <italic>bounded</italic> if for some . In fact, the “ideal” part of Theorem <xref ref-type=\"sec\" rid=\"FPar14\">1.14</xref> was derived in [##UREF##9##10##] from the following theorem, which will be essentially used also in this paper:</p>", "<title>Theorem 1.18</title>", "<p id=\"Par94\">(Banakh–Bardyla) If a semigroup <italic>X</italic> is ideally -closed, then the set is finite.</p>", "<p>The following theorem gives a partial answer to Problem <xref ref-type=\"sec\" rid=\"FPar15\">1.15</xref> for the class .</p>", "<title>Theorem 1.19</title>", "<p id=\"Par96\">If a semigroup <italic>X</italic> is absolutely -closed (and <italic>Z</italic>-viable), then the set is finite (and the semigroup <italic>Z</italic>(<italic>X</italic>) is absolutely -closed).</p>", "<p>For classes with a partial answer to Problem <xref ref-type=\"sec\" rid=\"FPar15\">1.15</xref> looks as follows.</p>", "<title>Theorem 1.20</title>", "<p id=\"Par98\">Let <italic>X</italic> be an absolutely -closed semigroup and . Assume that for any infinite countable subset and the subsemigroup of <italic>X</italic>, one of the following conditions is satisfied: <list list-type=\"order\"><list-item><p id=\"Par99\">for every the subsemigroup <italic>Ce</italic> of is commutative;</p></list-item><list-item><p id=\"Par100\"><italic>C</italic> is countable;</p></list-item><list-item><p id=\"Par101\">, and for every the subsemigroup <italic>Ce</italic> of is countable.</p></list-item><list-item><p id=\"Par102\"> and for every the subsemigroup <italic>Ce</italic> of is bounded.</p></list-item></list>Then the set is bounded, and every subsemigroup of of <italic>X</italic> is absolutely -closed.</p>", "<p>The cardinal appearing in Theorem <xref ref-type=\"sec\" rid=\"FPar20\">1.20</xref>(3) is defined as the smallest cardinality of a cover of the real line by nowhere dense subsets. The Baire Theorem implies that . It is well-known that under Martin’s Axiom. By [##UREF##16##17##, 7.13], the equality is equivalent to Martin’s Axiom for countable posets.</p>", "<p>By Theorem <xref ref-type=\"sec\" rid=\"FPar13\">1.13</xref>(1), the center <italic>Z</italic>(<italic>X</italic>) of any -closed semigroup is chain-finite. In fact, this is an order property of the poset <italic>E</italic>(<italic>X</italic>) endowed with the natural partial order defined by iff . In turns out that stronger closedness properties (like the ideal or projective -closedness) impose stronger restrictions on the partial order of the set <italic>E</italic>(<italic>X</italic>) and also on the partial order of the semilattice reflection of <italic>X</italic>.</p>", "<p>A congruence on a semigroup <italic>X</italic> is called a <italic>semilattice congruence</italic> if the quotient semigroup is a <italic>semilattice</italic>, i.e., a commutative semigroup of idempotents. The intersection of all semilattice congruences on a semigroup <italic>X</italic> is called the <italic>smallest semilattice congruence</italic> on <italic>X</italic> and the quotient semigroup is called the <italic>semilattice reflection</italic> of <italic>X</italic>. The smallest semilattice congruence was studied in the monographs [##UREF##17##18##, ##UREF##52##53##], surveys [##UREF##53##54##, ##UREF##54##55##] and papers [##UREF##2##3##, ##UREF##56##57##–##UREF##59##60##, ##UREF##64##65##–##UREF##66##67##].</p>", "<p>A partially ordered set is called<list list-type=\"bullet\"><list-item><p id=\"Par107\"><italic>chain-finite</italic> if each infinite subset contains two elements <italic>x</italic>, <italic>y</italic> such that and ;</p></list-item><list-item><p id=\"Par108\"><italic>well-founded</italic> if each nonempty set contains an element <italic>a</italic> such that .</p></list-item></list>It is easy to see that for every chain-finite semigroup <italic>X</italic> the poset <italic>E</italic>(<italic>X</italic>) is chain-finite. The converse holds if <italic>E</italic>(<italic>X</italic>) is a commutative subsemigroup of <italic>X</italic>.</p>", "<title>Theorem 1.21</title>", "<p id=\"Par109\">Let <italic>X</italic> be a semigroup. <list list-type=\"order\"><list-item><p id=\"Par110\">If <italic>X</italic> is ideally -closed, then the posets and are well-founded.</p></list-item><list-item><p id=\"Par111\">If <italic>X</italic> is projectively -closed, then and are chain-finite.</p></list-item><list-item><p id=\"Par112\">If <italic>X</italic> is projectively -closed and projectively -discrete, then and are finite;</p></list-item><list-item><p id=\"Par113\">If <italic>X</italic> is absolutely -closed, then and are finite.</p></list-item></list></p>", "<p>Theorem <xref ref-type=\"sec\" rid=\"FPar21\">1.21</xref> will be proved in Sect. <xref rid=\"Sec10\" ref-type=\"sec\">3</xref>. In Sect. <xref rid=\"Sec11\" ref-type=\"sec\">4</xref> we prove a general version of Theorem <xref ref-type=\"sec\" rid=\"FPar9\">1.9</xref> and in Sect. <xref rid=\"Sec12\" ref-type=\"sec\">5</xref> we prove Lemma <xref ref-type=\"sec\" rid=\"FPar45\">5.2</xref> giving a sufficient condition of the absolute -closedness. In Sect. <xref rid=\"Sec13\" ref-type=\"sec\">6</xref> we introduce the notion of an <italic>A</italic>-centrobounded semigroup and use this notion for characterizing bounded set of form in absolutely -closed semigroups. In Sect. <xref rid=\"Sec14\" ref-type=\"sec\">7</xref> we prove Theorem <xref ref-type=\"sec\" rid=\"FPar62\">7.1</xref> giving some sufficient conditions of the <italic>A</italic>-centroboundedness and implying Corollary <xref ref-type=\"sec\" rid=\"FPar64\">7.2</xref>, which is a more general version of Theorem <xref ref-type=\"sec\" rid=\"FPar20\">1.20</xref>. In Sects. <xref rid=\"Sec15\" ref-type=\"sec\">8</xref> and <xref rid=\"Sec16\" ref-type=\"sec\">9</xref> we prove Theorems <xref ref-type=\"sec\" rid=\"FPar16\">1.16</xref> and <xref ref-type=\"sec\" rid=\"FPar10\">1.10</xref>, respectively.</p>" ]

[]

[ "<title>Introduction and main results</title>", "<p id=\"Par2\">In many cases, completeness properties of various objects of General Topology or Topological Algebra can be characterized externally as closedness in ambient objects. For example, a metric space <italic>X</italic> is complete if and only if <italic>X</italic> is closed in any metric space containing <italic>X</italic> as a subspace. A uniform space <italic>X</italic> is complete if and only if <italic>X</italic> is closed in any uniform space containing <italic>X</italic> as a uniform subspace. A topological group <italic>G</italic> is Raĭkov complete if and only if it is closed in any topological group containing <italic>G</italic> as a subgroup.</p>", "<p id=\"Par3\">On the other hand, for topological semigroups there are no reasonable notions of (inner) completeness. Nonetheless we can define many completeness properties of semigroups via their closedness in ambient topological semigroups.</p>", "<p id=\"Par4\">A <italic>topological semigroup</italic> is a topological space <italic>X</italic> endowed with a continuous associative binary operation , .</p>", "<title>Definition 1.1</title>", "<p id=\"Par5\">Let be a class of topological semigroups. A topological semigroup <italic>X</italic> is called<list list-type=\"bullet\"><list-item><p id=\"Par6\">-<italic>closed</italic> if for any isomorphic topological embedding to a topological semigroup the image <italic>h</italic>[<italic>X</italic>] is closed in <italic>Y</italic>;</p></list-item><list-item><p id=\"Par7\"><italic>injectively</italic>\n<italic>-closed</italic> if for any injective continuous homomorphism to a topological semigroup the image <italic>h</italic>[<italic>X</italic>] is closed in <italic>Y</italic>;</p></list-item><list-item><p id=\"Par8\"><italic>absolutely</italic>\n<italic>-closed</italic> if for any continuous homomorphism to a topological semigroup the image <italic>h</italic>[<italic>X</italic>] is closed in <italic>Y</italic>.</p></list-item></list></p>", "<p>For any topological semigroup we have the implications:</p>", "<title>Definition 1.2</title>", "<p id=\"Par10\">A semigroup <italic>X</italic> is defined to be (<italic>injectively, absolutely</italic>) -<italic>closed</italic> if <italic>X</italic> endowed with the discrete topology has the corresponding closedness property.</p>", "<p>We will be interested in the (absolute, injective) -closedness for the classes:<list list-type=\"bullet\"><list-item><p id=\"Par12\"> of topological semigroups satisfying the separation axiom ;</p></list-item><list-item><p id=\"Par13\"> of Hausdorff topological semigroups;</p></list-item><list-item><p id=\"Par14\"> of Tychonoff zero-dimensional topological semigroups.</p></list-item></list>A topological space satisfies the separation axiom if all its finite subsets are closed. A topological space is <italic>zero-dimensional</italic> if it has a base of the topology consisting of <italic>clopen</italic> (= closed-and-open) sets. It is well-known (and easy to see) that every zero-dimensional topological space is Tychonoff.</p>", "<p>Since , for every semigroup we have the implications:Many examples distinguishing various categorical closedness properties can be found in [##UREF##8##9##].</p>", "<title>Historical Remark 1.3</title>", "<p id=\"Par16\">Similar complete objects in different classes have been investigated under different names. For instance:<list list-type=\"bullet\"><list-item><p id=\"Par17\">H-closed spaces (the class of Hausdorff topological spaces, by Velichko [##UREF##69##70##]);</p></list-item><list-item><p id=\"Par18\">complete topological groups (the class of Hausdorff topological groups, by Raikov [##UREF##60##61##]);</p></list-item><list-item><p id=\"Par19\">absolutely maximal topological semigroups (the class of Hausdorff topological semigroups, by Stepp [##UREF##62##63##]);</p></list-item><list-item><p id=\"Par20\">absolutely H-closed semilattices (the class of Hausdorff topological semilattices, by Gutik and Repovš [##UREF##42##43##]);</p></list-item><list-item><p id=\"Par21\">h-complete topological groups (the class of Hausdorff topological groups, by Dikranjan and Tonolo [##UREF##30##31##]);</p></list-item><list-item><p id=\"Par22\">categorically compact topological groups (the class of Hausdorff topological groups, by Dikranjan and Uspenskij [##UREF##31##32##]);</p></list-item><list-item><p id=\"Par23\">sealed topological groups (the class of Hausdorff topological groups, by Bader and Leibtag [##UREF##1##2##]).</p></list-item></list></p>", "<p>-Closed topological groups for various classes were investigated in [##UREF##0##1##–##UREF##2##3##, ##UREF##14##15##, ##UREF##31##32##, ##UREF##39##40##, ##UREF##44##45##, ##UREF##50##51##]. In particular, the closedness of commutative topological groups in the class of Hausdorff topological semigroups was investigated in [##UREF##46##47##, ##UREF##72##73##]; -closed topological semilattices were investigated in [##UREF##5##6##, ##UREF##6##7##, ##UREF##41##42##, ##UREF##42##43##, ##UREF##62##63##]. Completeness in Category Theory was investigated in [##UREF##23##24##, ##UREF##24##25##, ##UREF##33##34##, ##UREF##34##35##, ##UREF##36##37##, ##UREF##37##38##, ##UREF##49##50##]. In particular, closure operators in different categories were studied in [##UREF##18##19##, ##UREF##20##21##–##UREF##22##23##, ##UREF##27##28##, ##UREF##28##29##, ##UREF##38##39##, ##UREF##67##68##, ##UREF##70##71##]. This paper is a continuation of the papers [##UREF##7##8##–##UREF##9##10##, ##UREF##12##13##] providing inner characterizations of various closedness properties of (discrete topological) semigroups. In order to formulate such inner characterizations, let us recall some properties of semigroups.</p>", "<p>A semigroup <italic>X</italic> is called<list list-type=\"bullet\"><list-item><p id=\"Par26\"><italic>chain-finite</italic> if any infinite set contains elements such that ;</p></list-item><list-item><p id=\"Par27\"><italic>singular</italic> if there exists an infinite set such that <italic>AA</italic> is a singleton;</p></list-item><list-item><p id=\"Par28\"><italic>periodic</italic> if for every there exists such that is an idempotent;</p></list-item><list-item><p id=\"Par29\"><italic>bounded</italic> if there exists such that for every the <italic>n</italic>-th power is an idempotent;</p></list-item><list-item><p id=\"Par30\"><italic>group-finite</italic> if every subgroup of <italic>X</italic> is finite;</p></list-item><list-item><p id=\"Par31\"><italic>group-bounded</italic> if every subgroup of <italic>X</italic> is bounded;</p></list-item><list-item><p id=\"Par32\"><italic>group-commutative</italic> if every subgroup of <italic>X</italic> is commutative.</p></list-item></list>The following theorem (proved in [##UREF##7##8##]) characterizes -closed commutative semigroups.</p>", "<title>Theorem 1.4</title>", "<p id=\"Par33\">(Banakh–Bardyla) Let be a class of topological semigroups such that . A commutative semigroup <italic>X</italic> is -closed if and only if <italic>X</italic> is chain-finite, nonsingular, periodic, and group-bounded.</p>", "<p>A subset <italic>I</italic> of a semigroup <italic>X</italic> is called an <italic>ideal</italic> in <italic>X</italic> if . Every ideal determines the congruence on <italic>X</italic>. The quotient semigroup of <italic>X</italic> by this congruence is denoted by <italic>X</italic>/<italic>I</italic> and called the <italic>quotient semigroup</italic> of <italic>X</italic> by the ideal <italic>I</italic>. If , then the quotient semigroup can be identified with the semigroup <italic>X</italic>.</p>", "<p>Theorem <xref ref-type=\"sec\" rid=\"FPar4\">1.4</xref> implies that each subsemigroup of a -closed commutative semigroup is -closed. On the other hand, quotient semigroups of -closed commutative semigroups are not necessarily -closed, see Example 1.8 in [##UREF##7##8##]. This motivates the following notions.</p>", "<title>Definition 1.5</title>", "<p id=\"Par36\">A semigroup <italic>X</italic> is called<list list-type=\"bullet\"><list-item><p id=\"Par37\"><italic>projectively</italic>\n<italic>-closed</italic> if for any congruence on <italic>X</italic> the quotient semigroup is -closed;</p></list-item><list-item><p id=\"Par38\"><italic>ideally</italic>\n<italic>-closed</italic> if for any ideal the quotient semigroup <italic>X</italic>/<italic>I</italic> is -closed.</p></list-item></list></p>", "<p>It is easy to see that for every semigroup the following implications hold:Observe that a semigroup <italic>X</italic> is absolutely -closed if and only if for any congruence on <italic>X</italic> the semigroup is injectively -closed.</p>", "<p>For a semigroup <italic>X</italic>, letbe the set of idempotents of <italic>X</italic>.</p>", "<p>For an idempotent <italic>e</italic> of a semigroup <italic>X</italic>, let be the maximal subgroup of <italic>X</italic> that contains <italic>e</italic>. The union of all subgroups of <italic>X</italic> is called the <italic>Clifford part</italic> of <italic>S</italic>. A semigroup <italic>X</italic> is called<list list-type=\"bullet\"><list-item><p id=\"Par42\"><italic>Clifford</italic> if ;</p></list-item><list-item><p id=\"Par43\"><italic>Clifford + finite</italic> if is finite.</p></list-item></list>Ideally and projectively -closed commutative semigroups were characterized in [##UREF##7##8##] as follows.</p>", "<title>Theorem 1.6</title>", "<p id=\"Par44\">(Banakh–Bardyla) Let be a class of topological semigroups such that . For a commutative semigroup <italic>X</italic> the following conditions are equivalent: <list list-type=\"order\"><list-item><p id=\"Par45\"><italic>X</italic> is projectively -closed;</p></list-item><list-item><p id=\"Par46\"><italic>X</italic> is ideally -closed;</p></list-item><list-item><p id=\"Par47\">the semigroup <italic>X</italic> is chain-finite, group-bounded and Clifford + finite.</p></list-item></list></p>", "<p>In [##UREF##8##9##] it is shown that the injective (and absolute) -closedness is tightly related to the (projective) -discreteness.</p>", "<title>Definition 1.7</title>", "<p id=\"Par49\">Let be a class of topological semigroups. A semigroup <italic>X</italic> is called<list list-type=\"bullet\"><list-item><p id=\"Par50\">-<italic>discrete</italic> (or else -<italic>nontopologizable</italic>) if for any injective homomorphism to a topological semigroup the image <italic>h</italic>[<italic>X</italic>] is a discrete subspace of <italic>Y</italic>;</p></list-item><list-item><p id=\"Par51\">-<italic>topologizable</italic> if <italic>X</italic> is not -discrete;</p></list-item><list-item><p id=\"Par52\"><italic>projectively </italic><italic>-discrete</italic> if for every homomorphism to a topological semigroup the image <italic>h</italic>[<italic>X</italic>] is a discrete subspace of <italic>Y</italic>.</p></list-item></list></p>", "<p>The study of topologizable and nontopologizable semigroups is a classical topic in Topological Algebra that traces its history back to Markov’s problem [##UREF##51##52##] of topologizability of infinite groups, which was resolved in [##UREF##43##44##, ##UREF##61##62##] and [##UREF##55##56##] by constructing examples of nontopologizable infinite groups. An interplay between ultrafilters and topologizability of groups was expounded in the monograph [##UREF##71##72##]. For some other results on topologizability of semigroups, see [##UREF##4##5##, ##UREF##10##11##, ##UREF##25##26##, ##UREF##26##27##, ##UREF##29##30##, ##UREF##32##33##, ##UREF##35##36##, ##UREF##47##48##, ##UREF##48##49##, ##UREF##63##64##, ##UREF##68##69##].</p>", "<p>In Propositions 3.2 and 3.3 of [##UREF##8##9##] the following two characterizations are proved.</p>", "<title>Theorem 1.8</title>", "<p id=\"Par55\">(Banakh–Bardyla) A semigroup <italic>X</italic> is <list list-type=\"order\"><list-item><p id=\"Par56\">injectively -closed if and only if <italic>X</italic> is -closed and -discrete;</p></list-item><list-item><p id=\"Par57\">absolutely -closed if and only if <italic>X</italic> is projectively -closed and projectively -discrete.</p></list-item></list></p>", "<p>The following two theorems characterizing absolutely -closed commutative semigroups are the main results of this paper. In contrast to Theorems <xref ref-type=\"sec\" rid=\"FPar4\">1.4</xref> and <xref ref-type=\"sec\" rid=\"FPar6\">1.6</xref>, the characterizations of absolutely -closed semigroups essentially depend on the class , where we distinguish two cases: and .</p>", "<title>Theorem 1.9</title>", "<p id=\"Par59\">For a commutative semigroup <italic>X</italic> the following conditions are equivalent: <list list-type=\"order\"><list-item><p id=\"Par60\"><italic>X</italic> is absolutely -closed;</p></list-item><list-item><p id=\"Par61\"><italic>X</italic> is projectively -closed and projectively -discrete;</p></list-item><list-item><p id=\"Par62\"><italic>X</italic> is projectively -closed and projectively -discrete;</p></list-item><list-item><p id=\"Par63\"><italic>X</italic> is finite.</p></list-item></list></p>", "<title>Theorem 1.10</title>", "<p id=\"Par64\">Let be a class of topological semigroups such that . For a commutative semigroup <italic>X</italic> the following conditions are equivalent: <list list-type=\"order\"><list-item><p id=\"Par65\"><italic>X</italic> is absolutely -closed;</p></list-item><list-item><p id=\"Par66\"><italic>X</italic> is ideally -closed, injectively -closed and bounded;</p></list-item><list-item><p id=\"Par67\"><italic>X</italic> is ideally -closed, group-finite and bounded;</p></list-item><list-item><p id=\"Par68\"><italic>X</italic> is chain-finite, bounded, group-finite and Clifford + finite.</p></list-item></list></p>", "<p>Theorems <xref ref-type=\"sec\" rid=\"FPar9\">1.9</xref> and <xref ref-type=\"sec\" rid=\"FPar10\">1.10</xref> imply that the absolute -closedness of commutative semigroups is inherited by subsemigroups:</p>", "<title>Corollary 1.11</title>", "<p id=\"Par70\">Let be a class of topological semigroups such that either or . Every subsemigroup of an absolutely -closed commutative semigroup is absolutely -closed.</p>", "<title>Historical Remark 1.12</title>", "<p id=\"Par71\">Corollary <xref ref-type=\"sec\" rid=\"FPar11\">1.11</xref> does not generalize to noncommutative groups: by Theorem 1.17 in [##UREF##8##9##], every countable bounded group <italic>G</italic> without elements of order 2 is a subgroup of an absolutely -closed countable simple bounded group <italic>X</italic>. If the group <italic>G</italic> has infinite center, then <italic>G</italic> is not injectively -closed by Theorem <xref ref-type=\"sec\" rid=\"FPar13\">1.13</xref>(2) below. On the other hand, <italic>G</italic> is a subgroup of the absolutely -closed group <italic>X</italic>. This example also shows that the equivalences in Theorems <xref ref-type=\"sec\" rid=\"FPar9\">1.9</xref> and <xref ref-type=\"sec\" rid=\"FPar10\">1.10</xref> do not hold for non-commutative groups.</p>", "<p>For a semigroup <italic>X</italic> letbe the <italic>center</italic> of <italic>X</italic>, andbe the <italic>ideal center</italic> of <italic>X</italic>. Every commutative semigroup <italic>X</italic> has .</p>", "<p>The following theorem proved in [##UREF##7##8##, Sect. 5] and [##UREF##9##10##] describes some properties of the center of a semigroup possessing various closedness properties.</p>", "<title>Theorem 1.13</title>", "<p id=\"Par74\">(Banakh–Bardyla) Let <italic>X</italic> be a semigroup. <list list-type=\"order\"><list-item><p id=\"Par75\">If <italic>X</italic> is -closed, then the center <italic>Z</italic>(<italic>X</italic>) is chain-finite, periodic and nonsingular.</p></list-item><list-item><p id=\"Par76\">If <italic>X</italic> is -discrete or injectively -closed, then <italic>Z</italic>(<italic>X</italic>) is group-finite.</p></list-item><list-item><p id=\"Par77\">If <italic>X</italic> is ideally -closed, then <italic>Z</italic>(<italic>X</italic>) is group-bounded.</p></list-item></list></p>", "<p>In [##UREF##9##10##] it was proved that the (ideal) -closedness is inherited by the ideal center:</p>", "<title>Theorem 1.14</title>", "<p id=\"Par79\">(Banakh–Bardyla) Let be a class of topological semigroups such that . For any (ideally) -closed semigroup <italic>X</italic>, its ideal center is (ideally) -closed.</p>", "<p>Theorem <xref ref-type=\"sec\" rid=\"FPar14\">1.14</xref> suggests the following problem.</p>", "<title>Problem 1.15</title>", "<p id=\"Par81\">Let be a class of topological semigroups. Is the (ideal) center of any absolutely -closed semigroup <italic>X</italic> absolutely -closed?</p>", "<p>The “ideal” version of Problem <xref ref-type=\"sec\" rid=\"FPar15\">1.15</xref> has an affirmative answer.</p>", "<title>Theorem 1.16</title>", "<p id=\"Par83\">Let be a class of topological semigroups such that either or . Every absolutely -closed semigroup <italic>X</italic> has absolutely -closed ideal center .</p>", "<p>The “non-ideal” version of Problem <xref ref-type=\"sec\" rid=\"FPar15\">1.15</xref> has an affirmative answer for group-commutative <italic>Z</italic>-viable semigroups.</p>", "<p>Following Putcha and Weissglass [##UREF##59##60##] we call semigroup <italic>X</italic>\n<italic>viable</italic> if for any with we have . This notion can be localized using the notion of a viable idempotent.</p>", "<p>An idempotent <italic>e</italic> in a semigroup <italic>X</italic> is defined to be <italic>viable</italic> if the setis a <italic>coideal</italic> in <italic>X</italic> in the sense that is an ideal in <italic>X</italic>. By we denote the set of viable idempotents of a semigroup <italic>X</italic>.</p>", "<p>By Theorem 3.2 of [##UREF##3##4##], a semigroup <italic>X</italic> is viable if and only if each idempotent of <italic>X</italic> is viable if and only if for every with we have and . This characterization implies that every semigroup <italic>X</italic> with is viable. In particular, every commutative semigroup is viable.</p>", "<p>For ideally (absolutely) -closed semigroups we have the following description of the structure of maximal subgroups of viable idempotents, see [##UREF##9##10##, Theorem 1.7].</p>", "<title>Theorem 1.17</title>", "<p id=\"Par89\">(Banakh–Bardyla) Let <italic>e</italic> be a viable idempotent of a semigroup <italic>X</italic> and be the maximal subgroup of <italic>e</italic> in <italic>X</italic>. <list list-type=\"order\"><list-item><p id=\"Par90\">If <italic>X</italic> is ideally -closed, then the group is bounded.</p></list-item><list-item><p id=\"Par91\">If <italic>X</italic> is absolutely -closed, then the group is finite.</p></list-item></list></p>", "<p>A semigroup <italic>X</italic> is called <italic>Z</italic>-<italic>viable</italic> if , i.e., if each central idempotent of <italic>X</italic> is viable. It is clear that each viable semigroup is <italic>Z</italic>-viable. On the other hand, there exist semigroups which are not <italic>Z</italic>-viable, see Remark <xref ref-type=\"sec\" rid=\"FPar30\">2.6</xref>.</p>", "<p>For a subset <italic>A</italic> of semigroup <italic>X</italic> letA subset <italic>B</italic> of a semigroup <italic>X</italic> is called <italic>bounded</italic> if for some . In fact, the “ideal” part of Theorem <xref ref-type=\"sec\" rid=\"FPar14\">1.14</xref> was derived in [##UREF##9##10##] from the following theorem, which will be essentially used also in this paper:</p>", "<title>Theorem 1.18</title>", "<p id=\"Par94\">(Banakh–Bardyla) If a semigroup <italic>X</italic> is ideally -closed, then the set is finite.</p>", "<p>The following theorem gives a partial answer to Problem <xref ref-type=\"sec\" rid=\"FPar15\">1.15</xref> for the class .</p>", "<title>Theorem 1.19</title>", "<p id=\"Par96\">If a semigroup <italic>X</italic> is absolutely -closed (and <italic>Z</italic>-viable), then the set is finite (and the semigroup <italic>Z</italic>(<italic>X</italic>) is absolutely -closed).</p>", "<p>For classes with a partial answer to Problem <xref ref-type=\"sec\" rid=\"FPar15\">1.15</xref> looks as follows.</p>", "<title>Theorem 1.20</title>", "<p id=\"Par98\">Let <italic>X</italic> be an absolutely -closed semigroup and . Assume that for any infinite countable subset and the subsemigroup of <italic>X</italic>, one of the following conditions is satisfied: <list list-type=\"order\"><list-item><p id=\"Par99\">for every the subsemigroup <italic>Ce</italic> of is commutative;</p></list-item><list-item><p id=\"Par100\"><italic>C</italic> is countable;</p></list-item><list-item><p id=\"Par101\">, and for every the subsemigroup <italic>Ce</italic> of is countable.</p></list-item><list-item><p id=\"Par102\"> and for every the subsemigroup <italic>Ce</italic> of is bounded.</p></list-item></list>Then the set is bounded, and every subsemigroup of of <italic>X</italic> is absolutely -closed.</p>", "<p>The cardinal appearing in Theorem <xref ref-type=\"sec\" rid=\"FPar20\">1.20</xref>(3) is defined as the smallest cardinality of a cover of the real line by nowhere dense subsets. The Baire Theorem implies that . It is well-known that under Martin’s Axiom. By [##UREF##16##17##, 7.13], the equality is equivalent to Martin’s Axiom for countable posets.</p>", "<p>By Theorem <xref ref-type=\"sec\" rid=\"FPar13\">1.13</xref>(1), the center <italic>Z</italic>(<italic>X</italic>) of any -closed semigroup is chain-finite. In fact, this is an order property of the poset <italic>E</italic>(<italic>X</italic>) endowed with the natural partial order defined by iff . In turns out that stronger closedness properties (like the ideal or projective -closedness) impose stronger restrictions on the partial order of the set <italic>E</italic>(<italic>X</italic>) and also on the partial order of the semilattice reflection of <italic>X</italic>.</p>", "<p>A congruence on a semigroup <italic>X</italic> is called a <italic>semilattice congruence</italic> if the quotient semigroup is a <italic>semilattice</italic>, i.e., a commutative semigroup of idempotents. The intersection of all semilattice congruences on a semigroup <italic>X</italic> is called the <italic>smallest semilattice congruence</italic> on <italic>X</italic> and the quotient semigroup is called the <italic>semilattice reflection</italic> of <italic>X</italic>. The smallest semilattice congruence was studied in the monographs [##UREF##17##18##, ##UREF##52##53##], surveys [##UREF##53##54##, ##UREF##54##55##] and papers [##UREF##2##3##, ##UREF##56##57##–##UREF##59##60##, ##UREF##64##65##–##UREF##66##67##].</p>", "<p>A partially ordered set is called<list list-type=\"bullet\"><list-item><p id=\"Par107\"><italic>chain-finite</italic> if each infinite subset contains two elements <italic>x</italic>, <italic>y</italic> such that and ;</p></list-item><list-item><p id=\"Par108\"><italic>well-founded</italic> if each nonempty set contains an element <italic>a</italic> such that .</p></list-item></list>It is easy to see that for every chain-finite semigroup <italic>X</italic> the poset <italic>E</italic>(<italic>X</italic>) is chain-finite. The converse holds if <italic>E</italic>(<italic>X</italic>) is a commutative subsemigroup of <italic>X</italic>.</p>", "<title>Theorem 1.21</title>", "<p id=\"Par109\">Let <italic>X</italic> be a semigroup. <list list-type=\"order\"><list-item><p id=\"Par110\">If <italic>X</italic> is ideally -closed, then the posets and are well-founded.</p></list-item><list-item><p id=\"Par111\">If <italic>X</italic> is projectively -closed, then and are chain-finite.</p></list-item><list-item><p id=\"Par112\">If <italic>X</italic> is projectively -closed and projectively -discrete, then and are finite;</p></list-item><list-item><p id=\"Par113\">If <italic>X</italic> is absolutely -closed, then and are finite.</p></list-item></list></p>", "<p>Theorem <xref ref-type=\"sec\" rid=\"FPar21\">1.21</xref> will be proved in Sect. <xref rid=\"Sec10\" ref-type=\"sec\">3</xref>. In Sect. <xref rid=\"Sec11\" ref-type=\"sec\">4</xref> we prove a general version of Theorem <xref ref-type=\"sec\" rid=\"FPar9\">1.9</xref> and in Sect. <xref rid=\"Sec12\" ref-type=\"sec\">5</xref> we prove Lemma <xref ref-type=\"sec\" rid=\"FPar45\">5.2</xref> giving a sufficient condition of the absolute -closedness. In Sect. <xref rid=\"Sec13\" ref-type=\"sec\">6</xref> we introduce the notion of an <italic>A</italic>-centrobounded semigroup and use this notion for characterizing bounded set of form in absolutely -closed semigroups. In Sect. <xref rid=\"Sec14\" ref-type=\"sec\">7</xref> we prove Theorem <xref ref-type=\"sec\" rid=\"FPar62\">7.1</xref> giving some sufficient conditions of the <italic>A</italic>-centroboundedness and implying Corollary <xref ref-type=\"sec\" rid=\"FPar64\">7.2</xref>, which is a more general version of Theorem <xref ref-type=\"sec\" rid=\"FPar20\">1.20</xref>. In Sects. <xref rid=\"Sec15\" ref-type=\"sec\">8</xref> and <xref rid=\"Sec16\" ref-type=\"sec\">9</xref> we prove Theorems <xref ref-type=\"sec\" rid=\"FPar16\">1.16</xref> and <xref ref-type=\"sec\" rid=\"FPar10\">1.10</xref>, respectively.</p>" ]

[]

[]

[ "<p id=\"Par1\">Let be a class of topological semigroups. A semigroup <italic>X</italic> is called <italic>absolutely </italic><italic>-closed</italic> if for any homomorphism to a topological semigroup , the image <italic>h</italic>[<italic>X</italic>] is closed in <italic>Y</italic>. Let , , and be the classes of , Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup <italic>X</italic> is absolutely -closed if and only if <italic>X</italic> is absolutely -closed if and only if <italic>X</italic> is chain-finite, bounded, group-finite and Clifford + finite. On the other hand, a commutative semigroup <italic>X</italic> is absolutely -closed if and only if <italic>X</italic> is finite. Also, for a given absolutely -closed semigroup <italic>X</italic> we detect absolutely -closed subsemigroups in the center of <italic>X</italic>.</p>", "<title>Keywords</title>", "<title>Mathematics Subject Classification</title>", "<p>Open access funding provided by The Ministry of Education, Science, Research and Sport of the Slovak Republic in cooperation with Centre for Scientific and Technical Information of the Slovak Republic</p>" ]

[ "<title>Preliminaries</title>", "<p id=\"Par115\">In this section we collect some auxiliary results and notions that will be used in the remaining part of the paper.</p>", "<p id=\"Par116\">We denote by the set of finite ordinals and by the set of positive integers. Each ordinal is identified with the set of smaller ordinals.</p>", "<title>Partially ordered sets</title>", "<p id=\"Par117\">A <italic>poset</italic> is a set endowed with a partial order . For an element <italic>p</italic> of a poset <italic>P</italic>, letbe the <italic>lower</italic> and <italic>upper sets</italic> of <italic>p</italic> in <italic>P</italic>, respectively.</p>", "<title>Cardinal characteristics of the continuum</title>", "<p id=\"Par118\">Let<list list-type=\"bullet\"><list-item><p id=\"Par119\"> be the smallest cardinality of a cover of the real line by nowhere dense sets,</p></list-item><list-item><p id=\"Par120\"> be the smallest cardinality of a cover of the real line by sets of Lebesgue measure zero,</p></list-item><list-item><p id=\"Par121\"> be the smallest cardinality of a cover of the real line by closed subsets of Lebesgue measure zero, and</p></list-item><list-item><p id=\"Par122\"> be the smallest cardinality of a cover of the Baire space by compact sets.</p></list-item></list>By [##UREF##15##16##, 4.1],Martin’s Axiom implies that , see [##UREF##16##17##, §7]. By Theorem 7.13 in [##UREF##16##17##], the equality is equivalent to the Martin’s Axiom for countable posets. By [##UREF##15##16##, 5.6] and [##UREF##16##17##, 11.5], the strict inequalities and are consistent.</p>", "<title>Semigroup topologies and subinvariant metrics on semigroups</title>", "<p id=\"Par123\">A topology on a semigroup <italic>X</italic> is called a <italic>semigroup topology</italic> if is a topological semigroup.</p>", "<p id=\"Par124\">A metric <italic>d</italic> on a semigroup <italic>X</italic> is <italic>subinvariant</italic> if for every and we haveIt is easy to see that every subinvariant metric on a semigroup generates a semigroup topology.</p>", "<title>Zero-closed semigroups</title>", "<p id=\"Par125\">For a semigroup <italic>X</italic> its<list list-type=\"bullet\"><list-item><p id=\"Par126\"><italic>0-extension</italic> is the semigroup where is any element such that for every ;</p></list-item><list-item><p id=\"Par127\"><italic>1-extension</italic> is the semigroup where is any element such that for every .</p></list-item></list>Following [##UREF##8##9##], we call a semigroup <italic>X</italic>\n<italic>zero-closed</italic> if <italic>X</italic> is closed in its 0-extension endowed with any Hausdorff semigroup topology.</p>", "<p id=\"Par128\">A topological semigroup <italic>X</italic> is called 0-<italic>discrete</italic> if <italic>X</italic> contains a unique non-isolated point 0 such that for all . It is easy to see that every 0-discrete topological semigroup is zero-dimensional.</p>", "<title>Lemma 2.1</title>", "<p id=\"Par129\">Let be a class of topological semigroups containing all 0-discrete semigroups. Every -closed semigroup <italic>X</italic> is zero-closed.</p>", "<title>Proof</title>", "<p id=\"Par130\">Assuming that <italic>X</italic> is not zero-closed, we can find a Hausdorff semigroup topology on such that <italic>X</italic> is not closed in the topological space . Consider the topology on , generated by the base , and observe that is a 0-discrete Hausdorff topological semigroup containing <italic>X</italic> as a non-closed subsemigroup and proving that <italic>X</italic> is not -closed. </p>", "<title>Polybounded and polyfinite semigroups</title>", "<p id=\"Par131\">A <italic>semigroup polynomial</italic> on a semigroup <italic>X</italic> is a function of the form for some and some elements . The number <italic>n</italic> is called the <italic>degree</italic> of the polynomial <italic>f</italic> and is denoted by .</p>", "<p id=\"Par132\">A semigroup <italic>X</italic> is called -<italic>polybounded</italic> for a cardinal if for some elements and semigroup polynomials on <italic>X</italic>. A semigroup <italic>X</italic> is <italic>polybounded</italic> if <italic>X</italic> is <italic>n</italic>-polybounded for some .</p>", "<p id=\"Par133\">Polybounded semigroups were introduced in [##UREF##8##9##], where it was proved that countable zero-closed semigroups are polybounded and polybounded groups are absolutely -closed.</p>", "<p id=\"Par134\">A semigroup <italic>X</italic> is called <italic>polyfinite</italic> if there exist and a finite set such that for any there exists a semigroup polynomial of degree such that .</p>", "<title>Lemma 2.2</title>", "<p id=\"Par135\">Every polybounded semigroup <italic>X</italic> is polyfinite.</p>", "<title>Proof</title>", "<p id=\"Par136\">Since <italic>X</italic> is polybounded, there exist elements and semigroup polynomials on <italic>X</italic> such that . LetGiven any elements , find such that and then find such that . The semigroup polynomial has degree and , proving that <italic>X</italic> is polyfinite. </p>", "<p>The following theorem was proved in [##UREF##11##12##].</p>", "<title>Theorem 2.3</title>", "<p id=\"Par138\">Let <italic>X</italic> be a zero-closed semigroup. Then <list list-type=\"order\"><list-item><p id=\"Par139\"><italic>X</italic> is -polybounded for some .</p></list-item><list-item><p id=\"Par140\">If and <italic>X</italic> admits a subinvariant separable complete metric, then <italic>X</italic> is polybounded.</p></list-item><list-item><p id=\"Par141\">If and <italic>X</italic> admits a compact Hausdorff semigroup topology, then <italic>X</italic> is polybounded.</p></list-item><list-item><p id=\"Par142\">If and <italic>X</italic> admits a compact Hausdorff semigroup topology, then <italic>X</italic> is polyfinite.</p></list-item></list></p>", "<title>Prime coideals in semigroups</title>", "<p id=\"Par143\">A subset <italic>C</italic> of a semigroup <italic>X</italic> is called a (<italic>prime</italic>) <italic>coideal</italic> if is an ideal in <italic>X</italic> (and <italic>C</italic> is a subsemigroup of <italic>X</italic>). A subset is a prime coideal in <italic>X</italic> if and only if its characteristic functionis a homomorphism from <italic>X</italic> to the semilattice endowed with the operation of minimum.</p>", "<title>Lemma 2.4</title>", "<p id=\"Par144\">If a semigroup <italic>X</italic> is absolutely (resp. projectively) -closed, then any prime coideal in <italic>X</italic> is absolutely (resp. projectively) -closed.</p>", "<title>Proof</title>", "<p id=\"Par145\">Assume that a semigroup <italic>X</italic> is absolutely (resp. projectively) -closed and let <italic>C</italic> be a prime coideal in <italic>X</italic>. To prove that the semigroup <italic>C</italic> is absolutely (resp. projectively) -closed, take any homomorphism to a topological semigroup (such that the image <italic>h</italic>[<italic>C</italic>] is discrete in <italic>Y</italic>). Since <italic>C</italic> is a prime coideal in <italic>X</italic>, the mapis a homomorphism from <italic>X</italic> to the 0-extension of the topological semigroup <italic>Y</italic>, endowed with the topology . It follows from that . By the absolute (resp. projective) -closedness of <italic>X</italic>, the image is closed in and then the set is closed in , proving that the semigroup <italic>C</italic> is absolutely (resp. projectively) -closed. </p>", "<title>Viable idempotents in semigroups</title>", "<p id=\"Par146\">We recall that an idempotent <italic>e</italic> of a semigroup <italic>X</italic> is <italic>viable</italic> if the subsemigroup is a prime coideal in <italic>X</italic>. By we denote the set of viable idempotents in <italic>X</italic>.</p>", "<p id=\"Par147\">The following lemma was proved in [##UREF##9##10##, 2.5].</p>", "<title>Lemma 2.5</title>", "<p id=\"Par148\">For any semigroup <italic>X</italic> we have .</p>", "<title>Historical Remark 2.6</title>", "<p id=\"Par149\">The inclusion in Lemma <xref ref-type=\"sec\" rid=\"FPar29\">2.5</xref> cannot be improved to the inclusion : by [##UREF##13##14##] or [##UREF##19##20##] there exist infinite congruence-free monoids. In every congruence-free monoid the idempotent 1 is central but not viable.</p>", "<title>Proof of Theorem <xref ref-type=\"sec\" rid=\"FPar21\">1.21</xref></title>", "<p id=\"Par150\">In this section, for any semigroup <italic>X</italic> we study the order properties of the posets and and prove Theorem <xref ref-type=\"sec\" rid=\"FPar21\">1.21</xref>. By we denote the two-element semilattice endowed with the operation of minimum.</p>", "<title>Proposition 3.1</title>", "<p id=\"Par151\">Let <italic>X</italic> be a semigroup and be the quotient homomorphism onto its semilattice reflection. The restriction is injective and hence is an isomorphic embedding of the poset into the poset .</p>", "<title>Proof</title>", "<p id=\"Par152\">Given two viable idempotents , assume that . For every , the definition of a viable idempotent ensures that the semigroup is a coideal in <italic>X</italic>. Then the map defined byis a homomorphism. The equality implies thatThus, , which implies , and proves that the restriction is injective. </p>", "<p>In the following four lemmas we prove the statements of Theorem <xref ref-type=\"sec\" rid=\"FPar21\">1.21</xref>.</p>", "<title>Lemma 3.2</title>", "<p id=\"Par154\">For any ideally -closed semigroup <italic>X</italic>, the posets and are well-founded.</p>", "<title>Proof</title>", "<p id=\"Par155\">By Proposition <xref ref-type=\"sec\" rid=\"FPar31\">3.1</xref>, the poset embeds into the semilattice reflection of <italic>X</italic>, so it suffices to prove that the poset is well-founded. Assuming that <italic>Y</italic> is not well-founded, we can find a strictly decreasing sequence in <italic>Y</italic>. For every consider the upper set and observe that is a prime coideal in <italic>Y</italic>. Consequently, its preimage is a prime coideal in <italic>X</italic>.</p>", "<p id=\"Par156\">It is easy to see that is a subsemigroup of <italic>X</italic> and the complement is an ideal in <italic>X</italic>. Consider the semigroup endowed with the semigroup operation defined byEndow the semigroup <italic>S</italic> with the topology generated by the baseand observe that is a Hausdorff zero-dimensional topological semigroup with a unique non-isolated point <italic>P</italic>. Since contains the quotient semigroup as a discrete subsemigroup, the semigroup <italic>X</italic> is not ideally -closed, which contradicts our assumption. </p>", "<title>Lemma 3.3</title>", "<p id=\"Par157\">For any projectively -closed semigroup <italic>X</italic>, the posets and are chain-finite.</p>", "<title>Proof</title>", "<p id=\"Par158\">Let be the quotient homomorphism of <italic>X</italic> onto its semilattice reflection. If the semigroup <italic>X</italic> is projectively -closed, then its semilattice reflection is projectively -closed and hence -closed. By Theorem <xref ref-type=\"sec\" rid=\"FPar4\">1.4</xref>, the semilattice is chain-finite. Then is also chain-finite as a poset. By Proposition <xref ref-type=\"sec\" rid=\"FPar31\">3.1</xref>, the poset is chain-finite, being order isomorphic to a subset of the chain-finite poset . </p>", "<title>Lemma 3.4</title>", "<p id=\"Par159\">If a semigroup <italic>X</italic> is projectively -closed and projectively -discrete, then the sets and are finite.</p>", "<title>Proof</title>", "<p id=\"Par160\">Let be the quotient homomorphism of <italic>X</italic> onto its semilattice reflection. Consider the set <italic>H</italic> of all homomorphisms from to the two-element semilattice . Since homomorphisms to separate points of semilattices, the homomorphism , , is injective. Since the semilattice is -closed and -discrete, the image is a closed discrete subsemilattice of the compact topological semilattice . Hence is finite and so is the set . </p>", "<p>Theorem <xref ref-type=\"sec\" rid=\"FPar8\">1.8</xref> and Lemma <xref ref-type=\"sec\" rid=\"FPar37\">3.4</xref> imply the following lemma.</p>", "<title>Lemma 3.5</title>", "<p id=\"Par162\">For any absolutely -closed semigroup <italic>X</italic>, the sets and are finite.</p>", "<title>Absolutely -closed semigroups</title>", "<p id=\"Par163\">In this section we establish some properties of absolutely -closed semigroups and prove the following theorem that implies the characterization Theorem <xref ref-type=\"sec\" rid=\"FPar9\">1.9</xref> announced in the introduction.</p>", "<title>Theorem 4.1</title>", "<p id=\"Par164\">For any semigroup <italic>X</italic> we have implications of the following statements: <list list-type=\"order\"><list-item><p id=\"Par165\"><italic>X</italic> is finite;</p></list-item><list-item><p id=\"Par166\"><italic>X</italic> is absolutely -closed;</p></list-item><list-item><p id=\"Par167\"><italic>X</italic> is projectively -closed and projectively -discrete;</p></list-item><list-item><p id=\"Par168\"><italic>X</italic> is projectively -closed and projectively -discrete;</p></list-item><list-item><p id=\"Par169\"><italic>X</italic> is projectively -closed and is finite;</p></list-item><list-item><p id=\"Par170\"><italic>Z</italic>(<italic>X</italic>) is periodic and is finite.</p></list-item></list>If <italic>X</italic> is commutative, then the conditions (1)–(6) are equivalent.</p>", "<title>Proof</title>", "<p id=\"Par171\">The implication is trivial, the equivalence was proved in Theorem <xref ref-type=\"sec\" rid=\"FPar8\">1.8</xref>, and the implication is trivial.</p>", "<p id=\"Par172\">To prove that , assume that the semigroup <italic>X</italic> is projectively -closed and projectively -discrete. By Lemma <xref ref-type=\"sec\" rid=\"FPar37\">3.4</xref> the set is finite and by Theorem <xref ref-type=\"sec\" rid=\"FPar13\">1.13</xref>(1,2), the semigroup <italic>Z</italic>(<italic>X</italic>) is periodic and group-finite. Then for every the intersection is either empty or a finite subgroup of <italic>Z</italic>(<italic>X</italic>). In both cases, the set is finite. Then the setis finite, being the union of finitely many finite sets.</p>", "<p id=\"Par173\">To prove that , assume that the semigroup <italic>X</italic> is projectively -closed and the set is finite. By Theorem <xref ref-type=\"sec\" rid=\"FPar13\">1.13</xref>(1), the semigroup <italic>Z</italic>(<italic>X</italic>) is periodic. By Theorem <xref ref-type=\"sec\" rid=\"FPar18\">1.18</xref>, the set is finite and then the setis finite, too.</p>", "<p id=\"Par174\">Now assuming that <italic>X</italic> is commutative, we shall prove that . So, assume that the semigroup <italic>Z</italic>(<italic>X</italic>) is periodic and is finite. Being commutative, the semigroup <italic>X</italic> is viable and hence . The periodicity of <italic>Z</italic>(<italic>X</italic>) implies that and hence the commutative semigroup is finite. </p>", "<title>Corollary 4.2</title>", "<p id=\"Par175\">If a <italic>Z</italic>-viable semigroup is absolutely -closed, then its center <italic>Z</italic>(<italic>X</italic>) is finite and hence -closed.</p>", "<title>Proof</title>", "<p id=\"Par176\">The <italic>Z</italic>-viability of <italic>X</italic> yields . By Theorem <xref ref-type=\"sec\" rid=\"FPar13\">1.13</xref>(1), the semigroup <italic>Z</italic>(<italic>X</italic>) is periodic and hence is finite, by Theorem <xref ref-type=\"sec\" rid=\"FPar40\">4.1</xref>. </p>", "<title>A sufficient condition of the absolute -closedness</title>", "<p id=\"Par177\">In this section we shall prove a sufficient condition of the absolute -closedness. We shall use the following theorem, proved by Stepp in [##UREF##62##63##, Theorem 9].</p>", "<title>Theorem 5.1</title>", "<p id=\"Par178\">(Stepp) Every chain-finite semilattice is absolutely -closed.</p>", "<p>A semigroup <italic>X</italic> is called <italic>E</italic>-<italic>commutative</italic> if for any idempotents .</p>", "<title>Lemma 5.2</title>", "<p id=\"Par180\">Each chain-finite group-finite bounded Clifford + finite <italic>E</italic>-commutative semigroup <italic>X</italic> is absolutely -closed.</p>", "<title>Proof</title>", "<p id=\"Par181\">To show that <italic>X</italic> is absolutely -closed, take any homomorphism to a Hausdorff topological semigroup <italic>Y</italic>. We should prove that the semigroup <italic>h</italic>[<italic>X</italic>] is closed in <italic>Y</italic>. Replacing <italic>Y</italic> by , we can assume that <italic>h</italic>[<italic>X</italic>] is dense in <italic>Y</italic>. Since <italic>X</italic> is bounded, there exists such that and hence for every . Taking into account that <italic>h</italic> is a homomorphism, we conclude that for all . The closed subset of <italic>Y</italic> contains the dense set <italic>h</italic>[<italic>X</italic>] and hence coincides with <italic>Y</italic>. Therefore, for all . It follows that the continuous map , , is well-defined. Consider the function , , and observe that for every .</p>", "<p id=\"Par182\">Since <italic>X</italic> is a chain-finite <italic>E</italic>-commutative semigroup, the set <italic>E</italic>(<italic>X</italic>) is a chain-finite subsemilattice of <italic>X</italic>. By Theorem <xref ref-type=\"sec\" rid=\"FPar44\">5.1</xref>, the chain-finite semilattice <italic>E</italic>(<italic>X</italic>) is absolutely -closed and hence its image <italic>h</italic>[<italic>E</italic>(<italic>X</italic>)] is closed in the Hausdorff topological semigroup <italic>Y</italic>. The continuity of the map , , implies thatHence . The choice of <italic>n</italic> implies that for all . Since <italic>X</italic> is Clifford + finite, the set is finite. Then . By the Hausdorff property of <italic>Y</italic>, the set is closed in <italic>Y</italic> and contains the set . Then for any .</p>", "<p id=\"Par183\">Assuming that <italic>h</italic>[<italic>X</italic>] is not closed in <italic>Y</italic>, take any point and consider the idempotent . Since the semilattice <italic>E</italic>(<italic>X</italic>) is chain-finite, we can apply Theorem <xref ref-type=\"sec\" rid=\"FPar6\">1.6</xref> and conclude that the semilattice is chain-finite and so is the subsemilattice of <italic>E</italic>(<italic>Y</italic>). By Theorem <xref ref-type=\"sec\" rid=\"FPar44\">5.1</xref>, <italic>L</italic> is closed in <italic>E</italic>(<italic>Y</italic>). Then its complement is open in <italic>E</italic>(<italic>Y</italic>) and its preimage is an open neighborhood of <italic>y</italic> in <italic>Y</italic>. Since and the semilattice <italic>E</italic>(<italic>X</italic>) is chain-finite, the nonempty subsemilattice has a unique minimal element . Since the semigroup <italic>X</italic> is group-finite, the maximal subgroup is finite and so is the set . Since , there exists a neighborhood of <italic>y</italic> in <italic>Y</italic> such that . It implies that , as . Since , we can additionally assume that .</p>", "<p id=\"Par184\">Sincewe can choose an element and observe that and hence . The minimality of in ensures that . On the other hand, implies that and hence and . On the other hand, and hence . This contradiction shows that the set <italic>h</italic>[<italic>X</italic>] is closed in <italic>Y</italic>. </p>", "<title>Bounded sets in absolutely -closed semigroups</title>", "<p id=\"Par185\">In this section, given an absolutely -closed semigroup <italic>X</italic>, we characterize subsets for which the set is bounded in <italic>X</italic>. We recall that a subset is <italic>bounded</italic> if for some .</p>", "<p id=\"Par186\">The following notion plays a crucial role in our subsequent results.</p>", "<title>Definition 6.1</title>", "<p id=\"Par187\">A semigroup <italic>X</italic> is defined to be <italic>A</italic>-<italic>centrobounded</italic> over a set if there exists such that for every and with we have .</p>", "<p>In the following theorem we endow the set with the natural partial order considered in Sect. <xref rid=\"Sec10\" ref-type=\"sec\">3</xref>. A subset is called an <italic>antichain</italic> if for any distinct elements .</p>", "<title>Theorem 6.2</title>", "<p id=\"Par189\">Let <italic>X</italic> be an absolutely -closed semigroup. For a subset the following conditions are equivalent: <list list-type=\"order\"><list-item><p id=\"Par190\"> is bounded in <italic>X</italic>;</p></list-item><list-item><p id=\"Par191\"> is bounded in <italic>X</italic>;</p></list-item><list-item><p id=\"Par192\"><italic>X</italic> is <italic>B</italic>-centrobounded over every countable infinite antichain .</p></list-item></list></p>", "<title>Proof</title>", "<p id=\"Par193\">Replacing the semigroup <italic>X</italic> by its 1-extension , we lose no generality assuming that the semigroup <italic>X</italic> contains a two-sided unit 1. By Theorem <xref ref-type=\"sec\" rid=\"FPar13\">1.13</xref>(1,2), the semigroup <italic>Z</italic>(<italic>X</italic>) is chain-finite, periodic, nonsingular, and group-finite.</p>", "<p id=\"Par194\">The equivalence follows from Theorem <xref ref-type=\"sec\" rid=\"FPar18\">1.18</xref> and is trivial. Indeed, by (2), there exists such that . We claim that the number <italic>n</italic> witnesses that <italic>X</italic> is <italic>B</italic>-centrobounded over any set . Indeed, given any idempotent and elements with , by the periodicity of <italic>Z</italic>(<italic>X</italic>) and the choice of <italic>n</italic>, we haveand hence .</p>", "<p>It remains to prove the implication . Let be the map assigning to each element a unique idempotent in the monogenic semigroup . To derive a contradiction, assume that the condition (3) is satisfied but (2) is not.</p>", "<title>Claim 6.3</title>", "<p id=\"Par196\">There exists a sequence in such that <list list-type=\"order\"><list-item><p id=\"Par197\"> for any distinct numbers ;</p></list-item><list-item><p id=\"Par198\"> for any and distinct numbers .</p></list-item></list></p>", "<title>Proof</title>", "<p id=\"Par199\">Since the set is unbounded in <italic>X</italic>, for every there exists an element such that for any distinct positive numbers we have . Let be the family of two-element subsets of . Consider the function defined byBy the Ramsey Theorem (see [##UREF##40##41##, Theorem 5]), there exists an infinite set such that for some . If , then the set contains a unique idempotent <italic>u</italic> and hence the set is finite (since <italic>Z</italic>(<italic>X</italic>) is periodic and group-finite). By the Pigeonhole Principle, for any there are two numbers such that , which contradicts the choice of . Therefore, . If , then the set is an infinite chain in which is not possible as <italic>Z</italic>(<italic>X</italic>) is chain-finite. Therefore, and hence is an infinite antichain in <italic>E</italic>(<italic>Z</italic>(<italic>X</italic>)). Write the infinite set as for some strictly increasing sequence . For every put and observe that the sequence satisfies the conditions (1), (2) of Claim <xref ref-type=\"sec\" rid=\"FPar50\">6.3</xref>. </p>", "<p>Let be the quotient homomorphism of <italic>X</italic> onto its semilattice reflection.</p>", "<p>Let be the sequence from Claim <xref ref-type=\"sec\" rid=\"FPar50\">6.3</xref>. For every let . The inclusion and the periodicity of <italic>Z</italic>(<italic>X</italic>) imply that the idempotent is viable. Then the setis a prime coideal in <italic>X</italic> and moreover, , see Proposition 2.15 in [##UREF##3##4##].</p>", "<p>Since the semigroup <italic>X</italic> is absolutely -closed, for the idealin <italic>X</italic>, the quotient semigroup <italic>X</italic>/<italic>I</italic> is absolutely -closed.</p>", "<p>For convenience, by 0 we denote the element . The injectivity of the restriction implies that for any distinct . This implies that the ideal <italic>I</italic> is not empty and the element of the semigroup <italic>X</italic>/<italic>I</italic> is well-defined.</p>", "<p>Now we introduce a 0-discrete Hausdorff semigroup topology on the semigroup .</p>", "<p>Fix any free ultrafilter on . LetNote that the set <italic>Q</italic> is nonempty, as (we assumed that <italic>X</italic> contains a unit exactly to omit the easier case ).</p>", "<p>For any there exist such that for each , . Then for each . It follows that <italic>Q</italic> is a subsemilattice of . By Lemma <xref ref-type=\"sec\" rid=\"FPar35\">3.3</xref>, the semilattice is chain-finite and so is its subsemilattice <italic>Q</italic>. Thus, the semilattice <italic>Q</italic> contains the smallest element <italic>s</italic>. Since , there exists a set such that for all . Consider the prime coideal</p>", "<title>Claim 6.4</title>", "<p id=\"Par207\"> and hence .</p>", "<title>Proof</title>", "<p id=\"Par208\">The inclusion follows from the choice of . Now take any and observe that and hence and by the choice of <italic>s</italic>. Then</p>", "<p>To introduce the topology on <italic>Y</italic>, we need the following notations. For a real number <italic>r</italic> by we denote the integer part of <italic>r</italic>. For each and letThe definition of the set <italic>A</italic>(<italic>n</italic>, <italic>k</italic>) implies that if . For every and consider the subsetof <italic>Y</italic>. On the semigroup consider the topology generated by the base</p>", "<title>Claim 6.5</title>", "<p id=\"Par210\"> is a Hausdorff zero-dimensional topological semigroup.</p>", "<title>Proof</title>", "<p id=\"Par211\">To see that the topology is Hausdorff, it suffices to show that for any there exist and such that . If , then . If for some , then . Therefore, the topology is Hausdorff. Since 0 is a unique non-isolated point of , the topology is zero-dimensional.</p>", "<p id=\"Par212\">It remains to prove that is a topological semigroup. Given any two points and a neighborhood of their product , we need to find neighborhoods of , respectively, such that . If , then the neighborhoods and have the required property: .</p>", "<p id=\"Par213\">So, it remains to consider three cases: <list list-type=\"order\"><list-item><p id=\"Par214\"> and ;</p></list-item><list-item><p id=\"Par215\"> and ;</p></list-item><list-item><p id=\"Par216\">.</p></list-item></list>In each of these cases, , so we can find and such that and .</p>", "<p id=\"Par217\">(1) Assume that and . If , thenSo, we can put and .</p>", "<p id=\"Par218\">If , then and hence the set does not belong to the ultrafilter and then the set belongs to the ultrafilter . For every and we have , implying and . So we can put and .</p>", "<p id=\"Par219\">(2) The case and can be treated by analogy with the preceding case.</p>", "<p id=\"Par220\">(3) If , then we can put . Let us show that . Indeed, take any . If or <italic>a</italic> and <italic>b</italic> do not belong to the same subgroup , then . Otherwise, there exists such that and hence and for some numbers . Then and henceTaking into account that , we obtain . </p>", "<p>Note that , being a continuous homomorphic image of the absolutely -closed semigroup <italic>X</italic>, is itself -closed. But, as we will show later, this is not the case.</p>", "<p>Let be the ultrafilter on <italic>Y</italic> generated by the base where for . Note that for any the filter generated by the base is an ultrafilter on <italic>Y</italic>. Also, since we get that , where is the ultrafilter generated by the base .</p>", "<title>Claim 6.6</title>", "<p id=\"Par223\">For any the ultrafilter is the principal ultrafilter at 0.</p>", "<title>Proof</title>", "<p id=\"Par224\">The claim is obvious if . So, assume that . Since , the set does not belong to the ultrafilter . Then the set belongs to . Now observe that and hence . Therefore, the ultrafilter is principal at 0. </p>", "<title>Claim 6.7</title>", "<p id=\"Par225\">There exists such that for every .</p>", "<title>Proof</title>", "<p id=\"Par226\">Since <italic>X</italic> is -centrobounded, there exists such that for every and with we have .</p>", "<p id=\"Par227\">We claim that for every , the set does not belong to the ultrafilter . In the opposite case, the set has non-empty intersection with the set . Then there exists such that and hence for some and with . It follows from that . Then the equality implies that and hence . Now the choice of <italic>m</italic> ensures that . Thenwhich contradicts the choice of the point in Claim <xref ref-type=\"sec\" rid=\"FPar50\">6.3</xref>(2) as . </p>", "<p>Let . Extend the semigroup operation from <italic>Y</italic> to the set <italic>T</italic> by the formula:Let be the topology on the semigroup <italic>T</italic> which satisfies the following conditions:<list list-type=\"bullet\"><list-item><p id=\"Par229\"> is an open subspace of ;</p></list-item><list-item><p id=\"Par230\">if for some , then there exists such that .</p></list-item></list></p>", "<title>Claim 6.8</title>", "<p id=\"Par231\">The topology on <italic>T</italic> is Hausdorff and zero-dimensional.</p>", "<title>Proof</title>", "<p id=\"Par232\">First we show that the topological space is zero-dimensional. Given an open set and a point , we need to find a clopen set <italic>V</italic> in such that . We consider three possible cases.</p>", "<p id=\"Par233\">1. If , then <italic>u</italic> is an isolated point of <italic>Y</italic> and <italic>T</italic>. So, we can take . The definition of the topology ensures that is a clopen neighborhood of <italic>u</italic> in .</p>", "<p id=\"Par234\">2. If , then we can apply Claim <xref ref-type=\"sec\" rid=\"FPar58\">6.7</xref> and find and such that and for all . The definition of the topology ensures that is a clopen neighborhood of in .</p>", "<p id=\"Par235\">3. If for some , then by the definition of the topology , there exists such that and . Moreover, by Claim <xref ref-type=\"sec\" rid=\"FPar58\">6.7</xref>, we can assume that for some . By the definition of the topology , the set is a neighborhood of in . It remains to show that the set <italic>V</italic> is closed in . Given any , we should find a neighborhood of <italic>t</italic> such that . If , then the neighborhood of <italic>t</italic> is disjoint with <italic>V</italic>. If , then the neighborhood of is disjoint with <italic>V</italic>. Finally assume that for some . Consider the set . If , then , which contradicts the choice of . Therefore, and the set belongs to the ultrafilter . Then is a neighborhood of such that , proving that the set <italic>V</italic> is clopen.</p>", "<p id=\"Par236\">Therefore the topology is zero-dimensional and being , it is Hausdorff. </p>", "<p>To check the continuity of the semigroup operation in , take any elements and choose any neighborhood of their product <italic>ab</italic>. We must find neighborhoods of <italic>a</italic>, <italic>b</italic> such that . If , then such neighborhoods exist by the continuity of the semigroup operation in the topological semigroup .</p>", "<p>So, it remains to consider three cases:</p>", "<p>1. and for some . This case has three subcases.</p>", "<p>(1a) . In this case there exists a set such that , and then the neighborhoods and have the required property .</p>", "<p>(1b) . Since , the set does not belong to the ultrafilter and hence the set belongs to . Then for the neighborhoods and we have .</p>", "<p>(1c) . In this case and we can find and such that . We claim that the neighborhoods and satisfy . Given any elements and , we should check that . This is clear if . If , then for some , and . If , then . So, assume that for some . If , then and hence . Suppose that . Then andas .</p>", "<p>2. for some and . This case can be considered by analogy with the preceding case.</p>", "<p>3. and for some . In this case and we can find and such that . Let . Then the neighborhoods and have the required property: .</p>", "<p>Observe that the continuity of the binary operation in implies that it is associative, as <italic>Y</italic> is a dense subsemigroup of <italic>T</italic>. Thus, is a Hausdorff zero-dimensional topological semigroup which contains as a non-closed subsemigroup. But this contradicts the absolute -closedness of and <italic>X</italic>. The obtained contradiction completes the proof of the implication and also the proof of Theorem <xref ref-type=\"sec\" rid=\"FPar48\">6.2</xref>. </p>", "<title>Some sufficient conditions of centroboundedness</title>", "<p id=\"Par246\">In this section we shall find some sufficient conditions for centroboundedness, which will be combined with Theorem <xref ref-type=\"sec\" rid=\"FPar48\">6.2</xref> in order to obtain the boundedness of certain sets in absolutely -closed semigroups.</p>", "<p id=\"Par247\">The following theorem is the main result of this section.</p>", "<title>Theorem 7.1</title>", "<p id=\"Par248\">Let <italic>X</italic> be a semigroup, be a countable set of viable idempotents. Consider the prime coideal in <italic>X</italic>, the group , and the homomorphism , . The semigroup <italic>X</italic> is <italic>A</italic>-centrobounded if one of the following conditions is satisfied: <list list-type=\"order\"><list-item><p id=\"Par249\"><italic>X</italic> is projectively -closed and for every the subsemigroup <italic>Ce</italic> of is commutative;</p></list-item><list-item><p id=\"Par250\">the semigroup <italic>h</italic>[<italic>C</italic>] is polyfinite;</p></list-item><list-item><p id=\"Par251\"><italic>X</italic> is projectively -closed and <italic>h</italic>[<italic>C</italic>] is countable;</p></list-item><list-item><p id=\"Par252\"><italic>X</italic> is absolutely -closed, , and for every the subsemigroup is countable.</p></list-item><list-item><p id=\"Par253\"><italic>X</italic> is absolutely -closed, and for every the subsemigroup <italic>Ce</italic> of is bounded.</p></list-item></list></p>", "<title>Proof</title>", "<p id=\"Par254\">By our assumption, the set <italic>A</italic> is countable and hence admits an injective function . Consider the group endowed with the Tychonoff product topology, where the groups , are discrete. This topology is generated by the complete invariant metric defined byFor every , consider the homomorphismThe homomorphisms create the homomorphismFor every , let , , be the <italic>a</italic>th coordinate projection. The definition of the homomorphism <italic>h</italic> implies that for every .</p>", "<p id=\"Par255\">1. Assume that <italic>X</italic> is projectively -closed and for every the subsemigroup <italic>Ce</italic> of is commutative. Then the semigroup is commutative. By Lemma <xref ref-type=\"sec\" rid=\"FPar27\">2.4</xref>, the prime coideal <italic>C</italic> of <italic>X</italic> is projectively -closed and so is its homomorphic image <italic>h</italic>[<italic>C</italic>]. By Theorem <xref ref-type=\"sec\" rid=\"FPar13\">1.13</xref>(1), the -closed commutative semigroup is periodic and hence is a subgroup of the group <italic>H</italic>. By Theorem <xref ref-type=\"sec\" rid=\"FPar4\">1.4</xref>, the -closed commutative group <italic>h</italic>[<italic>C</italic>] is bounded. Then there exists such that for every . Consequently, for every and we have that , proving that <italic>X</italic> is <italic>A</italic>-centrobounded.</p>", "<p id=\"Par256\">2. Assume that the semigroup <italic>h</italic>[<italic>C</italic>] is polyfinite. Then there exist and a finite set such that for any there exists a semigroup polynomial of degree such that . To show that <italic>X</italic> is <italic>A</italic>-centrobounded, it suffices to check that for any and with we have where .</p>", "<p id=\"Par257\">For every the assumption implies and hence . By the choice of <italic>n</italic> and <italic>F</italic>, there exists a semigroup polynomial of degree such that . Find elements such that for all . For every , consider the element of the group . Let be the semigroup polynomial defined by for . It is easy to see that and hence . Thenand hence . By the Pigeonhole Principle, there exists a triple and two positive numbers such thatIt follows from and thatSimilarly we can prove that . Since is a group, the equality implies and . Since divides , .</p>", "<p id=\"Par258\">3. Assume that the semigroup <italic>X</italic> is projectively -closed and the set <italic>h</italic>[<italic>C</italic>] is countable. By Lemma <xref ref-type=\"sec\" rid=\"FPar27\">2.4</xref>, the prime coideal <italic>C</italic> in <italic>X</italic> is projectively -closed and so is its homomorphic image <italic>h</italic>[<italic>C</italic>]. Being -closed, the semigroup <italic>h</italic>[<italic>C</italic>] is zero-closed, see Lemma <xref ref-type=\"sec\" rid=\"FPar22\">2.1</xref>. By Theorem <xref ref-type=\"sec\" rid=\"FPar26\">2.3</xref>(1), the zero-closed countable semigroup <italic>h</italic>[<italic>C</italic>] is polybounded and by Lemma <xref ref-type=\"sec\" rid=\"FPar24\">2.2</xref>, <italic>h</italic>[<italic>C</italic>] is polyfinite. By the preceding statement, the semigroup <italic>X</italic> is <italic>A</italic>-centrobounded.</p>", "<p id=\"Par259\">4. Assume that the semigroup <italic>X</italic> is absolutely -closed, , and for every the subsemigroup is countable. Since , the subsemigroup <italic>h</italic>[<italic>C</italic>] of the metric group is separable. By Lemma <xref ref-type=\"sec\" rid=\"FPar27\">2.4</xref>, the prime coideal <italic>C</italic> is absolutely -closed and so is its homomorphic image <italic>h</italic>[<italic>C</italic>]. By the absolute -closedness of <italic>h</italic>[<italic>C</italic>], the semigroup <italic>h</italic>[<italic>C</italic>] is zero-closed and also <italic>h</italic>[<italic>C</italic>] is closed in the zero-dimensional topological group <italic>H</italic>. Then the metric is complete and hence <italic>h</italic>[<italic>C</italic>] is a Polish space.</p>", "<p id=\"Par260\">We claim that <italic>h</italic>[<italic>C</italic>] is polybounded. If , then the Polish space <italic>h</italic>[<italic>C</italic>] is countable, see [##UREF##45##46##, 6.5]. By Theorem <xref ref-type=\"sec\" rid=\"FPar26\">2.3</xref>(1), the countable zero-closed semigroup <italic>h</italic>[<italic>C</italic>] is polybounded. If , then the inequality implies and then <italic>h</italic>[<italic>C</italic>] is polybounded by Theorem <xref ref-type=\"sec\" rid=\"FPar26\">2.3</xref>(2). So, in both cases, the semigroup <italic>h</italic>[<italic>C</italic>] is polybounded. By Lemma <xref ref-type=\"sec\" rid=\"FPar24\">2.2</xref>, <italic>h</italic>[<italic>C</italic>] is polyfinite. By the second statement of this theorem, the semigroup <italic>X</italic> is <italic>A</italic>-centrobounded.</p>", "<p id=\"Par261\">5. Assume that and for every the semigroup of <italic>X</italic> is bounded. For a bounded subset , letFor a finite subset letbe the <italic>least common multiple</italic> of numbers in the set <italic>F</italic>.</p>", "<p id=\"Par262\">To derive a contradiction, assume that the semigroup <italic>X</italic> is not <italic>A</italic>-centrobounded. Writing down the negation of the <italic>A</italic>-centroboundedness, we obtain sequences , , and such that for every the following conditions are satisfied: <list list-type=\"simple\"><list-item><label>(i)</label><p id=\"Par263\">, , and ;</p></list-item><list-item><label>(ii)</label><p id=\"Par264\"> for every where .</p></list-item></list>Consider the group , and the homomorphism , . Observe that , where , , is the projection. For every , let , , be the <italic>n</italic>-th coordinate projection. Endow the group with the complete invariant metricBy Lemma <xref ref-type=\"sec\" rid=\"FPar27\">2.4</xref>, the prime coideal <italic>C</italic> of the absolutely -closed semigroup <italic>X</italic> is absolutely -closed and so is its homomorphic image . Since the Tychonoff product topology on the group is zero-dimensional, the absolutely -closed subsemigroup of is closed in . Being absolutely -closed, the semigroup is zero-closed by Lemma <xref ref-type=\"sec\" rid=\"FPar22\">2.1</xref>. By Theorem <xref ref-type=\"sec\" rid=\"FPar26\">2.3</xref>(1), the semigroup is -polybounded for some . Then for some elements and some semigroup polynomials .</p>", "<p id=\"Par265\">Recall that is the least common multiple of the numbers , , and observe that for every , its inverse in is the limit of the sequence , which implies that and means that is a subgroup of .</p>", "<p id=\"Par266\">For every consider the elementObserve that for every we havewhich means that the sequence converges to the identity element of the topological group .</p>", "<p id=\"Par267\">Let and . Define the family of elements of by the recursive formula:The definition of the ultrametric on and the convergence of imply that for every the sequence is Cauchy in the metric space and hence it converges to some element . Since and , there exists such that the set is uncountable and hence contains two distinct sequences such that . Let be the smallest number such that . Then and by the minimality of <italic>m</italic>. Let and observe that . We lose no generality assuming that and . It follows from that . Find elements such that for all .</p>", "<p id=\"Par268\">For every let . Let be the semigroup polynomial defined by for . It is clear that .</p>", "<p id=\"Par269\">It follows from and for all thatand hence , which contradicts the choice of . </p>", "<title>Corollary 7.2</title>", "<p id=\"Par270\">Let <italic>X</italic> be an absolutely -closed semigroup and . Assume that for any infinite countable antichain , the coideal and the homomorphism , , one of the following conditions is satisfied: <list list-type=\"order\"><list-item><p id=\"Par271\">for every the subsemigroup <italic>Ce</italic> of is commutative;</p></list-item><list-item><p id=\"Par272\">the semigroup <italic>h</italic>[<italic>C</italic>] is polyfinite;</p></list-item><list-item><p id=\"Par273\"><italic>h</italic>[<italic>C</italic>] is countable;</p></list-item><list-item><p id=\"Par274\">, and for every the subsemigroup is countable.</p></list-item><list-item><p id=\"Par275\"> and for every the subsemigroup <italic>Ce</italic> of is bounded.</p></list-item></list>Then the set is bounded, and every subsemigroup of of <italic>X</italic> is absolutely -closed.</p>", "<title>Proof</title>", "<p id=\"Par276\">By Theorems <xref ref-type=\"sec\" rid=\"FPar48\">6.2</xref> and <xref ref-type=\"sec\" rid=\"FPar62\">7.1</xref>, the set is bounded. Now let be any subsemigroup of <italic>X</italic>. By Theorem <xref ref-type=\"sec\" rid=\"FPar13\">1.13</xref>, the semigroup is chain-finite, group-finite, periodic and nonsingular. The periodicity of <italic>S</italic> implies that and hence . By Theorem <xref ref-type=\"sec\" rid=\"FPar18\">1.18</xref>, the set is finite, which implies that the semigroup <italic>S</italic> is Clifford + finite. By Lemma <xref ref-type=\"sec\" rid=\"FPar45\">5.2</xref>, the chain-finite group-finite bounded Clifford + finite commutative semigroup <italic>S</italic> is absolutely -closed. </p>", "<title>Corollary 7.3</title>", "<p id=\"Par277\">If a semigroup <italic>X</italic> is absolutely -closed, then its ideal center is bounded and absolutely -closed.</p>", "<title>Proof</title>", "<p id=\"Par278\">For every we have , which implies that the maximal subgroup is commutative. By Theorem <xref ref-type=\"sec\" rid=\"FPar13\">1.13</xref>, the semigroup <italic>Z</italic>(<italic>X</italic>) is chain-finite, group-finite, periodic, and nonsingular. By Lemma <xref ref-type=\"sec\" rid=\"FPar29\">2.5</xref>, and by the periodicity of , we obtain . By Corollary <xref ref-type=\"sec\" rid=\"FPar64\">7.2</xref>(1), the set is bounded in <italic>X</italic> and so is its subset . By the periodicity, . By Theorem <xref ref-type=\"sec\" rid=\"FPar18\">1.18</xref>, the setis finite, which means that the semigroup is Clifford + finite.</p>", "<p id=\"Par279\">Therefore, the commutative semigroup is chain-finite, group-finite, bounded, and Clifford + finite. By Lemma <xref ref-type=\"sec\" rid=\"FPar45\">5.2</xref>, is absolutely -closed. </p>", "<title>Proof of Theorem <xref ref-type=\"sec\" rid=\"FPar16\">1.16</xref></title>", "<p id=\"Par280\">Let be class of topological semigroups such that either or .</p>", "<p id=\"Par281\">Given an absolutely -closed semigroup, we should prove that the ideal center of <italic>X</italic> is absolutely -closed. By Lemma <xref ref-type=\"sec\" rid=\"FPar29\">2.5</xref>, . Since , the semigroup <italic>X</italic> is absolutely -closed. By Theorem <xref ref-type=\"sec\" rid=\"FPar13\">1.13</xref>, the semigroup <italic>Z</italic>(<italic>X</italic>) is periodic and henceIf , then by Theorem <xref ref-type=\"sec\" rid=\"FPar40\">4.1</xref>, the set is finite and so is its subset .</p>", "<p id=\"Par282\">If , then the semigroup is absolutely -closed by Corollary <xref ref-type=\"sec\" rid=\"FPar66\">7.3</xref>.</p>", "<title>Proof of Theorem <xref ref-type=\"sec\" rid=\"FPar10\">1.10</xref></title>", "<p id=\"Par283\">Given a class of topological semigroups with , and a commutative semigroup <italic>X</italic>, we shall prove the equivalence of the assertions (1)–(4) of the theorem.</p>", "<p id=\"Par284\">The implication follows from Corollary <xref ref-type=\"sec\" rid=\"FPar66\">7.3</xref> and the equality holding by the commutativity of <italic>X</italic>.</p>", "<p id=\"Par285\">The implication , , and follow from Theorems <xref ref-type=\"sec\" rid=\"FPar13\">1.13</xref>(2), <xref ref-type=\"sec\" rid=\"FPar6\">1.6</xref>, and Lemma <xref ref-type=\"sec\" rid=\"FPar45\">5.2</xref>, respectively.</p>" ]

[ "<title>Funding</title>", "<p>Open access funding provided by The Ministry of Education, Science, Research and Sport of the Slovak Republic in cooperation with Centre for Scientific and Technical Information of the Slovak Republic.</p>", "<title>Data Availability</title>", "<p>Not applicable.</p>" ]

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\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y\\in {\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M8\"><mml:mrow><mml:mi>Y</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq5\"><alternatives><tex-math id=\"M9\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M10\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq6\"><alternatives><tex-math id=\"M11\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M12\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq7\"><alternatives><tex-math id=\"M13\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M14\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq8\"><alternatives><tex-math id=\"M15\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T_1$$\\end{document}</tex-math><mml:math id=\"M16\"><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq9\"><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M18\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq10\"><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M20\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq11\"><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M22\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq12\"><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M24\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq13\"><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M26\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq14\"><alternatives><tex-math id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M28\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq15\"><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X\\times X\\rightarrow X$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:mrow><mml:mi>X</mml:mi><mml:mo>×</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq16\"><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(x,y)\\mapsto xy$$\\end{document}</tex-math><mml:math id=\"M32\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>↦</mml:mo><mml:mi>x</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq17\"><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M34\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq18\"><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M36\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq19\"><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h:X\\rightarrow Y$$\\end{document}</tex-math><mml:math id=\"M38\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq20\"><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y\\in {\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M40\"><mml:mrow><mml:mi>Y</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq21\"><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M42\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h:X\\rightarrow Y$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y\\in {\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M46\"><mml:mrow><mml:mi>Y</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M48\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h:X\\rightarrow Y$$\\end{document}</tex-math><mml:math id=\"M50\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y\\in {\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:mrow><mml:mi>Y</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\text{ absolutely } \\, {\\mathcal {C}}\\text{-closed } \\Rightarrow \\text{ injectively }\\,{\\mathcal {C}}\\text{-closed } \\Rightarrow {\\mathcal {C}}\\text{-closed }. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M54\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>absolutely</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"0.166667em\"/><mml:mi mathvariant=\"script\">C</mml:mi><mml:mtext>-closed</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mspace width=\"0.333333em\"/><mml:mtext>injectively</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"0.166667em\"/><mml:mi mathvariant=\"script\">C</mml:mi><mml:mtext>-closed</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mtext>-closed</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M58\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq30\"><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T_1$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq31\"><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M64\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq32\"><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M66\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq33\"><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T_1$$\\end{document}</tex-math><mml:math id=\"M68\"><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq34\"><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T_1$$\\end{document}</tex-math><mml:math id=\"M70\"><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq35\"><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathsf {T_{\\!z}S}\\subseteq \\mathsf {T_{\\!2}S}\\subseteq \\mathsf {T_{\\!1}S}$$\\end{document}</tex-math><mml:math id=\"M72\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ52\"></disp-formula>", "<inline-formula id=\"IEq36\"><alternatives><tex-math id=\"M73\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M74\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq37\"><alternatives><tex-math id=\"M75\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M76\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq38\"><alternatives><tex-math id=\"M77\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M78\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq39\"><alternatives><tex-math id=\"M79\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\subseteq X$$\\end{document}</tex-math><mml:math id=\"M80\"><mml:mrow><mml:mi>I</mml:mi><mml:mo>⊆</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq40\"><alternatives><tex-math id=\"M81\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y\\in I$$\\end{document}</tex-math><mml:math id=\"M82\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq41\"><alternatives><tex-math id=\"M83\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$xy\\notin \\{x,y\\}$$\\end{document}</tex-math><mml:math id=\"M84\"><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mo>∉</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq42\"><alternatives><tex-math id=\"M85\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$A\\subseteq X$$\\end{document}</tex-math><mml:math id=\"M86\"><mml:mrow><mml:mi>A</mml:mi><mml:mo>⊆</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq43\"><alternatives><tex-math id=\"M87\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\in X$$\\end{document}</tex-math><mml:math id=\"M88\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq44\"><alternatives><tex-math id=\"M89\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M90\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq45\"><alternatives><tex-math id=\"M91\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x^n$$\\end{document}</tex-math><mml:math id=\"M92\"><mml:msup><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq46\"><alternatives><tex-math id=\"M93\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M94\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq47\"><alternatives><tex-math id=\"M95\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\in X$$\\end{document}</tex-math><mml:math id=\"M96\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq48\"><alternatives><tex-math id=\"M97\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x^n$$\\end{document}</tex-math><mml:math id=\"M98\"><mml:msup><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq49\"><alternatives><tex-math id=\"M99\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M100\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq50\"><alternatives><tex-math id=\"M101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M102\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq51\"><alternatives><tex-math id=\"M103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathsf {T_{\\!z}S}\\subseteq {\\mathcal {C}}\\subseteq \\mathsf {T_{\\!1}S}$$\\end{document}</tex-math><mml:math id=\"M104\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>⊆</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq52\"><alternatives><tex-math id=\"M105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M106\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq53\"><alternatives><tex-math id=\"M107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$IX\\cup XI\\subseteq I$$\\end{document}</tex-math><mml:math id=\"M108\"><mml:mrow><mml:mi>I</mml:mi><mml:mi>X</mml:mi><mml:mo>∪</mml:mo><mml:mi>X</mml:mi><mml:mi>I</mml:mi><mml:mo>⊆</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq54\"><alternatives><tex-math id=\"M109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\subseteq X$$\\end{document}</tex-math><mml:math id=\"M110\"><mml:mrow><mml:mi>I</mml:mi><mml:mo>⊆</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq55\"><alternatives><tex-math id=\"M111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(I\\times I)\\cup \\{(x,y)\\in X\\times X:x=y\\}$$\\end{document}</tex-math><mml:math id=\"M112\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>I</mml:mi><mml:mo>×</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∪</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mo>×</mml:mo><mml:mi>X</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq56\"><alternatives><tex-math id=\"M113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I=\\emptyset $$\\end{document}</tex-math><mml:math id=\"M114\"><mml:mrow><mml:mi>I</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">∅</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq57\"><alternatives><tex-math id=\"M115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/\\emptyset $$\\end{document}</tex-math><mml:math id=\"M116\"><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi mathvariant=\"normal\">∅</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq58\"><alternatives><tex-math id=\"M117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M118\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq59\"><alternatives><tex-math id=\"M119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M120\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq60\"><alternatives><tex-math id=\"M121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M122\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq61\"><alternatives><tex-math id=\"M123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M124\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq62\"><alternatives><tex-math id=\"M125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M126\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq63\"><alternatives><tex-math id=\"M127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\approx $$\\end{document}</tex-math><mml:math id=\"M128\"><mml:mo>≈</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq64\"><alternatives><tex-math id=\"M129\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\approx }$$\\end{document}</tex-math><mml:math id=\"M130\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>≈</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq65\"><alternatives><tex-math id=\"M131\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M132\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq66\"><alternatives><tex-math id=\"M133\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M134\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq67\"><alternatives><tex-math id=\"M135\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\subseteq X$$\\end{document}</tex-math><mml:math id=\"M136\"><mml:mrow><mml:mi>I</mml:mi><mml:mo>⊆</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq68\"><alternatives><tex-math id=\"M137\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M138\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ2\"><alternatives><tex-math id=\"M139\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\text{ absolutely }\\, {\\mathcal {C}}\\text{-closed } \\Rightarrow \\text{ projectively }\\, {\\mathcal {C}}\\text{-closed } \\Rightarrow \\text{ ideally }\\, {\\mathcal {C}}\\text{-closed } \\Rightarrow {\\mathcal {C}}\\text{-closed. } \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M140\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>absolutely</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"0.166667em\"/><mml:mi mathvariant=\"script\">C</mml:mi><mml:mtext>-closed</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mspace width=\"0.333333em\"/><mml:mtext>projectively</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"0.166667em\"/><mml:mi mathvariant=\"script\">C</mml:mi><mml:mtext>-closed</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mspace width=\"0.333333em\"/><mml:mtext>ideally</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"0.166667em\"/><mml:mi mathvariant=\"script\">C</mml:mi><mml:mtext>-closed</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mtext>-closed.</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq69\"><alternatives><tex-math id=\"M141\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M142\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq70\"><alternatives><tex-math id=\"M143\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\approx $$\\end{document}</tex-math><mml:math id=\"M144\"><mml:mo>≈</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq71\"><alternatives><tex-math id=\"M145\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_\\approx $$\\end{document}</tex-math><mml:math id=\"M146\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>≈</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq72\"><alternatives><tex-math id=\"M147\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M148\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ3\"><alternatives><tex-math id=\"M149\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} E(X){\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{x\\in X:xx=x\\} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M150\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq73\"><alternatives><tex-math id=\"M151\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M152\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq74\"><alternatives><tex-math id=\"M153\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H(X)=\\bigcup _{e\\in E(X)}H_e$$\\end{document}</tex-math><mml:math id=\"M154\"><mml:mrow><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq75\"><alternatives><tex-math id=\"M155\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X=H(X)$$\\end{document}</tex-math><mml:math id=\"M156\"><mml:mrow><mml:mi>X</mml:mi><mml:mo>=</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq76\"><alternatives><tex-math id=\"M157\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X\\setminus H(X)$$\\end{document}</tex-math><mml:math id=\"M158\"><mml:mrow><mml:mi>X</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq77\"><alternatives><tex-math id=\"M159\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M160\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq78\"><alternatives><tex-math id=\"M161\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M162\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq79\"><alternatives><tex-math id=\"M163\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathsf {T_{\\!z}S}\\subseteq {\\mathcal {C}}\\subseteq \\mathsf {T_{\\!1}S}$$\\end{document}</tex-math><mml:math id=\"M164\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>⊆</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq80\"><alternatives><tex-math id=\"M165\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M166\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq81\"><alternatives><tex-math id=\"M167\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M168\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq82\"><alternatives><tex-math id=\"M169\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M170\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq83\"><alternatives><tex-math id=\"M171\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M172\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq84\"><alternatives><tex-math id=\"M173\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M174\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq85\"><alternatives><tex-math id=\"M175\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M176\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq86\"><alternatives><tex-math id=\"M177\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M178\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq87\"><alternatives><tex-math id=\"M179\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h:X\\rightarrow Y$$\\end{document}</tex-math><mml:math id=\"M180\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq88\"><alternatives><tex-math id=\"M181\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y\\in {\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M182\"><mml:mrow><mml:mi>Y</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq89\"><alternatives><tex-math id=\"M183\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M184\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq90\"><alternatives><tex-math id=\"M185\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M186\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq91\"><alternatives><tex-math id=\"M187\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M188\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq92\"><alternatives><tex-math id=\"M189\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h:X\\rightarrow Y$$\\end{document}</tex-math><mml:math id=\"M190\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq93\"><alternatives><tex-math id=\"M191\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y\\in {\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M192\"><mml:mrow><mml:mi>Y</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq94\"><alternatives><tex-math id=\"M193\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M194\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq95\"><alternatives><tex-math id=\"M195\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M196\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq96\"><alternatives><tex-math id=\"M197\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M198\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq97\"><alternatives><tex-math id=\"M199\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M200\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq98\"><alternatives><tex-math id=\"M201\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M202\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq99\"><alternatives><tex-math id=\"M203\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M204\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq100\"><alternatives><tex-math id=\"M205\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M206\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq101\"><alternatives><tex-math id=\"M207\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M208\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq102\"><alternatives><tex-math id=\"M209\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M210\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq103\"><alternatives><tex-math id=\"M211\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}=\\mathsf {T_{\\!1}S}$$\\end{document}</tex-math><mml:math id=\"M212\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq104\"><alternatives><tex-math id=\"M213\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathsf {T_{\\!z}S}\\subseteq {\\mathcal {C}}\\subseteq \\mathsf {T_{\\!2}S}$$\\end{document}</tex-math><mml:math id=\"M214\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>⊆</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq105\"><alternatives><tex-math id=\"M215\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M216\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq106\"><alternatives><tex-math id=\"M217\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M218\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq107\"><alternatives><tex-math id=\"M219\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M220\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq108\"><alternatives><tex-math id=\"M221\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M222\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq109\"><alternatives><tex-math id=\"M223\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M224\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq110\"><alternatives><tex-math id=\"M225\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M226\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq111\"><alternatives><tex-math id=\"M227\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathsf {T_{\\!z}S}\\subseteq {\\mathcal {C}}\\subseteq \\mathsf {T_{\\!2}S}$$\\end{document}</tex-math><mml:math id=\"M228\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>⊆</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq112\"><alternatives><tex-math id=\"M229\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M230\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq113\"><alternatives><tex-math id=\"M231\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M232\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq114\"><alternatives><tex-math id=\"M233\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M234\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq115\"><alternatives><tex-math id=\"M235\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M236\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq116\"><alternatives><tex-math id=\"M237\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M238\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq117\"><alternatives><tex-math id=\"M239\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M240\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq118\"><alternatives><tex-math id=\"M241\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}=\\mathsf {T_{\\!1}S}$$\\end{document}</tex-math><mml:math id=\"M242\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq119\"><alternatives><tex-math id=\"M243\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathsf {T_{\\!z}S}\\subseteq {\\mathcal {C}}\\subseteq \\mathsf {T_{\\!2}S}$$\\end{document}</tex-math><mml:math id=\"M244\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>⊆</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq120\"><alternatives><tex-math id=\"M245\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M246\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq121\"><alternatives><tex-math id=\"M247\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M248\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq122\"><alternatives><tex-math id=\"M249\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M250\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq123\"><alternatives><tex-math id=\"M251\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M252\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq124\"><alternatives><tex-math id=\"M253\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M254\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq125\"><alternatives><tex-math id=\"M255\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(1)\\Leftrightarrow (4)$$\\end{document}</tex-math><mml:math id=\"M256\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇔</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ4\"><alternatives><tex-math id=\"M257\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} Z(X){\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{z\\in X:\\forall x\\in X\\;\\;(zx=xz)\\} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M258\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>z</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mo>:</mml:mo><mml:mo>∀</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"0.277778em\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>z</mml:mi><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mi>z</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ5\"><alternatives><tex-math id=\"M259\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} I\\!Z(X){\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{z\\in Z(X):zX\\subseteq Z(X)\\} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M260\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>z</mml:mi><mml:mo>∈</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mi>z</mml:mi><mml:mi>X</mml:mi><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq126\"><alternatives><tex-math id=\"M261\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)=Z(X)=X$$\\end{document}</tex-math><mml:math id=\"M262\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq127\"><alternatives><tex-math id=\"M263\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M264\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq128\"><alternatives><tex-math id=\"M265\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M266\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq129\"><alternatives><tex-math id=\"M267\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M268\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq130\"><alternatives><tex-math id=\"M269\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M270\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq131\"><alternatives><tex-math id=\"M271\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M272\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq132\"><alternatives><tex-math id=\"M273\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M274\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq133\"><alternatives><tex-math id=\"M275\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathsf {T_{\\!z}S}\\subseteq {\\mathcal {C}}\\subseteq \\mathsf {T_{\\!1}S}$$\\end{document}</tex-math><mml:math id=\"M276\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>⊆</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq134\"><alternatives><tex-math id=\"M277\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M278\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq135\"><alternatives><tex-math id=\"M279\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)$$\\end{document}</tex-math><mml:math id=\"M280\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq136\"><alternatives><tex-math id=\"M281\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M282\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq137\"><alternatives><tex-math id=\"M283\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M284\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq138\"><alternatives><tex-math id=\"M285\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M286\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq139\"><alternatives><tex-math id=\"M287\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M288\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq140\"><alternatives><tex-math id=\"M289\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M290\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq141\"><alternatives><tex-math id=\"M291\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}=\\mathsf {T_{\\!1}S}$$\\end{document}</tex-math><mml:math id=\"M292\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq142\"><alternatives><tex-math id=\"M293\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathsf {T_{\\!z}S}\\subseteq {\\mathcal {C}}\\subseteq \\mathsf {T_{\\!2}S}$$\\end{document}</tex-math><mml:math id=\"M294\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>⊆</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq143\"><alternatives><tex-math id=\"M295\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M296\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq144\"><alternatives><tex-math id=\"M297\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M298\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq145\"><alternatives><tex-math id=\"M299\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)$$\\end{document}</tex-math><mml:math id=\"M300\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq146\"><alternatives><tex-math id=\"M301\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y\\in X$$\\end{document}</tex-math><mml:math id=\"M302\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq147\"><alternatives><tex-math id=\"M303\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{xy,yx\\}\\subseteq E(X)$$\\end{document}</tex-math><mml:math id=\"M304\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>⊆</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq148\"><alternatives><tex-math id=\"M305\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$xy=yx$$\\end{document}</tex-math><mml:math id=\"M306\"><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ6\"><alternatives><tex-math id=\"M307\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\tfrac{H_e}{e}{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{x\\in X:xe=ex\\in H_e\\} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M308\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mi>e</mml:mi></mml:mfrac></mml:mstyle><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq149\"><alternatives><tex-math id=\"M309\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X\\setminus \\frac{H_e}{e}$$\\end{document}</tex-math><mml:math id=\"M310\"><mml:mrow><mml:mi>X</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mi>e</mml:mi></mml:mfrac></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq150\"><alternatives><tex-math id=\"M311\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M312\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq151\"><alternatives><tex-math id=\"M313\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y\\in X$$\\end{document}</tex-math><mml:math id=\"M314\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq152\"><alternatives><tex-math id=\"M315\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$xy=e\\in E(X)$$\\end{document}</tex-math><mml:math id=\"M316\"><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq153\"><alternatives><tex-math id=\"M317\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$xe=ex$$\\end{document}</tex-math><mml:math id=\"M318\"><mml:mrow><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq154\"><alternatives><tex-math id=\"M319\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$ye=ey$$\\end{document}</tex-math><mml:math id=\"M320\"><mml:mrow><mml:mi>y</mml:mi><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq155\"><alternatives><tex-math id=\"M321\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E(X)\\subseteq Z(X)$$\\end{document}</tex-math><mml:math id=\"M322\"><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq156\"><alternatives><tex-math id=\"M323\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M324\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq157\"><alternatives><tex-math id=\"M325\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M326\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq158\"><alternatives><tex-math id=\"M327\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M328\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq159\"><alternatives><tex-math id=\"M329\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(H_e)$$\\end{document}</tex-math><mml:math id=\"M330\"><mml:mrow><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq160\"><alternatives><tex-math id=\"M331\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M332\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq161\"><alternatives><tex-math id=\"M333\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(H_e)$$\\end{document}</tex-math><mml:math id=\"M334\"><mml:mrow><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq162\"><alternatives><tex-math id=\"M335\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap E(X)\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M336\"><mml:mrow><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∩</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ7\"><alternatives><tex-math id=\"M337\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\!\\root {\\mathbb {N}} \\of {\\!A}{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\bigcup _{n\\in {\\mathbb {N}}}\\!\\root n \\of {\\!A}\\quad \\text{ where }\\quad \\!\\root n \\of {\\!A}{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{x\\in X:x^n\\in A\\}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M338\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:munder><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:munder><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:mroot><mml:mspace width=\"1em\"/><mml:mspace width=\"0.333333em\"/><mml:mtext>where</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"1em\"/><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:mroot><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mo>:</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>∈</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq163\"><alternatives><tex-math id=\"M339\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$B\\subseteq \\!\\root n \\of {\\!E(X)}$$\\end{document}</tex-math><mml:math id=\"M340\"><mml:mrow><mml:mi>B</mml:mi><mml:mo>⊆</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq164\"><alternatives><tex-math id=\"M341\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M342\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq165\"><alternatives><tex-math id=\"M343\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M344\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq166\"><alternatives><tex-math id=\"M345\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}\\setminus H(X)$$\\end{document}</tex-math><mml:math id=\"M346\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq167\"><alternatives><tex-math id=\"M347\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}=\\mathsf {T_{\\!1}S}$$\\end{document}</tex-math><mml:math id=\"M348\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq168\"><alternatives><tex-math id=\"M349\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M350\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq169\"><alternatives><tex-math id=\"M351\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}$$\\end{document}</tex-math><mml:math id=\"M352\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq170\"><alternatives><tex-math id=\"M353\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M354\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq171\"><alternatives><tex-math id=\"M355\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M356\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq172\"><alternatives><tex-math id=\"M357\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathsf {T_{\\!z}S}\\subseteq {\\mathcal {C}}\\subseteq \\mathsf {T_{\\!2}S}$$\\end{document}</tex-math><mml:math id=\"M358\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>⊆</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq173\"><alternatives><tex-math id=\"M359\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M360\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq174\"><alternatives><tex-math id=\"M361\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$A\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M362\"><mml:mrow><mml:mi>A</mml:mi><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq175\"><alternatives><tex-math id=\"M363\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$B\\subseteq A$$\\end{document}</tex-math><mml:math id=\"M364\"><mml:mrow><mml:mi>B</mml:mi><mml:mo>⊆</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq176\"><alternatives><tex-math id=\"M365\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\bigcap _{e\\in B}\\frac{H_e}{e}$$\\end{document}</tex-math><mml:math id=\"M366\"><mml:mrow><mml:mi>C</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:msub><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mi>e</mml:mi></mml:mfrac></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq177\"><alternatives><tex-math id=\"M367\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in B$$\\end{document}</tex-math><mml:math id=\"M368\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq178\"><alternatives><tex-math id=\"M369\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M370\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq179\"><alternatives><tex-math id=\"M371\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|C|\\le \\textrm{cov}({\\mathcal {M}})$$\\end{document}</tex-math><mml:math id=\"M372\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>≤</mml:mo><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq180\"><alternatives><tex-math id=\"M373\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M374\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq181\"><alternatives><tex-math id=\"M375\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M376\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq182\"><alternatives><tex-math id=\"M377\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|C|\\le {\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M378\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>≤</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq183\"><alternatives><tex-math id=\"M379\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M380\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq184\"><alternatives><tex-math id=\"M381\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M382\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq185\"><alternatives><tex-math id=\"M383\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M384\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq186\"><alternatives><tex-math id=\"M385\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$S\\subseteq Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M386\"><mml:mrow><mml:mi>S</mml:mi><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq187\"><alternatives><tex-math id=\"M387\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M388\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq188\"><alternatives><tex-math id=\"M389\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{cov}({\\mathcal {M}})$$\\end{document}</tex-math><mml:math id=\"M390\"><mml:mrow><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq189\"><alternatives><tex-math id=\"M391\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\omega _1\\le \\textrm{cov}({\\mathcal {M}})\\le {\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M392\"><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>≤</mml:mo><mml:mtext>cov</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq190\"><alternatives><tex-math id=\"M393\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{cov}({\\mathcal {M}})={\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M394\"><mml:mrow><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq191\"><alternatives><tex-math id=\"M395\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{cov}({\\mathcal {M}})={\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M396\"><mml:mrow><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq192\"><alternatives><tex-math id=\"M397\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M398\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq193\"><alternatives><tex-math id=\"M399\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\le $$\\end{document}</tex-math><mml:math id=\"M400\"><mml:mo>≤</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq194\"><alternatives><tex-math id=\"M401\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\le y$$\\end{document}</tex-math><mml:math id=\"M402\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>≤</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq195\"><alternatives><tex-math id=\"M403\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$xy=yx=x$$\\end{document}</tex-math><mml:math id=\"M404\"><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq196\"><alternatives><tex-math id=\"M405\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M406\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq197\"><alternatives><tex-math id=\"M407\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M408\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq198\"><alternatives><tex-math id=\"M409\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\approx $$\\end{document}</tex-math><mml:math id=\"M410\"><mml:mo>≈</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq199\"><alternatives><tex-math id=\"M411\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_\\approx $$\\end{document}</tex-math><mml:math id=\"M412\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>≈</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq200\"><alternatives><tex-math id=\"M413\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M414\"><mml:mo>⇕</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq201\"><alternatives><tex-math id=\"M415\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M416\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq202\"><alternatives><tex-math id=\"M417\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(P,\\le )$$\\end{document}</tex-math><mml:math id=\"M418\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>P</mml:mi><mml:mo>,</mml:mo><mml:mo>≤</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq203\"><alternatives><tex-math id=\"M419\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\subseteq P$$\\end{document}</tex-math><mml:math id=\"M420\"><mml:mrow><mml:mi>I</mml:mi><mml:mo>⊆</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq204\"><alternatives><tex-math id=\"M421\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\not \\le y$$\\end{document}</tex-math><mml:math id=\"M422\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>≰</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq205\"><alternatives><tex-math id=\"M423\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\not \\le x$$\\end{document}</tex-math><mml:math id=\"M424\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>≰</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq206\"><alternatives><tex-math id=\"M425\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$A\\subseteq P$$\\end{document}</tex-math><mml:math id=\"M426\"><mml:mrow><mml:mi>A</mml:mi><mml:mo>⊆</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq207\"><alternatives><tex-math id=\"M427\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{x\\in A:x\\le a\\}=\\{a\\}$$\\end{document}</tex-math><mml:math id=\"M428\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>≤</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq208\"><alternatives><tex-math id=\"M429\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M430\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq209\"><alternatives><tex-math id=\"M431\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M432\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq210\"><alternatives><tex-math id=\"M433\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(x)$$\\end{document}</tex-math><mml:math id=\"M434\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq211\"><alternatives><tex-math id=\"M435\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M436\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq212\"><alternatives><tex-math id=\"M437\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M438\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq213\"><alternatives><tex-math id=\"M439\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(x)$$\\end{document}</tex-math><mml:math id=\"M440\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq214\"><alternatives><tex-math id=\"M441\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M442\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq215\"><alternatives><tex-math id=\"M443\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M444\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq216\"><alternatives><tex-math id=\"M445\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M446\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq217\"><alternatives><tex-math id=\"M447\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(x)$$\\end{document}</tex-math><mml:math id=\"M448\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq218\"><alternatives><tex-math id=\"M449\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M450\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq219\"><alternatives><tex-math id=\"M451\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M452\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq220\"><alternatives><tex-math id=\"M453\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(x)$$\\end{document}</tex-math><mml:math id=\"M454\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq221\"><alternatives><tex-math id=\"M455\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M456\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq222\"><alternatives><tex-math id=\"M457\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M458\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq223\"><alternatives><tex-math id=\"M459\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M460\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq224\"><alternatives><tex-math id=\"M461\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\omega $$\\end{document}</tex-math><mml:math id=\"M462\"><mml:mi>ω</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq225\"><alternatives><tex-math id=\"M463\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathbb {N}}{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\omega {\\setminus }\\{0\\}$$\\end{document}</tex-math><mml:math id=\"M464\"><mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mi>ω</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq226\"><alternatives><tex-math id=\"M465\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in \\omega $$\\end{document}</tex-math><mml:math id=\"M466\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq227\"><alternatives><tex-math id=\"M467\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{k:k&lt;n\\}$$\\end{document}</tex-math><mml:math id=\"M468\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>k</mml:mi><mml:mo>:</mml:mo><mml:mi>k</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq228\"><alternatives><tex-math id=\"M469\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\le $$\\end{document}</tex-math><mml:math id=\"M470\"><mml:mo>≤</mml:mo></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ8\"><alternatives><tex-math id=\"M471\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} {\\downarrow }p{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{x\\in P:x\\le p\\}\\quad \\text{ and }\\quad {\\uparrow }p{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{x\\in P:p\\le x\\} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M472\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mo stretchy=\"false\">↓</mml:mo><mml:mi>p</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>P</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.333333em\"/><mml:mtext>and</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"1em\"/><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>p</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>P</mml:mi><mml:mo>:</mml:mo><mml:mi>p</mml:mi><mml:mo>≤</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq229\"><alternatives><tex-math id=\"M473\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{cov}({\\mathcal {M}})$$\\end{document}</tex-math><mml:math id=\"M474\"><mml:mrow><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq230\"><alternatives><tex-math id=\"M475\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{cov}({\\mathcal {N}})$$\\end{document}</tex-math><mml:math id=\"M476\"><mml:mrow><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq231\"><alternatives><tex-math id=\"M477\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{cov}(\\overline{{\\mathcal {N}}})$$\\end{document}</tex-math><mml:math id=\"M478\"><mml:mrow><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mover><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq232\"><alternatives><tex-math id=\"M479\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathfrak {d}}$$\\end{document}</tex-math><mml:math id=\"M480\"><mml:mi mathvariant=\"fraktur\">d</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq233\"><alternatives><tex-math id=\"M481\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\omega ^\\omega $$\\end{document}</tex-math><mml:math id=\"M482\"><mml:msup><mml:mi>ω</mml:mi><mml:mi>ω</mml:mi></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ9\"><alternatives><tex-math id=\"M483\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\max \\{\\textrm{cov}({\\mathcal {M}}),\\textrm{cov}({\\mathcal {N}})\\}\\le \\textrm{cov}(\\overline{{\\mathcal {N}}})\\le \\max \\{\\textrm{cov}({\\mathcal {N}}),{\\mathfrak {d}}\\}\\le {\\mathfrak {c}}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M484\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mo movablelimits=\"true\">max</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mtext>cov</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mtext>cov</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mtext>cov</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mo movablelimits=\"true\">max</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mtext>cov</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi mathvariant=\"fraktur\">d</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq234\"><alternatives><tex-math id=\"M485\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathfrak {d}}=\\textrm{cov}({\\mathcal {M}})=\\textrm{cov}({\\mathcal {N}})=\\textrm{cov}(\\overline{{\\mathcal {N}}})={\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M486\"><mml:mrow><mml:mi mathvariant=\"fraktur\">d</mml:mi><mml:mo>=</mml:mo><mml:mtext>cov</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mtext>cov</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mtext>cov</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq235\"><alternatives><tex-math id=\"M487\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{cov}({\\mathcal {M}})={\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M488\"><mml:mrow><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq236\"><alternatives><tex-math id=\"M489\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\max \\{\\textrm{cov}({\\mathcal {M}}),\\textrm{cov}({\\mathcal {N}})\\}&lt;\\textrm{cov}(\\overline{{\\mathcal {N}}})$$\\end{document}</tex-math><mml:math id=\"M490\"><mml:mrow><mml:mo movablelimits=\"true\">max</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mtext>cov</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mtext>cov</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>&lt;</mml:mo><mml:mtext>cov</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq237\"><alternatives><tex-math id=\"M491\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\max \\{\\textrm{cov}({\\mathcal {N}}),{\\mathfrak {d}}\\}&lt;{\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M492\"><mml:mrow><mml:mo movablelimits=\"true\">max</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mi mathvariant=\"fraktur\">d</mml:mi><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>&lt;</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq238\"><alternatives><tex-math id=\"M493\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau $$\\end{document}</tex-math><mml:math id=\"M494\"><mml:mi>τ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq239\"><alternatives><tex-math id=\"M495\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(X,\\tau )$$\\end{document}</tex-math><mml:math id=\"M496\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq240\"><alternatives><tex-math id=\"M497\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y,a\\in X$$\\end{document}</tex-math><mml:math id=\"M498\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ10\"><alternatives><tex-math id=\"M499\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} d(ax,ay)\\le d(x,y)\\quad \\text{ and }\\quad d(xa,ya)\\le d(x,y). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M500\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>a</mml:mi><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≤</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mspace width=\"1em\"/><mml:mspace width=\"0.333333em\"/><mml:mtext>and</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"1em\"/><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≤</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq241\"><alternatives><tex-math id=\"M501\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X^0=X\\cup \\{0\\}$$\\end{document}</tex-math><mml:math id=\"M502\"><mml:mrow><mml:msup><mml:mi>X</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mi>X</mml:mi><mml:mo>∪</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq242\"><alternatives><tex-math id=\"M503\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$0\\notin X$$\\end{document}</tex-math><mml:math id=\"M504\"><mml:mrow><mml:mn>0</mml:mn><mml:mo>∉</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq243\"><alternatives><tex-math id=\"M505\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$0x=0=x0$$\\end{document}</tex-math><mml:math id=\"M506\"><mml:mrow><mml:mn>0</mml:mn><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq244\"><alternatives><tex-math id=\"M507\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\in X^0$$\\end{document}</tex-math><mml:math id=\"M508\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>X</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq245\"><alternatives><tex-math id=\"M509\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X^1=X\\cup \\{1\\}$$\\end{document}</tex-math><mml:math id=\"M510\"><mml:mrow><mml:msup><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mi>X</mml:mi><mml:mo>∪</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq246\"><alternatives><tex-math id=\"M511\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$1\\notin X$$\\end{document}</tex-math><mml:math id=\"M512\"><mml:mrow><mml:mn>1</mml:mn><mml:mo>∉</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq247\"><alternatives><tex-math id=\"M513\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$1x=x=x1$$\\end{document}</tex-math><mml:math id=\"M514\"><mml:mrow><mml:mn>1</mml:mn><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq248\"><alternatives><tex-math id=\"M515\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\in X^1$$\\end{document}</tex-math><mml:math id=\"M516\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq249\"><alternatives><tex-math id=\"M517\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X^0=\\{0\\}\\cup X$$\\end{document}</tex-math><mml:math id=\"M518\"><mml:mrow><mml:msup><mml:mi>X</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq250\"><alternatives><tex-math id=\"M519\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x0=0=0x$$\\end{document}</tex-math><mml:math id=\"M520\"><mml:mrow><mml:mi>x</mml:mi><mml:mn>0</mml:mn><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mi>x</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq251\"><alternatives><tex-math id=\"M521\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\in X$$\\end{document}</tex-math><mml:math id=\"M522\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq252\"><alternatives><tex-math id=\"M523\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T_1$$\\end{document}</tex-math><mml:math id=\"M524\"><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq253\"><alternatives><tex-math id=\"M525\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M526\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq254\"><alternatives><tex-math id=\"M527\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M528\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq255\"><alternatives><tex-math id=\"M529\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau $$\\end{document}</tex-math><mml:math id=\"M530\"><mml:mi>τ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq256\"><alternatives><tex-math id=\"M531\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X^0$$\\end{document}</tex-math><mml:math id=\"M532\"><mml:msup><mml:mi>X</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq257\"><alternatives><tex-math id=\"M533\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(X^0,\\tau )$$\\end{document}</tex-math><mml:math id=\"M534\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>X</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq258\"><alternatives><tex-math id=\"M535\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau ^0$$\\end{document}</tex-math><mml:math id=\"M536\"><mml:msup><mml:mi>τ</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq259\"><alternatives><tex-math id=\"M537\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X^0$$\\end{document}</tex-math><mml:math id=\"M538\"><mml:msup><mml:mi>X</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq260\"><alternatives><tex-math id=\"M539\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\big \\{\\{x\\}:x\\in X\\big \\}\\cup \\tau $$\\end{document}</tex-math><mml:math id=\"M540\"><mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">{</mml:mo></mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq261\"><alternatives><tex-math id=\"M541\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(X^0,\\tau ^0)$$\\end{document}</tex-math><mml:math id=\"M542\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>X</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>τ</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq262\"><alternatives><tex-math id=\"M543\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M544\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq263\"><alternatives><tex-math id=\"M545\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M546\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq264\"><alternatives><tex-math id=\"M547\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f:X\\rightarrow X$$\\end{document}</tex-math><mml:math id=\"M548\"><mml:mrow><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq265\"><alternatives><tex-math id=\"M549\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f(x)=a_0xa_1x\\cdots xa_n$$\\end{document}</tex-math><mml:math id=\"M550\"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>x</mml:mi><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>x</mml:mi><mml:mo>⋯</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>a</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq266\"><alternatives><tex-math id=\"M551\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M552\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq267\"><alternatives><tex-math id=\"M553\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_0,\\dots , a_n\\in X^1$$\\end{document}</tex-math><mml:math id=\"M554\"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq268\"><alternatives><tex-math id=\"M555\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\deg (f)$$\\end{document}</tex-math><mml:math id=\"M556\"><mml:mrow><mml:mo>deg</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq269\"><alternatives><tex-math id=\"M557\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\kappa $$\\end{document}</tex-math><mml:math id=\"M558\"><mml:mi>κ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq270\"><alternatives><tex-math id=\"M559\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\kappa $$\\end{document}</tex-math><mml:math id=\"M560\"><mml:mi>κ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq271\"><alternatives><tex-math id=\"M561\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X=\\bigcup _{\\alpha \\in \\kappa }f_\\alpha ^{-1}(b_\\alpha )$$\\end{document}</tex-math><mml:math id=\"M562\"><mml:mrow><mml:mi>X</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mi>κ</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mi>f</mml:mi><mml:mi>α</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq272\"><alternatives><tex-math id=\"M563\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$b_\\alpha \\in X$$\\end{document}</tex-math><mml:math id=\"M564\"><mml:mrow><mml:msub><mml:mi>b</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq273\"><alternatives><tex-math id=\"M565\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f_\\alpha $$\\end{document}</tex-math><mml:math id=\"M566\"><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq274\"><alternatives><tex-math id=\"M567\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M568\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq275\"><alternatives><tex-math id=\"M569\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M570\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq276\"><alternatives><tex-math id=\"M571\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$d\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M572\"><mml:mrow><mml:mi>d</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq277\"><alternatives><tex-math id=\"M573\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\subseteq X$$\\end{document}</tex-math><mml:math id=\"M574\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>⊆</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq278\"><alternatives><tex-math id=\"M575\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y\\in X$$\\end{document}</tex-math><mml:math id=\"M576\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq279\"><alternatives><tex-math id=\"M577\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f:X\\rightarrow X$$\\end{document}</tex-math><mml:math id=\"M578\"><mml:mrow><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq280\"><alternatives><tex-math id=\"M579\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\le d$$\\end{document}</tex-math><mml:math id=\"M580\"><mml:mrow><mml:mo>≤</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq281\"><alternatives><tex-math id=\"M581\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{f(x),f(y)\\}\\subseteq F$$\\end{document}</tex-math><mml:math id=\"M582\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>⊆</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq282\"><alternatives><tex-math id=\"M583\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$b_0,\\dots ,b_{n-1}\\in X$$\\end{document}</tex-math><mml:math id=\"M584\"><mml:mrow><mml:msub><mml:mi>b</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq283\"><alternatives><tex-math id=\"M585\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f_0,\\dots ,f_{n-1}$$\\end{document}</tex-math><mml:math id=\"M586\"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq284\"><alternatives><tex-math id=\"M587\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X=\\bigcap _{i\\in n}f_i^{-1}(b_i)$$\\end{document}</tex-math><mml:math id=\"M588\"><mml:mrow><mml:mi>X</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mi>f</mml:mi><mml:mi>i</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ11\"><alternatives><tex-math id=\"M589\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} F=\\{b_i\\}_{i\\in n}\\cup \\{f_i(b_j):i,j\\in n\\}\\quad \\text{ and }\\quad d=\\max \\{\\deg (f_i\\circ f_j):i,j\\in n\\}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M590\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>∪</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>∈</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mspace width=\"1em\"/><mml:mspace width=\"0.333333em\"/><mml:mtext>and</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"1em\"/><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mo movablelimits=\"true\">max</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mo>deg</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>∘</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>∈</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq285\"><alternatives><tex-math id=\"M591\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y\\in X$$\\end{document}</tex-math><mml:math id=\"M592\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq286\"><alternatives><tex-math id=\"M593\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i\\in n$$\\end{document}</tex-math><mml:math id=\"M594\"><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq287\"><alternatives><tex-math id=\"M595\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f_i(x)=b_i$$\\end{document}</tex-math><mml:math id=\"M596\"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq288\"><alternatives><tex-math id=\"M597\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$j\\in n$$\\end{document}</tex-math><mml:math id=\"M598\"><mml:mrow><mml:mi>j</mml:mi><mml:mo>∈</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq289\"><alternatives><tex-math id=\"M599\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f_j(f_i(y))=b_j$$\\end{document}</tex-math><mml:math id=\"M600\"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq290\"><alternatives><tex-math id=\"M601\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f=f_j\\circ f_i:X\\rightarrow X$$\\end{document}</tex-math><mml:math id=\"M602\"><mml:mrow><mml:mi>f</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>∘</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq291\"><alternatives><tex-math id=\"M603\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\le d$$\\end{document}</tex-math><mml:math id=\"M604\"><mml:mrow><mml:mo>≤</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq292\"><alternatives><tex-math id=\"M605\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{f(x),f(y)\\}=\\{f_j(f_i(x)),f_j(f_i(y))\\}=\\{f_j(b_i),b_j\\}\\subseteq F$$\\end{document}</tex-math><mml:math id=\"M606\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq293\"><alternatives><tex-math id=\"M607\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M608\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq294\"><alternatives><tex-math id=\"M609\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\kappa $$\\end{document}</tex-math><mml:math id=\"M610\"><mml:mi>κ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq295\"><alternatives><tex-math id=\"M611\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\kappa &lt;\\max \\{2,|X|\\}$$\\end{document}</tex-math><mml:math id=\"M612\"><mml:mrow><mml:mi>κ</mml:mi><mml:mo>&lt;</mml:mo><mml:mo movablelimits=\"true\">max</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq296\"><alternatives><tex-math id=\"M613\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{cov}({\\mathcal {M}})={\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M614\"><mml:mrow><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq297\"><alternatives><tex-math id=\"M615\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{cov}({\\mathcal {M}})={\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M616\"><mml:mrow><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq298\"><alternatives><tex-math id=\"M617\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{cov}(\\overline{{\\mathcal {N}}})={\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M618\"><mml:mrow><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mover><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq299\"><alternatives><tex-math id=\"M619\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X\\setminus C$$\\end{document}</tex-math><mml:math id=\"M620\"><mml:mrow><mml:mi>X</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq300\"><alternatives><tex-math id=\"M621\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C\\subseteq X$$\\end{document}</tex-math><mml:math id=\"M622\"><mml:mrow><mml:mi>C</mml:mi><mml:mo>⊆</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ12\"><alternatives><tex-math id=\"M623\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\chi _C:X\\rightarrow \\{0,1\\},\\quad \\chi _C:x\\mapsto {\\left\\{ \\begin{array}{ll}1&amp;{}\\hbox { if}\\ x\\in C;\\\\ 0&amp;{}\\hbox { if}\\ x\\in X\\setminus C; \\end{array}\\right. } \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M624\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>χ</mml:mi><mml:mi>C</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msub><mml:mi>χ</mml:mi><mml:mi>C</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>↦</mml:mo><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mn>1</mml:mn></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>if</mml:mtext><mml:mspace width=\"4pt\"/><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi><mml:mo>;</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mn>0</mml:mn></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>if</mml:mtext><mml:mspace width=\"4pt\"/><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>C</mml:mi><mml:mo>;</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq301\"><alternatives><tex-math id=\"M625\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{0,1\\}$$\\end{document}</tex-math><mml:math id=\"M626\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq302\"><alternatives><tex-math id=\"M627\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M628\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq303\"><alternatives><tex-math id=\"M629\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M630\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq304\"><alternatives><tex-math id=\"M631\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M632\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq305\"><alternatives><tex-math id=\"M633\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M634\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq306\"><alternatives><tex-math id=\"M635\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h:C\\rightarrow Y$$\\end{document}</tex-math><mml:math id=\"M636\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq307\"><alternatives><tex-math id=\"M637\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y,\\tau )\\in \\mathsf {T_{\\!z}S}$$\\end{document}</tex-math><mml:math id=\"M638\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ13\"><alternatives><tex-math id=\"M639\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} {{\\bar{h}}}:X\\rightarrow Y^0,\\quad {{\\bar{h}}}:x\\mapsto {\\left\\{ \\begin{array}{ll}h(x)&amp;{}\\hbox { if}\\ x\\in C,\\\\ 0&amp;{}\\hbox { if}\\ x\\in X\\setminus C, \\end{array}\\right. } \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M640\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:msup><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mover accent=\"true\"><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>↦</mml:mo><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>if</mml:mtext><mml:mspace width=\"4pt\"/><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mn>0</mml:mn></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>if</mml:mtext><mml:mspace width=\"4pt\"/><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>C</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq308\"><alternatives><tex-math id=\"M641\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y^0$$\\end{document}</tex-math><mml:math id=\"M642\"><mml:msup><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq309\"><alternatives><tex-math id=\"M643\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau ^0=\\{U\\subseteq Y^0:U\\cap Y\\in \\tau \\}$$\\end{document}</tex-math><mml:math id=\"M644\"><mml:mrow><mml:msup><mml:mi>τ</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>U</mml:mi><mml:mo>⊆</mml:mo><mml:msup><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>:</mml:mo><mml:mi>U</mml:mi><mml:mo>∩</mml:mo><mml:mi>Y</mml:mi><mml:mo>∈</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq310\"><alternatives><tex-math id=\"M645\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y,\\tau )\\in \\mathsf {T_{\\!z}S}$$\\end{document}</tex-math><mml:math id=\"M646\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq311\"><alternatives><tex-math id=\"M647\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y^0,\\tau ^0)\\in \\mathsf {T_{\\!z}S}$$\\end{document}</tex-math><mml:math id=\"M648\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>τ</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq312\"><alternatives><tex-math id=\"M649\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M650\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq313\"><alternatives><tex-math id=\"M651\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{\\bar{h}}}[X]$$\\end{document}</tex-math><mml:math id=\"M652\"><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq314\"><alternatives><tex-math id=\"M653\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y^0,\\tau ^0)$$\\end{document}</tex-math><mml:math id=\"M654\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>τ</mml:mi><mml:mn>0</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq315\"><alternatives><tex-math id=\"M655\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h[C]=h[X]\\cap Y$$\\end{document}</tex-math><mml:math id=\"M656\"><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>=</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>∩</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq316\"><alternatives><tex-math id=\"M657\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y,\\tau )$$\\end{document}</tex-math><mml:math id=\"M658\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq317\"><alternatives><tex-math id=\"M659\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M660\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq318\"><alternatives><tex-math id=\"M661\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M662\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq319\"><alternatives><tex-math id=\"M663\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\frac{H_e}{e}{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{x\\in X:xe=ex\\in H_e\\}$$\\end{document}</tex-math><mml:math id=\"M664\"><mml:mrow><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mi>e</mml:mi></mml:mfrac><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq320\"><alternatives><tex-math id=\"M665\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M666\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq321\"><alternatives><tex-math id=\"M667\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E(I\\!Z(X))=E(Z)\\cap I\\!Z(X)\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M668\"><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∩</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq322\"><alternatives><tex-math id=\"M669\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E(Z)\\cap I\\!Z(X)\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M670\"><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∩</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq323\"><alternatives><tex-math id=\"M671\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E(X)\\cap Z(X)\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M672\"><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∩</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq324\"><alternatives><tex-math id=\"M673\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X\\ne \\{1\\}$$\\end{document}</tex-math><mml:math id=\"M674\"><mml:mrow><mml:mi>X</mml:mi><mml:mo>≠</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq325\"><alternatives><tex-math id=\"M675\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M676\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq326\"><alternatives><tex-math id=\"M677\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M678\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq327\"><alternatives><tex-math id=\"M679\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathbb {2}$$\\end{document}</tex-math><mml:math id=\"M680\"><mml:mn mathvariant=\"double-struck\">2</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq328\"><alternatives><tex-math id=\"M681\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{0,1\\}$$\\end{document}</tex-math><mml:math id=\"M682\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq329\"><alternatives><tex-math id=\"M683\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q:X\\rightarrow X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M684\"><mml:mrow><mml:mi>q</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq330\"><alternatives><tex-math id=\"M685\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q{\\restriction }_{V\\!E(X)}$$\\end{document}</tex-math><mml:math id=\"M686\"><mml:mrow><mml:mi>q</mml:mi><mml:msub><mml:mo>↾</mml:mo><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq331\"><alternatives><tex-math id=\"M687\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M688\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq332\"><alternatives><tex-math id=\"M689\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M690\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq333\"><alternatives><tex-math id=\"M691\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e_1,e_2\\in V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M692\"><mml:mrow><mml:msub><mml:mi>e</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>∈</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq334\"><alternatives><tex-math id=\"M693\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q(e_1)=q(e_2)$$\\end{document}</tex-math><mml:math id=\"M694\"><mml:mrow><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq335\"><alternatives><tex-math id=\"M695\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i\\in \\{1,2\\}$$\\end{document}</tex-math><mml:math id=\"M696\"><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq336\"><alternatives><tex-math id=\"M697\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\frac{H_{e_i}}{e_i}=\\{x\\in X:xe_i=e_ix\\in H_{e_i}\\}$$\\end{document}</tex-math><mml:math id=\"M698\"><mml:mrow><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfrac><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq337\"><alternatives><tex-math id=\"M699\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_i:X\\rightarrow \\mathbb {2}$$\\end{document}</tex-math><mml:math id=\"M700\"><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn mathvariant=\"double-struck\">2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ14\"><alternatives><tex-math id=\"M701\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} h_i(x)={\\left\\{ \\begin{array}{ll}1&amp;{}\\hbox { if}\\ x\\in \\frac{H_{e_i}}{e_i},\\\\ 0,&amp;{}\\text{ otherwise }, \\end{array}\\right. } \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M702\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mn>1</mml:mn></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>if</mml:mtext><mml:mspace width=\"4pt\"/><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>otherwise</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq338\"><alternatives><tex-math id=\"M703\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q(e_1)=q(e_2)$$\\end{document}</tex-math><mml:math id=\"M704\"><mml:mrow><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ15\"><alternatives><tex-math id=\"M705\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} h_1(e_{2})=h_1(e_{1})=1= h_2(e_{2})=h_2(e_{1}). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M706\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>=</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq339\"><alternatives><tex-math id=\"M707\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e_1e_2=e_2e_1\\in H_{e_1}\\cap H_{e_2}$$\\end{document}</tex-math><mml:math id=\"M708\"><mml:mrow><mml:msub><mml:mi>e</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>∈</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:msub><mml:mo>∩</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq340\"><alternatives><tex-math id=\"M709\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e_1=e_2$$\\end{document}</tex-math><mml:math id=\"M710\"><mml:mrow><mml:msub><mml:mi>e</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq341\"><alternatives><tex-math id=\"M711\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q{\\restriction }_{V\\!E(X)}$$\\end{document}</tex-math><mml:math id=\"M712\"><mml:mrow><mml:mi>q</mml:mi><mml:msub><mml:mo>↾</mml:mo><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq342\"><alternatives><tex-math id=\"M713\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M714\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq343\"><alternatives><tex-math id=\"M715\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M716\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq344\"><alternatives><tex-math id=\"M717\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M718\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq345\"><alternatives><tex-math id=\"M719\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M720\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq346\"><alternatives><tex-math id=\"M721\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M722\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq347\"><alternatives><tex-math id=\"M723\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M724\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq348\"><alternatives><tex-math id=\"M725\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y{\\mathop {=}\\limits ^{{\\textsf{def}}}}X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M726\"><mml:mrow><mml:mi>Y</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq349\"><alternatives><tex-math id=\"M727\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(y_n)_{n\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M728\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq350\"><alternatives><tex-math id=\"M729\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in \\omega $$\\end{document}</tex-math><mml:math id=\"M730\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq351\"><alternatives><tex-math id=\"M731\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\uparrow }y_n=\\{y\\in Y:y_n\\le y\\}$$\\end{document}</tex-math><mml:math id=\"M732\"><mml:mrow><mml:mo stretchy=\"false\">↑</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo>:</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>≤</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq352\"><alternatives><tex-math id=\"M733\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\uparrow }y_n$$\\end{document}</tex-math><mml:math id=\"M734\"><mml:mrow><mml:mo stretchy=\"false\">↑</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq353\"><alternatives><tex-math id=\"M735\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P_n=q^{-1}({\\uparrow }y_n)$$\\end{document}</tex-math><mml:math id=\"M736\"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>q</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">↑</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq354\"><alternatives><tex-math id=\"M737\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P=\\bigcup _{n\\in \\omega }P_n$$\\end{document}</tex-math><mml:math id=\"M738\"><mml:mrow><mml:mi>P</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>P</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq355\"><alternatives><tex-math id=\"M739\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I{\\mathop {=}\\limits ^{{\\textsf{def}}}}X\\setminus P$$\\end{document}</tex-math><mml:math id=\"M740\"><mml:mrow><mml:mi>I</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mi>X</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>P</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq356\"><alternatives><tex-math id=\"M741\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$S=P\\cup \\{P\\}\\cup \\{I\\}$$\\end{document}</tex-math><mml:math id=\"M742\"><mml:mrow><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mi>P</mml:mi><mml:mo>∪</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>∪</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq357\"><alternatives><tex-math id=\"M743\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$*:S\\times S\\rightarrow S$$\\end{document}</tex-math><mml:math id=\"M744\"><mml:mrow><mml:mrow/><mml:mo>∗</mml:mo><mml:mo>:</mml:mo><mml:mi>S</mml:mi><mml:mo>×</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ16\"><alternatives><tex-math id=\"M745\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} x*y={\\left\\{ \\begin{array}{ll}xy&amp;{}\\text{ if } \\,\\, (x,y)\\in P\\times P;\\\\ I&amp;{}\\hbox { if}\\ (x,y)\\in (S\\times \\{I\\})\\cup (\\{I\\}\\times S);\\\\ P&amp;{}\\text{ otherwise }. \\end{array}\\right. } \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M746\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>x</mml:mi><mml:mrow/><mml:mo>∗</mml:mo><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mi mathvariant=\"italic\">xy</mml:mi></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>if</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"0.166667em\"/><mml:mspace width=\"0.166667em\"/><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∈</mml:mo><mml:mi>P</mml:mi><mml:mo>×</mml:mo><mml:mi>P</mml:mi><mml:mo>;</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mi>I</mml:mi></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>if</mml:mtext><mml:mspace width=\"4pt\"/><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>S</mml:mi><mml:mo>×</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy=\"false\">}</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∪</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>×</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>;</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mi>P</mml:mi></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>otherwise</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq358\"><alternatives><tex-math id=\"M747\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau $$\\end{document}</tex-math><mml:math id=\"M748\"><mml:mi>τ</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ17\"><alternatives><tex-math id=\"M749\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\big \\{\\{P\\}\\cup (P\\setminus P_n):n\\in \\omega \\big \\}\\cup \\big \\{\\{x\\}:x\\in P\\cup \\{I\\}\\big \\} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M750\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">{</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>P</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>P</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:msub><mml:mi>P</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">{</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>P</mml:mi><mml:mo>∪</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">}</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq359\"><alternatives><tex-math id=\"M751\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(S,\\tau )$$\\end{document}</tex-math><mml:math id=\"M752\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>S</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq360\"><alternatives><tex-math id=\"M753\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(S,\\tau )$$\\end{document}</tex-math><mml:math id=\"M754\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>S</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq361\"><alternatives><tex-math id=\"M755\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/I=P\\cup \\{I\\}$$\\end{document}</tex-math><mml:math id=\"M756\"><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>I</mml:mi><mml:mo>=</mml:mo><mml:mi>P</mml:mi><mml:mo>∪</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq362\"><alternatives><tex-math id=\"M757\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M758\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq363\"><alternatives><tex-math id=\"M759\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M760\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq364\"><alternatives><tex-math id=\"M761\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M762\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq365\"><alternatives><tex-math id=\"M763\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M764\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq366\"><alternatives><tex-math id=\"M765\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M766\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq367\"><alternatives><tex-math id=\"M767\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q:X\\rightarrow X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M768\"><mml:mrow><mml:mi>q</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq368\"><alternatives><tex-math id=\"M769\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M770\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq369\"><alternatives><tex-math id=\"M771\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M772\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq370\"><alternatives><tex-math id=\"M773\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M774\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq371\"><alternatives><tex-math id=\"M775\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M776\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq372\"><alternatives><tex-math id=\"M777\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M778\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq373\"><alternatives><tex-math id=\"M779\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M780\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq374\"><alternatives><tex-math id=\"M781\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M782\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq375\"><alternatives><tex-math id=\"M783\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M784\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq376\"><alternatives><tex-math id=\"M785\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M786\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq377\"><alternatives><tex-math id=\"M787\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M788\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq378\"><alternatives><tex-math id=\"M789\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M790\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq379\"><alternatives><tex-math id=\"M791\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M792\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq380\"><alternatives><tex-math id=\"M793\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M794\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq381\"><alternatives><tex-math id=\"M795\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q:X\\rightarrow X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M796\"><mml:mrow><mml:mi>q</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq382\"><alternatives><tex-math id=\"M797\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M798\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq383\"><alternatives><tex-math id=\"M799\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathbb {2}$$\\end{document}</tex-math><mml:math id=\"M800\"><mml:mn mathvariant=\"double-struck\">2</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq384\"><alternatives><tex-math id=\"M801\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathbb {2}$$\\end{document}</tex-math><mml:math id=\"M802\"><mml:mn mathvariant=\"double-struck\">2</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq385\"><alternatives><tex-math id=\"M803\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta :X/_{\\Updownarrow }\\rightarrow \\mathbb {2}^{H}$$\\end{document}</tex-math><mml:math id=\"M804\"><mml:mrow><mml:mi>δ</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:msup><mml:mn mathvariant=\"double-struck\">2</mml:mn><mml:mi>H</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq386\"><alternatives><tex-math id=\"M805\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta :x\\mapsto (h(x))_{h\\in H}$$\\end{document}</tex-math><mml:math id=\"M806\"><mml:mrow><mml:mi>δ</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>↦</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>h</mml:mi><mml:mo>∈</mml:mo><mml:mi>H</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq387\"><alternatives><tex-math id=\"M807\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M808\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq388\"><alternatives><tex-math id=\"M809\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M810\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq389\"><alternatives><tex-math id=\"M811\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M812\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq390\"><alternatives><tex-math id=\"M813\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta [X/_{\\Updownarrow }]$$\\end{document}</tex-math><mml:math id=\"M814\"><mml:mrow><mml:mi>δ</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq391\"><alternatives><tex-math id=\"M815\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathbb {2}^H$$\\end{document}</tex-math><mml:math id=\"M816\"><mml:msup><mml:mn mathvariant=\"double-struck\">2</mml:mn><mml:mi>H</mml:mi></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq392\"><alternatives><tex-math id=\"M817\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M818\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq393\"><alternatives><tex-math id=\"M819\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M820\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq394\"><alternatives><tex-math id=\"M821\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M822\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq395\"><alternatives><tex-math id=\"M823\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M824\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq396\"><alternatives><tex-math id=\"M825\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M826\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq397\"><alternatives><tex-math id=\"M827\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M828\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq398\"><alternatives><tex-math id=\"M829\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M830\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq399\"><alternatives><tex-math id=\"M831\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M832\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq400\"><alternatives><tex-math id=\"M833\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(1)\\Rightarrow (2)\\Leftrightarrow (3)\\Rightarrow (4)\\Rightarrow (5)\\Rightarrow (6)$$\\end{document}</tex-math><mml:math id=\"M834\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇔</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>6</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq401\"><alternatives><tex-math id=\"M835\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M836\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq402\"><alternatives><tex-math id=\"M837\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M838\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq403\"><alternatives><tex-math id=\"M839\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M840\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq404\"><alternatives><tex-math id=\"M841\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M842\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq405\"><alternatives><tex-math id=\"M843\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M844\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq406\"><alternatives><tex-math id=\"M845\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M846\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq407\"><alternatives><tex-math id=\"M847\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap H(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}$$\\end{document}</tex-math><mml:math id=\"M848\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq408\"><alternatives><tex-math id=\"M849\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}$$\\end{document}</tex-math><mml:math id=\"M850\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq409\"><alternatives><tex-math id=\"M851\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(1)\\Rightarrow (2)$$\\end{document}</tex-math><mml:math id=\"M852\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq410\"><alternatives><tex-math id=\"M853\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(2)\\Leftrightarrow (3)$$\\end{document}</tex-math><mml:math id=\"M854\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇔</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq411\"><alternatives><tex-math id=\"M855\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(3)\\Rightarrow (4)$$\\end{document}</tex-math><mml:math id=\"M856\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq412\"><alternatives><tex-math id=\"M857\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(4)\\Rightarrow (5)$$\\end{document}</tex-math><mml:math id=\"M858\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq413\"><alternatives><tex-math id=\"M859\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M860\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq414\"><alternatives><tex-math id=\"M861\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M862\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq415\"><alternatives><tex-math id=\"M863\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M864\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq416\"><alternatives><tex-math id=\"M865\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in E(X)$$\\end{document}</tex-math><mml:math id=\"M866\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq417\"><alternatives><tex-math id=\"M867\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap H_e$$\\end{document}</tex-math><mml:math id=\"M868\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq418\"><alternatives><tex-math id=\"M869\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap H_e$$\\end{document}</tex-math><mml:math id=\"M870\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ18\"><alternatives><tex-math id=\"M871\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} Z(X)\\cap H(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}=\\bigcup _{e\\in V\\!E(X)}(Z(X)\\cap H_e) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M872\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo>=</mml:mo><mml:munder><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:munder><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq419\"><alternatives><tex-math id=\"M873\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(5)\\Rightarrow (6)$$\\end{document}</tex-math><mml:math id=\"M874\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>6</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq420\"><alternatives><tex-math id=\"M875\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M876\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq421\"><alternatives><tex-math id=\"M877\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap H(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}$$\\end{document}</tex-math><mml:math id=\"M878\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq422\"><alternatives><tex-math id=\"M879\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root \\infty \\of {V\\!E(X)}\\setminus H(X)$$\\end{document}</tex-math><mml:math id=\"M880\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>∞</mml:mi></mml:mroot><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ19\"><alternatives><tex-math id=\"M881\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}=\\big (Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}\\setminus H(X)\\big )\\cup \\big (Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}\\cap H(X)\\big ) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M882\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo>=</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo>∩</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq423\"><alternatives><tex-math id=\"M883\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(6)\\Rightarrow (1)$$\\end{document}</tex-math><mml:math id=\"M884\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>6</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq424\"><alternatives><tex-math id=\"M885\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}$$\\end{document}</tex-math><mml:math id=\"M886\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq425\"><alternatives><tex-math id=\"M887\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)=E(X)$$\\end{document}</tex-math><mml:math id=\"M888\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq426\"><alternatives><tex-math id=\"M889\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)=Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {E(X)}=Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}$$\\end{document}</tex-math><mml:math id=\"M890\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo>=</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq427\"><alternatives><tex-math id=\"M891\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X=Z(X)$$\\end{document}</tex-math><mml:math id=\"M892\"><mml:mrow><mml:mi>X</mml:mi><mml:mo>=</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq428\"><alternatives><tex-math id=\"M893\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M894\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq429\"><alternatives><tex-math id=\"M895\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M896\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq430\"><alternatives><tex-math id=\"M897\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {1}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M898\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq431\"><alternatives><tex-math id=\"M899\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E(X)\\cap Z(X)\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M900\"><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∩</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq432\"><alternatives><tex-math id=\"M901\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)=Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {E(X)\\cap Z(X)}\\subseteq Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}$$\\end{document}</tex-math><mml:math id=\"M902\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∩</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq433\"><alternatives><tex-math id=\"M903\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M904\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq434\"><alternatives><tex-math id=\"M905\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M906\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq435\"><alternatives><tex-math id=\"M907\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M908\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq436\"><alternatives><tex-math id=\"M909\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M910\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq437\"><alternatives><tex-math id=\"M911\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$xy=yx$$\\end{document}</tex-math><mml:math id=\"M912\"><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq438\"><alternatives><tex-math id=\"M913\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y\\in E(X)$$\\end{document}</tex-math><mml:math id=\"M914\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq439\"><alternatives><tex-math id=\"M915\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M916\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq440\"><alternatives><tex-math id=\"M917\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M918\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq441\"><alternatives><tex-math id=\"M919\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h:X\\rightarrow Y$$\\end{document}</tex-math><mml:math id=\"M920\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq442\"><alternatives><tex-math id=\"M921\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{h[X]}$$\\end{document}</tex-math><mml:math id=\"M922\"><mml:mover><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq443\"><alternatives><tex-math id=\"M923\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M924\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq444\"><alternatives><tex-math id=\"M925\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x^n\\in E(X)$$\\end{document}</tex-math><mml:math id=\"M926\"><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq445\"><alternatives><tex-math id=\"M927\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x^{2n}=x^n$$\\end{document}</tex-math><mml:math id=\"M928\"><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq446\"><alternatives><tex-math id=\"M929\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\in X$$\\end{document}</tex-math><mml:math id=\"M930\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq447\"><alternatives><tex-math id=\"M931\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y^{2n}=y^n$$\\end{document}</tex-math><mml:math id=\"M932\"><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq448\"><alternatives><tex-math id=\"M933\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in h[X]$$\\end{document}</tex-math><mml:math id=\"M934\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq449\"><alternatives><tex-math id=\"M935\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{y\\in Y:y^{2n}=y^n\\}$$\\end{document}</tex-math><mml:math id=\"M936\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo>:</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq450\"><alternatives><tex-math id=\"M937\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y^{n}\\in E(Y)$$\\end{document}</tex-math><mml:math id=\"M938\"><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq451\"><alternatives><tex-math id=\"M939\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in Y$$\\end{document}</tex-math><mml:math id=\"M940\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq452\"><alternatives><tex-math id=\"M941\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi :Y\\rightarrow E(Y)$$\\end{document}</tex-math><mml:math id=\"M942\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo>:</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq453\"><alternatives><tex-math id=\"M943\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi :y\\mapsto y^n$$\\end{document}</tex-math><mml:math id=\"M944\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo>:</mml:mo><mml:mi>y</mml:mi><mml:mo>↦</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq454\"><alternatives><tex-math id=\"M945\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi :X\\rightarrow E(X)$$\\end{document}</tex-math><mml:math id=\"M946\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq455\"><alternatives><tex-math id=\"M947\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi :x\\mapsto x^n$$\\end{document}</tex-math><mml:math id=\"M948\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>↦</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq456\"><alternatives><tex-math id=\"M949\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h\\circ \\psi (x)=h(x^n)=(h(x))^n=\\phi \\circ h(x)$$\\end{document}</tex-math><mml:math id=\"M950\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>∘</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>∘</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq457\"><alternatives><tex-math id=\"M951\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\in X$$\\end{document}</tex-math><mml:math id=\"M952\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq458\"><alternatives><tex-math id=\"M953\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M954\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq459\"><alternatives><tex-math id=\"M955\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi :Y\\rightarrow E(Y)$$\\end{document}</tex-math><mml:math id=\"M956\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo>:</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq460\"><alternatives><tex-math id=\"M957\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi :y\\mapsto y^n$$\\end{document}</tex-math><mml:math id=\"M958\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo>:</mml:mo><mml:mi>y</mml:mi><mml:mo>↦</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ20\"><alternatives><tex-math id=\"M959\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} h[E(X)]\\subseteq E(Y)= \\phi [Y]=\\phi [\\overline{h[X]}]\\subseteq \\overline{\\phi [h[X]]}=\\overline{h[\\psi [X]]}=\\overline{h[E(X)]}=h[E(X)]. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M960\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mover><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:mover><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq461\"><alternatives><tex-math id=\"M961\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h[E(X)]=E(Y)=\\phi [Y]$$\\end{document}</tex-math><mml:math id=\"M962\"><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq462\"><alternatives><tex-math id=\"M963\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x=x^{n+1}$$\\end{document}</tex-math><mml:math id=\"M964\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq463\"><alternatives><tex-math id=\"M965\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\in H(X)$$\\end{document}</tex-math><mml:math id=\"M966\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq464\"><alternatives><tex-math id=\"M967\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F=X\\setminus H(X)$$\\end{document}</tex-math><mml:math id=\"M968\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>=</mml:mo><mml:mi>X</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq465\"><alternatives><tex-math id=\"M969\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y=\\overline{h[X]}=\\overline{h[H(X)\\cup F]}=\\overline{h[H(X)]}\\cup h[F]$$\\end{document}</tex-math><mml:math id=\"M970\"><mml:mrow><mml:mi>Y</mml:mi><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∪</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>∪</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq466\"><alternatives><tex-math id=\"M971\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{y\\in Y:y=y^{n+1}\\}\\supseteq h[H(X)]$$\\end{document}</tex-math><mml:math id=\"M972\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo>:</mml:mo><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>⊇</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq467\"><alternatives><tex-math id=\"M973\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{h[H(X)]}=\\overline{h[X{\\setminus } F]}\\supseteq \\overline{h[X]}{\\setminus } h[F]=Y{\\setminus } h[F]$$\\end{document}</tex-math><mml:math id=\"M974\"><mml:mrow><mml:mover><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>⊇</mml:mo><mml:mover><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq468\"><alternatives><tex-math id=\"M975\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y^{n+1}=y$$\\end{document}</tex-math><mml:math id=\"M976\"><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq469\"><alternatives><tex-math id=\"M977\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in Y\\setminus h[F]$$\\end{document}</tex-math><mml:math id=\"M978\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq470\"><alternatives><tex-math id=\"M979\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in Y{\\setminus } h[X]\\subseteq Y{\\setminus } h[F]$$\\end{document}</tex-math><mml:math id=\"M980\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>⊆</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq471\"><alternatives><tex-math id=\"M981\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e=y^n=\\phi (y)\\in \\phi [Y]=E(Y)=h[E(X)]$$\\end{document}</tex-math><mml:math id=\"M982\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq472\"><alternatives><tex-math id=\"M983\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h[E(X)]=E(Y)$$\\end{document}</tex-math><mml:math id=\"M984\"><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>=</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq473\"><alternatives><tex-math id=\"M985\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$L=\\{f\\in E(Y):ef\\ne e\\}$$\\end{document}</tex-math><mml:math id=\"M986\"><mml:mrow><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>f</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>:</mml:mo><mml:mi>e</mml:mi><mml:mi>f</mml:mi><mml:mo>≠</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq474\"><alternatives><tex-math id=\"M987\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\uparrow }e=\\{f\\in E(Y):ef=e\\}$$\\end{document}</tex-math><mml:math id=\"M988\"><mml:mrow><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>f</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>:</mml:mo><mml:mi>e</mml:mi><mml:mi>f</mml:mi><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq475\"><alternatives><tex-math id=\"M989\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$U=\\phi ^{-1}[{\\uparrow }e]$$\\end{document}</tex-math><mml:math id=\"M990\"><mml:mrow><mml:mi>U</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq476\"><alternatives><tex-math id=\"M991\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in h[E(X)]$$\\end{document}</tex-math><mml:math id=\"M992\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq477\"><alternatives><tex-math id=\"M993\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h^{-1}[{\\uparrow }e]\\cap E(X)$$\\end{document}</tex-math><mml:math id=\"M994\"><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq478\"><alternatives><tex-math id=\"M995\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mu \\in h^{-1}(e)$$\\end{document}</tex-math><mml:math id=\"M996\"><mml:mrow><mml:mi>μ</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq479\"><alternatives><tex-math id=\"M997\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{\\mu }$$\\end{document}</tex-math><mml:math id=\"M998\"><mml:msub><mml:mi>H</mml:mi><mml:mi>μ</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq480\"><alternatives><tex-math id=\"M999\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h[H_{\\mu }\\cup F]$$\\end{document}</tex-math><mml:math id=\"M1000\"><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>μ</mml:mi></mml:msub><mml:mo>∪</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq481\"><alternatives><tex-math id=\"M1001\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y=y^{n+1}=ey\\notin h[H_{\\mu }\\cup F]$$\\end{document}</tex-math><mml:math id=\"M1002\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:mi>y</mml:mi><mml:mo>∉</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>μ</mml:mi></mml:msub><mml:mo>∪</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq482\"><alternatives><tex-math id=\"M1003\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_y\\subseteq U$$\\end{document}</tex-math><mml:math id=\"M1004\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>⊆</mml:mo><mml:mi>U</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq483\"><alternatives><tex-math id=\"M1005\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_y^{n+1}\\subseteq U\\setminus h[H_{\\mu }\\cup F]$$\\end{document}</tex-math><mml:math id=\"M1006\"><mml:mrow><mml:msubsup><mml:mi>O</mml:mi><mml:mi>y</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>⊆</mml:mo><mml:mi>U</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>μ</mml:mi></mml:msub><mml:mo>∪</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq484\"><alternatives><tex-math id=\"M1007\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$eO_y\\cap h[H_{\\mu }\\cup F]=\\emptyset $$\\end{document}</tex-math><mml:math id=\"M1008\"><mml:mrow><mml:mi>e</mml:mi><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>∩</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>μ</mml:mi></mml:msub><mml:mo>∪</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">∅</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq485\"><alternatives><tex-math id=\"M1009\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$eO_y\\subset O_y^{n+1}$$\\end{document}</tex-math><mml:math id=\"M1010\"><mml:mrow><mml:mi>e</mml:mi><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>⊂</mml:mo><mml:msubsup><mml:mi>O</mml:mi><mml:mi>y</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq486\"><alternatives><tex-math id=\"M1011\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi (ey)=(ey)^n=(y^{n+1})^n=(y^n)^{n+1}=e^{n+1}=e\\in {\\uparrow }e$$\\end{document}</tex-math><mml:math id=\"M1012\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>e</mml:mi><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>e</mml:mi><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>e</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq487\"><alternatives><tex-math id=\"M1013\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi [eO_y]\\subseteq {\\uparrow }e$$\\end{document}</tex-math><mml:math id=\"M1014\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>e</mml:mi><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>⊆</mml:mo><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>e</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ21\"><alternatives><tex-math id=\"M1015\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} y\\in Y\\setminus h[X]=(\\overline{h[H(X)]}\\cup h[F])\\setminus h[X]=\\overline{h[H(X)]}\\setminus h[X], \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1016\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>∪</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq488\"><alternatives><tex-math id=\"M1017\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\in H(X)\\cap h^{-1}[O_y]$$\\end{document}</tex-math><mml:math id=\"M1018\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq489\"><alternatives><tex-math id=\"M1019\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h((\\mu x)^n)=(h(\\mu )h(x))^n=\\phi (eh(x))\\in \\phi [eO_y]\\subseteq {\\uparrow }e$$\\end{document}</tex-math><mml:math id=\"M1020\"><mml:mrow><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>μ</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>e</mml:mi><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>e</mml:mi><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>e</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq490\"><alternatives><tex-math id=\"M1021\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(\\mu x)^n\\in E(X)\\cap h^{-1}[{\\uparrow }e]$$\\end{document}</tex-math><mml:math id=\"M1022\"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>μ</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq491\"><alternatives><tex-math id=\"M1023\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mu $$\\end{document}</tex-math><mml:math id=\"M1024\"><mml:mi>μ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq492\"><alternatives><tex-math id=\"M1025\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E(X)\\cap h^{-1}[{\\uparrow }e]$$\\end{document}</tex-math><mml:math id=\"M1026\"><mml:mrow><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq493\"><alternatives><tex-math id=\"M1027\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mu \\le (\\mu x)^n$$\\end{document}</tex-math><mml:math id=\"M1028\"><mml:mrow><mml:mi>μ</mml:mi><mml:mo>≤</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>μ</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq494\"><alternatives><tex-math id=\"M1029\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mu (\\mu x)^n=(\\mu x)^n$$\\end{document}</tex-math><mml:math id=\"M1030\"><mml:mrow><mml:mi>μ</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>μ</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>μ</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq495\"><alternatives><tex-math id=\"M1031\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(\\mu x)^n\\le \\mu $$\\end{document}</tex-math><mml:math id=\"M1032\"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>μ</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>≤</mml:mo><mml:mi>μ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq496\"><alternatives><tex-math id=\"M1033\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mu =(\\mu x)^n$$\\end{document}</tex-math><mml:math id=\"M1034\"><mml:mrow><mml:mi>μ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>μ</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq497\"><alternatives><tex-math id=\"M1035\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mu x\\in H_\\mu \\cup F$$\\end{document}</tex-math><mml:math id=\"M1036\"><mml:mrow><mml:mi>μ</mml:mi><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>μ</mml:mi></mml:msub><mml:mo>∪</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq498\"><alternatives><tex-math id=\"M1037\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h(\\mu x)=h(\\mu )h(x)\\in eO_y$$\\end{document}</tex-math><mml:math id=\"M1038\"><mml:mrow><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>μ</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mi>e</mml:mi><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq499\"><alternatives><tex-math id=\"M1039\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h(\\mu x)\\in h[H_{\\mu }\\cup F]\\cap eO_y=\\emptyset $$\\end{document}</tex-math><mml:math id=\"M1040\"><mml:mrow><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>μ</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>μ</mml:mi></mml:msub><mml:mo>∪</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mi>e</mml:mi><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">∅</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq500\"><alternatives><tex-math id=\"M1041\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M1042\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq501\"><alternatives><tex-math id=\"M1043\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1044\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq502\"><alternatives><tex-math id=\"M1045\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1046\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq503\"><alternatives><tex-math id=\"M1047\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$A\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M1048\"><mml:mrow><mml:mi>A</mml:mi><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq504\"><alternatives><tex-math id=\"M1049\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M1050\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq505\"><alternatives><tex-math id=\"M1051\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$B\\subseteq X$$\\end{document}</tex-math><mml:math id=\"M1052\"><mml:mrow><mml:mi>B</mml:mi><mml:mo>⊆</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq506\"><alternatives><tex-math id=\"M1053\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$B\\subseteq \\!\\root n \\of {E(X)}{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{x\\in X:x^n\\in E(X)\\}$$\\end{document}</tex-math><mml:math id=\"M1054\"><mml:mrow><mml:mi>B</mml:mi><mml:mo>⊆</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:mroot><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mo>:</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq507\"><alternatives><tex-math id=\"M1055\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1056\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq508\"><alternatives><tex-math id=\"M1057\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$A\\subseteq E(X)$$\\end{document}</tex-math><mml:math id=\"M1058\"><mml:mrow><mml:mi>A</mml:mi><mml:mo>⊆</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq509\"><alternatives><tex-math id=\"M1059\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1060\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq510\"><alternatives><tex-math id=\"M1061\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M1062\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq511\"><alternatives><tex-math id=\"M1063\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y\\in \\bigcap _{a\\in A}\\frac{H_a}{a}$$\\end{document}</tex-math><mml:math id=\"M1064\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mi>a</mml:mi></mml:mfrac></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq512\"><alternatives><tex-math id=\"M1065\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(xe)(ye)^{-1}\\in H_e\\cap Z(X)$$\\end{document}</tex-math><mml:math id=\"M1066\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>∈</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>∩</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq513\"><alternatives><tex-math id=\"M1067\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$((xe)(ye)^{-1})^n\\in E(X)$$\\end{document}</tex-math><mml:math id=\"M1068\"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq514\"><alternatives><tex-math id=\"M1069\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M1070\"><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq515\"><alternatives><tex-math id=\"M1071\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\le $$\\end{document}</tex-math><mml:math id=\"M1072\"><mml:mo>≤</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq516\"><alternatives><tex-math id=\"M1073\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$A\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M1074\"><mml:mrow><mml:mi>A</mml:mi><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq517\"><alternatives><tex-math id=\"M1075\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\not \\le y$$\\end{document}</tex-math><mml:math id=\"M1076\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>≰</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq518\"><alternatives><tex-math id=\"M1077\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y\\in A$$\\end{document}</tex-math><mml:math id=\"M1078\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq519\"><alternatives><tex-math id=\"M1079\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1080\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq520\"><alternatives><tex-math id=\"M1081\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$A\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M1082\"><mml:mrow><mml:mi>A</mml:mi><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq521\"><alternatives><tex-math id=\"M1083\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M1084\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq522\"><alternatives><tex-math id=\"M1085\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap H(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M1086\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq523\"><alternatives><tex-math id=\"M1087\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$B\\subseteq A$$\\end{document}</tex-math><mml:math id=\"M1088\"><mml:mrow><mml:mi>B</mml:mi><mml:mo>⊆</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq524\"><alternatives><tex-math id=\"M1089\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X^1$$\\end{document}</tex-math><mml:math id=\"M1090\"><mml:msup><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq525\"><alternatives><tex-math id=\"M1091\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(1)\\Leftrightarrow (2)$$\\end{document}</tex-math><mml:math id=\"M1092\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇔</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq526\"><alternatives><tex-math id=\"M1093\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(2)\\Rightarrow (3)$$\\end{document}</tex-math><mml:math id=\"M1094\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq527\"><alternatives><tex-math id=\"M1095\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1096\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq528\"><alternatives><tex-math id=\"M1097\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap H(X)\\cap \\!\\root {\\mathbb {N}} \\of {A}\\subseteq \\!\\root n \\of {E(X)}$$\\end{document}</tex-math><mml:math id=\"M1098\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mi>A</mml:mi><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo>⊆</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq529\"><alternatives><tex-math id=\"M1099\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$B\\subseteq A$$\\end{document}</tex-math><mml:math id=\"M1100\"><mml:mrow><mml:mi>B</mml:mi><mml:mo>⊆</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq530\"><alternatives><tex-math id=\"M1101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in B$$\\end{document}</tex-math><mml:math id=\"M1102\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq531\"><alternatives><tex-math id=\"M1103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y\\in \\bigcap _{b\\in B}\\frac{H_b}{b}$$\\end{document}</tex-math><mml:math id=\"M1104\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>b</mml:mi><mml:mo>∈</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mi>b</mml:mi></mml:mfrac></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq532\"><alternatives><tex-math id=\"M1105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(xe)(ye)^{-1}\\in Z(X)$$\\end{document}</tex-math><mml:math id=\"M1106\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>∈</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ22\"><alternatives><tex-math id=\"M1107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} (xe)(ye)^{-1}\\in Z(X)\\cap H_e\\cap \\!\\root {\\mathbb {N}} \\of {e}\\subseteq Z(X)\\cap H(X)\\cap \\!\\root {\\mathbb {N}} \\of {A}\\subseteq \\!\\root n \\of {E(X)} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1108\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>∈</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mi>e</mml:mi><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mi>A</mml:mi><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo>⊆</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:mroot></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq533\"><alternatives><tex-math id=\"M1109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$((xe)(ye)^{-1})^n\\in E(X)$$\\end{document}</tex-math><mml:math id=\"M1110\"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq534\"><alternatives><tex-math id=\"M1111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(3)\\Rightarrow (2)$$\\end{document}</tex-math><mml:math id=\"M1112\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq535\"><alternatives><tex-math id=\"M1113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pi :\\!\\root {\\mathbb {N}} \\of {\\!E(X)}\\rightarrow E(X)$$\\end{document}</tex-math><mml:math id=\"M1114\"><mml:mrow><mml:mi>π</mml:mi><mml:mo>:</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq536\"><alternatives><tex-math id=\"M1115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\in \\!\\root {\\mathbb {N}} \\of {\\!E(X)}$$\\end{document}</tex-math><mml:math id=\"M1116\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq537\"><alternatives><tex-math id=\"M1117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pi (x)$$\\end{document}</tex-math><mml:math id=\"M1118\"><mml:mrow><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq538\"><alternatives><tex-math id=\"M1119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x^{\\mathbb {N}}{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{x^n:n\\in {\\mathbb {N}}\\}$$\\end{document}</tex-math><mml:math id=\"M1120\"><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:msup><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>:</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq539\"><alternatives><tex-math id=\"M1121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(z_k)_{k\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M1122\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq540\"><alternatives><tex-math id=\"M1123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap H(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M1124\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq541\"><alternatives><tex-math id=\"M1125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pi (z_k)\\not \\le \\pi (z_n)$$\\end{document}</tex-math><mml:math id=\"M1126\"><mml:mrow><mml:mi>π</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≰</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq542\"><alternatives><tex-math id=\"M1127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k,n\\in \\omega $$\\end{document}</tex-math><mml:math id=\"M1128\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq543\"><alternatives><tex-math id=\"M1129\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_k^i\\ne z_k^j$$\\end{document}</tex-math><mml:math id=\"M1130\"><mml:mrow><mml:msubsup><mml:mi>z</mml:mi><mml:mi>k</mml:mi><mml:mi>i</mml:mi></mml:msubsup><mml:mo>≠</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>k</mml:mi><mml:mi>j</mml:mi></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq544\"><alternatives><tex-math id=\"M1131\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k\\in \\omega $$\\end{document}</tex-math><mml:math id=\"M1132\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq545\"><alternatives><tex-math id=\"M1133\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i,j\\in \\{1,\\dots ,2^k\\}$$\\end{document}</tex-math><mml:math id=\"M1134\"><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:msup><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq546\"><alternatives><tex-math id=\"M1135\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap H(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M1136\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq547\"><alternatives><tex-math id=\"M1137\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k\\in \\omega $$\\end{document}</tex-math><mml:math id=\"M1138\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq548\"><alternatives><tex-math id=\"M1139\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$g_k\\in Z(X)\\cap H(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M1140\"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq549\"><alternatives><tex-math id=\"M1141\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i,j\\le 2^k$$\\end{document}</tex-math><mml:math id=\"M1142\"><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq550\"><alternatives><tex-math id=\"M1143\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$g_k^i\\ne g_k^j$$\\end{document}</tex-math><mml:math id=\"M1144\"><mml:mrow><mml:msubsup><mml:mi>g</mml:mi><mml:mi>k</mml:mi><mml:mi>i</mml:mi></mml:msubsup><mml:mo>≠</mml:mo><mml:msubsup><mml:mi>g</mml:mi><mml:mi>k</mml:mi><mml:mi>j</mml:mi></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq551\"><alternatives><tex-math id=\"M1145\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$[\\omega ]^2$$\\end{document}</tex-math><mml:math id=\"M1146\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq552\"><alternatives><tex-math id=\"M1147\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\omega $$\\end{document}</tex-math><mml:math id=\"M1148\"><mml:mi>ω</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq553\"><alternatives><tex-math id=\"M1149\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\chi :[\\omega ]^2\\rightarrow \\{0,1,2\\}$$\\end{document}</tex-math><mml:math id=\"M1150\"><mml:mrow><mml:mi>χ</mml:mi><mml:mo>:</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">→</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ23\"><alternatives><tex-math id=\"M1151\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\chi (\\{n,m\\})={\\left\\{ \\begin{array}{ll}0&amp;{}\\hbox { if}\\,\\,\\ \\pi (g_n)=\\pi (g_m);\\\\ 1&amp;{}\\hbox {if}\\,\\, \\pi (g_n)&lt;\\pi (g_m) \\hbox {or} \\pi (g_m)&lt;\\pi (g_n);\\\\ 2&amp;{}\\text{ otherwise }. \\end{array}\\right. } \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1152\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>χ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mn>0</mml:mn></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>if</mml:mtext><mml:mspace width=\"0.166667em\"/><mml:mspace width=\"0.166667em\"/><mml:mspace width=\"4pt\"/><mml:mi>π</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>;</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mn>1</mml:mn></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mtext>if</mml:mtext><mml:mspace width=\"0.166667em\"/><mml:mspace width=\"0.166667em\"/><mml:mi>π</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>&lt;</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mtext>or</mml:mtext><mml:mi>π</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>&lt;</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>;</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mn>2</mml:mn></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>otherwise</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq554\"><alternatives><tex-math id=\"M1153\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega \\subseteq \\omega $$\\end{document}</tex-math><mml:math id=\"M1154\"><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo>⊆</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq555\"><alternatives><tex-math id=\"M1155\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\chi [[\\Omega ]^2]=\\{c\\}$$\\end{document}</tex-math><mml:math id=\"M1156\"><mml:mrow><mml:mi>χ</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>c</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq556\"><alternatives><tex-math id=\"M1157\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$c\\in \\{0,1,2\\}$$\\end{document}</tex-math><mml:math id=\"M1158\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq557\"><alternatives><tex-math id=\"M1159\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$c=0$$\\end{document}</tex-math><mml:math id=\"M1160\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq558\"><alternatives><tex-math id=\"M1161\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{\\pi (g_n)\\}_{n\\in \\Omega }$$\\end{document}</tex-math><mml:math id=\"M1162\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq559\"><alternatives><tex-math id=\"M1163\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{g_k\\}_{k\\in \\Omega }\\subseteq Z(X)\\cap H_u$$\\end{document}</tex-math><mml:math id=\"M1164\"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow></mml:msub><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>u</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq560\"><alternatives><tex-math id=\"M1165\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k&gt;|Z(X)\\cap H_u|$$\\end{document}</tex-math><mml:math id=\"M1166\"><mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>&gt;</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo></mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>u</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq561\"><alternatives><tex-math id=\"M1167\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i&lt;j\\le k$$\\end{document}</tex-math><mml:math id=\"M1168\"><mml:mrow><mml:mi>i</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq562\"><alternatives><tex-math id=\"M1169\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$g_k^i=g_k^j$$\\end{document}</tex-math><mml:math id=\"M1170\"><mml:mrow><mml:msubsup><mml:mi>g</mml:mi><mml:mi>k</mml:mi><mml:mi>i</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mi>g</mml:mi><mml:mi>k</mml:mi><mml:mi>j</mml:mi></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq563\"><alternatives><tex-math id=\"M1171\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$g_k$$\\end{document}</tex-math><mml:math id=\"M1172\"><mml:msub><mml:mi>g</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq564\"><alternatives><tex-math id=\"M1173\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$c\\ne 0$$\\end{document}</tex-math><mml:math id=\"M1174\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq565\"><alternatives><tex-math id=\"M1175\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$c=1$$\\end{document}</tex-math><mml:math id=\"M1176\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq566\"><alternatives><tex-math id=\"M1177\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{\\pi (g_k)\\}_{k\\in \\Omega }$$\\end{document}</tex-math><mml:math id=\"M1178\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq567\"><alternatives><tex-math id=\"M1179\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E(X)\\cap Z(X)$$\\end{document}</tex-math><mml:math id=\"M1180\"><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∩</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq568\"><alternatives><tex-math id=\"M1181\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$c=2$$\\end{document}</tex-math><mml:math id=\"M1182\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq569\"><alternatives><tex-math id=\"M1183\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{\\pi (g_k)\\}_{k\\in \\Omega }$$\\end{document}</tex-math><mml:math id=\"M1184\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq570\"><alternatives><tex-math id=\"M1185\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega $$\\end{document}</tex-math><mml:math id=\"M1186\"><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq571\"><alternatives><tex-math id=\"M1187\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{n_k\\}_{k\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M1188\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq572\"><alternatives><tex-math id=\"M1189\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(n_k)_{k\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M1190\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq573\"><alternatives><tex-math id=\"M1191\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k\\in \\omega $$\\end{document}</tex-math><mml:math id=\"M1192\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq574\"><alternatives><tex-math id=\"M1193\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_k=g_{n_k}$$\\end{document}</tex-math><mml:math id=\"M1194\"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:msub><mml:mi>n</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq575\"><alternatives><tex-math id=\"M1195\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(z_k)_{k\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M1196\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq576\"><alternatives><tex-math id=\"M1197\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M1198\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq577\"><alternatives><tex-math id=\"M1199\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q:X\\rightarrow X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M1200\"><mml:mrow><mml:mi>q</mml:mi><mml:mo>:</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq578\"><alternatives><tex-math id=\"M1201\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(z_n)_{n\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M1202\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq579\"><alternatives><tex-math id=\"M1203\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in \\omega $$\\end{document}</tex-math><mml:math id=\"M1204\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq580\"><alternatives><tex-math id=\"M1205\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e_n=\\pi (z_n)$$\\end{document}</tex-math><mml:math id=\"M1206\"><mml:mrow><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq581\"><alternatives><tex-math id=\"M1207\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_n\\in Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M1208\"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq582\"><alternatives><tex-math id=\"M1209\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e_n=\\pi (z_n)\\in Z(X)\\cap A\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M1210\"><mml:mrow><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>π</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mi>A</mml:mi><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ24\"><alternatives><tex-math id=\"M1211\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\tfrac{H_{e_n}}{e_n}=\\{x\\in X:xe_n=e_nx\\in H_{e_n}\\} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1212\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mfrac></mml:mstyle><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq583\"><alternatives><tex-math id=\"M1213\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\frac{H_{e_n}}{e_n}=q^{-1}[{\\uparrow }q(e_n)]$$\\end{document}</tex-math><mml:math id=\"M1214\"><mml:mrow><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mfrac><mml:mo>=</mml:mo><mml:msup><mml:mi>q</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq584\"><alternatives><tex-math id=\"M1215\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1216\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ25\"><alternatives><tex-math id=\"M1217\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\textstyle I{\\mathop {=}\\limits ^{{\\textsf{def}}}}X\\setminus \\bigcup _{n\\in \\omega }\\tfrac{H_{e_n}}{e_n}=X\\setminus q^{-1}\\left[ \\bigcup _{n\\in \\omega }{\\uparrow }q(e_n)\\right] \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1218\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mrow><mml:mi>I</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mi>X</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:msub><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mfrac></mml:mstyle><mml:mo>=</mml:mo><mml:mi>X</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:msup><mml:mi>q</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mfenced close=\"]\" open=\"[\"><mml:msub><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:mstyle></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq585\"><alternatives><tex-math id=\"M1219\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1220\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq586\"><alternatives><tex-math id=\"M1221\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\in X/I$$\\end{document}</tex-math><mml:math id=\"M1222\"><mml:mrow><mml:mi>I</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq587\"><alternatives><tex-math id=\"M1223\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q{\\restriction }_{V\\!E(X)}$$\\end{document}</tex-math><mml:math id=\"M1224\"><mml:mrow><mml:mi>q</mml:mi><mml:msub><mml:mo>↾</mml:mo><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq588\"><alternatives><tex-math id=\"M1225\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e_ne_m\\in I$$\\end{document}</tex-math><mml:math id=\"M1226\"><mml:mrow><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq589\"><alternatives><tex-math id=\"M1227\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n,m\\in \\omega $$\\end{document}</tex-math><mml:math id=\"M1228\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq590\"><alternatives><tex-math id=\"M1229\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$0=I$$\\end{document}</tex-math><mml:math id=\"M1230\"><mml:mrow><mml:mn>0</mml:mn><mml:mo>=</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq591\"><alternatives><tex-math id=\"M1231\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau $$\\end{document}</tex-math><mml:math id=\"M1232\"><mml:mi>τ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq592\"><alternatives><tex-math id=\"M1233\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y{\\mathop {=}\\limits ^{{\\textsf{def}}}}X/I$$\\end{document}</tex-math><mml:math id=\"M1234\"><mml:mrow><mml:mi>Y</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq593\"><alternatives><tex-math id=\"M1235\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1236\"><mml:mi mathvariant=\"script\">F</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq594\"><alternatives><tex-math id=\"M1237\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1238\"><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ26\"><alternatives><tex-math id=\"M1239\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} Q=\\{y\\in X/_{\\Updownarrow }:\\exists F\\in {\\mathcal {F}} \\;\\forall n\\in F\\;\\; q(e_n)\\le y\\}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1240\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>Q</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub><mml:mo>:</mml:mo><mml:mo>∃</mml:mo><mml:mi>F</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi><mml:mspace width=\"0.277778em\"/><mml:mo>∀</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>F</mml:mi><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"0.277778em\"/><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq595\"><alternatives><tex-math id=\"M1241\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q(1)\\in Q$$\\end{document}</tex-math><mml:math id=\"M1242\"><mml:mrow><mml:mi>q</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∈</mml:mo><mml:mi>Q</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq596\"><alternatives><tex-math id=\"M1243\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Q=\\emptyset $$\\end{document}</tex-math><mml:math id=\"M1244\"><mml:mrow><mml:mi>Q</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">∅</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq597\"><alternatives><tex-math id=\"M1245\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_1,y_2\\in Q$$\\end{document}</tex-math><mml:math id=\"M1246\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>∈</mml:mo><mml:mi>Q</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq598\"><alternatives><tex-math id=\"M1247\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F_1, F_2\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1248\"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq599\"><alternatives><tex-math id=\"M1249\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q(e_n)\\le y_i$$\\end{document}</tex-math><mml:math id=\"M1250\"><mml:mrow><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq600\"><alternatives><tex-math id=\"M1251\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in F_i$$\\end{document}</tex-math><mml:math id=\"M1252\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq601\"><alternatives><tex-math id=\"M1253\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i\\in \\{1,2\\}$$\\end{document}</tex-math><mml:math id=\"M1254\"><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq602\"><alternatives><tex-math id=\"M1255\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q(e_n)\\le y_1y_2$$\\end{document}</tex-math><mml:math id=\"M1256\"><mml:mrow><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq603\"><alternatives><tex-math id=\"M1257\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in F_1\\cap F_2\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1258\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>∩</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq604\"><alternatives><tex-math id=\"M1259\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M1260\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq605\"><alternatives><tex-math id=\"M1261\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X/_{\\Updownarrow }$$\\end{document}</tex-math><mml:math id=\"M1262\"><mml:mrow><mml:mi>X</mml:mi><mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>⇕</mml:mo></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq606\"><alternatives><tex-math id=\"M1263\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$s\\in Q$$\\end{document}</tex-math><mml:math id=\"M1264\"><mml:mrow><mml:mi>s</mml:mi><mml:mo>∈</mml:mo><mml:mi>Q</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq607\"><alternatives><tex-math id=\"M1265\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F_s\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1266\"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq608\"><alternatives><tex-math id=\"M1267\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q(e_n)\\le s$$\\end{document}</tex-math><mml:math id=\"M1268\"><mml:mrow><mml:mi>q</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≤</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq609\"><alternatives><tex-math id=\"M1269\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in F_s$$\\end{document}</tex-math><mml:math id=\"M1270\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ27\"><alternatives><tex-math id=\"M1271\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} C{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\bigcap _{n\\in F_s}\\tfrac{H_{e_n}}{e_n}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1272\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>C</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:munder><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:munder><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mfrac></mml:mstyle><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq610\"><alternatives><tex-math id=\"M1273\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\uparrow }s=\\bigcap _{n\\in F_s}{\\uparrow }q(e_n)$$\\end{document}</tex-math><mml:math id=\"M1274\"><mml:mrow><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq611\"><alternatives><tex-math id=\"M1275\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q^{-1}({\\uparrow }s)=C$$\\end{document}</tex-math><mml:math id=\"M1276\"><mml:mrow><mml:msup><mml:mi>q</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq612\"><alternatives><tex-math id=\"M1277\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\uparrow }s\\subseteq \\bigcap _{n\\in F_s}{\\uparrow }q(e_n)$$\\end{document}</tex-math><mml:math id=\"M1278\"><mml:mrow><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>s</mml:mi><mml:mo>⊆</mml:mo><mml:msub><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq613\"><alternatives><tex-math id=\"M1279\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F_s$$\\end{document}</tex-math><mml:math id=\"M1280\"><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq614\"><alternatives><tex-math id=\"M1281\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in \\bigcap _{n\\in F_s}{\\uparrow }q(e_n)$$\\end{document}</tex-math><mml:math id=\"M1282\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq615\"><alternatives><tex-math id=\"M1283\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{n\\in \\omega :q(e_n)\\le y\\}\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1284\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi><mml:mo>:</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq616\"><alternatives><tex-math id=\"M1285\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in Q$$\\end{document}</tex-math><mml:math id=\"M1286\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Q</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq617\"><alternatives><tex-math id=\"M1287\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$s\\le y$$\\end{document}</tex-math><mml:math id=\"M1288\"><mml:mrow><mml:mi>s</mml:mi><mml:mo>≤</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ28\"><alternatives><tex-math id=\"M1289\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} q^{-1}[{\\uparrow }s]=q^{-1}\\left[ \\bigcap _{n\\in F_s}{\\uparrow }q(e_n)\\right] =\\bigcap _{n\\in F_s}q^{-1}[{\\uparrow }q(e_n)]=\\bigcap _{n\\in F_s}\\tfrac{H_{e_n}}{e_n}=C. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1290\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msup><mml:mi>q</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>q</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mfenced close=\"]\" open=\"[\"><mml:munder><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:munder><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:munder><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:munder><mml:msup><mml:mi>q</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">↑</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:munder><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mfrac></mml:mstyle><mml:mo>=</mml:mo><mml:mi>C</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq618\"><alternatives><tex-math id=\"M1291\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M1292\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq619\"><alternatives><tex-math id=\"M1293\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau $$\\end{document}</tex-math><mml:math id=\"M1294\"><mml:mi>τ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq620\"><alternatives><tex-math id=\"M1295\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\lfloor r\\rfloor {\\mathop {=}\\limits ^{{\\textsf{def}}}}\\max \\{n\\in {\\mathbb {Z}}:n\\le r\\}$$\\end{document}</tex-math><mml:math id=\"M1296\"><mml:mrow><mml:mrow><mml:mo>⌊</mml:mo><mml:mi>r</mml:mi><mml:mo>⌋</mml:mo></mml:mrow><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mo movablelimits=\"true\">max</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">Z</mml:mi><mml:mo>:</mml:mo><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq621\"><alternatives><tex-math id=\"M1297\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1298\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq622\"><alternatives><tex-math id=\"M1299\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in F_s$$\\end{document}</tex-math><mml:math id=\"M1300\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ29\"><alternatives><tex-math id=\"M1301\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} A(n,k)=\\bigcup _{2\\le p\\le \\lfloor 2^n/k\\rfloor }z_n^pC\\subseteq H_{e_n}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1302\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mo>⋃</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>≤</mml:mo><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>k</mml:mi><mml:mo>⌋</mml:mo></mml:mrow></mml:munder><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:msubsup><mml:mi>C</mml:mi><mml:mo>⊆</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq623\"><alternatives><tex-math id=\"M1303\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$A(n,k)=\\emptyset $$\\end{document}</tex-math><mml:math id=\"M1304\"><mml:mrow><mml:mi>A</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">∅</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq624\"><alternatives><tex-math id=\"M1305\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$2^n/k&lt;2$$\\end{document}</tex-math><mml:math id=\"M1306\"><mml:mrow><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>k</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq625\"><alternatives><tex-math id=\"M1307\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1308\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq626\"><alternatives><tex-math id=\"M1309\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1310\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ30\"><alternatives><tex-math id=\"M1311\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} U_{k,F}{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{0\\}\\cup \\bigcup _{n\\in F\\cap F_s}A(n,k) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1312\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:munder><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>F</mml:mi><mml:mo>∩</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:munder><mml:mi>A</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq627\"><alternatives><tex-math id=\"M1313\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y=X/I$$\\end{document}</tex-math><mml:math id=\"M1314\"><mml:mrow><mml:mi>Y</mml:mi><mml:mo>=</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq628\"><alternatives><tex-math id=\"M1315\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau $$\\end{document}</tex-math><mml:math id=\"M1316\"><mml:mi>τ</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ31\"><alternatives><tex-math id=\"M1317\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\big \\{\\{y\\}:y\\in Y\\setminus \\{0\\}\\big \\}\\cup \\{U_{k,F}:k\\in {\\mathbb {N}},\\;F\\in {\\mathcal {F}}\\}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1318\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">{</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"0.277778em\"/><mml:mi>F</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq629\"><alternatives><tex-math id=\"M1319\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y,\\tau )$$\\end{document}</tex-math><mml:math id=\"M1320\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq630\"><alternatives><tex-math id=\"M1321\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau $$\\end{document}</tex-math><mml:math id=\"M1322\"><mml:mi>τ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq631\"><alternatives><tex-math id=\"M1323\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in Y\\setminus \\{0\\}$$\\end{document}</tex-math><mml:math id=\"M1324\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq632\"><alternatives><tex-math id=\"M1325\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1326\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq633\"><alternatives><tex-math id=\"M1327\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1328\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq634\"><alternatives><tex-math id=\"M1329\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\notin U_{k,F}$$\\end{document}</tex-math><mml:math id=\"M1330\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∉</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq635\"><alternatives><tex-math id=\"M1331\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\notin \\bigcup _{n\\in F_s}H_{e_n}$$\\end{document}</tex-math><mml:math id=\"M1332\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∉</mml:mo><mml:msub><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq636\"><alternatives><tex-math id=\"M1333\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\notin U_{1,F_s}$$\\end{document}</tex-math><mml:math id=\"M1334\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∉</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq637\"><alternatives><tex-math id=\"M1335\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in H_{e_n}$$\\end{document}</tex-math><mml:math id=\"M1336\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq638\"><alternatives><tex-math id=\"M1337\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in F_s$$\\end{document}</tex-math><mml:math id=\"M1338\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq639\"><alternatives><tex-math id=\"M1339\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\notin U_{1,F_s\\setminus \\{n\\}}$$\\end{document}</tex-math><mml:math id=\"M1340\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∉</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq640\"><alternatives><tex-math id=\"M1341\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau $$\\end{document}</tex-math><mml:math id=\"M1342\"><mml:mi>τ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq641\"><alternatives><tex-math id=\"M1343\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y,\\tau )$$\\end{document}</tex-math><mml:math id=\"M1344\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq642\"><alternatives><tex-math id=\"M1345\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau $$\\end{document}</tex-math><mml:math id=\"M1346\"><mml:mi>τ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq643\"><alternatives><tex-math id=\"M1347\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y,\\tau )$$\\end{document}</tex-math><mml:math id=\"M1348\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq644\"><alternatives><tex-math id=\"M1349\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y,y'\\in Y$$\\end{document}</tex-math><mml:math id=\"M1350\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq645\"><alternatives><tex-math id=\"M1351\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_{yy'}\\in \\tau $$\\end{document}</tex-math><mml:math id=\"M1352\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi>y</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq646\"><alternatives><tex-math id=\"M1353\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yy'\\in Y$$\\end{document}</tex-math><mml:math id=\"M1354\"><mml:mrow><mml:mi>y</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq647\"><alternatives><tex-math id=\"M1355\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_y,O_{y'}\\in \\tau $$\\end{document}</tex-math><mml:math id=\"M1356\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub><mml:mo>∈</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq648\"><alternatives><tex-math id=\"M1357\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y,y'$$\\end{document}</tex-math><mml:math id=\"M1358\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq649\"><alternatives><tex-math id=\"M1359\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_yO_{y'}\\subseteq O_{yy'}$$\\end{document}</tex-math><mml:math id=\"M1360\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:msub><mml:mi>O</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub><mml:mo>⊆</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi>y</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq650\"><alternatives><tex-math id=\"M1361\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y,y'\\in Y{\\setminus } \\{0\\}$$\\end{document}</tex-math><mml:math id=\"M1362\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq651\"><alternatives><tex-math id=\"M1363\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_{y}=\\{y\\}$$\\end{document}</tex-math><mml:math id=\"M1364\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq652\"><alternatives><tex-math id=\"M1365\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_{y'}=\\{y'\\}$$\\end{document}</tex-math><mml:math id=\"M1366\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq653\"><alternatives><tex-math id=\"M1367\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_yO_{y'}=\\{yy'\\}\\subseteq O_{yy'}$$\\end{document}</tex-math><mml:math id=\"M1368\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:msub><mml:mi>O</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi>y</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq654\"><alternatives><tex-math id=\"M1369\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\ne 0$$\\end{document}</tex-math><mml:math id=\"M1370\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq655\"><alternatives><tex-math id=\"M1371\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y'=0$$\\end{document}</tex-math><mml:math id=\"M1372\"><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq656\"><alternatives><tex-math id=\"M1373\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y=0$$\\end{document}</tex-math><mml:math id=\"M1374\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq657\"><alternatives><tex-math id=\"M1375\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y'\\ne 0$$\\end{document}</tex-math><mml:math id=\"M1376\"><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq658\"><alternatives><tex-math id=\"M1377\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y=0=y'$$\\end{document}</tex-math><mml:math id=\"M1378\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>=</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq659\"><alternatives><tex-math id=\"M1379\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yy'=0$$\\end{document}</tex-math><mml:math id=\"M1380\"><mml:mrow><mml:mi>y</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq660\"><alternatives><tex-math id=\"M1381\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1382\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq661\"><alternatives><tex-math id=\"M1383\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1384\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq662\"><alternatives><tex-math id=\"M1385\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\subseteq F_s$$\\end{document}</tex-math><mml:math id=\"M1386\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>⊆</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq663\"><alternatives><tex-math id=\"M1387\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$U_{k,F}\\subseteq O_{yy'}$$\\end{document}</tex-math><mml:math id=\"M1388\"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub><mml:mo>⊆</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi>y</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq664\"><alternatives><tex-math id=\"M1389\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\ne 0$$\\end{document}</tex-math><mml:math id=\"M1390\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq665\"><alternatives><tex-math id=\"M1391\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y'=0$$\\end{document}</tex-math><mml:math id=\"M1392\"><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq666\"><alternatives><tex-math id=\"M1393\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in C$$\\end{document}</tex-math><mml:math id=\"M1394\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ32\"><alternatives><tex-math id=\"M1395\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} yU_{k,F}=\\{0\\}\\cup \\bigcup _{n\\in F}yA(n,k){} &amp; {} =\\{0\\}\\cup \\bigcup _{n\\in F}\\bigcup _{2\\le p\\le \\lfloor 2^n/k\\rfloor }\\\\ yz_n^pC{} &amp; {} = \\{0\\}\\cup \\bigcup _{n\\in F}\\bigcup _{2\\le p\\le \\lfloor 2^n/k\\rfloor }z_n^pyC\\subseteq \\{0\\}\\cup \\bigcup _{n\\in F}\\bigcup _{2\\le p\\le \\lfloor 2^n/k\\rfloor }\\\\ z_n^pC{} &amp; {} =U_{k,F}\\subseteq O_{yy'}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1396\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:munder><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:munder><mml:mi>y</mml:mi><mml:mi>A</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow/></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:munder><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:munder><mml:munder><mml:mo>⋃</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>≤</mml:mo><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>k</mml:mi><mml:mo>⌋</mml:mo></mml:mrow></mml:munder></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:mi>y</mml:mi><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:msubsup><mml:mi>C</mml:mi><mml:mrow/></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:munder><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:munder><mml:munder><mml:mo>⋃</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>≤</mml:mo><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>k</mml:mi><mml:mo>⌋</mml:mo></mml:mrow></mml:munder><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:msubsup><mml:mi>y</mml:mi><mml:mi>C</mml:mi><mml:mo>⊆</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:munder><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:munder><mml:munder><mml:mo>⋃</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>≤</mml:mo><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>k</mml:mi><mml:mo>⌋</mml:mo></mml:mrow></mml:munder></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:msubsup><mml:mi>C</mml:mi><mml:mrow/></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mo>=</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub><mml:mo>⊆</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi>y</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq667\"><alternatives><tex-math id=\"M1397\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_y=\\{y\\}$$\\end{document}</tex-math><mml:math id=\"M1398\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq668\"><alternatives><tex-math id=\"M1399\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_{y'}=U_{k,F}$$\\end{document}</tex-math><mml:math id=\"M1400\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq669\"><alternatives><tex-math id=\"M1401\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\notin C$$\\end{document}</tex-math><mml:math id=\"M1402\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∉</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq670\"><alternatives><tex-math id=\"M1403\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q(y)\\notin Q$$\\end{document}</tex-math><mml:math id=\"M1404\"><mml:mrow><mml:mi>q</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∉</mml:mo><mml:mi>Q</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq671\"><alternatives><tex-math id=\"M1405\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{n\\in {\\mathbb {N}}:q(e_n)\\le q(y)\\}$$\\end{document}</tex-math><mml:math id=\"M1406\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi><mml:mo>:</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq672\"><alternatives><tex-math id=\"M1407\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1408\"><mml:mi mathvariant=\"script\">F</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq673\"><alternatives><tex-math id=\"M1409\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$G{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{n\\in F_s:q(e_n)\\not \\le q(y)\\}$$\\end{document}</tex-math><mml:math id=\"M1410\"><mml:mrow><mml:mi>G</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≰</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq674\"><alternatives><tex-math id=\"M1411\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1412\"><mml:mi mathvariant=\"script\">F</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq675\"><alternatives><tex-math id=\"M1413\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in G$$\\end{document}</tex-math><mml:math id=\"M1414\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>G</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq676\"><alternatives><tex-math id=\"M1415\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1416\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq677\"><alternatives><tex-math id=\"M1417\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q(yz_n^p)=q(y)q(e_n)\\ne q(e_n)$$\\end{document}</tex-math><mml:math id=\"M1418\"><mml:mrow><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≠</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq678\"><alternatives><tex-math id=\"M1419\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yz_n^p\\in I$$\\end{document}</tex-math><mml:math id=\"M1420\"><mml:mrow><mml:mi>y</mml:mi><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:msubsup><mml:mo>∈</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq679\"><alternatives><tex-math id=\"M1421\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yU_{1,G}=\\{0\\}$$\\end{document}</tex-math><mml:math id=\"M1422\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>G</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq680\"><alternatives><tex-math id=\"M1423\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_y=\\{y\\}$$\\end{document}</tex-math><mml:math id=\"M1424\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq681\"><alternatives><tex-math id=\"M1425\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_{y'}=U_{1,G}$$\\end{document}</tex-math><mml:math id=\"M1426\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>G</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq682\"><alternatives><tex-math id=\"M1427\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y=0$$\\end{document}</tex-math><mml:math id=\"M1428\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq683\"><alternatives><tex-math id=\"M1429\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y'\\ne 0$$\\end{document}</tex-math><mml:math id=\"M1430\"><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq684\"><alternatives><tex-math id=\"M1431\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y=0=y'$$\\end{document}</tex-math><mml:math id=\"M1432\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>=</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq685\"><alternatives><tex-math id=\"M1433\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_y=O_{y'}=U_{4k,F}$$\\end{document}</tex-math><mml:math id=\"M1434\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq686\"><alternatives><tex-math id=\"M1435\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_yO_{y'}\\subseteq U_{k,F}$$\\end{document}</tex-math><mml:math id=\"M1436\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:msub><mml:mi>O</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub><mml:mo>⊆</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq687\"><alternatives><tex-math id=\"M1437\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a,b\\in U_{4k,F}$$\\end{document}</tex-math><mml:math id=\"M1438\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mn>4</mml:mn><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq688\"><alternatives><tex-math id=\"M1439\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$0\\in \\{a,b\\}$$\\end{document}</tex-math><mml:math id=\"M1440\"><mml:mrow><mml:mn>0</mml:mn><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq689\"><alternatives><tex-math id=\"M1441\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{e_n}$$\\end{document}</tex-math><mml:math id=\"M1442\"><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq690\"><alternatives><tex-math id=\"M1443\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$ab=0\\in U_{k,F}$$\\end{document}</tex-math><mml:math id=\"M1444\"><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>∈</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq691\"><alternatives><tex-math id=\"M1445\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1446\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq692\"><alternatives><tex-math id=\"M1447\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a,b\\in A(n,4k)$$\\end{document}</tex-math><mml:math id=\"M1448\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq693\"><alternatives><tex-math id=\"M1449\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a\\in z_n^mC$$\\end{document}</tex-math><mml:math id=\"M1450\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:msubsup><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq694\"><alternatives><tex-math id=\"M1451\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$b=z_n^tC$$\\end{document}</tex-math><mml:math id=\"M1452\"><mml:mrow><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>t</mml:mi></mml:msubsup><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq695\"><alternatives><tex-math id=\"M1453\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$m,t\\in \\{2,\\dots ,\\lfloor 2^n/4k\\rfloor \\}$$\\end{document}</tex-math><mml:math id=\"M1454\"><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>4</mml:mn><mml:mi>k</mml:mi><mml:mo>⌋</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq696\"><alternatives><tex-math id=\"M1455\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$2\\le \\lfloor 2^n/4k\\rfloor \\le 2^n/4k$$\\end{document}</tex-math><mml:math id=\"M1456\"><mml:mrow><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>4</mml:mn><mml:mi>k</mml:mi><mml:mo>⌋</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>4</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ33\"><alternatives><tex-math id=\"M1457\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} 2\\le m+t\\le 2\\lfloor 2^n/4k\\rfloor \\le 2^n/2k=2^n/k-2^n/2k\\le 2^n/k-4\\le \\lfloor 2^n/k\\rfloor . \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1458\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mi>t</mml:mi><mml:mo>≤</mml:mo><mml:mn>2</mml:mn><mml:mrow><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>4</mml:mn><mml:mi>k</mml:mi><mml:mo>⌋</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>≤</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>4</mml:mn><mml:mo>≤</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>k</mml:mi><mml:mo>⌋</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq697\"><alternatives><tex-math id=\"M1459\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_n\\in Z(X)$$\\end{document}</tex-math><mml:math id=\"M1460\"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq698\"><alternatives><tex-math id=\"M1461\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$ab\\in z_n^mCz_n^tC\\subseteq z_n^{m+t}C\\subseteq A(n,k)\\subseteq U_{k,F}$$\\end{document}</tex-math><mml:math id=\"M1462\"><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mo>∈</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:msubsup><mml:mi>C</mml:mi><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>t</mml:mi></mml:msubsup><mml:mi>C</mml:mi><mml:mo>⊆</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msubsup><mml:mi>C</mml:mi><mml:mo>⊆</mml:mo><mml:mi>A</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq699\"><alternatives><tex-math id=\"M1463\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M1464\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq700\"><alternatives><tex-math id=\"M1465\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y,\\tau )$$\\end{document}</tex-math><mml:math id=\"M1466\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq701\"><alternatives><tex-math id=\"M1467\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1468\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq702\"><alternatives><tex-math id=\"M1469\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1470\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq703\"><alternatives><tex-math id=\"M1471\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_{{\\mathcal {F}}}$$\\end{document}</tex-math><mml:math id=\"M1472\"><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq704\"><alternatives><tex-math id=\"M1473\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{z_F:F\\in {\\mathcal {F}}\\}$$\\end{document}</tex-math><mml:math id=\"M1474\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>F</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq705\"><alternatives><tex-math id=\"M1475\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_F{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{z_n:n\\in F\\}$$\\end{document}</tex-math><mml:math id=\"M1476\"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq706\"><alternatives><tex-math id=\"M1477\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1478\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq707\"><alternatives><tex-math id=\"M1479\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in Y$$\\end{document}</tex-math><mml:math id=\"M1480\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq708\"><alternatives><tex-math id=\"M1481\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y z_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1482\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq709\"><alternatives><tex-math id=\"M1483\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{yG: G\\in z_{\\mathcal {F}}\\}$$\\end{document}</tex-math><mml:math id=\"M1484\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:mi>G</mml:mi><mml:mo>:</mml:mo><mml:mi>G</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq710\"><alternatives><tex-math id=\"M1485\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{z_n:n\\in {\\mathbb {N}}\\}\\subseteq Z(X)$$\\end{document}</tex-math><mml:math id=\"M1486\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq711\"><alternatives><tex-math id=\"M1487\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yz_{\\mathcal {F}}=z_{\\mathcal {F}}y$$\\end{document}</tex-math><mml:math id=\"M1488\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq712\"><alternatives><tex-math id=\"M1489\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_{\\mathcal {F}}y$$\\end{document}</tex-math><mml:math id=\"M1490\"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq713\"><alternatives><tex-math id=\"M1491\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{Gy: G\\in z_{\\mathcal {F}}\\}$$\\end{document}</tex-math><mml:math id=\"M1492\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>G</mml:mi><mml:mi>y</mml:mi><mml:mo>:</mml:mo><mml:mi>G</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq714\"><alternatives><tex-math id=\"M1493\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in Y\\setminus C$$\\end{document}</tex-math><mml:math id=\"M1494\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq715\"><alternatives><tex-math id=\"M1495\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y z_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1496\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq716\"><alternatives><tex-math id=\"M1497\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y=0$$\\end{document}</tex-math><mml:math id=\"M1498\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq717\"><alternatives><tex-math id=\"M1499\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in Y{\\setminus }\\{0\\}=X{\\setminus } I$$\\end{document}</tex-math><mml:math id=\"M1500\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>=</mml:mo><mml:mi>X</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq718\"><alternatives><tex-math id=\"M1501\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in Y\\setminus C$$\\end{document}</tex-math><mml:math id=\"M1502\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq719\"><alternatives><tex-math id=\"M1503\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{n\\in {\\mathbb {N}}:q(e_n)\\le q(y)\\}$$\\end{document}</tex-math><mml:math id=\"M1504\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi><mml:mo>:</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq720\"><alternatives><tex-math id=\"M1505\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1506\"><mml:mi mathvariant=\"script\">F</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq721\"><alternatives><tex-math id=\"M1507\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F=\\{n\\in F_s:q(e_n)\\not \\le q(y)\\}$$\\end{document}</tex-math><mml:math id=\"M1508\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≰</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq722\"><alternatives><tex-math id=\"M1509\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1510\"><mml:mi mathvariant=\"script\">F</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq723\"><alternatives><tex-math id=\"M1511\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yz_F\\subseteq I$$\\end{document}</tex-math><mml:math id=\"M1512\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>⊆</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq724\"><alternatives><tex-math id=\"M1513\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yz_F=\\{0\\}$$\\end{document}</tex-math><mml:math id=\"M1514\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq725\"><alternatives><tex-math id=\"M1515\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yz_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1516\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq726\"><alternatives><tex-math id=\"M1517\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M1518\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq727\"><alternatives><tex-math id=\"M1519\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$m\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1520\"><mml:mrow><mml:mi>m</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq728\"><alternatives><tex-math id=\"M1521\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$U_{m,F_s}\\notin y {x}_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1522\"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub><mml:mo>∉</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq729\"><alternatives><tex-math id=\"M1523\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in C$$\\end{document}</tex-math><mml:math id=\"M1524\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq730\"><alternatives><tex-math id=\"M1525\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{e_n\\}_{n\\in F_s}$$\\end{document}</tex-math><mml:math id=\"M1526\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq731\"><alternatives><tex-math id=\"M1527\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$m\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1528\"><mml:mrow><mml:mi>m</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq732\"><alternatives><tex-math id=\"M1529\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in F_s$$\\end{document}</tex-math><mml:math id=\"M1530\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq733\"><alternatives><tex-math id=\"M1531\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y\\in C=\\bigcap _{k\\in F_s}\\frac{H_{e_k}}{e_k}$$\\end{document}</tex-math><mml:math id=\"M1532\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mfrac></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq734\"><alternatives><tex-math id=\"M1533\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(xe_n)(ye_n)^{-1}\\in Z(X)$$\\end{document}</tex-math><mml:math id=\"M1534\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>∈</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq735\"><alternatives><tex-math id=\"M1535\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\big ((xe_n)(ye_n)^{-1}\\big )^m\\in E(X)$$\\end{document}</tex-math><mml:math id=\"M1536\"><mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mi>m</mml:mi></mml:msup><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq736\"><alternatives><tex-math id=\"M1537\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in C$$\\end{document}</tex-math><mml:math id=\"M1538\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq737\"><alternatives><tex-math id=\"M1539\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$U_{m,F_s}$$\\end{document}</tex-math><mml:math id=\"M1540\"><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq738\"><alternatives><tex-math id=\"M1541\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yz_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1542\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq739\"><alternatives><tex-math id=\"M1543\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$U_{m,F_s}$$\\end{document}</tex-math><mml:math id=\"M1544\"><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq740\"><alternatives><tex-math id=\"M1545\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yz_{F_s}$$\\end{document}</tex-math><mml:math id=\"M1546\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq741\"><alternatives><tex-math id=\"M1547\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in F_s$$\\end{document}</tex-math><mml:math id=\"M1548\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq742\"><alternatives><tex-math id=\"M1549\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yz_n\\in U_{m,F_s}$$\\end{document}</tex-math><mml:math id=\"M1550\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq743\"><alternatives><tex-math id=\"M1551\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_ny=yz_n=z_n^pc$$\\end{document}</tex-math><mml:math id=\"M1552\"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:msubsup><mml:mi>c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq744\"><alternatives><tex-math id=\"M1553\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$c\\in C$$\\end{document}</tex-math><mml:math id=\"M1554\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq745\"><alternatives><tex-math id=\"M1555\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1556\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq746\"><alternatives><tex-math id=\"M1557\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$2\\le p\\le \\lfloor 2^n/m\\rfloor $$\\end{document}</tex-math><mml:math id=\"M1558\"><mml:mrow><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>≤</mml:mo><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>m</mml:mi><mml:mo>⌋</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq747\"><alternatives><tex-math id=\"M1559\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$c,y\\in C\\subseteq \\frac{H_{e_n}}{e_n}$$\\end{document}</tex-math><mml:math id=\"M1560\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi><mml:mo>⊆</mml:mo><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mfrac></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq748\"><alternatives><tex-math id=\"M1561\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$ce_n,ye_n\\in H_{e_n}$$\\end{document}</tex-math><mml:math id=\"M1562\"><mml:mrow><mml:mi>c</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq749\"><alternatives><tex-math id=\"M1563\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_ny=z_n^pc$$\\end{document}</tex-math><mml:math id=\"M1564\"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:msubsup><mml:mi>c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq750\"><alternatives><tex-math id=\"M1565\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_nye_n=z_n^pce_n\\in H_{e_n}$$\\end{document}</tex-math><mml:math id=\"M1566\"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mi>y</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:msubsup><mml:mi>c</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq751\"><alternatives><tex-math id=\"M1567\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(ce_n)(ye_n)^{-1}=z_n^{-(p-1)}\\in Z(X)\\cap H_{e_n}$$\\end{document}</tex-math><mml:math id=\"M1568\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>c</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>∈</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq752\"><alternatives><tex-math id=\"M1569\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$((ce_n)(ye_n)^{-1})^m=e_n$$\\end{document}</tex-math><mml:math id=\"M1570\"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>c</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>m</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ34\"><alternatives><tex-math id=\"M1571\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} z_n^m=(z_n^p(ce_n)(ye_n)^{-1})^m=z_n^{pm}((ce_n)(ye_n)^{-1})^m=z_n^{pm}e_n=z_n^{pm}, \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1572\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>c</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>m</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pm</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>c</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>m</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pm</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pm</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq753\"><alternatives><tex-math id=\"M1573\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_n$$\\end{document}</tex-math><mml:math id=\"M1574\"><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq754\"><alternatives><tex-math id=\"M1575\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$m&lt;pm\\le \\lfloor 2^n/m\\rfloor m\\le 2^n$$\\end{document}</tex-math><mml:math id=\"M1576\"><mml:mrow><mml:mi>m</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>p</mml:mi><mml:mi>m</mml:mi><mml:mo>≤</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>m</mml:mi><mml:mo>⌋</mml:mo></mml:mrow><mml:mi>m</mml:mi><mml:mo>≤</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq755\"><alternatives><tex-math id=\"M1577\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M1578\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq756\"><alternatives><tex-math id=\"M1579\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T=Y\\cup \\{yz_{\\mathcal {F}}:y\\in C\\}$$\\end{document}</tex-math><mml:math id=\"M1580\"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mi>Y</mml:mi><mml:mo>∪</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ35\"><alternatives><tex-math id=\"M1581\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} ab= {\\left\\{ \\begin{array}{ll} ab&amp;{}\\hbox { if }a,b\\in Y;\\\\ ayz_{\\mathcal {F}}&amp;{}\\hbox { if } a\\in C\\hbox { and }b=yz_{\\mathcal {F}}\\hbox { for some} y\\in C;\\\\ ybz_{\\mathcal {F}}&amp;{}\\hbox { if } a=yz_{\\mathcal {F}}\\hbox { for some }y\\in C\\hbox { and }b\\in C;\\\\ 0&amp;{} \\text{ in } \\text{ all } \\text{ other } \\text{ cases }. \\end{array}\\right. } \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1582\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mi mathvariant=\"italic\">ab</mml:mi></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>if</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo>;</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mi>a</mml:mi><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>if</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi><mml:mspace width=\"0.333333em\"/><mml:mtext>and</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mspace width=\"0.333333em\"/><mml:mtext>for some</mml:mtext><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi><mml:mo>;</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mi>y</mml:mi><mml:mi>b</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>if</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mspace width=\"0.333333em\"/><mml:mtext>for some</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi><mml:mspace width=\"0.333333em\"/><mml:mtext>and</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mi>b</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi><mml:mo>;</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mn>0</mml:mn></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mspace width=\"0.333333em\"/><mml:mtext>in</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"0.333333em\"/><mml:mtext>all</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"0.333333em\"/><mml:mtext>other</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"0.333333em\"/><mml:mtext>cases</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq757\"><alternatives><tex-math id=\"M1583\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta $$\\end{document}</tex-math><mml:math id=\"M1584\"><mml:mi>θ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq758\"><alternatives><tex-math id=\"M1585\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y,\\tau )$$\\end{document}</tex-math><mml:math id=\"M1586\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq759\"><alternatives><tex-math id=\"M1587\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(T,\\theta )$$\\end{document}</tex-math><mml:math id=\"M1588\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq760\"><alternatives><tex-math id=\"M1589\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yz_{\\mathcal {F}}\\in U\\in \\theta $$\\end{document}</tex-math><mml:math id=\"M1590\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>U</mml:mi><mml:mo>∈</mml:mo><mml:mi>θ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq761\"><alternatives><tex-math id=\"M1591\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in C$$\\end{document}</tex-math><mml:math id=\"M1592\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq762\"><alternatives><tex-math id=\"M1593\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1594\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq763\"><alternatives><tex-math id=\"M1595\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yz_F\\subseteq U$$\\end{document}</tex-math><mml:math id=\"M1596\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>⊆</mml:mo><mml:mi>U</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq764\"><alternatives><tex-math id=\"M1597\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta $$\\end{document}</tex-math><mml:math id=\"M1598\"><mml:mi>θ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq765\"><alternatives><tex-math id=\"M1599\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(T,\\theta )$$\\end{document}</tex-math><mml:math id=\"M1600\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq766\"><alternatives><tex-math id=\"M1601\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$U\\in \\theta $$\\end{document}</tex-math><mml:math id=\"M1602\"><mml:mrow><mml:mi>U</mml:mi><mml:mo>∈</mml:mo><mml:mi>θ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq767\"><alternatives><tex-math id=\"M1603\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$u\\in U$$\\end{document}</tex-math><mml:math id=\"M1604\"><mml:mrow><mml:mi>u</mml:mi><mml:mo>∈</mml:mo><mml:mi>U</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq768\"><alternatives><tex-math id=\"M1605\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(T,\\theta )$$\\end{document}</tex-math><mml:math id=\"M1606\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq769\"><alternatives><tex-math id=\"M1607\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$u\\in V\\subseteq U$$\\end{document}</tex-math><mml:math id=\"M1608\"><mml:mrow><mml:mi>u</mml:mi><mml:mo>∈</mml:mo><mml:mi>V</mml:mi><mml:mo>⊆</mml:mo><mml:mi>U</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq770\"><alternatives><tex-math id=\"M1609\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$u\\in Y\\setminus \\{0\\}$$\\end{document}</tex-math><mml:math id=\"M1610\"><mml:mrow><mml:mi>u</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq771\"><alternatives><tex-math id=\"M1611\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V=\\{u\\}$$\\end{document}</tex-math><mml:math id=\"M1612\"><mml:mrow><mml:mi>V</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq772\"><alternatives><tex-math id=\"M1613\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta $$\\end{document}</tex-math><mml:math id=\"M1614\"><mml:mi>θ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq773\"><alternatives><tex-math id=\"M1615\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V=\\{u\\}$$\\end{document}</tex-math><mml:math id=\"M1616\"><mml:mrow><mml:mi>V</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq774\"><alternatives><tex-math id=\"M1617\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(T,\\theta )$$\\end{document}</tex-math><mml:math id=\"M1618\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq775\"><alternatives><tex-math id=\"M1619\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$u=0$$\\end{document}</tex-math><mml:math id=\"M1620\"><mml:mrow><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq776\"><alternatives><tex-math id=\"M1621\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$m\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1622\"><mml:mrow><mml:mi>m</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq777\"><alternatives><tex-math id=\"M1623\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1624\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq778\"><alternatives><tex-math id=\"M1625\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$U_{m,F}\\subseteq U$$\\end{document}</tex-math><mml:math id=\"M1626\"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub><mml:mo>⊆</mml:mo><mml:mi>U</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq779\"><alternatives><tex-math id=\"M1627\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$U_{m,F}\\notin yz_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1628\"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub><mml:mo>∉</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq780\"><alternatives><tex-math id=\"M1629\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in C$$\\end{document}</tex-math><mml:math id=\"M1630\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq781\"><alternatives><tex-math id=\"M1631\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta $$\\end{document}</tex-math><mml:math id=\"M1632\"><mml:mi>θ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq782\"><alternatives><tex-math id=\"M1633\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V=U_{m,F}$$\\end{document}</tex-math><mml:math id=\"M1634\"><mml:mrow><mml:mi>V</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq783\"><alternatives><tex-math id=\"M1635\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$u=0$$\\end{document}</tex-math><mml:math id=\"M1636\"><mml:mrow><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq784\"><alternatives><tex-math id=\"M1637\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(T,\\theta )$$\\end{document}</tex-math><mml:math id=\"M1638\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq785\"><alternatives><tex-math id=\"M1639\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$u=yz_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1640\"><mml:mrow><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq786\"><alternatives><tex-math id=\"M1641\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in C$$\\end{document}</tex-math><mml:math id=\"M1642\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq787\"><alternatives><tex-math id=\"M1643\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta $$\\end{document}</tex-math><mml:math id=\"M1644\"><mml:mi>θ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq788\"><alternatives><tex-math id=\"M1645\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1646\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq789\"><alternatives><tex-math id=\"M1647\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\subseteq F_s$$\\end{document}</tex-math><mml:math id=\"M1648\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>⊆</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq790\"><alternatives><tex-math id=\"M1649\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yx_F\\subseteq U$$\\end{document}</tex-math><mml:math id=\"M1650\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>⊆</mml:mo><mml:mi>U</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq791\"><alternatives><tex-math id=\"M1651\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$yx_F\\cap U_{m,F_s}=\\emptyset $$\\end{document}</tex-math><mml:math id=\"M1652\"><mml:mrow><mml:mi>y</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>∩</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">∅</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq792\"><alternatives><tex-math id=\"M1653\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$m\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1654\"><mml:mrow><mml:mi>m</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq793\"><alternatives><tex-math id=\"M1655\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta $$\\end{document}</tex-math><mml:math id=\"M1656\"><mml:mi>θ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq794\"><alternatives><tex-math id=\"M1657\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$V{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{yz_{\\mathcal {F}}\\}\\cup yx_F\\subseteq U$$\\end{document}</tex-math><mml:math id=\"M1658\"><mml:mrow><mml:mi>V</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>⊆</mml:mo><mml:mi>U</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq795\"><alternatives><tex-math id=\"M1659\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$u=yz_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1660\"><mml:mrow><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq796\"><alternatives><tex-math id=\"M1661\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(T,\\theta )$$\\end{document}</tex-math><mml:math id=\"M1662\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq797\"><alternatives><tex-math id=\"M1663\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(T,\\theta )$$\\end{document}</tex-math><mml:math id=\"M1664\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq798\"><alternatives><tex-math id=\"M1665\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t\\in T\\setminus V$$\\end{document}</tex-math><mml:math id=\"M1666\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mi>T</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>V</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq799\"><alternatives><tex-math id=\"M1667\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_t\\in \\theta $$\\end{document}</tex-math><mml:math id=\"M1668\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>θ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq800\"><alternatives><tex-math id=\"M1669\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_t\\cap V=\\emptyset $$\\end{document}</tex-math><mml:math id=\"M1670\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>∩</mml:mo><mml:mi>V</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">∅</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq801\"><alternatives><tex-math id=\"M1671\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t\\in Y{\\setminus }\\{0\\}$$\\end{document}</tex-math><mml:math id=\"M1672\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq802\"><alternatives><tex-math id=\"M1673\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_t{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{t\\}\\in \\theta $$\\end{document}</tex-math><mml:math id=\"M1674\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mi>θ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq803\"><alternatives><tex-math id=\"M1675\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t=0$$\\end{document}</tex-math><mml:math id=\"M1676\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq804\"><alternatives><tex-math id=\"M1677\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_t{\\mathop {=}\\limits ^{{\\textsf{def}}}}U_{m,F_s}$$\\end{document}</tex-math><mml:math id=\"M1678\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq805\"><alternatives><tex-math id=\"M1679\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t=0$$\\end{document}</tex-math><mml:math id=\"M1680\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq806\"><alternatives><tex-math id=\"M1681\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t=c z_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1682\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq807\"><alternatives><tex-math id=\"M1683\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$c\\in C$$\\end{document}</tex-math><mml:math id=\"M1684\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq808\"><alternatives><tex-math id=\"M1685\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E=\\{n\\in F:yz_n=c z_n\\}$$\\end{document}</tex-math><mml:math id=\"M1686\"><mml:mrow><mml:mi>E</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>F</mml:mi><mml:mo>:</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq809\"><alternatives><tex-math id=\"M1687\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1688\"><mml:mrow><mml:mi>E</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq810\"><alternatives><tex-math id=\"M1689\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$u=yz_{\\mathcal {F}}=c z_{\\mathcal {F}}=t$$\\end{document}</tex-math><mml:math id=\"M1690\"><mml:mrow><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq811\"><alternatives><tex-math id=\"M1691\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t\\notin V$$\\end{document}</tex-math><mml:math id=\"M1692\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>∉</mml:mo><mml:mi>V</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq812\"><alternatives><tex-math id=\"M1693\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E\\notin {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1694\"><mml:mrow><mml:mi>E</mml:mi><mml:mo>∉</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq813\"><alternatives><tex-math id=\"M1695\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$G{\\mathop {=}\\limits ^{{\\textsf{def}}}}F{\\setminus } E=\\{n\\in F:yz_n\\ne c z_n\\}$$\\end{document}</tex-math><mml:math id=\"M1696\"><mml:mrow><mml:mi>G</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mi>F</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>E</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>F</mml:mi><mml:mo>:</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>≠</mml:mo><mml:mi>c</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq814\"><alternatives><tex-math id=\"M1697\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1698\"><mml:mi mathvariant=\"script\">F</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq815\"><alternatives><tex-math id=\"M1699\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_t=\\{t\\}\\cup c z_{G}$$\\end{document}</tex-math><mml:math id=\"M1700\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:mi>c</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>G</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq816\"><alternatives><tex-math id=\"M1701\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t=c z_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1702\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq817\"><alternatives><tex-math id=\"M1703\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_t\\cap V=\\emptyset $$\\end{document}</tex-math><mml:math id=\"M1704\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>∩</mml:mo><mml:mi>V</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">∅</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq818\"><alternatives><tex-math id=\"M1705\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta $$\\end{document}</tex-math><mml:math id=\"M1706\"><mml:mi>θ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq819\"><alternatives><tex-math id=\"M1707\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T_1$$\\end{document}</tex-math><mml:math id=\"M1708\"><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq820\"><alternatives><tex-math id=\"M1709\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M1710\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq821\"><alternatives><tex-math id=\"M1711\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(T,\\theta )$$\\end{document}</tex-math><mml:math id=\"M1712\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq822\"><alternatives><tex-math id=\"M1713\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a,b\\in T$$\\end{document}</tex-math><mml:math id=\"M1714\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq823\"><alternatives><tex-math id=\"M1715\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_{ab}\\in \\theta $$\\end{document}</tex-math><mml:math id=\"M1716\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ab</mml:mi></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:mi>θ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq824\"><alternatives><tex-math id=\"M1717\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_a,O_b\\in \\theta $$\\end{document}</tex-math><mml:math id=\"M1718\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>θ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq825\"><alternatives><tex-math id=\"M1719\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_aO_b\\subseteq O_{ab}$$\\end{document}</tex-math><mml:math id=\"M1720\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>O</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>⊆</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ab</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq826\"><alternatives><tex-math id=\"M1721\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a,b\\in Y$$\\end{document}</tex-math><mml:math id=\"M1722\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq827\"><alternatives><tex-math id=\"M1723\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y,\\tau )$$\\end{document}</tex-math><mml:math id=\"M1724\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq828\"><alternatives><tex-math id=\"M1725\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a\\in Y$$\\end{document}</tex-math><mml:math id=\"M1726\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq829\"><alternatives><tex-math id=\"M1727\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$b=yz_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1728\"><mml:mrow><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq830\"><alternatives><tex-math id=\"M1729\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in C$$\\end{document}</tex-math><mml:math id=\"M1730\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq831\"><alternatives><tex-math id=\"M1731\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a\\in C$$\\end{document}</tex-math><mml:math id=\"M1732\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq832\"><alternatives><tex-math id=\"M1733\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1734\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq833\"><alternatives><tex-math id=\"M1735\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$ayx_F\\subseteq O_{ab}$$\\end{document}</tex-math><mml:math id=\"M1736\"><mml:mrow><mml:mi>a</mml:mi><mml:mi>y</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>⊆</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ab</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq834\"><alternatives><tex-math id=\"M1737\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_a=\\{a\\}$$\\end{document}</tex-math><mml:math id=\"M1738\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq835\"><alternatives><tex-math id=\"M1739\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_b=yx_F\\cup \\{yx_{{\\mathcal {F}}}\\}$$\\end{document}</tex-math><mml:math id=\"M1740\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>∪</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq836\"><alternatives><tex-math id=\"M1741\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_aO_b=ayx_F\\cup \\{ayx_{{\\mathcal {F}}}\\}\\subseteq O_{ab}$$\\end{document}</tex-math><mml:math id=\"M1742\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>O</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mi>y</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>∪</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>a</mml:mi><mml:mi>y</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ab</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq837\"><alternatives><tex-math id=\"M1743\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a\\in Y\\setminus (\\{0\\}\\cup C)=X\\setminus (I\\cup C)$$\\end{document}</tex-math><mml:math id=\"M1744\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>∪</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>X</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>I</mml:mi><mml:mo>∪</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq838\"><alternatives><tex-math id=\"M1745\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a\\notin C$$\\end{document}</tex-math><mml:math id=\"M1746\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>∉</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq839\"><alternatives><tex-math id=\"M1747\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{n\\in \\omega :q(e_n)\\le q(a)\\}$$\\end{document}</tex-math><mml:math id=\"M1748\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi><mml:mo>:</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq840\"><alternatives><tex-math id=\"M1749\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1750\"><mml:mi mathvariant=\"script\">F</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq841\"><alternatives><tex-math id=\"M1751\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{n\\in F_{s}:q(e_n)\\not \\le q(a)\\}$$\\end{document}</tex-math><mml:math id=\"M1752\"><mml:mrow><mml:mi>F</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≰</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq842\"><alternatives><tex-math id=\"M1753\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1754\"><mml:mi mathvariant=\"script\">F</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq843\"><alternatives><tex-math id=\"M1755\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_a=\\{a\\}$$\\end{document}</tex-math><mml:math id=\"M1756\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq844\"><alternatives><tex-math id=\"M1757\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_b=yz_F\\cup \\{yz_{\\mathcal {F}}\\}$$\\end{document}</tex-math><mml:math id=\"M1758\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>∪</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq845\"><alternatives><tex-math id=\"M1759\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_aO_b=\\{0\\}=\\{ab\\}\\subseteq O_{ab}$$\\end{document}</tex-math><mml:math id=\"M1760\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>O</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ab</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq846\"><alternatives><tex-math id=\"M1761\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a=0$$\\end{document}</tex-math><mml:math id=\"M1762\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq847\"><alternatives><tex-math id=\"M1763\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$ab=0$$\\end{document}</tex-math><mml:math id=\"M1764\"><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq848\"><alternatives><tex-math id=\"M1765\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1766\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq849\"><alternatives><tex-math id=\"M1767\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1768\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq850\"><alternatives><tex-math id=\"M1769\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$U_{k,F}\\subseteq O_{ab}$$\\end{document}</tex-math><mml:math id=\"M1770\"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub><mml:mo>⊆</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ab</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq851\"><alternatives><tex-math id=\"M1771\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_a=U_{2k,F}$$\\end{document}</tex-math><mml:math id=\"M1772\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq852\"><alternatives><tex-math id=\"M1773\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_b=yz_{F}\\cup \\{yz_{{\\mathcal {F}}}\\}$$\\end{document}</tex-math><mml:math id=\"M1774\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>F</mml:mi></mml:msub><mml:mo>∪</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq853\"><alternatives><tex-math id=\"M1775\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_aO_b\\subseteq O_{ab}$$\\end{document}</tex-math><mml:math id=\"M1776\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>O</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>⊆</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ab</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq854\"><alternatives><tex-math id=\"M1777\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a'\\in O_a$$\\end{document}</tex-math><mml:math id=\"M1778\"><mml:mrow><mml:msup><mml:mi>a</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>∈</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq855\"><alternatives><tex-math id=\"M1779\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$b'\\in O_b$$\\end{document}</tex-math><mml:math id=\"M1780\"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>∈</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mi>b</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq856\"><alternatives><tex-math id=\"M1781\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a'b'\\in O_{ab}$$\\end{document}</tex-math><mml:math id=\"M1782\"><mml:mrow><mml:msup><mml:mi>a</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msup><mml:mi>b</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>∈</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ab</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq857\"><alternatives><tex-math id=\"M1783\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a'=0$$\\end{document}</tex-math><mml:math id=\"M1784\"><mml:mrow><mml:msup><mml:mi>a</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq858\"><alternatives><tex-math id=\"M1785\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a'\\ne 0$$\\end{document}</tex-math><mml:math id=\"M1786\"><mml:mrow><mml:msup><mml:mi>a</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq859\"><alternatives><tex-math id=\"M1787\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a'=z_n^py'$$\\end{document}</tex-math><mml:math id=\"M1788\"><mml:mrow><mml:msup><mml:mi>a</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:msubsup><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq860\"><alternatives><tex-math id=\"M1789\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in F$$\\end{document}</tex-math><mml:math id=\"M1790\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq861\"><alternatives><tex-math id=\"M1791\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$2\\le p\\le \\lfloor 2^n/2k\\rfloor $$\\end{document}</tex-math><mml:math id=\"M1792\"><mml:mrow><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>≤</mml:mo><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>⌋</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq862\"><alternatives><tex-math id=\"M1793\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y'\\in C$$\\end{document}</tex-math><mml:math id=\"M1794\"><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq863\"><alternatives><tex-math id=\"M1795\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$b'=yz_{{\\mathcal {F}}}$$\\end{document}</tex-math><mml:math id=\"M1796\"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq864\"><alternatives><tex-math id=\"M1797\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a'b'=0\\in O_{ab}$$\\end{document}</tex-math><mml:math id=\"M1798\"><mml:mrow><mml:msup><mml:mi>a</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msup><mml:mi>b</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>∈</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ab</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq865\"><alternatives><tex-math id=\"M1799\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$b'=yz_{n'}$$\\end{document}</tex-math><mml:math id=\"M1800\"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq866\"><alternatives><tex-math id=\"M1801\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n'\\in F$$\\end{document}</tex-math><mml:math id=\"M1802\"><mml:mrow><mml:msup><mml:mi>n</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>∈</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq867\"><alternatives><tex-math id=\"M1803\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\ne n'$$\\end{document}</tex-math><mml:math id=\"M1804\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>≠</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq868\"><alternatives><tex-math id=\"M1805\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a'b'=z_n^py'yz_{n'}\\in I$$\\end{document}</tex-math><mml:math id=\"M1806\"><mml:mrow><mml:msup><mml:mi>a</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msup><mml:mi>b</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:msubsup><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub><mml:mo>∈</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq869\"><alternatives><tex-math id=\"M1807\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a'b'=0=ab\\in O_{ab}$$\\end{document}</tex-math><mml:math id=\"M1808\"><mml:mrow><mml:msup><mml:mi>a</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msup><mml:mi>b</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ab</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq870\"><alternatives><tex-math id=\"M1809\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n=n'$$\\end{document}</tex-math><mml:math id=\"M1810\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq871\"><alternatives><tex-math id=\"M1811\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a'b'=z_n^py'yz_n=z_n^{p+1}y'y$$\\end{document}</tex-math><mml:math id=\"M1812\"><mml:mrow><mml:msup><mml:mi>a</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msup><mml:mi>b</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:msubsup><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ36\"><alternatives><tex-math id=\"M1813\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} 2\\le p+1\\le \\lfloor 2^n/2k\\rfloor +1\\le 2^n/2k+1=2^n/k-2^n/2k+1\\le 2^n/k-2+1\\le \\lfloor 2^n/k\\rfloor \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1814\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>⌋</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>=</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>k</mml:mi><mml:mo>⌋</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq872\"><alternatives><tex-math id=\"M1815\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$2\\le p\\le \\lfloor 2^n/2k\\rfloor \\le 2^n/2k$$\\end{document}</tex-math><mml:math id=\"M1816\"><mml:mrow><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>≤</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>⌋</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq873\"><alternatives><tex-math id=\"M1817\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a=yz_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1818\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq874\"><alternatives><tex-math id=\"M1819\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\in C$$\\end{document}</tex-math><mml:math id=\"M1820\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq875\"><alternatives><tex-math id=\"M1821\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$b\\in Y$$\\end{document}</tex-math><mml:math id=\"M1822\"><mml:mrow><mml:mi>b</mml:mi><mml:mo>∈</mml:mo><mml:mi>Y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq876\"><alternatives><tex-math id=\"M1823\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a=yz_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1824\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq877\"><alternatives><tex-math id=\"M1825\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$b=y'z_{\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1826\"><mml:mrow><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq878\"><alternatives><tex-math id=\"M1827\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y,y'\\in C$$\\end{document}</tex-math><mml:math id=\"M1828\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq879\"><alternatives><tex-math id=\"M1829\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$ab=0$$\\end{document}</tex-math><mml:math id=\"M1830\"><mml:mrow><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq880\"><alternatives><tex-math id=\"M1831\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1832\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq881\"><alternatives><tex-math id=\"M1833\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\in {\\mathcal {F}}$$\\end{document}</tex-math><mml:math id=\"M1834\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq882\"><alternatives><tex-math id=\"M1835\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$U_{k,F}\\subseteq O_{ab}$$\\end{document}</tex-math><mml:math id=\"M1836\"><mml:mrow><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub><mml:mo>⊆</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ab</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq883\"><alternatives><tex-math id=\"M1837\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E=\\{n\\in F\\cap F_s:2\\le \\lfloor 2^n/k\\rfloor \\}$$\\end{document}</tex-math><mml:math id=\"M1838\"><mml:mrow><mml:mi>E</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>F</mml:mi><mml:mo>∩</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mn>2</mml:mn><mml:mo>≤</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>k</mml:mi><mml:mo>⌋</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq884\"><alternatives><tex-math id=\"M1839\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_a=yz_E\\cup \\{yz_{{\\mathcal {F}}}\\}$$\\end{document}</tex-math><mml:math id=\"M1840\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi>E</mml:mi></mml:msub><mml:mo>∪</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>y</mml:mi><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq885\"><alternatives><tex-math id=\"M1841\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_b=y'z_E\\cup \\{y'z_{\\mathcal {F}}\\}$$\\end{document}</tex-math><mml:math id=\"M1842\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mi>z</mml:mi><mml:mi>E</mml:mi></mml:msub><mml:mo>∪</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mi>z</mml:mi><mml:mi mathvariant=\"script\">F</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq886\"><alternatives><tex-math id=\"M1843\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$O_aO_b\\subseteq U_{k,F}\\subseteq O_{ab}$$\\end{document}</tex-math><mml:math id=\"M1844\"><mml:mrow><mml:msub><mml:mi>O</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>O</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>⊆</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:msub><mml:mo>⊆</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ab</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq887\"><alternatives><tex-math id=\"M1845\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(T,\\theta )$$\\end{document}</tex-math><mml:math id=\"M1846\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq888\"><alternatives><tex-math id=\"M1847\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(T,\\theta )$$\\end{document}</tex-math><mml:math id=\"M1848\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq889\"><alternatives><tex-math id=\"M1849\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y,\\tau )$$\\end{document}</tex-math><mml:math id=\"M1850\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq890\"><alternatives><tex-math id=\"M1851\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1852\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq891\"><alternatives><tex-math id=\"M1853\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y,\\tau )$$\\end{document}</tex-math><mml:math id=\"M1854\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq892\"><alternatives><tex-math id=\"M1855\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(3)\\Rightarrow (2)$$\\end{document}</tex-math><mml:math id=\"M1856\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq893\"><alternatives><tex-math id=\"M1857\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M1858\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq894\"><alternatives><tex-math id=\"M1859\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1860\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq895\"><alternatives><tex-math id=\"M1861\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$A\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M1862\"><mml:mrow><mml:mi>A</mml:mi><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq896\"><alternatives><tex-math id=\"M1863\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\bigcap _{e\\in A}\\frac{H_e}{e}$$\\end{document}</tex-math><mml:math id=\"M1864\"><mml:mrow><mml:mi>C</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:msub><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mi>e</mml:mi></mml:mfrac></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq897\"><alternatives><tex-math id=\"M1865\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\prod _{e\\in A}H_e$$\\end{document}</tex-math><mml:math id=\"M1866\"><mml:mrow><mml:mi>H</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:msub><mml:mo>∏</mml:mo><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq898\"><alternatives><tex-math id=\"M1867\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h:C\\rightarrow H$$\\end{document}</tex-math><mml:math id=\"M1868\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>H</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq899\"><alternatives><tex-math id=\"M1869\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h:x\\mapsto (xe)_{e\\in A}$$\\end{document}</tex-math><mml:math id=\"M1870\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>↦</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq900\"><alternatives><tex-math id=\"M1871\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1872\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq901\"><alternatives><tex-math id=\"M1873\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M1874\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq902\"><alternatives><tex-math id=\"M1875\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M1876\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq903\"><alternatives><tex-math id=\"M1877\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1878\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq904\"><alternatives><tex-math id=\"M1879\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1880\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq905\"><alternatives><tex-math id=\"M1881\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|h[C]|\\le \\textrm{cov}({\\mathcal {M}})$$\\end{document}</tex-math><mml:math id=\"M1882\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>≤</mml:mo><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq906\"><alternatives><tex-math id=\"M1883\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M1884\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq907\"><alternatives><tex-math id=\"M1885\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Ce\\subseteq H_e$$\\end{document}</tex-math><mml:math id=\"M1886\"><mml:mrow><mml:mi>C</mml:mi><mml:mi>e</mml:mi><mml:mo>⊆</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq908\"><alternatives><tex-math id=\"M1887\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1888\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq909\"><alternatives><tex-math id=\"M1889\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|h[C]|\\le {\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M1890\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>≤</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq910\"><alternatives><tex-math id=\"M1891\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M1892\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq911\"><alternatives><tex-math id=\"M1893\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M1894\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq912\"><alternatives><tex-math id=\"M1895\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\lambda :A\\rightarrow \\omega $$\\end{document}</tex-math><mml:math id=\"M1896\"><mml:mrow><mml:mi>λ</mml:mi><mml:mo>:</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq913\"><alternatives><tex-math id=\"M1897\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\prod _{e\\in A}H_e$$\\end{document}</tex-math><mml:math id=\"M1898\"><mml:mrow><mml:mi>H</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:msub><mml:mo>∏</mml:mo><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq914\"><alternatives><tex-math id=\"M1899\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M1900\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq915\"><alternatives><tex-math id=\"M1901\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M1902\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq916\"><alternatives><tex-math id=\"M1903\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho :H\\times H\\rightarrow {\\mathbb {R}}$$\\end{document}</tex-math><mml:math id=\"M1904\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo>:</mml:mo><mml:mi>H</mml:mi><mml:mo>×</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ37\"><alternatives><tex-math id=\"M1905\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\rho ((x_a)_{a\\in A},(y_a)_{a\\in A})=\\max \\left( \\{0\\}\\cup \\left\\{ \\tfrac{1}{2^{\\lambda (a)}}:a\\in A,\\;x_a\\ne y_a\\right\\} \\right) . \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1906\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mo movablelimits=\"true\">max</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:mfenced close=\"}\" open=\"{\"><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:mn>1</mml:mn><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mi>λ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:mfrac></mml:mstyle><mml:mo>:</mml:mo><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"0.277778em\"/><mml:msub><mml:mi>x</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>≠</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mfenced></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq917\"><alternatives><tex-math id=\"M1907\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M1908\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ38\"><alternatives><tex-math id=\"M1909\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\textstyle h_e:C\\rightarrow H_e,\\quad h:x\\mapsto xe=ex. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1910\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>↦</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:mi>x</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:mstyle></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq918\"><alternatives><tex-math id=\"M1911\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_e$$\\end{document}</tex-math><mml:math id=\"M1912\"><mml:msub><mml:mi>h</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ39\"><alternatives><tex-math id=\"M1913\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\textstyle h:C\\rightarrow H,\\quad h:x\\mapsto (h_e(x))_{e\\in A}=(xe)_{e\\in A}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M1914\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>H</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>↦</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:mstyle></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq919\"><alternatives><tex-math id=\"M1915\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a\\in A$$\\end{document}</tex-math><mml:math id=\"M1916\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq920\"><alternatives><tex-math id=\"M1917\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{pr}_a:H\\rightarrow H_a$$\\end{document}</tex-math><mml:math id=\"M1918\"><mml:mrow><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>a</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq921\"><alternatives><tex-math id=\"M1919\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{pr}_a:(x_e)_{e\\in A}\\mapsto x_a$$\\end{document}</tex-math><mml:math id=\"M1920\"><mml:mrow><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>a</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mo>↦</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq922\"><alternatives><tex-math id=\"M1921\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{pr}_a\\circ h=h_a$$\\end{document}</tex-math><mml:math id=\"M1922\"><mml:mrow><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>a</mml:mi></mml:msub><mml:mo>∘</mml:mo><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq923\"><alternatives><tex-math id=\"M1923\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a\\in A$$\\end{document}</tex-math><mml:math id=\"M1924\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq924\"><alternatives><tex-math id=\"M1925\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1926\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq925\"><alternatives><tex-math id=\"M1927\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M1928\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq926\"><alternatives><tex-math id=\"M1929\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M1930\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq927\"><alternatives><tex-math id=\"M1931\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h[C]\\subseteq \\prod _{e\\in A}Ce$$\\end{document}</tex-math><mml:math id=\"M1932\"><mml:mrow><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:msub><mml:mo>∏</mml:mo><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mi>C</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq928\"><alternatives><tex-math id=\"M1933\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1934\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq929\"><alternatives><tex-math id=\"M1935\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1936\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq930\"><alternatives><tex-math id=\"M1937\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h[C]\\subseteq H$$\\end{document}</tex-math><mml:math id=\"M1938\"><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>⊆</mml:mo><mml:mi>H</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq931\"><alternatives><tex-math id=\"M1939\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M1940\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq932\"><alternatives><tex-math id=\"M1941\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1942\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq933\"><alternatives><tex-math id=\"M1943\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z^n\\in E(H)$$\\end{document}</tex-math><mml:math id=\"M1944\"><mml:mrow><mml:msup><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq934\"><alternatives><tex-math id=\"M1945\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z\\in h[C]$$\\end{document}</tex-math><mml:math id=\"M1946\"><mml:mrow><mml:mi>z</mml:mi><mml:mo>∈</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq935\"><alternatives><tex-math id=\"M1947\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y\\in C$$\\end{document}</tex-math><mml:math id=\"M1948\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq936\"><alternatives><tex-math id=\"M1949\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M1950\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq937\"><alternatives><tex-math id=\"M1951\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$((xe)(ye)^{-1})^n=\\textrm{pr}_e((h(x)h(y)^{-1})^n)=e$$\\end{document}</tex-math><mml:math id=\"M1952\"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>e</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>h</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>e</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq938\"><alternatives><tex-math id=\"M1953\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1954\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq939\"><alternatives><tex-math id=\"M1955\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\subseteq h[C]$$\\end{document}</tex-math><mml:math id=\"M1956\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>⊆</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq940\"><alternatives><tex-math id=\"M1957\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y\\in h[C]$$\\end{document}</tex-math><mml:math id=\"M1958\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq941\"><alternatives><tex-math id=\"M1959\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f:h[C]\\rightarrow h[C]$$\\end{document}</tex-math><mml:math id=\"M1960\"><mml:mrow><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq942\"><alternatives><tex-math id=\"M1961\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\le n$$\\end{document}</tex-math><mml:math id=\"M1962\"><mml:mrow><mml:mo>≤</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq943\"><alternatives><tex-math id=\"M1963\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{f(x),f(y)\\}\\subseteq F$$\\end{document}</tex-math><mml:math id=\"M1964\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>⊆</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq944\"><alternatives><tex-math id=\"M1965\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M1966\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq945\"><alternatives><tex-math id=\"M1967\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x,y\\in C$$\\end{document}</tex-math><mml:math id=\"M1968\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq946\"><alternatives><tex-math id=\"M1969\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z{\\mathop {=}\\limits ^{{\\textsf{def}}}}(xe)(ye)^{-1}\\in Z(X)$$\\end{document}</tex-math><mml:math id=\"M1970\"><mml:mrow><mml:mi>z</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>∈</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq947\"><alternatives><tex-math id=\"M1971\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z^{m}=e$$\\end{document}</tex-math><mml:math id=\"M1972\"><mml:mrow><mml:msup><mml:mi>z</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mi>e</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq948\"><alternatives><tex-math id=\"M1973\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$m=(|F|n)^2!$$\\end{document}</tex-math><mml:math id=\"M1974\"><mml:mrow><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>!</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq949\"><alternatives><tex-math id=\"M1975\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M1976\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq950\"><alternatives><tex-math id=\"M1977\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z=(xe)(ye)^{-1}\\in H_e\\cap Z(X)$$\\end{document}</tex-math><mml:math id=\"M1978\"><mml:mrow><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>∈</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>∩</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq951\"><alternatives><tex-math id=\"M1979\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$xe=zye$$\\end{document}</tex-math><mml:math id=\"M1980\"><mml:mrow><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mi>z</mml:mi><mml:mi>y</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq952\"><alternatives><tex-math id=\"M1981\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(xe)^k=(zye)^k=z^k(ye)^k$$\\end{document}</tex-math><mml:math id=\"M1982\"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>z</mml:mi><mml:mi>y</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq953\"><alternatives><tex-math id=\"M1983\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f_k:h[C]\\rightarrow h[C]$$\\end{document}</tex-math><mml:math id=\"M1984\"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq954\"><alternatives><tex-math id=\"M1985\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\le n$$\\end{document}</tex-math><mml:math id=\"M1986\"><mml:mrow><mml:mo>≤</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq955\"><alternatives><tex-math id=\"M1987\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{f_k(h(x^k)),f_k(h(y^k))\\}\\subseteq F$$\\end{document}</tex-math><mml:math id=\"M1988\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>⊆</mml:mo><mml:mi>F</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq956\"><alternatives><tex-math id=\"M1989\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_0,\\dots ,a_{\\deg (f_k)}\\in h[C]$$\\end{document}</tex-math><mml:math id=\"M1990\"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mo>deg</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq957\"><alternatives><tex-math id=\"M1991\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f_k(v)=a_0va_1v\\cdots va_{\\deg (f_k)}$$\\end{document}</tex-math><mml:math id=\"M1992\"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>v</mml:mi><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>v</mml:mi><mml:mo>⋯</mml:mo><mml:mi>v</mml:mi><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mo>deg</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq958\"><alternatives><tex-math id=\"M1993\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v\\in h[C]$$\\end{document}</tex-math><mml:math id=\"M1994\"><mml:mrow><mml:mi>v</mml:mi><mml:mo>∈</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq959\"><alternatives><tex-math id=\"M1995\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i\\in \\{0,\\dots ,\\deg (f_k)\\}$$\\end{document}</tex-math><mml:math id=\"M1996\"><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:mo>deg</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq960\"><alternatives><tex-math id=\"M1997\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\check{a}}_i=\\textrm{pr}_e(a_i)$$\\end{document}</tex-math><mml:math id=\"M1998\"><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>e</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq961\"><alternatives><tex-math id=\"M1999\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M2000\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq962\"><alternatives><tex-math id=\"M2001\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\check{f}}_k:H_e\\rightarrow H_e$$\\end{document}</tex-math><mml:math id=\"M2002\"><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>k</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq963\"><alternatives><tex-math id=\"M2003\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\check{f}}_k(v)={\\check{a}}_0v{\\check{a}}_1v\\dots v{\\check{a}}_{\\deg (f_k)}$$\\end{document}</tex-math><mml:math id=\"M2004\"><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:msub><mml:mi>v</mml:mi><mml:msub><mml:mover accent=\"true\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:msub><mml:mi>v</mml:mi><mml:mo>⋯</mml:mo><mml:mi>v</mml:mi><mml:msub><mml:mover accent=\"true\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mrow><mml:mo>deg</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq964\"><alternatives><tex-math id=\"M2005\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v\\in H_e$$\\end{document}</tex-math><mml:math id=\"M2006\"><mml:mrow><mml:mi>v</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq965\"><alternatives><tex-math id=\"M2007\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{pr}_e\\circ f_k=({\\check{f}}_k\\circ \\textrm{pr}_e){\\restriction }_{h[C]}$$\\end{document}</tex-math><mml:math id=\"M2008\"><mml:mrow><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>e</mml:mi></mml:msub><mml:mo>∘</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>k</mml:mi></mml:msub><mml:mo>∘</mml:mo><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>e</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mo>↾</mml:mo><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq966\"><alternatives><tex-math id=\"M2009\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{pr}_e\\circ f_k\\circ h={\\check{f}}_k\\circ \\textrm{pr}_e\\circ h={\\check{f}}_k\\circ h_e$$\\end{document}</tex-math><mml:math id=\"M2010\"><mml:mrow><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>e</mml:mi></mml:msub><mml:mo>∘</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>∘</mml:mo><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>k</mml:mi></mml:msub><mml:mo>∘</mml:mo><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>e</mml:mi></mml:msub><mml:mo>∘</mml:mo><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>k</mml:mi></mml:msub><mml:mo>∘</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ40\"><alternatives><tex-math id=\"M2011\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\{\\textrm{pr}_e(f_k(h(x^k))),\\textrm{pr}_e(f_k(h(y^k)))\\}=\\{{\\check{f}}_k(h_e(x^k)),{\\check{f}}_k(h_e(y^k))\\}=\\{{\\check{f}}_k(x^ke),{\\check{f}}_k(y^ke)\\} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M2012\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>e</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>e</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq967\"><alternatives><tex-math id=\"M2013\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{{\\check{f}}_k(x^ke),{\\check{f}}_k(y^ke)\\}\\subseteq \\textrm{pr}_e[F]$$\\end{document}</tex-math><mml:math id=\"M2014\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>k</mml:mi></mml:msup><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>e</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq968\"><alternatives><tex-math id=\"M2015\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(a,b,d)\\in \\textrm{pr}_e[F]\\times \\textrm{pr}_e[F]\\times \\{1,\\dots ,n\\}$$\\end{document}</tex-math><mml:math id=\"M2016\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>e</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>×</mml:mo><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>e</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>×</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq969\"><alternatives><tex-math id=\"M2017\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i&lt;j\\le 1+n|\\textrm{pr}_e[F]|^2$$\\end{document}</tex-math><mml:math id=\"M2018\"><mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>j</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mtext>pr</mml:mtext><mml:mi>e</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ41\"><alternatives><tex-math id=\"M2019\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} ({\\check{f}}_i(x^ie),{\\check{f}}_i(y^ie),\\deg ({\\check{f}}_i))=(a,b,d)=({\\check{f}}_j(x^je),{\\check{f}}_j(y^je),\\deg ({\\check{f}}_j)). \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M2020\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mo>deg</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>j</mml:mi></mml:msup><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>j</mml:mi></mml:msup><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mo>deg</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq970\"><alternatives><tex-math id=\"M2021\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x^ie=z^iy^ie$$\\end{document}</tex-math><mml:math id=\"M2022\"><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msup><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mi>e</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq971\"><alternatives><tex-math id=\"M2023\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z\\in Z(X)$$\\end{document}</tex-math><mml:math id=\"M2024\"><mml:mrow><mml:mi>z</mml:mi><mml:mo>∈</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ42\"><alternatives><tex-math id=\"M2025\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} a={\\check{f}}_i(x^ie)={\\check{f}}_{i}(z^iy^ie)=z^{i\\deg ({\\check{f}}_i)}{\\check{f}}_i(y^ie)=z^{id}b. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M2026\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msup><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>deg</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">id</mml:mi></mml:mrow></mml:msup><mml:mi>b</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq972\"><alternatives><tex-math id=\"M2027\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a=z^{jd}b$$\\end{document}</tex-math><mml:math id=\"M2028\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">jd</mml:mi></mml:mrow></mml:msup><mml:mi>b</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq973\"><alternatives><tex-math id=\"M2029\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M2030\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq974\"><alternatives><tex-math id=\"M2031\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z^{id}b=z^{jd}b$$\\end{document}</tex-math><mml:math id=\"M2032\"><mml:mrow><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">id</mml:mi></mml:mrow></mml:msup><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">jd</mml:mi></mml:mrow></mml:msup><mml:mi>b</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq975\"><alternatives><tex-math id=\"M2033\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z^{id}=z^{jd}$$\\end{document}</tex-math><mml:math id=\"M2034\"><mml:mrow><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">id</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">jd</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq976\"><alternatives><tex-math id=\"M2035\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z^{(j-i)d}=e$$\\end{document}</tex-math><mml:math id=\"M2036\"><mml:mrow><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>e</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq977\"><alternatives><tex-math id=\"M2037\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(j-i)d\\le |F|^2 n^2$$\\end{document}</tex-math><mml:math id=\"M2038\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mo>≤</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq978\"><alternatives><tex-math id=\"M2039\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$m=(|F|n)^2!$$\\end{document}</tex-math><mml:math id=\"M2040\"><mml:mrow><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>!</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq979\"><alternatives><tex-math id=\"M2041\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z^m=e$$\\end{document}</tex-math><mml:math id=\"M2042\"><mml:mrow><mml:msup><mml:mi>z</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mi>e</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq980\"><alternatives><tex-math id=\"M2043\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M2044\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq981\"><alternatives><tex-math id=\"M2045\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M2046\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq982\"><alternatives><tex-math id=\"M2047\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M2048\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq983\"><alternatives><tex-math id=\"M2049\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M2050\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq984\"><alternatives><tex-math id=\"M2051\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|h[C]|\\le \\textrm{cov}({\\mathcal {M}})$$\\end{document}</tex-math><mml:math id=\"M2052\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>≤</mml:mo><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq985\"><alternatives><tex-math id=\"M2053\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M2054\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq986\"><alternatives><tex-math id=\"M2055\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Ce\\subseteq H_e$$\\end{document}</tex-math><mml:math id=\"M2056\"><mml:mrow><mml:mi>C</mml:mi><mml:mi>e</mml:mi><mml:mo>⊆</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq987\"><alternatives><tex-math id=\"M2057\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h[C]\\subseteq \\prod _{e\\in A}Ce\\subseteq H$$\\end{document}</tex-math><mml:math id=\"M2058\"><mml:mrow><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:msub><mml:mo>∏</mml:mo><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mi>C</mml:mi><mml:mi>e</mml:mi><mml:mo>⊆</mml:mo><mml:mi>H</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq988\"><alternatives><tex-math id=\"M2059\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(H,\\rho )$$\\end{document}</tex-math><mml:math id=\"M2060\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>H</mml:mi><mml:mo>,</mml:mo><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq989\"><alternatives><tex-math id=\"M2061\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M2062\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq990\"><alternatives><tex-math id=\"M2063\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M2064\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq991\"><alternatives><tex-math id=\"M2065\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho {\\restriction }_{h[C]\\times h[C]}$$\\end{document}</tex-math><mml:math id=\"M2066\"><mml:mrow><mml:mi>ρ</mml:mi><mml:msub><mml:mo>↾</mml:mo><mml:mrow><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>×</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq992\"><alternatives><tex-math id=\"M2067\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|h[C]|&lt;{\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M2068\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>&lt;</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq993\"><alternatives><tex-math id=\"M2069\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|h[C]|={\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M2070\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq994\"><alternatives><tex-math id=\"M2071\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|h[C]|\\le \\textrm{cov}({\\mathcal {M}})$$\\end{document}</tex-math><mml:math id=\"M2072\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>≤</mml:mo><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq995\"><alternatives><tex-math id=\"M2073\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{cov}({\\mathcal {M}})={\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M2074\"><mml:mrow><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq996\"><alternatives><tex-math id=\"M2075\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|h[C]|\\le {\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M2076\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>≤</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq997\"><alternatives><tex-math id=\"M2077\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M2078\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq998\"><alternatives><tex-math id=\"M2079\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Ce\\subseteq H_e$$\\end{document}</tex-math><mml:math id=\"M2080\"><mml:mrow><mml:mi>C</mml:mi><mml:mi>e</mml:mi><mml:mo>⊆</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq999\"><alternatives><tex-math id=\"M2081\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$B\\subseteq X$$\\end{document}</tex-math><mml:math id=\"M2082\"><mml:mrow><mml:mi>B</mml:mi><mml:mo>⊆</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ43\"><alternatives><tex-math id=\"M2083\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\exp (B){\\mathop {=}\\limits ^{{\\textsf{def}}}}\\min \\{n\\in {\\mathbb {N}}:B\\subseteq \\!\\root n \\of {\\!E(X)}\\}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M2084\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mo>exp</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mo movablelimits=\"true\">min</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi><mml:mo>:</mml:mo><mml:mi>B</mml:mi><mml:mo>⊆</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:mroot><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq1000\"><alternatives><tex-math id=\"M2085\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\subseteq {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M2086\"><mml:mrow><mml:mi>F</mml:mi><mml:mo>⊆</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ44\"><alternatives><tex-math id=\"M2087\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\textsf {lcm}(F){\\mathop {=}\\limits ^{{\\textsf{def}}}}\\min \\{n\\in {\\mathbb {N}}:\\forall x\\in F\\;\\exists k\\in {\\mathbb {N}}\\;\\;(n=xk)\\} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M2088\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi mathvariant=\"sans-serif\">lcm</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>F</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mo movablelimits=\"true\">min</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi><mml:mo>:</mml:mo><mml:mo>∀</mml:mo><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>F</mml:mi><mml:mspace width=\"0.277778em\"/><mml:mo>∃</mml:mo><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"0.277778em\"/><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq1001\"><alternatives><tex-math id=\"M2089\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(x^+_n)_{n\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M2090\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>+</mml:mo></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1002\"><alternatives><tex-math id=\"M2091\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(x^-_n)_{n\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M2092\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>-</mml:mo></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1003\"><alternatives><tex-math id=\"M2093\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(z_n)_{n\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M2094\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1004\"><alternatives><tex-math id=\"M2095\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(e_n)_{n\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M2096\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1005\"><alternatives><tex-math id=\"M2097\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in \\omega $$\\end{document}</tex-math><mml:math id=\"M2098\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1006\"><alternatives><tex-math id=\"M2099\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x^+_n,x^-_n\\in C$$\\end{document}</tex-math><mml:math id=\"M2100\"><mml:mrow><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>+</mml:mo></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>-</mml:mo></mml:msubsup><mml:mo>∈</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1007\"><alternatives><tex-math id=\"M2101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e_n=Z(X)\\cap A$$\\end{document}</tex-math><mml:math id=\"M2102\"><mml:mrow><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1008\"><alternatives><tex-math id=\"M2103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_n=(x^+_ne_n)(x^-_ne_n)^{-1}\\in Z(X)\\cap H_{e_n}$$\\end{document}</tex-math><mml:math id=\"M2104\"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>+</mml:mo></mml:msubsup><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>-</mml:mo></mml:msubsup><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>∈</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1009\"><alternatives><tex-math id=\"M2105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_n^i\\ne e_n$$\\end{document}</tex-math><mml:math id=\"M2106\"><mml:mrow><mml:msubsup><mml:mi>z</mml:mi><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msubsup><mml:mo>≠</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1010\"><alternatives><tex-math id=\"M2107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$1\\le i\\le n\\mu _n$$\\end{document}</tex-math><mml:math id=\"M2108\"><mml:mrow><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>i</mml:mi><mml:mo>≤</mml:mo><mml:mi>n</mml:mi><mml:msub><mml:mi>μ</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1011\"><alternatives><tex-math id=\"M2109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mu _n{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\textsf {lcm}\\{\\exp (Ce_k):k&lt;n\\}$$\\end{document}</tex-math><mml:math id=\"M2110\"><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mi mathvariant=\"sans-serif\">lcm</mml:mi><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mo>exp</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>C</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mi>k</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1012\"><alternatives><tex-math id=\"M2111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H'=\\prod _{n\\in \\omega }H_{e_n}$$\\end{document}</tex-math><mml:math id=\"M2112\"><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mo>∏</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1013\"><alternatives><tex-math id=\"M2113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h':C\\rightarrow H'$$\\end{document}</tex-math><mml:math id=\"M2114\"><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>:</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1014\"><alternatives><tex-math id=\"M2115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h':x\\mapsto (xe_n)_{n\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M2116\"><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>↦</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1015\"><alternatives><tex-math id=\"M2117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h'=\\textrm{pr}'\\circ h$$\\end{document}</tex-math><mml:math id=\"M2118\"><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mtext>pr</mml:mtext><mml:mo>′</mml:mo></mml:msup><mml:mo>∘</mml:mo><mml:mi>h</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1016\"><alternatives><tex-math id=\"M2119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{pr}':H\\rightarrow H'$$\\end{document}</tex-math><mml:math id=\"M2120\"><mml:mrow><mml:msup><mml:mtext>pr</mml:mtext><mml:mo>′</mml:mo></mml:msup><mml:mo>:</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1017\"><alternatives><tex-math id=\"M2121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{pr}':(x_a)_{a\\in A}\\mapsto (x_{e_n})_{n\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M2122\"><mml:mrow><mml:msup><mml:mtext>pr</mml:mtext><mml:mo>′</mml:mo></mml:msup><mml:mo>:</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:msub><mml:mo>↦</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1018\"><alternatives><tex-math id=\"M2123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in \\omega $$\\end{document}</tex-math><mml:math id=\"M2124\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1019\"><alternatives><tex-math id=\"M2125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{pr}'_n:H'\\rightarrow H_{e_n}$$\\end{document}</tex-math><mml:math id=\"M2126\"><mml:mrow><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>n</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mo>:</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">→</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1020\"><alternatives><tex-math id=\"M2127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{pr}'_n:(x_k)_{k\\in \\omega }\\mapsto x_n$$\\end{document}</tex-math><mml:math id=\"M2128\"><mml:mrow><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>n</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mo>:</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub><mml:mo>↦</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1021\"><alternatives><tex-math id=\"M2129\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H'$$\\end{document}</tex-math><mml:math id=\"M2130\"><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ45\"><alternatives><tex-math id=\"M2131\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\rho ':H'\\times H'\\rightarrow {\\mathbb {R}},\\quad \\rho '((x_n)_{n\\in \\omega },(y_n)_{n\\in \\omega })\\mapsto \\left( \\{0\\}\\cup \\left\\{ \\tfrac{1}{2^n}:n\\in \\omega ,\\;x_n\\ne y_n\\right\\} \\right) . \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M2132\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msup><mml:mi>ρ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>:</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"double-struck\">R</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"1em\"/><mml:msup><mml:mi>ρ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>↦</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>∪</mml:mo><mml:mfenced close=\"}\" open=\"{\"><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:mn>1</mml:mn><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:mfrac></mml:mstyle><mml:mo>:</mml:mo><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"0.277778em\"/><mml:msub><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>≠</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mfenced></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq1022\"><alternatives><tex-math id=\"M2133\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M2134\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1023\"><alternatives><tex-math id=\"M2135\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M2136\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1024\"><alternatives><tex-math id=\"M2137\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h'[C]\\subseteq H'$$\\end{document}</tex-math><mml:math id=\"M2138\"><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1025\"><alternatives><tex-math id=\"M2139\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H'=\\prod _{n\\in \\omega }H_{e_n}$$\\end{document}</tex-math><mml:math id=\"M2140\"><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mo>∏</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1026\"><alternatives><tex-math id=\"M2141\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M2142\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1027\"><alternatives><tex-math id=\"M2143\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h'[C]$$\\end{document}</tex-math><mml:math id=\"M2144\"><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1028\"><alternatives><tex-math id=\"M2145\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H'$$\\end{document}</tex-math><mml:math id=\"M2146\"><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1029\"><alternatives><tex-math id=\"M2147\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H'$$\\end{document}</tex-math><mml:math id=\"M2148\"><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1030\"><alternatives><tex-math id=\"M2149\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M2150\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1031\"><alternatives><tex-math id=\"M2151\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h'[C]$$\\end{document}</tex-math><mml:math id=\"M2152\"><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1032\"><alternatives><tex-math id=\"M2153\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h'[C]$$\\end{document}</tex-math><mml:math id=\"M2154\"><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1033\"><alternatives><tex-math id=\"M2155\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\kappa $$\\end{document}</tex-math><mml:math id=\"M2156\"><mml:mi>κ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1034\"><alternatives><tex-math id=\"M2157\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\kappa &lt;|h'[C]|=|\\textrm{pr}'[h[C]|\\le |h[C]|\\le {\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M2158\"><mml:mrow><mml:mrow><mml:mi>κ</mml:mi><mml:mo>&lt;</mml:mo><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msup><mml:mtext>pr</mml:mtext><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1035\"><alternatives><tex-math id=\"M2159\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h'[C]=\\bigcup _{\\alpha \\in \\kappa }f_\\alpha ^{-1}(b_\\alpha )$$\\end{document}</tex-math><mml:math id=\"M2160\"><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mi>κ</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mi>f</mml:mi><mml:mi>α</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1036\"><alternatives><tex-math id=\"M2161\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$b_\\alpha \\in h'[C]$$\\end{document}</tex-math><mml:math id=\"M2162\"><mml:mrow><mml:msub><mml:mi>b</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1037\"><alternatives><tex-math id=\"M2163\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f_\\alpha :h'[C]\\rightarrow h'[C]$$\\end{document}</tex-math><mml:math id=\"M2164\"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1038\"><alternatives><tex-math id=\"M2165\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mu _n$$\\end{document}</tex-math><mml:math id=\"M2166\"><mml:msub><mml:mi>μ</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1039\"><alternatives><tex-math id=\"M2167\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\exp (C{e_k})$$\\end{document}</tex-math><mml:math id=\"M2168\"><mml:mrow><mml:mo>exp</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>C</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1040\"><alternatives><tex-math id=\"M2169\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k&lt;n$$\\end{document}</tex-math><mml:math id=\"M2170\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1041\"><alternatives><tex-math id=\"M2171\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\in h'[C]\\subseteq \\prod _{n\\in \\omega }Ce_n$$\\end{document}</tex-math><mml:math id=\"M2172\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:msub><mml:mo>∏</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub><mml:mi>C</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1042\"><alternatives><tex-math id=\"M2173\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x^{-1}$$\\end{document}</tex-math><mml:math id=\"M2174\"><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1043\"><alternatives><tex-math id=\"M2175\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H'$$\\end{document}</tex-math><mml:math id=\"M2176\"><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1044\"><alternatives><tex-math id=\"M2177\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(x^{1+\\mu _n})_{n\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M2178\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1045\"><alternatives><tex-math id=\"M2179\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x^{-1}\\in \\overline{h'[C]}=h'[C]$$\\end{document}</tex-math><mml:math id=\"M2180\"><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>∈</mml:mo><mml:mover><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1046\"><alternatives><tex-math id=\"M2181\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h'[C]$$\\end{document}</tex-math><mml:math id=\"M2182\"><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1047\"><alternatives><tex-math id=\"M2183\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H'$$\\end{document}</tex-math><mml:math id=\"M2184\"><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1048\"><alternatives><tex-math id=\"M2185\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n\\in \\omega $$\\end{document}</tex-math><mml:math id=\"M2186\"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ46\"><alternatives><tex-math id=\"M2187\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} x^\\pm _n=(h'(x^+_n)h'(x^-_n)^{-1})^{\\mu _n}\\in h'[C]\\subseteq H'. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M2188\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>±</mml:mo></mml:msubsup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>+</mml:mo></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>-</mml:mo></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msup><mml:mo>∈</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq1049\"><alternatives><tex-math id=\"M2189\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k&lt;n$$\\end{document}</tex-math><mml:math id=\"M2190\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ47\"><alternatives><tex-math id=\"M2191\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\textrm{pr}'_k(x^\\pm _n)= &amp; {} \\textrm{pr}'_k\\big ((h'(x^+_n)h'(x^-_n)^{-1})^{\\mu _n}\\big )=((x^+_ne_k)(x^-_ne_k)^{-1})^{\\mu _n}\\\\= &amp; {} \\big (((x_n^+e_k)(x^-_ne_k)^{-1})^{\\exp (C{e_k})}\\big )^{\\mu _n/\\exp (C{e_k})}= e_k^{\\mu _n/\\exp (C{e_k})}=e_k, \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M2192\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>k</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>±</mml:mo></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>k</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>+</mml:mo></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>-</mml:mo></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msup><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>+</mml:mo></mml:msubsup><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>-</mml:mo></mml:msubsup><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msup></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:mo>=</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>+</mml:mo></mml:msubsup><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>-</mml:mo></mml:msubsup><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo>exp</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>C</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>exp</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>C</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mi>e</mml:mi><mml:mi>k</mml:mi><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mo>exp</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>C</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq1050\"><alternatives><tex-math id=\"M2193\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(x^\\pm _n)_{n\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M2194\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>±</mml:mo></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1051\"><alternatives><tex-math id=\"M2195\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e'=(e_k)_{k\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M2196\"><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1052\"><alternatives><tex-math id=\"M2197\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H'=\\prod _{k\\in \\omega }H_{e_k}$$\\end{document}</tex-math><mml:math id=\"M2198\"><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msub><mml:mo>∏</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1053\"><alternatives><tex-math id=\"M2199\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$2=\\{0,1\\}$$\\end{document}</tex-math><mml:math id=\"M2200\"><mml:mrow><mml:mn>2</mml:mn><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1054\"><alternatives><tex-math id=\"M2201\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$2^{&lt;\\omega }{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\bigcup _{n\\in \\omega }2^n$$\\end{document}</tex-math><mml:math id=\"M2202\"><mml:mrow><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mo>&lt;</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msup><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:msub><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1055\"><alternatives><tex-math id=\"M2203\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(p_s)_{s\\in 2^{&lt;\\omega }}$$\\end{document}</tex-math><mml:math id=\"M2204\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mo>&lt;</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1056\"><alternatives><tex-math id=\"M2205\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H'$$\\end{document}</tex-math><mml:math id=\"M2206\"><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ48\"><alternatives><tex-math id=\"M2207\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} p_{\\emptyset }=e' \\text{ and } p_{s\\hat{\\;}0}=p_s, p_{s\\hat{\\;}1}=p_s\\cdot x^\\pm _n \\text{ for } \\text{ every } n\\in \\omega \\text{ and } s\\in 2^n. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M2208\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi mathvariant=\"normal\">∅</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mspace width=\"0.333333em\"/><mml:mtext>and</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mover accent=\"true\"><mml:mspace width=\"0.277778em\"/><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mover accent=\"true\"><mml:mspace width=\"0.277778em\"/><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>·</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>±</mml:mo></mml:msubsup><mml:mspace width=\"0.333333em\"/><mml:mtext>for</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"0.333333em\"/><mml:mtext>every</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi><mml:mspace width=\"0.333333em\"/><mml:mtext>and</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mi>s</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq1057\"><alternatives><tex-math id=\"M2209\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho '$$\\end{document}</tex-math><mml:math id=\"M2210\"><mml:msup><mml:mi>ρ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1058\"><alternatives><tex-math id=\"M2211\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H'$$\\end{document}</tex-math><mml:math id=\"M2212\"><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1059\"><alternatives><tex-math id=\"M2213\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x^\\pm _n\\rightarrow e'$$\\end{document}</tex-math><mml:math id=\"M2214\"><mml:mrow><mml:msubsup><mml:mi>x</mml:mi><mml:mi>n</mml:mi><mml:mo>±</mml:mo></mml:msubsup><mml:mo stretchy=\"false\">→</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1060\"><alternatives><tex-math id=\"M2215\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$s\\in 2^\\omega $$\\end{document}</tex-math><mml:math id=\"M2216\"><mml:mrow><mml:mi>s</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>ω</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1061\"><alternatives><tex-math id=\"M2217\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(p_{s{\\restriction }_n})_{n\\in \\omega }$$\\end{document}</tex-math><mml:math id=\"M2218\"><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:msub><mml:mo>↾</mml:mo><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ω</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1062\"><alternatives><tex-math id=\"M2219\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(H',\\rho ')$$\\end{document}</tex-math><mml:math id=\"M2220\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>ρ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1063\"><alternatives><tex-math id=\"M2221\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p_s\\in h'[C]\\subseteq H'$$\\end{document}</tex-math><mml:math id=\"M2222\"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1064\"><alternatives><tex-math id=\"M2223\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{p_s:s\\in 2^\\omega \\}\\subseteq h'[C]\\subseteq \\bigcup _{\\alpha \\in \\kappa }f_\\alpha ^{-1}(b_\\alpha )$$\\end{document}</tex-math><mml:math id=\"M2224\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:mi>s</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>ω</mml:mi></mml:msup><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:msub><mml:mo>⋃</mml:mo><mml:mrow><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mi>κ</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mi>f</mml:mi><mml:mi>α</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1065\"><alternatives><tex-math id=\"M2225\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\kappa &lt;{\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M2226\"><mml:mrow><mml:mi>κ</mml:mi><mml:mo>&lt;</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1066\"><alternatives><tex-math id=\"M2227\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha \\in \\kappa $$\\end{document}</tex-math><mml:math id=\"M2228\"><mml:mrow><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mi>κ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1067\"><alternatives><tex-math id=\"M2229\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$2^\\omega _\\alpha {\\mathop {=}\\limits ^{{\\textsf{def}}}}\\{s\\in 2^\\omega :p_s\\in f_\\alpha ^{-1}(b_\\alpha )\\}$$\\end{document}</tex-math><mml:math id=\"M2230\"><mml:mrow><mml:msubsup><mml:mn>2</mml:mn><mml:mi>α</mml:mi><mml:mi>ω</mml:mi></mml:msubsup><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>s</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>ω</mml:mi></mml:msup><mml:mo>:</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:msubsup><mml:mi>f</mml:mi><mml:mi>α</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1068\"><alternatives><tex-math id=\"M2231\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$s,s'\\in 2^\\omega _\\alpha $$\\end{document}</tex-math><mml:math id=\"M2232\"><mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>s</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>∈</mml:mo><mml:msubsup><mml:mn>2</mml:mn><mml:mi>α</mml:mi><mml:mi>ω</mml:mi></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1069\"><alternatives><tex-math id=\"M2233\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$s{\\restriction }_{\\deg (f_\\alpha )}=s'{\\restriction }_{\\deg (f_\\alpha )}$$\\end{document}</tex-math><mml:math id=\"M2234\"><mml:mrow><mml:mi>s</mml:mi><mml:msub><mml:mo>↾</mml:mo><mml:mrow><mml:mo>deg</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>s</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mo>↾</mml:mo><mml:mrow><mml:mo>deg</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1070\"><alternatives><tex-math id=\"M2235\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$m\\in {\\mathbb {N}}$$\\end{document}</tex-math><mml:math id=\"M2236\"><mml:mrow><mml:mi>m</mml:mi><mml:mo>∈</mml:mo><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1071\"><alternatives><tex-math id=\"M2237\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$s(m)\\ne s'(m)$$\\end{document}</tex-math><mml:math id=\"M2238\"><mml:mrow><mml:mi>s</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≠</mml:mo><mml:msup><mml:mi>s</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>m</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1072\"><alternatives><tex-math id=\"M2239\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$m\\ge \\deg (f_\\alpha )\\ge 1$$\\end{document}</tex-math><mml:math id=\"M2240\"><mml:mrow><mml:mi>m</mml:mi><mml:mo>≥</mml:mo><mml:mo>deg</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1073\"><alternatives><tex-math id=\"M2241\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$s{\\restriction }_m=s'{\\restriction }_m$$\\end{document}</tex-math><mml:math id=\"M2242\"><mml:mrow><mml:mi>s</mml:mi><mml:msub><mml:mo>↾</mml:mo><mml:mi>m</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>s</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mo>↾</mml:mo><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1074\"><alternatives><tex-math id=\"M2243\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t=s{\\restriction }_m=s'{\\restriction }_m$$\\end{document}</tex-math><mml:math id=\"M2244\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mi>s</mml:mi><mml:msub><mml:mo>↾</mml:mo><mml:mi>m</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>s</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msub><mml:mo>↾</mml:mo><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1075\"><alternatives><tex-math id=\"M2245\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{p_{s{\\restriction }(m+1)},p_{s'{\\restriction }(m+1)}\\}=\\{p_{t\\hat{\\;}0},p_{t\\hat{\\;}1}\\}$$\\end{document}</tex-math><mml:math id=\"M2246\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mo>↾</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:msup><mml:mi>s</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>↾</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mover accent=\"true\"><mml:mspace width=\"0.277778em\"/><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mover accent=\"true\"><mml:mspace width=\"0.277778em\"/><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1076\"><alternatives><tex-math id=\"M2247\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p_{s{\\restriction }(m+1)}=p_{t\\hat{\\;}0}=p_{t}$$\\end{document}</tex-math><mml:math id=\"M2248\"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mo>↾</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mover accent=\"true\"><mml:mspace width=\"0.277778em\"/><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1077\"><alternatives><tex-math id=\"M2249\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p_{s'{\\restriction }(m+1)}=p_{t\\hat{\\;}1}=p_tx_m^\\pm $$\\end{document}</tex-math><mml:math id=\"M2250\"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:msup><mml:mi>s</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>↾</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mover accent=\"true\"><mml:mspace width=\"0.277778em\"/><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msubsup><mml:mi>x</mml:mi><mml:mi>m</mml:mi><mml:mo>±</mml:mo></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1078\"><alternatives><tex-math id=\"M2251\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f_\\alpha (p_s)=b_\\alpha =f_\\alpha (p_{s'})$$\\end{document}</tex-math><mml:math id=\"M2252\"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:msup><mml:mi>s</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1079\"><alternatives><tex-math id=\"M2253\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{pr}'_m(f_\\alpha (p_s))=\\textrm{pr}'_m(f_\\alpha (p_{s'}))$$\\end{document}</tex-math><mml:math id=\"M2254\"><mml:mrow><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:msup><mml:mi>s</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1080\"><alternatives><tex-math id=\"M2255\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_0,a_1,\\dots , a_{\\deg (f_\\alpha )}\\in h'[C]$$\\end{document}</tex-math><mml:math id=\"M2256\"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mo>deg</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1081\"><alternatives><tex-math id=\"M2257\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f_\\alpha (x)=a_0xa_1x\\cdots xa_{\\deg (f_\\alpha )}$$\\end{document}</tex-math><mml:math id=\"M2258\"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>x</mml:mi><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>x</mml:mi><mml:mo>⋯</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mo>deg</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1082\"><alternatives><tex-math id=\"M2259\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\in h'[C]$$\\end{document}</tex-math><mml:math id=\"M2260\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1083\"><alternatives><tex-math id=\"M2261\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i\\in \\{0,\\dots ,\\deg (f_\\alpha )\\}$$\\end{document}</tex-math><mml:math id=\"M2262\"><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:mo>deg</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1084\"><alternatives><tex-math id=\"M2263\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\check{a}}_i=\\textrm{pr}'_m(a_i)$$\\end{document}</tex-math><mml:math id=\"M2264\"><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1085\"><alternatives><tex-math id=\"M2265\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\check{f}}_\\alpha :H_{e_m}\\rightarrow H_{e_m}$$\\end{document}</tex-math><mml:math id=\"M2266\"><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mo>:</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1086\"><alternatives><tex-math id=\"M2267\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\check{f}}_\\alpha (x)={\\check{a}}_0x{\\check{a}}_1x\\cdots x{\\check{a}}_{\\deg (f_\\alpha )}$$\\end{document}</tex-math><mml:math id=\"M2268\"><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:msub><mml:mi>x</mml:mi><mml:msub><mml:mover accent=\"true\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:msub><mml:mi>x</mml:mi><mml:mo>⋯</mml:mo><mml:mi>x</mml:mi><mml:msub><mml:mover accent=\"true\"><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mrow><mml:mo>deg</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1087\"><alternatives><tex-math id=\"M2269\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\in H_{e_m}$$\\end{document}</tex-math><mml:math id=\"M2270\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1088\"><alternatives><tex-math id=\"M2271\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{pr}'_m\\circ f_\\alpha ={\\check{f}}_\\alpha \\circ \\textrm{pr}'_m$$\\end{document}</tex-math><mml:math id=\"M2272\"><mml:mrow><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mo>∘</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mo>∘</mml:mo><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1089\"><alternatives><tex-math id=\"M2273\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{pr}'_m(x_m^\\pm )=z_m^{\\mu _m}\\in Z(X)$$\\end{document}</tex-math><mml:math id=\"M2274\"><mml:mrow><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>m</mml:mi><mml:mo>±</mml:mo></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>m</mml:mi><mml:msub><mml:mi>μ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:msubsup><mml:mo>∈</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1090\"><alternatives><tex-math id=\"M2275\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textrm{pr}'_m(x_k^\\pm )=e_m$$\\end{document}</tex-math><mml:math id=\"M2276\"><mml:mrow><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>k</mml:mi><mml:mo>±</mml:mo></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>e</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1091\"><alternatives><tex-math id=\"M2277\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$m&lt;k$$\\end{document}</tex-math><mml:math id=\"M2278\"><mml:mrow><mml:mi>m</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ49\"><alternatives><tex-math id=\"M2279\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} {\\check{f}}_\\alpha (\\textrm{pr}'_m(p_t))= &amp; {} {\\check{f}}_\\alpha (\\textrm{pr}'_m(p_s))=\\textrm{pr}'_m(f_\\alpha (p_s))=\\textrm{pr}'_m(f_{\\alpha }(p_{s'}))={\\check{f}}_\\alpha (\\textrm{pr}'_m(p_{s'}))={\\check{f}}_\\alpha (\\textrm{pr}'_m(p_{t\\hat{\\;}1}))\\\\= &amp; {} {\\check{f}}_\\alpha (\\textrm{pr}'_m(p_t)\\textrm{pr}'_m(x_m^\\pm ))={\\check{f}}_\\alpha (\\textrm{pr}'_m(p_t)z_m^{\\mu _m})={\\check{f}}_\\alpha (\\textrm{pr}'_m(p_t))z_m^{\\mu _m\\deg {f_\\alpha }} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M2280\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:msup><mml:mi>s</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:msup><mml:mi>s</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mover accent=\"true\"><mml:mspace width=\"0.277778em\"/><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:mo>=</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>x</mml:mi><mml:mi>m</mml:mi><mml:mo>±</mml:mo></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mi>z</mml:mi><mml:mi>m</mml:mi><mml:msub><mml:mi>μ</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">ˇ</mml:mo></mml:mover><mml:mi>α</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mtext>pr</mml:mtext><mml:mi>m</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mi>z</mml:mi><mml:mi>m</mml:mi><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo>deg</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mrow></mml:msubsup></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq1092\"><alternatives><tex-math id=\"M2281\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e_m=z_m^{\\mu _m\\deg {f_\\alpha }}$$\\end{document}</tex-math><mml:math id=\"M2282\"><mml:mrow><mml:msub><mml:mi>e</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mi>m</mml:mi><mml:mrow><mml:msub><mml:mi>μ</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo>deg</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>α</mml:mi></mml:msub></mml:mrow></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1093\"><alternatives><tex-math id=\"M2283\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z_m$$\\end{document}</tex-math><mml:math id=\"M2284\"><mml:msub><mml:mi>z</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1094\"><alternatives><tex-math id=\"M2285\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M2286\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1095\"><alternatives><tex-math id=\"M2287\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M2288\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1096\"><alternatives><tex-math id=\"M2289\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$A\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M2290\"><mml:mrow><mml:mi>A</mml:mi><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1097\"><alternatives><tex-math id=\"M2291\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$B\\subseteq A$$\\end{document}</tex-math><mml:math id=\"M2292\"><mml:mrow><mml:mi>B</mml:mi><mml:mo>⊆</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1098\"><alternatives><tex-math id=\"M2293\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\bigcap _{e\\in B}\\frac{H_e}{e}$$\\end{document}</tex-math><mml:math id=\"M2294\"><mml:mrow><mml:mi>C</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:msub><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:mfrac><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mi>e</mml:mi></mml:mfrac></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1099\"><alternatives><tex-math id=\"M2295\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h:C\\rightarrow H{\\mathop {=}\\limits ^{{\\textsf{def}}}}\\prod _{b\\in B}H_b$$\\end{document}</tex-math><mml:math id=\"M2296\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>H</mml:mi><mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"sans-serif\">def</mml:mi></mml:mover><mml:msub><mml:mo>∏</mml:mo><mml:mrow><mml:mi>b</mml:mi><mml:mo>∈</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>H</mml:mi><mml:mi>b</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1100\"><alternatives><tex-math id=\"M2297\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h:x\\mapsto (xe)_{e\\in B} $$\\end{document}</tex-math><mml:math id=\"M2298\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>:</mml:mo><mml:mi>x</mml:mi><mml:mo>↦</mml:mo><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mi>e</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1101\"><alternatives><tex-math id=\"M2299\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in B$$\\end{document}</tex-math><mml:math id=\"M2300\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1102\"><alternatives><tex-math id=\"M2301\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M2302\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1103\"><alternatives><tex-math id=\"M2303\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|h[C]|\\le \\textrm{cov}({\\mathcal {M}})$$\\end{document}</tex-math><mml:math id=\"M2304\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>≤</mml:mo><mml:mtext>cov</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1104\"><alternatives><tex-math id=\"M2305\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M2306\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1105\"><alternatives><tex-math id=\"M2307\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Ce\\subseteq H_e$$\\end{document}</tex-math><mml:math id=\"M2308\"><mml:mrow><mml:mi>C</mml:mi><mml:mi>e</mml:mi><mml:mo>⊆</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1106\"><alternatives><tex-math id=\"M2309\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|h[C]|\\le {\\mathfrak {c}}$$\\end{document}</tex-math><mml:math id=\"M2310\"><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>h</mml:mi><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>≤</mml:mo><mml:mi mathvariant=\"fraktur\">c</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1107\"><alternatives><tex-math id=\"M2311\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in A$$\\end{document}</tex-math><mml:math id=\"M2312\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1108\"><alternatives><tex-math id=\"M2313\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M2314\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1109\"><alternatives><tex-math id=\"M2315\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M2316\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1110\"><alternatives><tex-math id=\"M2317\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$S\\subseteq Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M2318\"><mml:mrow><mml:mi>S</mml:mi><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1111\"><alternatives><tex-math id=\"M2319\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M2320\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1112\"><alternatives><tex-math id=\"M2321\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M2322\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1113\"><alternatives><tex-math id=\"M2323\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$S\\subseteq Z(X)\\cap \\!\\root \\infty \\of {\\!A}$$\\end{document}</tex-math><mml:math id=\"M2324\"><mml:mrow><mml:mi>S</mml:mi><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi>∞</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1114\"><alternatives><tex-math id=\"M2325\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$S\\subseteq Z(X)$$\\end{document}</tex-math><mml:math id=\"M2326\"><mml:mrow><mml:mi>S</mml:mi><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1115\"><alternatives><tex-math id=\"M2327\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H(S)=S\\cap H(X)$$\\end{document}</tex-math><mml:math id=\"M2328\"><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>S</mml:mi><mml:mo>∩</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1116\"><alternatives><tex-math id=\"M2329\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$S{\\setminus } H(S)\\subseteq Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {\\!A}{\\setminus } H(X)$$\\end{document}</tex-math><mml:math id=\"M2330\"><mml:mrow><mml:mi>S</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1117\"><alternatives><tex-math id=\"M2331\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root \\infty \\of {\\!A}{\\setminus } H(X)\\supseteq S{\\setminus } H(S)$$\\end{document}</tex-math><mml:math id=\"M2332\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi>A</mml:mi></mml:mrow><mml:mi>∞</mml:mi></mml:mroot><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>⊇</mml:mo><mml:mi>S</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>S</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1118\"><alternatives><tex-math id=\"M2333\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M2334\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1119\"><alternatives><tex-math id=\"M2335\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M2336\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1120\"><alternatives><tex-math id=\"M2337\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M2338\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1121\"><alternatives><tex-math id=\"M2339\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)$$\\end{document}</tex-math><mml:math id=\"M2340\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1122\"><alternatives><tex-math id=\"M2341\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M2342\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1123\"><alternatives><tex-math id=\"M2343\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\in E(I\\!Z(X))$$\\end{document}</tex-math><mml:math id=\"M2344\"><mml:mrow><mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1124\"><alternatives><tex-math id=\"M2345\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e=H_ee\\subseteq X\\cdot I\\!Z(X)\\subseteq Z(X)$$\\end{document}</tex-math><mml:math id=\"M2346\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mi>e</mml:mi><mml:mo>⊆</mml:mo><mml:mi>X</mml:mi><mml:mo>·</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1125\"><alternatives><tex-math id=\"M2347\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_e$$\\end{document}</tex-math><mml:math id=\"M2348\"><mml:msub><mml:mi>H</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1126\"><alternatives><tex-math id=\"M2349\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E(I\\!Z(X))\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M2350\"><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1127\"><alternatives><tex-math id=\"M2351\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)$$\\end{document}</tex-math><mml:math id=\"M2352\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1128\"><alternatives><tex-math id=\"M2353\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)=I\\!Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {E(I\\!Z(X))}$$\\end{document}</tex-math><mml:math id=\"M2354\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1129\"><alternatives><tex-math id=\"M2355\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {E(I\\!Z(X))}$$\\end{document}</tex-math><mml:math id=\"M2356\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1130\"><alternatives><tex-math id=\"M2357\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)$$\\end{document}</tex-math><mml:math id=\"M2358\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1131\"><alternatives><tex-math id=\"M2359\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H(I\\!Z(X))=H(X)\\cap I\\!Z(X)$$\\end{document}</tex-math><mml:math id=\"M2360\"><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>∩</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ50\"><alternatives><tex-math id=\"M2361\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} I\\!Z(X)\\setminus H(I\\!Z(X))=I\\!Z(X)\\setminus H(X)= I\\!Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {E(I\\!Z(X))}\\setminus H(X)\\\\ \\subseteq Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}\\setminus H(X) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M2362\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi>H</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq1132\"><alternatives><tex-math id=\"M2363\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)$$\\end{document}</tex-math><mml:math id=\"M2364\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1133\"><alternatives><tex-math id=\"M2365\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)$$\\end{document}</tex-math><mml:math id=\"M2366\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1134\"><alternatives><tex-math id=\"M2367\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)$$\\end{document}</tex-math><mml:math id=\"M2368\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1135\"><alternatives><tex-math id=\"M2369\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {2}}\\textsf {S}$$\\end{document}</tex-math><mml:math id=\"M2370\"><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1136\"><alternatives><tex-math id=\"M2371\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\square $$\\end{document}</tex-math><mml:math id=\"M2372\"><mml:mo>□</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1137\"><alternatives><tex-math id=\"M2373\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M2374\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1138\"><alternatives><tex-math id=\"M2375\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}=\\mathsf {T_{\\!1}S}$$\\end{document}</tex-math><mml:math id=\"M2376\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1139\"><alternatives><tex-math id=\"M2377\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathsf {T_{\\!z}S}\\subseteq {\\mathcal {C}}\\subseteq \\mathsf {T_{\\!2}S}$$\\end{document}</tex-math><mml:math id=\"M2378\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>⊆</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1140\"><alternatives><tex-math id=\"M2379\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M2380\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1141\"><alternatives><tex-math id=\"M2381\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)$$\\end{document}</tex-math><mml:math id=\"M2382\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1142\"><alternatives><tex-math id=\"M2383\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M2384\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1143\"><alternatives><tex-math id=\"M2385\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E(I\\!Z(X))\\subseteq V\\!E(X)$$\\end{document}</tex-math><mml:math id=\"M2386\"><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>⊆</mml:mo><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1144\"><alternatives><tex-math id=\"M2387\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathsf {T_{\\!z}S}\\subseteq {\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M2388\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1145\"><alternatives><tex-math id=\"M2389\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textsf {T}_{\\!\\textsf {z}\\textsf {S}}$$\\end{document}</tex-math><mml:math id=\"M2390\"><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ51\"><alternatives><tex-math id=\"M2391\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} I\\!Z(X)= I\\!Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {E(I\\!Z(X))}\\subseteq Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M2392\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo>⊆</mml:mo><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq1146\"><alternatives><tex-math id=\"M2393\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}=\\mathsf {T_{\\!1}S}$$\\end{document}</tex-math><mml:math id=\"M2394\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1147\"><alternatives><tex-math id=\"M2395\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z(X)\\cap \\!\\root {\\mathbb {N}} \\of {V\\!E(X)}$$\\end{document}</tex-math><mml:math id=\"M2396\"><mml:mrow><mml:mi>Z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mspace width=\"-0.166667em\"/><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"double-struck\">N</mml:mi></mml:mroot></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1148\"><alternatives><tex-math id=\"M2397\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)$$\\end{document}</tex-math><mml:math id=\"M2398\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1149\"><alternatives><tex-math id=\"M2399\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathsf {T_{\\!z}S}\\subseteq {\\mathcal {C}}\\subseteq \\mathsf {T_{\\!2}S}$$\\end{document}</tex-math><mml:math id=\"M2400\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>⊆</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1150\"><alternatives><tex-math id=\"M2401\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)$$\\end{document}</tex-math><mml:math id=\"M2402\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1151\"><alternatives><tex-math id=\"M2403\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M2404\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1152\"><alternatives><tex-math id=\"M2405\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {C}}$$\\end{document}</tex-math><mml:math id=\"M2406\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1153\"><alternatives><tex-math id=\"M2407\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathsf {T_{\\!z}S}\\subseteq {\\mathcal {C}}\\subseteq \\mathsf {T_{\\!2}S}$$\\end{document}</tex-math><mml:math id=\"M2408\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mi mathvariant=\"sans-serif\">z</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow><mml:mo>⊆</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>⊆</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant=\"sans-serif\">T</mml:mi><mml:mrow><mml:mspace width=\"-0.166667em\"/><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"sans-serif\">S</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1154\"><alternatives><tex-math id=\"M2409\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(1)\\Rightarrow (2)$$\\end{document}</tex-math><mml:math id=\"M2410\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1155\"><alternatives><tex-math id=\"M2411\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\!Z(X)=Z(X)=X$$\\end{document}</tex-math><mml:math id=\"M2412\"><mml:mrow><mml:mi>I</mml:mi><mml:mspace width=\"-0.166667em\"/><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1156\"><alternatives><tex-math id=\"M2413\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(2)\\Rightarrow (3)$$\\end{document}</tex-math><mml:math id=\"M2414\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1157\"><alternatives><tex-math id=\"M2415\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(3)\\Rightarrow (4)$$\\end{document}</tex-math><mml:math id=\"M2416\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1158\"><alternatives><tex-math id=\"M2417\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(4)\\Rightarrow (1)$$\\end{document}</tex-math><mml:math id=\"M2418\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">⇒</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>" ]

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[ "<fn-group><fn><p>Serhii Bardyla was supported by the Slovak Research and Development Agency under the Contract no. APVV-21-0468 and by the Austrian Science Fund FWF (Grant I 5930).</p></fn><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<title>Background</title>", "<p id=\"Par27\">Erectile dysfunction (ED) is a vascular disease, which is estimated to affect 322 million men worldwide by 2025, and 30–75% of men experience ED, especially diabetes-related ED [##REF##10444124##1##]. Oral phosphodiesterase 5 (PDE5) inhibitors, which rely on endogenous nitric oxide (NO) bioavailability, constitute the first-line treatment for diabetes-induced ED [##REF##24049429##2##]. However, long-term microangiopathy in erectile tissues of patients with diabetes mellitus (DM) can induce hypoxia and structural changes, which cause endothelial-pericyte dysfunction and peripheral neuropathy [##REF##30098549##3##] and may lead to poor responses to oral PDE5 inhibitors [##REF##22611498##4##]. The clinical development of several novel targets, including COMP-Ang1 [##REF##21270241##5##], vascular endothelial growth factor [##REF##16140112##6##], and brain-derived neurotrophic factor [##UREF##0##7##], have been hindered due to insufficient efficacy, side effects, and protein engineering difficulties in preclinical studies. Therefore, an efficient therapeutic strategy with minimal adverse effects is needed in the complicated pathology of angiopathy and neuropathy in diabetes-induced ED.</p>", "<p id=\"Par28\">Extracellular vesicles (EVs) are released by almost all cell types [##UREF##1##8##] and comprise proteins, lipids, messenger RNAs (mRNAs), microRNAs (miRNAs), and nucleic acids, which play multiple roles in the physiological and pathological communication between donor and recipient cells [##REF##32457507##9##]. EV-mimetic nanovesicles (NV) isolated from mouse embryonic stem cells and mouse corpus cavernous pericytes (MCPs) significantly induce neurovascular regeneration in diabetes-induced and cavernous nerve injury (CNI)-induced ED mouse models [##REF##31882614##10##, ##REF##32855091##11##]. However, the detailed pathways and target genes regulated by MCP<italic>-</italic>derived EVs (MCPs-EVs) remain unelucidated. Moreover, miRNAs repress or degrade mRNA at the post-transcriptional level, thereby regulating various cellular processes, such as angiogenesis, proliferation, apoptosis, and neural regeneration [##UREF##2##12##, ##REF##27326408##13##]. We hypothesized that MCPs-EVs promote neurovascular regeneration by delivering miRNAs, thereby improving erectile function in diabetic mice.</p>", "<p id=\"Par29\">This study primary objectives were conducted to determine the type and content of miRNAs in MCPs-EVs as well as the neurovascular regenerative function under diabetic conditions; and secondary objectives was to determine the target genes of miRNAs in MCPs-EVs.</p>" ]

[ "<title>Methods</title>", "<title>Ethics statement and study design</title>", "<p id=\"Par30\">This animal model study included 75 adult male C57BL/6 J mice (8 weeks old; Orient Bio, Inc.): 15 for the MCPs primary culture, MCPs-EVs isolation, and small-RNA sequencing analysis; 20 for in vitro studies; and 40 for in vivo studies. All animal experiments were approved by the Ethics Committee at the Inha University (approval number: INHA 200309-691). Animals were monitored daily for health and behavior as previously described [##REF##35866351##14##, ##REF##35562586##15##]. In all studies, animals were anesthetized with an intramuscular injection of ketamine (100 mg/kg; Yuhan Corp., Seoul, Korea) and xylazine (5 mg/kg; Bayer Korea, Seoul, Korea). Animals were euthanized by 100% CO₂ gas replacement; cessation of heartbeat and respiration were confirmed prior to harvesting tissue. All animal studies were randomized and blinded. No mice died during the experimental procedures.</p>", "<title>Primary culture of MCPs and MCECs and treatment</title>", "<p id=\"Par31\">Primary MCPs and MCECs were prepared from mouse penile tissue as previously described [##REF##26044953##16##, ##REF##22548733##17##]. Briefly, 8 weeks old male C57BL/6 J mice were euthanized by 100% CO₂ gas replacement. Then, the penis tissues were harvested and maintained in sterile vials with HBSS (Gibco). After washing with PBS for 3 times, the urethra and dorsal neurovascular bundle were removed, and only the corpus cavernosum tissues were used. For MCPs culture, the corpus cavernosum tissues were cut into approximately 1-2 mm sections and settled via gravity into collagen I-coated 35 mm cell culture dishes with 300 μL complement DMEM (GIBCO) at 37 °C for 20 min in a 5% CO₂ atmosphere. Then, 900 μL of complement medium was added and incubated at 37 °C with 5% CO_2. The complement medium contained 20% FBS, 1% penicillin/streptomycin, and 10 nM human pigment epithelium-derived factor (PEDF; Sigma-Aldrich). The medium was changed every 2 days, and after approximately 10 days sprouting cells were sub-cultured into collagen I (Advanced BioMatrix, San Diego, CA, USA)-coated dishes. For MCECs culture, corpus cavernosum tissues were cut into approximately 1-2 mm sections, put into 60 mm dishes, and covered by Matrigel (Becton Dickinson, Mountain View, CA, USA). Tissues were cultured with the complement M199 medium (Gibco) containing 20% fetal bovine serum (FBS, Gibco), 0.5 mg/mL of heparin (Sigma-Aldrich), 5 ng/mL of recombinant human vascular endothelial growth factor (VEGF, R&amp;D Systems Inc., Minneapolis, MN, USA), and 1% penicillin/streptomycin (Gibco) in a 5% CO2 atmosphere incubator at 37 °C. After cells were confluent on the bottom of the 60 mm cell culture dishes (approximately 2 weeks of culture), sprouting cells were sub-cultured into other cell culture dishes coated with 0.2% gelatin (Sigma-Aldrich, St Louis, MO, USA). Cells from passages 2 to 3 were used for the experiments. Diabetes-induced angiopathy was mimicked by serum-starving cells (MCECs and MCPs) for 24 hours and then exposing them to normal-glucose (NG; 5 mM glucose, Sigma-Aldrich, St. Louis, MO, USA) or high-glucose (HG; 30 mM glucose) conditions for 3 days [##REF##30929324##18##].</p>", "<title>MCPs-EV isolation, quantitation, and identification</title>", "<p id=\"Par32\">MCPs-EVs were isolated from DMEM-cultured MCPs (Gibco; Carlsbad, CA, USA) in medium supplemented with 20% exosome-depleted FBS (Gibco), with 1% penicillin/streptomycin (Gibco), using a commercial EV isolation kit (ExoQuick-TC, System Biosciences, LLC., Palo Alto, CA, USA) according to the manufacturers’ instructions. The EXOCET exosome quantitation kit (System Biosciences, LLC.) was used to quantify MCPs-EVs, and their concentration was adjusted to 1 μg/μL before treatments.</p>", "<p id=\"Par33\">The MCPs-EV morphology was ascertained by transmission electron microscopy (TEM; Electron Microscopy Sciences, Fort Washington, PA, USA) and their characterization was validated based on the expression of one negative and three positive EV markers in Western blotting [##REF##32855091##11##] (negative marker: GM130 [1:1000; BD Biosciences, San Jose, CA, USA]; Three positive EV markers [1:1000]: CD9 [Abcam, Cambridge, MA, USA), CD63 [NOVUS Biologicals, Littleton, CO, USA], and CD81 [NOVUS Biologicals]).</p>", "<title>Fluorescence dye labeling of the MCPs-EVs for tracking analysis</title>", "<p id=\"Par34\">MCPs-EVs were labelled with 1,1′-dioctadecyl-3,3,3′,3′-tetramethylindodicarbocyanine, 4-chlorobenzenesulfonate salt (DiD) red-fluorescent dye (ThermoFisher Scientific, Inc., Carlsbad, CA, USA) according to the manufacturer’s instructions. After 6-hour DiD-labeled MCPs-EV treatment, MCECs were fixed by incubation with 4% formaldehyde for 15 minutes. DiD dye tracking was determined under a confocal fluorescence microscope (K1-Fluo, Nanoscope Systems, Inc., Daejeon, Korea).</p>", "<title>miRNA identification by small-RNA sequencing analysis</title>", "<p id=\"Par35\">The small-RNA-sequencing assay was performed by E-Biogen Inc. (Korea). For control and test RNAs, a library was constructed using the NEBNext Multiplex Small RNA Library Prep kit (New England BioLabs, Inc., USA) according to the manufacturer’ instructions. The small-RNA-sequencing data were deposited in the Gene Expression Omnibus database (<ext-link ext-link-type=\"uri\" xlink:href=\"http://www.ncbi.nlm.nih.gov/geo;\">www.ncbi.nlm.nih.gov/geo;</ext-link> accession no. GSE195533).</p>", "<title>Transfection of MCPs with miR148a-3p inhibitors</title>", "<p id=\"Par36\">Primary MCPs were transfected with 20 nM miR148a-3p inhibitor (a mirVana® miRNA inhibitor, Cat #, 4,464,084, Thermo Fisher, San Jose, CA, USA) using Lipofectamine 2000 (Invitrogen, Carlsbad, CA, USA). After 24 hours, the culture medium was replaced with DMEM supplemented with 20% exosome-depleted FBS with 1% penicillin/streptomycin for 3 days, and the culture medium was collected for conditioned MCPs-EV isolation.</p>", "<title>Extraction of RNA in MCPs-EVs</title>", "<p id=\"Par37\">Exosomal RNA was isolated from 150 ul of MCPs-EVs using the miRNeasy Serum/Plasma Kit (Qiagen, Hilden, Germany) according to the manufacturer’s protocol as described previously [##REF##28607588##19##].</p>", "<title>Real-time PCR (qPCR)</title>", "<p id=\"Par38\">Total RNA from MCPs was isolated using TRIzol (Invitrogen) according to the manufacturer’s instructions. miRNAs were reverse-transcribed using the Taqman microRNA Reverse Transcription kit (Applied Biosystems, Carlsbad, CA, USA) and the associated miRNA-specific stem-loop primers (Applied Biosystems). The RT reaction conditions were: 30 minutes at 16 °C to anneal primers, 30 minutes at 42 °C for extension of primers on miRNA and synthesis of the first cDNA strands, and 5 minutes at 85 °C to stop the reaction. qPCR was performed on the 7500 Fast Real-Time PCR System (Applied Biosystems) with the following conditions: one denaturing step at 95 °C for 10 minutes, followed by 40 denaturing cycles at 95 °C for 15 seconds; and annealing and elongation at 60 °C for 60 seconds. The data presented correspond to the mean of 2^-ddct from at least three independent experiments after normalization to U6.</p>", "<title>Measurement of nitric oxide (NO) levels</title>", "<p id=\"Par39\">The nitrite assay kit (MAK367, Sigma-Aldrich) was used to determine nitric oxide (NO) concentration in MCECs, according to the manufacturer’s protocol as described previously [##REF##35866351##14##]. MCECs were seeded in 6-well plates at a density of 5 × 10⁵ cells/well in 2 mL of M199 medium. After 24 hours, MCECs were exposed to glucose conditions with or without MCPs-EV or miR-148a-3p-depleted MCPs-EV for 72 hours at 37 °C in a humidified 5% CO₂ atmosphere. Then, cultured medium was collected for NO concentration measurement. Nitrite levels was measured at a wavelength of 540 nm, using a microplate spectrophotometer (BioTek Instruments Inc., Winooski, VT, USA). Each experiment contained six replicates and repeated four times.</p>", "<title>Tube-formation assay</title>", "<p id=\"Par40\">Tube-formation assays were performed as described previously [##REF##33258323##20##]. Tube formation was monitored for 18 hours under a phase-contrast microscope (CKX41, Olympus, Tokyo, Japan). The number of master junctions from four separate experiments were quantified using Image J (National Institutes of Health [NIH] 1.34, <ext-link ext-link-type=\"uri\" xlink:href=\"http://rsbweb.nih.gov/ij/\">http://rsbweb.nih.gov/ij/</ext-link>).</p>", "<title>Cell-migration assay</title>", "<p id=\"Par41\">The migration assay was performed with the SPLScar™Block system (SPL life sciences, Pocheon-si, Gyeonggi-do, Korea) on 60-mm culture dishes [##REF##33258323##20##]. Images were obtained using a phase-contrast microscope (Olympus), and cell migration was analyzed by determining the ratio of cells that moved into the frame line in the figures from four separate block systems using Image J.</p>", "<title>TUNEL assay</title>", "<p id=\"Par42\">The terminal deoxynucleotidyl transferase-mediated deoxyuridine triphosphate nick-end labeling (TUNEL) assay was performed using the ApopTag® Fluorescein In Situ Apoptosis Detection Kit (Chemicon, Temecula, CA, USA) as described previously [##REF##33258327##21##]. Samples were mounted in a solution (Vector Laboratories Inc., Burlingame, CA, USA) containing 4,6-diamidino-2-phenylindole (DAPI), a nuclear stain. Digital images were obtained using a confocal fluorescence microscope (K1-Fluo; Nanoscope Systems, Inc). The number of apoptotic cells was counted using Image J.</p>", "<title>Animal treatment and measurement of erectile function</title>", "<p id=\"Par43\">Diabetes was induced in 8-week-old C57BL/6 J mice by injecting them with streptozotocin (STZ; 50 mg/kg, Sigma-Aldrich) for 5 consecutive days [##REF##19732306##22##]. After 8 weeks, mice were randomly distributed into four groups: control nondiabetic mice and STZ-induced diabetic mice receiving two successive intracavernous injections of PBS (20 μL; days − 3 and 0), two successive intracavernous injections of MCPs-EV control reagent (5 μg in 20 μL PBS, days − 3 and 0), or two successive intracavernous injection of miR-148a-3p-depleted MCPs-EV (5 μg in 20 μL PBS, days − 3 and 0) into the midportion of the corpus cavernosum (<italic>n</italic> = 5 per group). For injections, mice were anesthetized with intramuscular injections of ketamine (100 mg/kg) and xylazine (5 mg/kg) and positioned supine on a thermoregulated surgical table and the base of the penis was compressed with a vascular clamp before injection. After injection, the clamp was left in place for 30 minutes to prevent backflow of blood from the penis.</p>", "<p id=\"Par44\">Two weeks later, the erectile function measurement was performed as previously described [##REF##19732306##22##]. Systemic blood pressure was measured continuously by using a noninvasive tail-cuff system (Visitech Systems, Apex, NC, USA) before the measurement of intracavernous pressure (ICP). The ratios of maximal ICP and total ICP to mean systolic blood pressure (MSBP) were calculated to normalize for variations in systemic blood pressure.</p>", "<title>Histological examination</title>", "<p id=\"Par45\">For fluorescence microscopy, mice penile tissue was fixed in 4% paraformaldehyde for 24 hours at 4 °C, and the frozen tissue sections (12-μm thick) were incubated overnight at 4 °C with primary antibodies including: CD31 (endothelial cell marker, 1:50; Millipore, Temecula, CA, USA), NG2 antibody (pericyte marker, 1:50; Millipore), β (III)-tubulin antibody (neuronal cell marker, 1:200; Abcam), nNOS (neuronal cell marker, 1:100; Santa Cruz Biotechnology Inc., Dallas, TX USA), or phospho-Histone H3 (PH3; Mitosis marker, Upstate Biotechnology Inc., Temecula, CA, USA). After several washes with PBS, samples were incubated with donkey anti-rabbit DyLight® 550 (1:200; Abcam), goat anti-Armenian hamster Fluorescein isothiocyanate (FITC; 1:200; Jackson ImmunoResearch Laboratories, West grove, PA, USA), donkey anti-rabbit FITC (1:200; Jackson ImmunoResearch Laboratories), and donkey anti-chicken Tetramethylrhodamine (TRITC) secondary antibodies (1:200; Jackson ImmunoResearch Laboratories) for 2 hours at room temperature. Using a DAPI-based solution (Vector Laboratories Inc.), samples were mounted for nuclear staining. Samples were visualized and images were obtained with a confocal microscope (Nanoscope Systems, Inc). Quantitative analysis was performed using Image J.</p>", "<title>Target prediction with bioinformatics</title>", "<p id=\"Par46\">Computational predictions of miRNA target genes were obtained using the following published algorithms: DIANA-microT-CDS (<ext-link ext-link-type=\"uri\" xlink:href=\"http://www.microrna.gr/microT-CDS\">http://www.microrna.gr/microT-CDS</ext-link>), TargetScan (<ext-link ext-link-type=\"uri\" xlink:href=\"http://www.targetscan.org\">http://www.targetscan.org</ext-link>), and miRDB (<ext-link ext-link-type=\"uri\" xlink:href=\"http://www.mirdb.org\">http://www.mirdb.org</ext-link>).</p>", "<title>Luciferase miRNA target reporting assays</title>", "<p id=\"Par47\">A PDK4 3’UTR target plasmid (50 ng, GeneCopoeia, Rockville, MD, USA) or a negative control vector (50 ng, GeneCopoeia) with miR148a-3p mimic (50 nM, Ambion) or a control mimic (50 nM, Ambion) were co-transfected into MCECs using Lipofectamine2000 transfection reagent. After 48 hours, the cells were lysed and luciferase activities were measured using the Dual-Luciferase Reporter Assay Kit 2.0 (GeneCopoeia) and a luminometer (BioTek Instruments Inc., Winooski, VT, USA). Relative luciferase activity was calculated by normalizing the <italic>Renilla</italic> Luciferase signal against that of firefly luciferase.</p>", "<title>Western blotting analysis</title>", "<p id=\"Par48\">Equal amounts of protein (30 μg per lane) were subjected to 4–20% SDS-PAGE and then transferred to PVDF membranes. After blocking with 5% non-fat dry milk for 1.5 hours at room temperature, the membranes were incubated at 4 °C overnight with the following primary antibodies: pyruvate dehydrogenase kinase-4 (<italic>PDK4</italic>; 1:1000; NOVUS Biologicals) and β-actin (1:4000; Santa Cruz Biotechnology). The membranes were washed thrice for 10 min each with PBST at room temperature. Subsequently, the membranes were incubated with goat anti-rabbit IgG H&amp;L (HRP; 1:1000; Abcam), donkey anti-goat IgG H&amp;L (HRP; 1:1000; Abcam), or goat anti-mouse IgG H&amp;L (HRP; 1:1000; Abcam) secondary antibodies for 2 hours at room temperature, and signals were visualized using an ECL detection system (Amersham Pharmacia Biotech, Inc.). The results were quantified by densitometric analysis using Image J.</p>", "<title>Statistical analysis</title>", "<p id=\"Par49\">All results are expressed as the mean ± SEM of at least four independent experiments. The unpaired <italic>t</italic>-test was used to compare two groups, and one-way ANOVA followed by Tukey’s post hoc test was used for four-group comparisons. The analysis was conducted using GraphPad Prism version 8 (Graph Pad Software, Inc.). <italic>P</italic> values less than 0.05 were considered statistically significant. Statistical sample sizes were determined based on our previous studies [##REF##35562586##15##, ##REF##35813481##23##]. In vivo functional evaluation requires at least 5 animals per group, and other experiments require at least 4 samples per group for more effective statistical analysis.</p>" ]

[ "<title>Results</title>", "<title>MCPs-EV characterization and tracking analysis in MCECs</title>", "<p id=\"Par50\">Based on the TEM images, the MCPs-EVs exhibited a unique cup-shaped morphology (diameter ~ 30 nm; Fig. ##FIG##0##1##a). Western blotting showed that positive EVs surface markers, such as CD9, CD63, and CD81, were present in MCPs-EVs but had low expression levels in MCP lysate. Conversely, negative EV surface markers, such as GM130, were not detected in purified MCPs-EVs (Fig. ##FIG##0##1##b). To determine whether the secreted MCPs-EVs were taken up by endothelial cells, DiD red dye-labeled MCPs-EVs were treated in MCECs and were detected inside the MCECs after 6 hours (Fig. ##FIG##0##1##c). These data suggest that MPCs-EVs are taken up by MCECs.</p>", "<title>MicroRNA profiling of MCPs-EVs</title>", "<p id=\"Par51\">To assess the composition of miRNAs in MCPs-EVs, we performed small-RNA sequencing analysis with MCPs-EVs. We detected only 90 miRNAs in MCPs-EVs, and the top 10 expressed miRNAs are listed in Table ##TAB##0##1##. We focused on miR-148a-3p, the most abundant miRNA in MCPs-EVs.\n</p>", "<title>Inhibition of miR-148a-3p expression reduces the angiogenic effect of MCPs-EVs under high-glucose conditions</title>", "<p id=\"Par52\">To assess the angiogenic effect of miR-148a-3p, which has the highest expression in MCPs-EVs, we transfected a miR-148a-3p inhibitor in MCPs and isolated MCPs-EVs, we found that the expression of miR-148a-3p was significantly reduced in MCPs and MCPs-EVs (Fig. ##FIG##1##2##a and b). Then, we evaluated the expression of miR-148a-3p and nitrite levels in MCECs, which was treated with PBS, MCPs-EVs-regent control (MCPs-EVs-RC), MCPs-EVs-miR-148a-3p inhibitor (MCPs-EVs-miR-148a-3p-i) under normal-glucose (NG) and high-glucose (HG) conditions. We found that the expression of miR-148a-3p (Fig. ##FIG##1##2##c) and nitrite levels (Fig. ##FIG##1##2##d) were significant reduced in MCECs under HG conditions, and MCPs-EVs treatment leads to the recovery of miR-148a-3p expression and nitrite levels. However, there was no significant change in the expression of miR-148a-3p and nitrite levels in the group treated with miR-148a-3p-depleted MCP-EVs under HG conditions. Next, tube-formation (Fig. ##FIG##1##2##e) and cell-migration (Fig. ##FIG##1##2##f) abilities were significantly reduced in MCECs exposed to the high-glucose PBS environment. Reagent-controlled MCPs-EVs can induce tube formation and migration of endothelial cells under high-glucose conditions, thereby promoting angiogenesis. However, these effects were significantly attenuated in the group treated with miR-148a-3p-depleted MCPs-EVs under high-glucose conditions (Fig. ##FIG##1##2##e–h). Furthermore, using TUNEL assay and PH3 staining, we found that miR-148a-3p-depleted MCPs-EVs were unable to reduce apoptosis (Fig. ##FIG##2##3##)a-b or induce proliferation (Fig. ##FIG##2##3##c and d) of MCECs under high-glucose conditions. Collectively, these results suggest that miR-148a-3p plays an important role in MCPs-EV-promoted angiogenesis by inducing MCEC migration, survival, and proliferation under high-glucose conditions.</p>", "<title>MCPs-EVs improve erectile function through miR-148a-3p in diabetic mice</title>", "<p id=\"Par53\">To investigate whether MCPs-EVs have beneficial effects via miR-148a-3p on erectile function in diabetic mice, we intracavernously injected the reagent-controlled or miR-148a-3p-depleted MCPs-EVs and evaluated erectile function 2 weeks later. During electrical stimulation, the ratios of maximal and total intracavernous pressure (ICP) to MSBP were significantly lower in PBS-treated diabetic mice than in age-matched non-diabetic controls. Interestingly, diabetic mice treated with MCPs-EVs had significantly improved erection parameters, reaching almost 93% of the values in controls. However, miR-148a-3p-depleted MCPs-EV treatment showed no such effects in diabetic mice (Fig. ##FIG##3##4##). Immunofluorescence staining for CD31 (Fig. ##FIG##4##5##a, c), NG2 (Fig. ##FIG##4##5## a, d) in cavernosum tissues and β (III)-tubulin (Fig. ##FIG##4##5## b, e) and neuronal NOS (nNOS; Fig. ##FIG##4##5## b, f) in dorsal nerve bundles demonstrated that MCPs-EVs significantly improved the endothelial cell, pericyte, and nerve composition in diabetic mice. Fasting and postprandial blood glucose concentrations were significantly higher in diabetic mice than in control mice. However, there were no significant differences in body weight and blood glucose levels in diabetic mice regardless of treatment (Table ##TAB##1##2##). No detectable differences in MSBP were observed between the three STZ-induced experimental groups. These results suggest that miR-148a-3p plays an important role in MCPs-EVs-induced improvement in erectile function by rescuing cavernous neurovascular regeneration in diabetic mice.</p>", "<title><italic>PDK4</italic> as a target gene of miR-148a-3p</title>", "<p id=\"Par54\">There were 98 target genes for miR-148a-3p that were predicted from the DIANA-microT-CDS, TargetScan, and miRDB according to the following parameters: target score &gt; 80, intersection in all three algorithms, and are detected in MCECs through the published RNA-sequencing result [##REF##33258323##20##]. From the literature review and removal of genes to identify a target of miR-148a-3p, we selected 5 predicted target genes: <italic>Ago2, S1pr1, PDK4, Prkaa1</italic>, and <italic>Adam10</italic>. Of these genes, we focused on <italic>PDK4</italic> expression which was significantly regulated by miR-148a-3p (Fig. ##FIG##5##6##a, b). Using TargetScan, the binding sequences of miR-148a-3p at position 655-662 of PDK4 3’UTR is shown in Fig. ##FIG##5##6##c. The luciferase reporter assay demonstrated that luciferase activity was significantly reduced in the PDK4 3’UTR plasmid and miR148a-3p-mimic co-transfected group. However, the luciferase activity in the control vector-transfected group showed no difference (Fig. ##FIG##5##6##d). These results suggest that miR-148a-3p inhibits PDK4 expression by directly binding to the 3’UTR of PDK4 mRNA.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par55\">Increasingly, evidence has suggested that miRNAs are widely involved in physiological and pathological processes, including in cancer, diabetes mellitus, and neurological disorders [##UREF##2##12##, ##REF##27326409##24##, ##REF##25714967##25##]. However, the detailed role of miRNAs in pericyte-derived EVs is unknown. Using small-RNA sequencing analysis, we found that miR-148a-3p had the highest expression in MCPs-EVs. High-glucose conditions promote pericyte dysfunction [##REF##33258327##21##] and reduce miR-148a-3p levels in human retinal microvascular endothelial cells [##REF##32661705##26##]. Furthermore, miR-148a-3p played different roles in angiogenesis due to different microenvironments [##REF##29921422##27##]. Therefore, we investigated whether high-glucose-induced pericyte dysfunction could reduce the secretion of miR-148a-3p-containing MCPs-EVs, thereby reducing angiogenesis under high-glucose conditions. To further investigate the role of miR-148a-3p in the promotion of angiogenesis by MCPs-EVs, we depleted the miR-148a-3p in MCPs-EVs and found that the angiogenic effect was significantly reduced under high-glucose conditions. Thus, miR-148a-3p may play an important role in MCPs-EV-induced angiogenesis.</p>", "<p id=\"Par56\">Next, we used a diabetes-induced ED mouse model to evaluate the neurovascular regenerative effects of MCPs-EVs as described previously [##REF##33258327##21##, ##REF##19732306##22##]. We found that MCPs-EVs significantly improved erectile function by inducing endothelial cell, pericyte, and neuronal cell content upregulation in the cavernosal tissues of diabetic mice. However, miR-148a-3p-depleted MCPs-EVs did not improve erectile function in diabetic mice. Therefore, MCPs-EVs can function as an intercellular delivery tool for miR-148a-3p transfer into recipient cells for improving erectile function in diabetic mice.</p>", "<p id=\"Par57\">To demonstrate how miR-148a-3p regulates erectile function in patients with diabetes, we reviewed the literature related to miR-148a-3p found in endothelial cells (HUVECs) (PMID: 31723119). Multiple targets have been identified, such as NRP-1, ROBO1, and ITGa5. But in order to find targets in MCECs, we used three miRNA target prediction programs and subsequent verification experiments, and we found that PDK4 is a regulated target by miR-148a-3p. The luciferase assay results demonstrated that miR-148a-3p directly targets the 3’UTR of PDK4, thereby reducing PDK4 expression. The PDK4 level is elevated in patients with diabetes, and a PDK inhibitor enhances insulin activity by promoting glucose oxidation [##REF##25003070##28##]. In addition, reducing PDK4 expression with siRNAs in pulmonary arterial hypertension-induced pericytes can enhance the endothelial–pericyte interaction [##REF##27456128##29##]. Thus, we hypothesized that miR-148a-3p transferred by MCPs-EVs might reduce PDK4 expression, thereby promoting neurovascular regeneration in diabetic mice. However, this study did not confirm whether decreased PDK4 expression in the MCPs-EVs treatment group directly affected the erection in diabetic ED mice. Nonetheless, this study provides a basis for understanding the detailed mechanisms and therapeutic value of MCPs-EVs in improving erection in diabetes-induced ED.</p>", "<p id=\"Par58\">In this study, we showed, for the first time, that MCPs-EVs improve erectile function in diabetic mice in a miR-148a-3p-dependent manner. However, our study has some limitations. First, we did not verify whether the known targets regulated by miR-148a-3p (NRP-1, ROBO1, and ITGa5) in endothelial cells (HUVEC) also apply to MCECs. In the future, studies of known targets of miR-148a-3p may be help us to further understood the specific mechanism of miR-148a-3p in diabetic ED. Second, we could not evaluate the expression of miR-148a-3p and nitrite levels in in vivo study. Third, we could not evaluate whether repression of PDK4 expression by miR-148a-3p had a direct effect on erection in diabetes-induced ED. Studies are needed to evaluate the detailed mechanism and function of PDK4 in ED and other vascular and/ or neurological disorders.</p>" ]

[ "<title>Conclusions</title>", "<p id=\"Par59\">Our study shows that miR-148a-3p, which is highly expressed in MCPs-EVs, plays an important role in enhancing neurovascular regeneration and ultimately improving erectile function by inhibiting PDK4 expression in diabetic mice.</p>" ]

[ "<title>Background</title>", "<p id=\"Par1\">To investigate the regulatory role of microRNA (miR)-148a-3p in mouse corpus cavernous pericyte (MCPs)-derived extracellular vesicles (EVs) in the treatment of diabetes-induced erectile dysfunction (ED).</p>", "<title>Methods</title>", "<p id=\"Par2\">Mouse corpus cavernous tissue was used for MCP primary culture and EV isolation. Small-RNA sequencing analysis was performed to assess the type and content of miRs in MCPs-EVs. Four groups of mice were used: control nondiabetic mice and streptozotocin-induced diabetic mice receiving two intracavernous injections (days − 3 and 0) of phosphate buffered saline, MCPs-EVs transfected with reagent control, or MCPs-EVs transfected with a miR-148a-3p inhibitor. miR-148a-3p function in MCPs-EVs was evaluated by tube-formation assay, migration assay, TUNEL assay, intracavernous pressure, immunofluorescence staining, and Western blotting.</p>", "<title>Results</title>", "<p id=\"Par3\">We extracted EVs from MCPs, and small-RNA sequencing analysis showed miR-148a-3p enrichment in MCPs-EVs. Exogenous MCPs-EV administration effectively promoted mouse cavernous endothelial cell (MCECs) tube formation, migration, and proliferation, and reduced MCECs apoptosis under high-glucose conditions. These effects were significantly attenuated in miR-148a-3p-depleted MCPs-EVs, which were extracted after inhibiting miR-148a-3p expression in MCPs. Repetitive intracavernous injections of MCPs-EVs improved erectile function by inducing cavernous neurovascular regeneration in diabetic mice. Using online bioinformatics databases and luciferase report assays, we predicted that pyruvate dehydrogenase kinase-4 (PDK4) is a potential target gene of miR-148a-3p.</p>", "<title>Conclusions</title>", "<p id=\"Par4\">Our findings provide new and reliable evidence that miR-148a-3p in MCPs-EVs significantly enhances cavernous neurovascular regeneration by inhibiting PDK4 expression in diabetic mice.</p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1186/s12894-023-01378-4.</p>", "<title>Keywords</title>" ]

[ "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Acknowledgements</title>", "<p>The manuscript is edited by a professional, native English-speaking editor at Wordvice ([email protected]; Kevin Heintz, Managing Editor, Wordvice).</p>", "<title>Authors’ contributions</title>", "<p>JO and GNY conceived and designed the experiments. JO, FYL, FRF, LN, MNV, YH and GNY performed the experiments. FYL, FRF, LN, MNV, YH collected the experimental specimens. JO, SGP, TZ and GNY wrote the manuscript. All authors are accountable for all aspects of work.</p>", "<title>Funding</title>", "<p>This research was supported by Inha University Research Grant (Guo Nan Yin).</p>", "<title>Availability of data and materials</title>", "<p> They are available from the corresponding author on special request.</p>", "<title>Declarations</title>", "<title>Ethics approval and consent to participate</title>", "<p id=\"Par60\">All study protocol for this research project and all male C57BL/6 J mice(8 weeks old, Orient Bio, Korea) used in this study were approved by the Ethics Committee at the Inha University (approval number: INHA 200309-691). All methods were carried out in accordance with relevant guidelines and regulations. ARRIVE guidelines for reporting animal research was followed.</p>", "<title>Consent for publication</title>", "<p id=\"Par61\">Not applicable.</p>", "<title>Competing interests</title>", "<p id=\"Par62\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>MCP-derived extracellular vesicle (MCPs-EV) characterization and tracking analysis in MCECs. <bold>a</bold> Representative transmission electron micrograph (TEM) phase images for detecting isolated MCPs-EVs as indicated by the arrows. Scale bar = 100 nm. <bold>b</bold> Representative Western blot for three positive EV markers (CD9, CD63, and CD81) and one negative EV marker (GM130) in MCP lysate and MCPs-EVs. <bold>c</bold> DiD-labeled MCPs-EVs (red) were treated with MCECs for 6 h. Scale bar = 50 μm. MCPs, mouse corpus cavernous pericyte; MCECs, mouse cavernous endothelial cells; DiD, 1,1′-dioctadecyl-3,3,3′,3′-tetramethylindodicarbocyanine, 4-chlorobenzenesulfonate salt</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>MCPs-EVs induces angiogenesis in MCECs through miR-148a-3p. <bold>a</bold> The mRNA levels of miR-148a-3p decreased in MCPs transfected with miR-148a-3p inhibitor compared to the microRNA control (miRcon). <bold>b</bold> The mRNA levels of miR-148a-3p decreased in MCPs-EVs isolated from conditioned MCPs after transfected with miR-148a-3p inhibitor compared to the microRNA control (miRcon). <bold>c</bold> and <bold>d</bold> The miR-148a-3p mRNA levels (<bold>c</bold>) and nitrite production (<bold>d</bold>) in MCECs treated with PBS, MCPs-EVs-regent control (MCPs-EVs-RC, 1 μg/mL), MCPs-EVs-miR-148a-3p inhibitor (MCPs-EVs-miR-148a-3p-i, 1 μg/mL) under normal-glucose (NG) and high-glucose (HG) conditions for 3 days. <bold>e</bold> Tube-formation assay was performed in MCECs treated with PBS, MCPs-EVs-regent control (MCPs-EVs-RC, 1 μg/mL), MCPs-EVs-miR-148a-3p inhibitor (MCPs-EVs-miR-148a-3p-i, 1 μg/mL) under normal-glucose (NG) and high-glucose (HG) conditions for 3 days; representative images obtained at 18 hours (screen magnification, 40×). <bold>f</bold> Migration assay was performed in MCECs with the same treatment conditions as for tube formation; representative images were obtained at 24 hours (screen magnification, 40×). <bold>g</bold> Number of master junctions were quantified using Image J and the results are presented as mean ± SEM (<italic>n</italic> = 4). <bold>h</bold> Ratio of cells that migrated into the red-dotted frame were quantified using Image J and the results are presented as mean ± SEM (<italic>n</italic> = 4). The value expressed as ratios of the NG group was set to 1. **<italic>p</italic> &lt; 0.01; ***<italic>p</italic> &lt; 0.001. MCPs, mouse corpus cavernous pericyte; MCECs, mouse cavernous endothelial cells; ns, not significant</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>MCPs-EVs decreased apoptosis and increased proliferation of MCECs through miR-148a-3p under high-glucose (HG) conditions. <bold>a</bold> TUNEL (green) immunostaining in MCECs treated with PBS, MCPs-EVs-reagent control (MCPs-EVs-RC, 1 μg/mL), MCPs-EVs-miR-148a-3p inhibitor (MCPs-EVs-miR-148a-3p-i, 1 μg/mL) under normal-glucose (NG) and HG conditions for 3 days. Scale bar = 100 μm. <bold>b</bold> Number of apoptotic cells were quantified by ImageJ and the results are presented as mean ± SEM (n = 4). <bold>c</bold> PH3 (red) immunostaining in MCECs with the same treatment conditions as for TUNEL immunostaining. Nuclear were labeled with DAPI (blue). <bold>d</bold> Number of PH3-positive cells were quantified by ImageJ and the results are presented as mean ± SEM (n = 4). ***<italic>p</italic> &lt; 0.001. TUNEL, terminal deoxynucleotidyl transferase-mediated deoxyuridine triphosphate nick end labeling; MCPs, mouse corpus cavernous pericyte; MCECs, mouse cavernous endothelial cells; PH3, phospho-Histone H3; ns, not significant; DAPI, 4.6-diamidino-2-phenylindole</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><p>MCPs-EVs improved erectile function through miR-148a-3p in STZ-induced diabetic mice. <bold>a</bold> Representative ICP responses for age-matched nondiabetic controls, diabetic mice stimulated at 2 weeks after intracavernous PBS, MCPs-EVs-reagent control (MCPs-EVs-RC, 5 μg/20 μL), and MCPs-EVs-miR-148a-3p inhibitor (MCPs-EVs-miR-148a-3p-i, 5 μg/20 μL) injection. The cavernous nerve was stimulated at 5 V, and stimulus time is indicated by a solid bar. <bold>b, c</bold> Ratios of mean maximal ICP and total ICP (area under the curve) versus MSBP were calculated for each group. Data in graphs are mean ± SEM (<italic>n</italic> = 5). ***<italic>p</italic> &lt; 0.001. STZ, streptozotocin; ICP, intracavernous pressure; MSBP, mean systolic blood pressure; MCPs, mouse corpus cavernous pericytes; ns, not significant</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><p>MCPs-EVs improve cavernous endothelial cell, pericytes, and neuronal cell content through miR-148a-3p in STZ-induced diabetic mice. <bold>a</bold> CD31 (green) and NG2 (red) immunostaining in cavernous tissue from age-matched nondiabetic controls, diabetic mice stimulated at 2 weeks after intracavernous PBS, MCPs-EVs-reagent control (MCPs-EVs-RC, 5 μg/20 μL), or MCPs-EVs-miR-148a-3p inhibitor (MCPs-EVs-miR-148a-3p-i, 5 μg/20 μL) injection; scale bar = 100 μm. <bold>b</bold> β (III)-tubulin (red) and nNOS (green) immunostaining in the same cavernous tissue section with the abovementioned CD31 staining groups; scale bar = 25 μm. Nuclear were labeled with DAPI (blue). <bold>c-f</bold> Quantitative analysis of cavernous endothelial cell, pericytes, and β (III)-tubulin- or nNOS-expressing neuronal cell contents using Image J. Data in graphs are presented as mean ± SEM (<italic>n</italic> = 4). The value expressed as ratios of the control group was set to 1. **<italic>p</italic> &lt; 0.01; ***<italic>p</italic> &lt; 0.001. STZ, streptozotocin; MCPs, mouse corpus cavernous pericytes; DAPI = 4,6-diamidino-2-phenylindole; ns, not significant</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><p><italic>PDK4</italic> was identified as a target gene of miR-148a-3p. <bold>a</bold> Representative Western blots for <italic>PDK4</italic> in MCECs treated with PBS, MCPs-EVs-reagent control (MCPs-EVs-RC, 1 μg/mL) and MCPs-EVs-miR-148a-3p inhibitor (MCPs-EVs-miR-148a-3p-i, 1 μg/mL) under normal-glucose (NG) and high-glucose (HG) conditions for 3 days. <bold>b</bold> Relative intensities of PDK4 and β-actin on Image J analysis. Data in graphs are presented as mean ± SEM (<italic>n</italic> = 4). Values expressed as ratios of the control group were set to 1. **<italic>p</italic> &lt; 0.01; ***<italic>p</italic> &lt; 0.001. <bold>c</bold> The binding sequences of miR-148a-3p on position 655-662 of PDK4 3’UTR. <bold>d</bold> Luciferase reporter assay was used to assess the binding capacity between miR-148a-3p and PDK4 in MCECs. Data in graphs are presented as mean ± SEM (n = 4). Values expressed as ratios of the NC mimics co-transfected with PDK4 3’UTR plasmid group was set to 1. ***<italic>p</italic> &lt; 0.001. MCPs, mouse corpus cavernous pericyte; MCECs, mouse cavernous endothelial cells; NC mimics, negative control mimics; ns, not significant</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Small RNA sequencing analysis of Mouse cavernous pericytes (MCPs)-derived extracellular vesicles (EVs)</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Gene symbol</th><th>MCPs-EVs Raw data (RC)</th><th>Sequence</th></tr></thead><tbody><tr><td><bold>mmu-miR-148a-3p</bold></td><td><bold>15,453</bold></td><td><bold>UCAGUGCACUACAGAACUUUGU</bold></td></tr><tr><td>mmu-miR-7051-5p</td><td>13,658</td><td>UCACCAGGAGGAAGUUGGGUCA</td></tr><tr><td>mmu-miR-125a-5p</td><td>3847</td><td>UCCCUGAGACCCUUUAACCUGUGA</td></tr><tr><td>mmu-miR-151-3p</td><td>3773</td><td>CUAGACUGAGGCUCCUUGAGG</td></tr><tr><td>mmu-miR-5110</td><td>3136</td><td>GGAGGAGGUAGAGGGUGGUGGAAUU</td></tr><tr><td>mmu-miR-486b-5p</td><td>1559</td><td>UCCUGUACUGAGCUGCCCCGAG</td></tr><tr><td>mmu-miR-486a-5p</td><td>1364</td><td>UCCUGUACUGAGCUGCCCCGAG</td></tr><tr><td>mmu-miR-21a-5p</td><td>609</td><td>UAGCUUAUCAGACUGAUGUUGA</td></tr><tr><td>mmu-miR-191-5p</td><td>546</td><td>CAACGGAAUCCCAAAAGCAGCUG</td></tr><tr><td>mmu-miR-5126</td><td>372</td><td>GCGGGCGGGGCCGGGGGCGGGG</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Physiologic and metabolic parameters: 2 weeks after treatment with PBS, MCPs-EVs (RC), MCPs-EVs (miR-148a-3p i)</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th colspan=\"2\"/><th colspan=\"3\">STZ-induced diabetic mice</th></tr><tr><th/><th>Control</th><th>PBS</th><th>MCPs-EVs (RC)</th><th>MCPs-EVs (miR-148a-3p i)</th></tr></thead><tbody><tr><td><bold>Body weight (g)</bold></td><td>31.5 ± 0.6</td><td>23.7 ± 1.0*</td><td>25.3 ± 0.3*</td><td>25.8 ± 0.5*</td></tr><tr><td><bold>Fasting glucose (mg/dl)</bold></td><td>103.6 ± 2.6</td><td>551.2 ± 19.8*</td><td>567.2 ± 19.3*</td><td>549.4 ± 18.2*</td></tr><tr><td><bold>Postprandial glucose (mg/dl)</bold></td><td>168.8 ± 18.0</td><td>590.0 ± 6.4*</td><td>595.2 ± 4.8*</td><td>594.0 ± 5.7*</td></tr><tr><td><bold>MSBP (mm Hg)</bold></td><td>102.0 ± 2.6</td><td>101.2 ± 2.7</td><td>100.6 ± 1.9</td><td>102.0 ± 1.2</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p>Values are the mean ± SEM for <italic>n</italic> = 5 animals per group. <italic>RC</italic> regent control; <italic>miR</italic>-148a-3p i miR-148a-3p inhibitor; <italic>STZ</italic> streptozotocin; <italic>MSBP</italic> Mean systolic blood pressure; *<italic>P</italic> &lt; 0.05 vs. Control group</p></table-wrap-foot>", "<fn-group><fn><p>The original online version of this article was revised: the author name \"Yin Guonan\" was incorrectly written as \"Yin Gounan\" and the same has been updated.</p></fn><fn><p><bold>Publisher’s Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p><bold>Change history</bold></p><p>1/13/2024</p><p>A Correction to this paper has been published: 10.1186/s12894-023-01399-z</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"12894_2023_1378_Fig1_HTML\" id=\"MO1\"/>", "<graphic xlink:href=\"12894_2023_1378_Fig2_HTML\" id=\"MO2\"/>", "<graphic xlink:href=\"12894_2023_1378_Fig3_HTML\" id=\"MO3\"/>", "<graphic xlink:href=\"12894_2023_1378_Fig4_HTML\" id=\"MO4\"/>", "<graphic xlink:href=\"12894_2023_1378_Fig5_HTML\" id=\"MO5\"/>", "<graphic xlink:href=\"12894_2023_1378_Fig6_HTML\" id=\"MO6\"/>" ]

[ "<media xlink:href=\"12894_2023_1378_MOESM1_ESM.pdf\"><caption><p><bold>Additional file 1.</bold>\n</p></caption></media>" ]

{ "acronym": [ "MCP", "EVs", "ED", "MCECs", "PDK4", "PDE5", "NO", "DM", "mRNAs", "miRNAs", "NV", "CNI", "TUNEL", "DAPI", "ICP", "MSBP", "PH3", "FITC", "TRITC", "NG", "HG", "STZ" ], "definition": [ "Mouse corpus cavernous pericyte", "Extracellular vesicles", "Erectile dysfunction", "Mouse cavernous endothelial cell", "Pyruvate dehydrogenase kinase-4", "Phosphodiesterase 5", "Nitric oxide", "Diabetes mellitus", "Messenger RNAs", "MicroRNAs", "Nanovesicles", "Cavernous nerve injury", "Terminal deoxynucleotidyl transferase-mediated deoxyuridine triphosphate nick-end labeling", "4,6-diamidino-2-phenylindole", "Intracavernous pressure", "Mean systolic blood pressure", "Phospho-Histone H3", "Fluorescein isothiocyanate", "Tetramethylrhodamine", "Normal-glucose", "High-glucose", "Streptozotocin" ] }

[ "<title>Background</title>", "<p id=\"Par7\">Unplanned pregnancies and the prevalence of sexually transmitted diseases are significant concerns in Sub-Saharan Africa, where approximately half of all pregnancies among young women in low-income areas occur without planning [##REF##25080352##1##–##REF##30727995##3##]. Moreover, two-thirds of new cases of sexually transmitted infections originate from this region [##REF##27347270##4##], and over 80% of individuals report experiencing technology-facilitated sexual violence, gender-based violence, or other forms of violence [##REF##27347270##4##, ##UREF##0##5##]. Given the challenging circumstances in Sub-Saharan Africa, it is imperative to reduce the incidence of unplanned pregnancies and the spread of diseases. This involves ensuring that adolescents have access to information about contraception [##UREF##0##5##] and learn about sexual health to be able to make educated choices.</p>", "<p id=\"Par8\">Access to knowledge empowers individuals to make informed and sustainable decisions regarding their bodies and health. Unfortunately, adolescent, and especially Kenyan women, often lack access to essential healthcare information [##REF##34034757##6##]. The lack of access to structured, verified, valid, and reliable sexual information, especially concerning contraception, menstruation, and female genital mutilation, leaves young women vulnerable to health risks [##UREF##1##7##]. This issue is multifaceted. Reasons being, the official national school curriculum in Kenya does not include comprehensive sex education, resulting in teachers being ill-prepared to address these topics in the classroom [##REF##31433599##8##]. Furthermore, sexual health matters are influenced by religious, tribal, and social affiliations, which can lead to varied or suppressed discussions [##UREF##1##7##, ##UREF##2##9##]. The societal taboo surrounding contraceptive practices can even lead to fatal abortions to prevent unwanted pregnancies [##UREF##3##10##]. This primarily affects young Kenyan women between the ages of 18–35 in resource-poor rural areas, creating a vicious cycle.</p>", "<p id=\"Par9\">Considering the challenges in lack of infrastructure and limited financial resources and access to education and sexual health information, where on the contrary the increased use of the internet and digital technologies presents an opportunity for behavioral change [##UREF##4##11##]. Literature shows a growing focus on digital tools to address these problems [##UREF##4##11##, ##UREF##3##10##, ##UREF##1##7##, ##UREF##5##12##–##REF##32254043##16##].</p>", "<p id=\"Par10\">The use of digital tools and social media has boosted confidence in this regard of allowing feedback and dispelling misconceptions [##UREF##1##7##, ##UREF##4##11##, ##UREF##5##12##]. Providing information can contribute to reducing the risks of sexually transmitted infections like HIV and unplanned pregnancies, with great potential for educating low-income and vulnerable communities [##REF##27649758##13##, ##UREF##6##14##]. Despite a potential accessibility of digital sexual health educative tools, and willingness to share and engage with sensitive information, lasting user engagement is crucial [##REF##27649758##13##]. Therefore, it is essential to explore innovative approaches to provide easily accessible knowledge for everyone via a digital tool in regarding sensitive information. The newly developed prototype, development in alignment with the double diamond established framework according to Soehnchen et al. [##REF##37921860##17##]. serves for on-site te testing with a predefined target group in Kenya. The tool aims to be a secure platform for sensitive and intimate information. The prototype appears in form of a web-based application focusing on sexual health educative, contraception, menstrual period, and female genital mutilation information, available in English and Swahili (Fig. ##FIG##0##1##) with audio files. The tool is designed to address limited and stigmatized knowledge, among 18–35-year-old. Recognizing that the target audience has a different cultural background, and the design process originates from a European cultural perspective, it is crucial to involving potential end-users in the process. User involvement is a prerequisite for high acceptance, sustainable development, and successful implementation [##UREF##6##14##].</p>", "<p id=\"Par98\">\n\n</p>", "<p id=\"Par11\">Therefore, the study employs a multimethod approach, combining a product-centered evaluation (usability test) with a user-centered evaluation (acceptance survey) to explore the acceptability and intention to use the developed digital tool.</p>", "<title>Objectives</title>", "<p id=\"Par12\">A product-centered evaluation (usability test) and a user-centered evaluation (acceptance survey) took place, onsite in volunteer cooperation with local community centers in rural regions in Kenya. Therefore, the study aims to investigate [##REF##25080352##1##] the acceptance of the given digital tool and the behavioral intention to use the tool for sexual health information of young women in Kenya and [##REF##29485986##2##] investigation of the usability of the given digital tool for sexual health information. To achieve the research objectives, the following research questions are derived: (RQ1) What factors influence the acceptance of and behavioral intention of the named target group to use the digital tool for sexual health information? And second, (RQ2) how is the usability of the digital tool for sexual health information assessed by the mentioned target group?</p>" ]

[ "<title>Methods</title>", "<title>Derivation of the used measurements</title>", "<p id=\"Par13\">Technology Acceptance Models, like UTAUT, aim to explain effects of acceptability and intention to use of new technologies. The UTAUT model, developed by Venkatesh [##UREF##7##18##], unifies eight established research models including the Theory of Reasoned Action [##UREF##8##19##], Technology Acceptance Model (TAM) [##UREF##9##20##], Motivational Mode (MM) [##UREF##10##21##], Theory of Planned Behavior (TPB) [##UREF##11##22##], combined TAM and TPB (C-TAM-TPB) [##UREF##12##23##], Model of PC Utilization (MPCU) [##UREF##13##24##], Innovations-Diffusions-Theory (IDT) [##UREF##14##25##] and the Social Cognitive Theory (SCT) [##UREF##15##26##]. Nevertheless theUTAUT model outperformed all mentioned individual models [##UREF##7##18##].</p>", "<p id=\"Par14\">The origin UTAUT model consists of four key constructs as exogenous variables, performance expectancy (PE), effort expectancy (EE), social influence (SI) and facilitating condition (FC), as well as the moderators age, gender, experience and voluntariness of use, which are described as direct determinants of the behavioral intention (BI) to use a system [##UREF##7##18##]. Venkatesh et al. considers the discriminance between the newly assembled constructs of the model. It shows that almost all of them are significantly correlated, but this correlation is in the low range. It can be concluded that the latent variables may influence each other, yet the constructs differ from each other [##UREF##7##18##]. The UTAUT model is applied in various studies, in context of new technologies and healthcare. The settings reach from health care service delivery [##UREF##16##27##], mobile health communication and/or information systems [##UREF##1##7##, ##UREF##17##28##, ##UREF##18##29##], acceptance and use of social media in the healthcare context [##UREF##4##11##] and mobile (sexual) health apps [##REF##32254043##16##].</p>", "<p id=\"Par15\">To cover the particularities and complexity of healthcare setting respectively, several modifications have been proposed in the UTAUT model by the former mentioned studies. This is also the case in this study. Inspired by Olamijuwon et al. [##UREF##4##11##], the modified revision of the UTAUT model proposed by Dwivedi [##UREF##19##30##] is chosen as an appropriate theoretical framework for the study. The intention behind this, is to consider individual characteristics (users’ attitude towards technology and technological anxiety) that are not seen as direct determinants of the intention to use a system in the origin UTAUT model. In addition, the modified model is specified to the context of sexual health education for a digital tool in Kenya. Figure ##FIG##1##2## shows the framework of the research model proposed to this study, based on UTAUT.</p>", "<p id=\"Par16\">\n\n</p>", "<p id=\"Par17\">The constructs of the model, which may determine the intention to use a new technology (BI) and the corresponding hypotheses are described below.</p>", "<p id=\"Par18\"><bold>Performance expectancy</bold> describes the factor of perceived usefulness (PU), which demonstrates the degree of a person’s belief that the usage of the new technology will help the user [##UREF##7##18##, ##UREF##9##20##]. Venkatesh et al. and Dwivedi et al. observed and underline the construct as a direct determinant influencing BI [##UREF##7##18##, ##UREF##19##30##]. Previous acceptance and usability studies, in the context of digital health, also indicate that the construct has a significant positive impact on BI [##UREF##1##7##, ##UREF##4##11##, ##UREF##18##29##]. Hence, the associated research hypothesis can be phrased as follows:</p>", "<title>H1</title>", "<p id=\"Par19\">PU has a significant impact on the behavioral intention to use the digital tool for sexuality information.</p>", "<p><bold>Effort expectancy</bold> (EE) describes the degree of perceived ease of use with the technology [##UREF##7##18##, ##UREF##9##20##], as this construct is determining to influence BI positively [##UREF##7##18##, ##UREF##19##30##]. In the context of the intention to use digital technologies in healthcare, some former studies, rejected a significant impact of EE on BI [##REF##31433599##8##, ##UREF##19##30##–##UREF##21##32##]. However, an in-depth literature research shows that studies, explicitly, in context of sexual information and e-health technology in developing countries, were able to demonstrate a significant positive impact of EE on BI [##UREF##4##11##, ##UREF##20##31##]. Following this, the associated research hypothesis can be phrased as follows:</p>", "<title>H2</title>", "<p id=\"Par21\">EE has a significant impact on the behavioral intention to use the digital tool for sexuality information.</p>", "<p><bold>Social influence</bold> (SI) describes the perceived extent to which an individual believes that another person or group expects the individual user to use the technology. SI is seen as a direct determinant of BI [##UREF##7##18##]. The dynamic of the nature of cultural sensitive sexual health information, and the associated role of religion, are both considered, within the SI construct [##UREF##4##11##]. Previous studies mostly have found a significant positive impact of SI on the intention to use digital technologies in health-related topics [##UREF##1##7##, ##UREF##4##11##, ##UREF##16##27##, ##UREF##18##29##, ##UREF##20##31##], but also a negative impact was found in a related study [##UREF##18##29##]. The associated research hypothesis can be phrased as follows:</p>", "<title>H3</title>", "<p id=\"Par23\">SI has a significant impact on the behavioral intention to use the digital tool for sexuality information.</p>", "<p><bold>Facilitating conditions</bold> (FC) describes the perceived extent of an individual, supporting the existence of organizational and technical infrastructure to use of technology [##UREF##7##18##]. The construct is seen as a direct determinant of BI [##UREF##19##30##]. Previous research studies show a significant positive impact of FC on BI in the setting of mobile health communication and/or information systems [##UREF##1##7##, ##UREF##17##28##, ##UREF##18##29##, ##UREF##20##31##] as well as acceptance and use of social media in the healthcare context [##UREF##4##11##, ##UREF##20##31##]. The associated research hypothesis can be phrased as follows:</p>", "<title>H4</title>", "<p id=\"Par25\">FC has a significant impact on the behavioral intention to use the digital tool for sexuality information.</p>", "<p><bold>Attitude towards technology use</bold> (ATU) describes the positive and negative feelings of an individual about performing the targeted behavior, and is seen as a construct with a direct positive impact on BI [##UREF##19##30##]. Other context relevant studies also depict ATU as a construct with significant positive impact on BI [##UREF##1##7##, ##UREF##4##11##, ##UREF##16##27##]. Another study found a significant negative impact [##UREF##1##7##]. The associated research hypothesis can be phrased as follows:</p>", "<title>H5</title>", "<p id=\"Par27\">ATU has a significant impact on the behavioral intention to use the digital tool for sexuality information.</p>", "<p>Supplemental to the UTAUT method, for a product-centered evaluation, the System Usability Scale (SUS) questionnaire is used. SUS was developed by John Brook in 1986 with the aim to create a simple standardized questionnaire to evaluate people’s perception of usability of a computer systems in a short time [##UREF##21##32##]. Since its creation, the questionnaire has been used in numerous scenarios and has become established as an effective, valid and reliable tool for assessing the usability of a wide variety of different products and services [##UREF##22##33##]. It functions as an industry standard and its widespread usage, makes it possible to compare the results, one has achieved with other products and systems [##UREF##21##32##]. In addition, a comparison of different versions of one product with itself, is also quick and inexpensive.</p>", "<title>Procedure and survey design</title>", "<p id=\"Par29\">To explore the acceptability and intention to use of the developed digital tool for sexual health information (focusing on contraception product information and menstrual period) in Kenya a multimethod cross-sectional survey is used to collect information from a convenience sample. The total population of Kenya is 54.56 million in January 2023, of which 50.4% are female. Nearly 30% of the population is between 18 and 34 years old [##UREF##23##34##], which is roughly the targeted end-users of the developed digital tool, and addressed in this survey. As mentioned above, the theoretical basis of the study is mainly UTAUT (addressing RQ1) and complementary SUS (addressing RQ2). The ethic committee of the Witten/Herdecke University approved the research project on 06.08.2022 under the protocol code: S-119/2022.</p>", "<p id=\"Par30\">For the procedure of quantitative research, all participants were offered a smartphone or a laptop, in case they do not have any digital device themselves. The volunteering participants had the opportunity to explore the developed prototype [##REF##37921860##17##] for 20 min. They were able to read through all content and listen to the video files. After 20 min, the participants filled out the survey via smartphone or laptop.</p>", "<p id=\"Par31\">The survey consists of three sections. Table ##TAB##0##1## displays an overview of the questions. Section A contains a general information survey with nine questions to get demographic information (e.g., sex, age, education level, work status) and information about electronic device ownership and type of use regarding to health information from respondents. In section B, six questions are used to elicit responses to different statements, appropriate to the six constructs of our specified model. The statements serve as multiple questionnaire items, adopted from Venkatesh et al. and rewritten for the purpose of this study [##UREF##7##18##]. There are between four and eight statements per construct. All constructs are measured on a 7-point Likert scale from 1 (strongly disagree) to 7 (strongly agree).</p>", "<p id=\"Par32\">\n\n</p>", "<p id=\"Par33\">Section C contains the standardized SUS questionnaire. It is a 10-item-questionnaire measuring on a 5-point Likert response format scale from 1 (strongly disagree) to 5 (strongly agree).</p>", "<p id=\"Par34\">Pre-testing was conducted, using a sample size of women presented in the community centers in Kenya, as well as women in Germany. The reporting of the e-survey results, the Checklist for Reporting Results of Internet E-surveys (CHERRIES) as suggested by Eysenbach, is considered, see Table ##TAB##1##2##.</p>", "<p id=\"Par35\">\n\n</p>", "<title>Participants</title>", "<p id=\"Par36\">The survey period was from 01.12.2022 to 31.01.2023. The online survey was available during the period and is distributed via contact persons in community centers, girl schools, and universities in rural areas in Kenya, using a convenience sampling technique. The target group of this study is predominantly female, from rural areas in Kenya between the age of 18–35 years. To support the participants and to achieve a higher response quote, the survey has been also accompanied in person from 11.01.2023 to 23.01.2023. For this purpose, various community centers in Eldoret, of Making More Health (MMH), as well as MOI University in Eldoret and Learning Lions have been visited for three days each. To create a pleasant atmosphere for the sensitive topic, the personal accompaniment consists of a woman from the Fraunhofer ISST in Germany and a trusted person from the respective institution on site. This procedure had the aim to counteract possible problems of comprehension in terms of content and language.</p>", "<title>Statistical analysis</title>", "<p id=\"Par37\">The statistical program R [##UREF##24##35##] and in particular the package lavaan [##UREF##25##36##] were used for data analysis. Based on our multimethod approach, the data analysis is split into three parts: [##REF##25080352##1##] a descriptive analysis [##REF##29485986##2##], a SEM, and [##REF##30727995##3##] an evaluation of SUS-model. First, a descriptive analysis is used to describe the sample characteristics of the 77 respondents. Second, the SEM, consisting of CFA and linear regression [##UREF##26##37##] Klicken oder tippen Sie hier, um Text einzugeben. is used to analyze the relationships in the UTAUT model and to test the hypotheses. Further, an evaluation of SUS is carried out according to Brooke [##UREF##21##32##]. For the five positively worded questions, the score contribution is the numerical scale position. Respectively, for the remaining five negatively worded questions the score contribution is the reverse scale position. The overall value of SUS amounts to a value between 0 and 100 [##UREF##21##32##]. As a rule of thumb, with a score of 68 or above, SUS is considered as good [##UREF##21##32##, ##UREF##22##33##].</p>" ]

[ "<title>Results</title>", "<title>Descriptive analysis</title>", "<p id=\"Par38\">A total of 96 people took part in the survey. About N = 77 responses were assessed as complete and consequently included in the analysis. The descriptive characteristics of the participants are shown in Table ##TAB##2##3##.</p>", "<p id=\"Par39\">\n\n</p>", "<p id=\"Par40\">As intended, the study main respondents are women, located in rural Kenya. Most participants preferred to fill out the survey in English, over their tribal dialect, as Swahili has many different forms, is constantly changing, and many are not able to understand it. Participants between 20 and 30 years constitute the largest group, of respondents, specifically, most of them were born in 2000 (17/77, 22%). In line with this observation, most participants report being a student, followed by a current working status as unemployed and working. Most of the participants have secondary or tertiary education. The respective relationship status is relatively evenly represented. In terms of owning technical items, it is noticed that, despite one person, every participant who owns technical devices has at least a smartphone. Regarding online search behavior it is shown, that the more specific the search intention towards sexual health, the less often an online search takes place. Most participants described their digital experience as, at least as good. Through this we assume that the participants can compare the given tool with other digital experiences and use their experiences to navigate through the tool. The items, working status, education, relationship status and digital experience will be examined in context to the latent construct of BI, more precisely later. As the items are aiming to be included in the selection for the following regression analysis.</p>", "<p id=\"Par41\">Table ##TAB##3##4## displays mean and standard deviation per item of the questionnaire. A 7-point scale was applied to all questions. With a mean value resulting in of 5.9 to 6.1, the answers to the eight questions of the first both constructs, perceived usefulness, and perceived ease of use, are close to each other and indicate good feedback on the digital tool. The four questions of the BI construct, also perform well. The averages are close to each other. The mean values of the questions on SI are in the lower half of the scale. The scatter is larger than for the previously mentioned constructs. The moments on facilitating conditions do not give a consistent pattern. At ATU, the answers to the first four questions are in the upper range of the scale. The negatively directed questions’ averages TA2 - TA4 are in the lower range. TA1 stands out with a comparatively high mean value of 4.0. However, the standard deviation is also relatively high.</p>", "<p id=\"Par42\">\n\n</p>", "<title>Structural equation model</title>", "<p id=\"Par43\">It was decided to use SEM in this case, which is a CFA, followed by regression to address the evaluation of the questionnaire, as a measuring instrument for the latent variables. Χ² test for model conformity, performing on the null hypothesis, that the predicted model and observed data are equal. Further Comparative Fit Index (CFI), Root Mean Square Error of Approximation (RMSEA) and Standardized Root Mean Residual (SRMR) were considered.</p>", "<p id=\"Par44\">First, the model was specified, according to the UTAUT model (see Fig. ##FIG##1##2##). To identify the model, the variances of the latent variables in the CFA were set to 1. 12 observations were removed, due to missing values. Hence the number of observations used was 65 (65/77, 84.4%).</p>", "<p id=\"Par45\">Considering factor loadings and internal correlations, it was noted that questions FC1 - FC4 (s. Table ##TAB##0##1##), which correspond to the FC construct, do not tend to indicate a common latent variable. Moreover, the standardized factor loading of FC3, leads to a numerical problem: the factor loading is estimated to be &gt; 1 and the resulting estimated variance, is negative. Additionally, this also clearly indicates a content error in the model. Therefore, the items FC1 - FC4 are removed from the model.</p>", "<p id=\"Par46\">The next step intends to improve the indices by extending the model, and thus arrive at a better-fitting model. For this purpose, the correlation between items, as well as the use of one item for several latent variables, will be allowed. Hence, the modification indices are computed. The proposals for respecification are selected step by step. For each decision, question whether the extension is meaningful, in terms of content, and select the variant that promises a large improvement in the Χ^2, distribution, test statistic for the hypothesis of model conformity. The use of the items BI1- BI4, within the other constructs, is avoided. As this is supposed to be our target variable. After several respecification steps, the model shown in Fig. ##FIG##2##3## is obtained. The corresponding CFA output, concerning factor loadings and interactions, are shown in Table ##TAB##4##5##.</p>", "<p id=\"Par102\">\n\n</p>", "<p id=\"Par47\">\n\n</p>", "<p id=\"Par48\">The Χ² test statistic amounts to χ²₂₅₉ <italic>=</italic> 532, <italic>p</italic> &lt; .001 and is thus, compared to the original model, is still highly significant. The CFI improves remarkably by 0.16 to 0.75, compared to the primary model. An improvement can also be seen in the other indices, leading to RMSEA = 0.13 and SRMR = 0.14. However, the criteria of fit indices according to Hu and Bentler [##UREF##27##38##], is still not fulfilled.</p>", "<p id=\"Par49\">The quality of the measurement model is evaluated by considering indicator reliability, internal reliability of the constructs, discriminant validity, as well as convergence validity.</p>", "<p id=\"Par50\">Indicator reliability is measured through standardized indicator loadings. Figure ##FIG##2##3## shows that for PU and EE, three out of four items are &gt; |0.7|. The remaining are still satisfactory with &gt; |0.5|. For EE, as mentioned, the item PePu4 is added, which has an acceptable charge. Four of five SI loadings SI are &gt; |0.7|, one is only slightly below. For ATU, half of the loadings are &lt; |0.4|, namely ATU2, ATU3, ATU4 and TA1. These items have negligible contribution to the measurement of the latent variable. With around |0.6| the loadings of ATU1 and TA2 can be classified as moderate. Finally, considering the factor loadings of BI. BI1 and BI2 load rather weakly on the latent variable, with values between |0.4| and |0.5|. BI3 and BI4 on the other hand are remarkably strong. Again, PePu4 has a moderate loading on BI, as an addition respecification.</p>", "<p id=\"Par51\">Internal reliability is examined, considering Cronbach α, as shown in Table ##TAB##5##6##. It turns out that PU, EE as well as SI, show a very good strength of association and the indicator for BI is classified as good, according to the rules of thumb [##UREF##28##39##]. ATU, on the other hand, is somewhat weak. One should keep in mind, that with eight questions, remarkably more items are chosen for this construct, compared to others.</p>", "<p id=\"Par52\">\n\n</p>", "<p id=\"Par53\">Convergent validity and discriminant validity are assessed by considering the correlation between the latent variables of Table ##TAB##6##7##.</p>", "<p id=\"Par54\">\n\n</p>", "<p id=\"Par55\">The discrimination between SI and ATU is rather unsatisfying. On one hand, a greater convergence between BI, and on the other latent variables would be desirable. The highest of these correlations is between BI and PU.</p>", "<p id=\"Par56\">The second step of SEM focuses on the question, of what influence the constructs and variables have on the latent target variable BI.</p>", "<p id=\"Par57\">First, some control variables were considered. Boxplots were used to examine whether the attributes of the categorial variables show a difference on the outcome of the latent construct BI. See Fig. ##FIG##3##4## for the control variable working status, Fig. ##FIG##4##5## for education, Fig. ##FIG##5##6## for relationship status and for digital experience see Fig. ##FIG##6##7##. No evident structural variations can be found, because the boxes overlap to a large extent. Therefore, these items were not used for further investigation.</p>", "<p id=\"Par104\">\n\n</p>", "<p id=\"Par106\">\n\n</p>", "<p id=\"Par108\">\n\n</p>", "<p id=\"Par110\">\n\n</p>", "<p id=\"Par58\">Subsequent to the factor analysis, a multiple linear regression is employed to identify relationships between the eight identified independent variables (PU, EE, ATU, SI, FC1, FC2, FC3, FC4) and one dependent variable (BI). The results, depicted in Table ##TAB##7##8##; Fig. ##FIG##7##8##, show that the constructs PU, EE, SI and ATU, as well as FC3, positively influence the behavioral intention to use the digital tool for sexual health information, while FC1 and FC2 influence BI negatively. The usefulness of the tool (PU) has a particularly strong influence on BI. Likewise, ATU, FC1, FC2 and FC3 have, by a level of 5%, a significant impact on the target. Item FC4, which is intended to collect information about whether the user would accept technical assistance from a person, has virtually no impact on the outcome of BI. Given the high variance of responses for item FC4 (M = 4.4, SD = 2.3), see Table ##TAB##3##4## above, it cannot be due to the fact that technical assistance was not considered by the participants.</p>", "<p id=\"Par112\">\n\n</p>", "<p id=\"Par59\">\n\n</p>", "<p id=\"Par60\">R² is for the overall model 0.51, which implies that half of the variance for the dependent construct BI, is explained through the eight independent variables in the regression model.</p>", "<p id=\"Par61\">The correlation between year of birth and BI amounts to − 0.11, which is quite weak. Adding year of birth as numerical predictors, does not improve the regression model and is therefore discarded.</p>", "<p id=\"Par62\">In response to RQ1, based on the results of the multiple linear regression analysis the following hypothesis are supported or not evidenced:</p>", "<title>H1</title>", "<p id=\"Par63\">Perceived usefulness has a significant impact on the behavioral intention to use the digital tool for sexual information. à Supported.</p>", "<title>H2</title>", "<p id=\"Par64\">Perceived ease of use has a significant impact on the behavioral intention to use the digital tool for sexual health information. à Not evidenced.</p>", "<title>H3</title>", "<p id=\"Par65\">Social Influence has a significant impact on the behavioral intention to use the digital tool for sexual health information. à Not evidenced.</p>", "<title>H4</title>", "<p id=\"Par66\">Facilitating conditions have a significant impact on the behavioral intention to use the digital tool for sexual health information. à Differentiated considerations necessary.</p>", "<title>H5</title>", "<p id=\"Par67\">Attitude to use has a significant impact on the behavioral intention to use the digital tool for sexual health information. à Supported.</p>", "<title>SUS</title>", "<p id=\"Par68\">There are 13 observations removed due to missing values. Hence the number of observations used for the SUS is 64 (64/77, 83.1%). Ten questions pertain to SUS, see Section C of the Questionnaire in Table ##TAB##0##1##, (s. Table ##TAB##0##1##) each with a 5-point scale. According to Brook [##UREF##21##32##], for positively worded questions (question 1, 3, 5, 7, 9), the score contribution is the scale position. Consequently, for negatively worded questions (question 2, 4, 6, 8, 10), the score contribution is the reversed scale position. To interpret the SUS score, it can be plotted on a scale from 0 to 100, with 100 being the best possible score. A SUS score of 67.3 is determined. The rating of the scale position can be done via acceptability scores or adjective ratings. Regarding acceptability, a score below 50 is described as unacceptable, above 70 as acceptable and in between as marginal. In terms of adjective ratings, from a score of 52, the SUS is considered as “okay” and from above average at 71 as “good” [##UREF##21##32##, ##UREF##22##33##]. Thus, the SUS score of this study can be classified as “high marginal” to nearly “acceptable” and in terms of adjective ratings described as “okay”, nearly “good”.</p>", "<p id=\"Par69\">The RQ2, “How is the usability of the digital tool for sexual health information assessed?”, can therefore be answered with: The usability of the digital tool for sexual health information is assessed as ok and high marginal. This implies that there is still potential for improvement.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par70\">This study investigated the acceptance to use and usability of a newly developed digital tool for sexual health education of young adults in Kenya, concerning the utilization of a modified UTAUT model. The analysis and a comparison of the results with prior studies confirm the applicability of the modified revision of UTAUT model proposed by Dwivedi with additional specifications to meet the context of sexual health education throughout the given digital tool in Kenya.</p>", "<p id=\"Par71\">Among the significant explanatory features, perceived usefulness has the greatest influence on the behavioral intention. It has also been depicted as a construct with significant effect on the intention to use, in prior studies [##UREF##1##7##, ##UREF##7##18##, ##UREF##18##29##, ##UREF##19##30##]. Specifically, aligning with former studies, it has been identified as the construct with the strongest impact on BI, compared to the other constructs [##UREF##1##7##, ##UREF##7##18##, ##UREF##18##29##]. Hence, the usefulness of the tool is in the first place for usage behavior. Participating women consider the tool to be useful for providing health information, which could lead to a contribution of their sexual and reproductive health improvement, as well as enhancing users to make better and more informed decisions for their mental and physical health state.</p>", "<p id=\"Par72\">To consider the individual characteristics of the respondents (users’ attitude towards technology and technology anxiety), the origin UTAUT model of Venkatesh is extended by the construct ATU [##UREF##4##11##, ##UREF##19##30##]. The significant effects of the construct attitude to use on BI, conforms with results of Chilliers, Olamijuwon, and Dwiwedi [##UREF##1##7##, ##UREF##4##11##, ##UREF##19##30##]. The results indicate the positive and negative feelings of the women using digital tools for sexual health information, which influences the actual use. In addition, a rather weak internal consistency of ATU was observed. Therefore, for further research it is proposed to reduce the number of items, as the questions ATU2, ATU3, ATU4 and TA1 are less suitable for measuring the latent variable. It may be helpful to combine ATU2, ATU3 and ATU4 to one item, as they turn out to be highly correlated. Furthermore, one could consider reformulating the questions to improve the discrimination between the constructs ATU and SI.</p>", "<p id=\"Par73\">Supplementary research could examine construct ATU more specifically. Potential research directions could include the investigation of the interactions between ATU and the other constructs. As well as examining the attitudes of young women in Sub-Sharan Africa towards digital sexual health information through an application what kind of attitude the participants have towards the digital tool for sexual health information tools. As well as how deep religious and cultural believes determine the level of education. Further the acceptability and intention to use a digital tool by men, to inform themselves about female sexual health, could be explored. This could contribute to a holistic sexual education approach, strengthen an understanding of female well-being, and create sensitivity towards sexual health as well as takeing over responsibility to make choices without religion and cultural stigmata.</p>", "<p id=\"Par74\">Regarding the construct, facilitating conditions, a differentiated consideration of the results is necessary. This can be explained with the finding, of the not well represented latent construct by the underlying questions. The questions are ambiguous and represent several aspects. This can be explained by the heterogeneous nature of the questions within this construct. Therefore, the construct is not considered in the CFA, but the individual questions are included in the further analysis. Upcoming research could construct FC in a more differentiated way and divide it into several constructs.</p>", "<p id=\"Par75\">Our results show that three out of four measured FC-items, have a significant impact on BI. Prior similar studies have also identified a significant impact of FC on BI [##UREF##4##11##, ##UREF##17##28##–##UREF##19##30##, ##UREF##26##37##, ##UREF##29##40##]. However, in contrast to the former mentioned studies, negative effects are evident. Based on the content of the question, one possible explanation for the negative effect of FC1 and FC2 could be, that people who have the necessary digital resources for getting informed available (smartphone, Wi-Fi, etc.), have the necessary knowledge to use a digital tool (e.g., basic digital knowledge). They do not have a high need for the developed web-based application, as they can get information from other digital sources. However, no direct effect between the degree of education and the intention to use the tool was apparent, as shown in Fig. ##FIG##4##5##.</p>", "<p id=\"Par76\">The not evidenced statistically significant impact of effort expectancy on the participants BI, aligns with the results of Akinnuwesi, Chilliers and Khatun [##UREF##1##7##, ##UREF##17##28##, ##UREF##18##29##].</p>", "<p id=\"Par77\">Regarding the significant impact of social influence, the literature shows opposing results. In contrast to some earlier studies [##UREF##18##29##], in this study, social influence has no significant impact on BI. In terms of acceptance and the use of a new cloud-based mHealth technology, Khatuns results also show, that SI has no significant effect on BI [##UREF##17##28##]. To interpret the results, it is necessary to consider the context and cultural surrounding of the participants in this study. There might be a selection bias, as people who were willing to participate in such a study in the first place do not mind talking about sexual health information and may tend to be more open-minded and progressive. This suggests that participants may feel less constrained by factors like family, friends, people with the same religious beliefs or religious belief itself, when using the given tool for sexual health information. The findings show a correlation between SI and the construct ATU, which indicates that SI and ATU influence each other, which is consistent with literature [##UREF##4##11##, ##UREF##19##30##], and with the observation during the procedure of the questionnaire. It was observed that the participants in rural regions of Kenya where willing and interested in using the tool, but perceived with a potential bias.</p>", "<p id=\"Par78\">The fit indices CFI, SRMR, and RMSEA, as well as Cronbach α indicate a rough direction, but they should be viewed with caution for small data sets, as is the case here. Some rules of thumb may not be applicable.</p>", "<p id=\"Par79\">Regarding RQ2, (How is the usability of the digital tool for sexual health information assessed by the mentioned target group?), the usability of the developed web-based application for sexual health information can be, with a SUS score of 67.3, interpreted as okay and high marginal. In context of digital tools for female sexual health, literature shows, SUS scores which are above a score of 70, for average usability [##UREF##30##41##, ##REF##34076578##42##]. For example, Dubinskaya et al. investigated existing applications addressing female sexual health, focusing on the educational content and indicating for one application, a SUS score of 70, and for five further apps, a score of 97.5 [##UREF##30##41##]. In addition, SUS is also applied in terms of gamified applications digital interventions focusing on sexual health education. Here, literature reveals for a sexual health education mobile game a SUS score of 77 [##REF##32012041##15##] and a score of 68.4 for a digital infertility prevention training [##UREF##31##43##]. When comparing the SUS scores of those studies with those in literature, it should be noted that the digital tool of this study is a prototype. Therefore, the obtained SUS score helps to rank the usability and identifies a potential for usability improvement. Further studies should examine the comparison between the prior mentioned digital tools and a SMS based sexual health information platform.</p>", "<p id=\"Par80\">During the study, additional content was requested by the participants, after viewing the prototype, concerning cervical and breast cancer, as well as a parenting style and guidelines for young parents. The parenting and reproductive health knowledge shall be illustrated, easy and comprehendible, preferred in illustrated video format in English.</p>", "<p id=\"Par81\">Nevertheless, there are limitations within the study that need to be taken into consideration. It should be considered that the generalization of results of the study regarding digital tools for sexual health information is limited to the specifically developed web-based application prototype, as well as too the defined constructs in Chap. 2.1, of the UTAUT model. In addition, a generalization should be tempered with caution, because a convenience sample is used and therefore findings may not be generalizable to a larger population. Throughout the use of SUS, a frequently used measurement, the lack of generalization is addressed, as well as a standard outcome to compare with other studies. Further research should consider, standardized measurements, like SUS, in addition to specified methods to the study context, in order to create comparability [##UREF##32##44##].</p>", "<p id=\"Par82\">Based on the findings of this study, theoretical and practical implications can be derived. Starting with the theoretical implications, the results of the study contribute to theory in several ways.</p>", "<p id=\"Par83\">Firstly, the research makes a fundamental contribution to theory, by enriching e-health research in the context of a very sensitive topic in a developing country and rural areas. By doing so, the results provide insights of theoretical of predictors with a significant impact on the behavioral intention to use e-health technology in this case, a digital tool for sexuality information by young adults, primarily women, between 18 and 35 in developing countries. Secondly, the study concludes, the modified revision of the UTAUT model, proposed by Dwivedi [##UREF##19##30##] and SUS, are applicable to digital health research and accordingly extend the theoretical application of the validated methods, to the intention to use a digital tool for sexual health education by young adults in Kenya. In addition, by using these methods the comparison basis in digital health research is extended. Nevertheless, it should be considered that the origin UTAUT model was not developed for the context of e-health research, and therefore adaptations to the own study context are advisable. Following this, inspired by Olamijuwon et al. [##UREF##4##11##], the origin UTAUT model was extended according to Dwivedi with determinants considering the individual characteristics, and further specify the modified model to the context of sexual health education through cultural norms in Kenya.</p>", "<p id=\"Par84\">Derived from these studies, there is also a practical implication for the adoption of digital tools in context of sexual health education, in developing countries. The study findings provide practical information for the design and implementation of a digital health tool delivering sexual health information in a developing country, besides Soehnchen et al. [##REF##37921860##17##]. Especially for e-health technology designers and researchers, the findings are beneficial to understand challenges of a specific context. As discussed before, considering the theoretical constructs PE, FC and ATU is imperative for the acceptance and intention to use a digital tool for sexual health education by young Kenyan adults.</p>" ]

[ "<title>Conclusions</title>", "<p id=\"Par85\">This research study describes a validated approach assessing acceptance and usability of a newly developed digital tool designed for sexual health education in Kenya. The study adopted the Unified Theory of Acceptance and Use of Technology (UTAUT) and the System Usability Scale (SUS) to this specific context, recognizing the need to account for unique cultural, religious, and educational factors in Kenya. The study places a distinctive focus on the acceptability and intention to use of the developed digital tool, which covers sensitive topics, such as contraception information, menstrual period education, and video educative materials concerning female genital mutilation. In doing so, the study hints the profound impact of the correlation on expectancy, social influence, facilitating conditions, behavior, and overall technology affiliation while shedding light on the specific context of local culture, religion, and educational levels on technology acceptance.</p>", "<p id=\"Par86\">The findings reveal a high intention to use the digital tool, among participants, scoring approximately 6.1 to 6.2 on a 7-point scale to use. Highlighting that the newly developed digital tool, effectively fulfills its purpose of delivering comprehensive sexual health education.</p>", "<p id=\"Par87\">Perceived expectancy (PU), and attitude to use (ATU) underline the verified positive impact on an individual user in its behavioral intention (BI) to use the digital tool for sexuality information, and show its potential, while significantly impacting the social influence (SI) factor. Social influence (SI) did not demonstrate statistical significance in this study, nevertheless, its presence should not be underestimated. Notably, participants situated in the rural community center, where the prototype was tested, exhibited a higher level of receptiveness to innovative tools, due to their pre-existing access to digital infrastructure. However, factors of facilitating conditions (FC) and effort expectancy (EE) differ in significance of impact in behavioral intention to use the digital tool. This specific context adds to the novelty of findings.</p>", "<p id=\"Par88\">In conclusion, the study tailored survey methods and approach for acceptance and usability for sexual health education in Kenya, as well as the high intention to use, scores the tool’s potential effectiveness in addressing crucial sexual health topics, in this specific cultural and educational setting. Additionally, it paves the way for further investigation into the influence of cultural and religious factors on technology and digital educational tools.</p>" ]

[ "<title>Background</title>", "<p id=\"Par1\">Unplanned pregnancies and sexually transmitted diseases are a concern in Sub-Saharan Africa, particularly in low-income areas. Access to sexual health information is limited, partly due to the absence of comprehensive sex education in the national school curriculum and social taboos. In response to these challenges, this study introduces a web-based prototype, designed to provide essential sexual health information, targeting 18 to 35-year-old Kenyans, focusing on contraception, menstruation, and female genital mutilation.</p>", "<title>Method</title>", "<p id=\"Par2\">Aiming to investigate young adults’ behavioral intention to use a digital tool for sexuality education, by analyzing factors affecting acceptance and usability in low-income and resource-poor regions in Kenya. To explore the acceptability and use of the developed digital tool, this study used a modified version of the Unified Theory of Acceptance and Use of Technology (UTAUT), complemented by the System Usability Scale (SUS) questionnaire. For statistical analysis, a Structural Equation Model (SEM) including Confirmatory Factor Analysis (CFA) and Linear Regression was used. Regarding the reporting of the E-survey results, the Checklist for Reporting Results of Internet E-surveys (CHERRIES), was considered.</p>", "<title>Results</title>", "<p id=\"Par3\">Survey information from 77 persons (69 female, 7 male, 1 diverse) were collected. A modified UTAUT appears as an appropriate model for measuring the constructs and integrating evidence-based approaches to advanced and safe sexual healthcare information. Results from the SEM showed perceived usefulness, attitude towards healthcare integrated evidence technology, and usability as well as having a significant positive impact on the acceptance, the intention to use as well as wellbeing. Having the resources and knowledge necessary for the usage of a digital tool turns out to have a significant negative impact. A SUS score of 67.3 indicates the usability of the tool for sexual health information, assessed as okay.</p>", "<title>Conclusions</title>", "<p id=\"Par4\">The study adopts validated methods to assess the acceptability and usability of a digital sexual health education tool in Kenya. Emphasizing its potential effectiveness and highlighting the influence of cultural and contextual factors on technology adoption.</p>", "<title>Keywords</title>" ]

[]

[ "<title>Acknowledgements</title>", "<p>We would like to acknowledge the support given in the community centers in Eldoret, Kisumu, Lodwar and Nairobi for contributing and welcoming us. Additional we would like to thank all volunteers for participating in the study.</p>", "<title>Author contributions</title>", "<p>Conceptualization, Clarissa Soehnchen; methodology Clarissa Soehnchen, Vera Weirauch.; validation, Clarissa Soehnchen, Vera Weirauch; formal analysis, Clarissa Soehnchen, Vera Weirauch, Rebecca Schmook; investigation, Clarissa Soehnchen; resources, Clarissa Soehnchen; data curation, Clarissa Soehnchen, Vera Weirauch; writing—original draft preparation, Clarissa Soehnchen, Vera Weirauch, Rebecca Schmook.; writing—review and editing, Clarissa Soehnchen, Vera Weirauch, Rebecca Schmook, Sven. Meister; supervision, Maike Henningsen, Sven Meister.</p>", "<title>Funding</title>", "<p>Open Access funding enabled and organized by Projekt DEAL.</p>", "<title>Data Availability</title>", "<p>According to the guidelines for good scientific practice and research data management of Witten/Herdecke University, data can be requested by contacting the corresponding author.</p>", "<title>Declarations</title>", "<title>Ethics approval and consent to participate</title>", "<p id=\"Par301\">The study is conducted in accordance with the Declaration of Helsinki and approved by the Institutional Review Board (or Ethics Committee) of Witten/Herdecke University with protocol code: S-119/2022 and date of approval: 06.08.2022. Informed consent to participate in the study was obtained from all subjects involved in the study through the LimeSurvey software, participating patients cannot be identified.</p>", "<title>Consent of publication</title>", "<p id=\"Par302\">Not applicable</p>", "<title>Competing interests</title>", "<p id=\"Par115\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Prototype screenshot</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>UTAUT: Research model of this study</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>Final respecified factor model</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><p>Boxplots of the latent target variable BI depending on the working status of the study participants</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><p>Boxplots of the latent target variable BI depending on the education of the study participants</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><p>Boxplots of the latent target variable BI depending on the relationship status of the study participants</p></caption></fig>", "<fig id=\"Fig7\"><label>Fig. 7</label><caption><p>Boxplots of the latent target variable BI depending on the study participants’ digital experience with the tool</p></caption></fig>", "<fig id=\"Fig8\"><label>Fig. 8</label><caption><p>Regression Model: Architectural illustration</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Questionnaire of the survey</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr><td align=\"left\" colspan=\"4\">\n<bold>Section A</bold>\n</td></tr><tr><td align=\"left\" colspan=\"3\">\n<bold>Question English</bold>\n</td><td align=\"left\">\n<bold>Question Swahili</bold>\n</td></tr><tr><td align=\"left\" colspan=\"3\">My gender is ...</td><td align=\"left\">Jinsia yangu ni ...</td></tr><tr><td align=\"left\" colspan=\"3\">Where are you located?</td><td align=\"left\">Mahali uliko?</td></tr><tr><td align=\"left\" colspan=\"3\">What is your year of birth?</td><td align=\"left\">Mwaka wa kuzali?</td></tr><tr><td align=\"left\" colspan=\"3\">What is your current working status?</td><td align=\"left\">Hali yako ya ajira?</td></tr><tr><td align=\"left\" colspan=\"3\">What is your highest educational level?</td><td align=\"left\">Kiwango chako cha juu cha elimu?</td></tr><tr><td align=\"left\" colspan=\"3\">What is your relationship status?</td><td align=\"left\">Hali ya Uhusiano wako ni gani?</td></tr><tr><td align=\"left\" colspan=\"3\"><p>Do you own any of the following items?</p><p>Computer; Smartphone, Tablet, None</p></td><td align=\"left\"><p>Je, unamiliki mojawapo ya hivi vifaa?</p><p>Tarakilishi; Simu ya rununu ya ‘Android’; Simu ya Tabuleti; La</p></td></tr><tr><td align=\"left\" colspan=\"3\"><p>If you want to get specific information. How often do you ...</p><p>Search online for information; Search online for health information; Search online for sexual health information</p></td><td align=\"left\"><p>Ikiwa unataka kupata habari maalum. Ni mara ngapi wewe...</p><p>kutafuta ujumbe kwa mtandao; tafuta ujumbe wa afya mtandaoni; kutafuta ujumbe wa afya ya kujumuiana kingono mtandaoni</p></td></tr><tr><td align=\"left\" colspan=\"3\">Please rate your digital experience.</td><td align=\"left\">Je, matumizi yako ya kidijitali ni mazuri kwa kiasi gani? Tafadhali pima utajriba wako wa ubunifu wa kisasa.</td></tr><tr><td align=\"left\" colspan=\"4\">\n<bold>Section B - UTAUT</bold>\n</td></tr><tr><td align=\"left\">\n<bold>Construct</bold>\n</td><td align=\"left\">\n<bold>Item</bold>\n</td><td align=\"left\">\n<bold>Question English</bold>\n</td><td align=\"left\">\n<bold>Question Swahili</bold>\n</td></tr><tr><td align=\"left\" rowspan=\"4\">PU (perceived usefulness- performance expectancy)</td><td align=\"left\">PePu1</td><td align=\"left\">I find the given digital tool useful for providing sexual health information.</td><td align=\"left\">Naona kwamba kifaa hiki cha kidijitali ni cha muhimu kwa kusaidia kwa habari ya afya ya uzazi/kujumuiana.</td></tr><tr><td align=\"left\">PePu2</td><td align=\"left\">Using the given digital tool would make it easier to inform myself and others about sexual and reproductive health.</td><td align=\"left\">Kutumia kifaa hiki cha kidijitali kunarahisisha mimi na wengine kujipatia habari (ujumbe) kuhusu afya ya kijinsia na uzazi.</td></tr><tr><td align=\"left\">PePu3</td><td align=\"left\">Using the given digital tool would contribute to improvements in my sexual and reproductive health.</td><td align=\"left\">Kwa kutumia kifaa hiki cha kidijitali kunawezesha kuchangia kuboresha afya yangu ya kijinsia na uzazi.</td></tr><tr><td align=\"left\">PePu4</td><td align=\"left\">Using the given digital tool for sexual health information would enhance myself to make better and more informed decisions about my sexual and reproductive health.</td><td align=\"left\">Kwa kutumia kifaa hiki cha kidijitali kwa ujumbe wa afya ya ngono kunaniwezesha kufanya uamuzi kuhusu afya yangu ya kijinsia na uzazi.</td></tr><tr><td align=\"left\" rowspan=\"4\">EE (effort expectancy- perceived ease of use)</td><td align=\"left\">EePeu1</td><td align=\"left\">Learning to operate the digital tool for sexual health information would be easy for me.</td><td align=\"left\">Kujifunza jinsi ya kutumia kifaa cha kidijitali kwa kupata habari (ujumbe) kuhusu afya ya kijinsia ni rahisi kwangu.</td></tr><tr><td align=\"left\">EePeu2</td><td align=\"left\">I find the digital tool is easy to use.</td><td align=\"left\">Napata kifaa hiki cha kidijitali ni rahisi kutumia.</td></tr><tr><td align=\"left\">EePeu3</td><td align=\"left\">My interaction with the digital tool to access sexual health information would be/is clear and understandable.</td><td align=\"left\">Kutangamana kwangu na kifaa hiki cha kidijitali ili kupata habari kuhusu afya ya kijinsia ni dhahiri na ya kueleweka vizuri.</td></tr><tr><td align=\"left\">EePeu4</td><td align=\"left\">It would be easy for me to become skillful at using the digital tool for sexual health information.</td><td align=\"left\">Ni rahisi kwangu kuwa mjuzi wa kutumia kifaa cha kidijitali cha kupata habari (ujumbe) kuhusu afya ya kijinsia.</td></tr><tr><td align=\"left\" rowspan=\"5\">SI (social influence)</td><td align=\"left\">SI1</td><td align=\"left\">People who are important to me would disapprove of me using the digital tool for sexual health information.</td><td align=\"left\">Watu walio muhimu kwangu wanaweza kutokubaliana na mimi kutumia kifaa cha kidijitali cha kupata habari (ujumbe) kuhusu afya ya kijinsia.</td></tr><tr><td align=\"left\">SI2</td><td align=\"left\">People who influence my behaviour would disapprove of me using the digital tool for sexual health information.</td><td align=\"left\">Watu wanao shawishi na kuathiri tabia yangu wanaweza kutokubaliana na mimi kutumia kifaa cha kidijitali kuhusu habari (ujumbe) wa afya ya kijinsia.</td></tr><tr><td align=\"left\">SI3</td><td align=\"left\">My family would disapprove of me using the digital tool for sexual health information.</td><td align=\"left\">Familia yangu inaweza kutokubaliana na mimi kutumia kifaa cha kidijitali kuhusu habari (ujumbe) wa afya ya kijinsia.</td></tr><tr><td align=\"left\">SI4</td><td align=\"left\">People who share the same religious belief will disapprove of me using the digital tool for sexual helath information.</td><td align=\"left\">Watu wenye Imani sawa na mimi wanaweza kutokubaliana na mimi kutumia kifaa cha kidijitali kuhusu habari (ujumbe) wa afya ya kijinsia.</td></tr><tr><td align=\"left\">SI5</td><td align=\"left\">My religious belief does not support interacting with sexual health information on a digital tool even if it is made by a credible organization.</td><td align=\"left\">Imani yangu ya kidini yangu hainiruhusu kutangamana na habari (ujumbe) za afya ya kijinsia kwa kifaa cha kidijitali hata kama kimetengenezwa na shirika tajika.</td></tr><tr><td align=\"left\" rowspan=\"4\">FC (facilitating conditions)</td><td align=\"left\">FC1</td><td align=\"left\">I have the resources necessary to use the digital tool (e.g. wifi, laptop, smartphone).</td><td align=\"left\">Niko na raslimali muhimu za kutumia kwa kifaa cha kidijitali (kwa mfaano[k.m] uwezo wa mtandao, tarakilishi, simu ya mkono/rununu).</td></tr><tr><td align=\"left\">FC2</td><td align=\"left\">I have the knowledge necessary to use the digital tool (e.g. basic digital knowledge).</td><td align=\"left\">Niko na maarifa ya kutumia kifaa cha digitali kama vile maarifa ya kimsingi ya kidigitali.</td></tr><tr><td align=\"left\">FC3</td><td align=\"left\">Using the digital tool to get sexual health information fits well with the way I like to get information.</td><td align=\"left\">Kutumia kifaa cha kidijitali cha kupata habari (ujumbe) kuhusu afya ya kijinsia inaambatana na vile mimi hutaka kupata habari (ujumbe).</td></tr><tr><td align=\"left\">FC4</td><td align=\"left\">I think that I would use the help of a technical person to use the digital tool for sexual health information.</td><td align=\"left\">Nadhani naweza kuhitaji huduma za mtaalam mwenye ujuzi aweze kunisaidia kutumia kifaa cha kidijitali kupata habari (ujumbe) kuhusu afya ya kijinsia.</td></tr><tr><td align=\"left\" rowspan=\"8\">ATU (attitude towards technology use)</td><td align=\"left\">ATU1</td><td align=\"left\">Using the digital tool for sexual health information is a good idea.</td><td align=\"left\">Kutumia kifaa cha kidijitali ili kupata habari (ujumbe) kuhusu afya ya kijinsia ni wazo zuri.</td></tr><tr><td align=\"left\">ATU2</td><td align=\"left\">Accessing sexual health information through a digital tool is more comfortable than searching other internet sources like google.</td><td align=\"left\">Kupata habari (ujumbe) kuhusu afya ya kijinisia kupitia kifaa cha kidijitali ni bora zaidi kuliko kutumia mitandao mingine ya kutafuta habari kama google.</td></tr><tr><td align=\"left\">ATU3</td><td align=\"left\">I would have fun using the digital tool for sexual health information.</td><td align=\"left\">Naweza kuwa na furaha kwa kutumia kifaa cha digitali kupata habari (ujumbe) kuhusu afya ya ngono.</td></tr><tr><td align=\"left\">ATU4</td><td align=\"left\">Using the digital tool for sexual health information is interesting.</td><td align=\"left\">Kutumia kifaa cha kidijitali kupata habari (ujumbe) kuhusu afya ya kijinsia inavutia.</td></tr><tr><td align=\"left\">TA1</td><td align=\"left\">Using the digital tool to seek sexual health information would make me very nervous.</td><td align=\"left\">Kutumia kifaa cha kidijitali kupata habari (ujumbe) kuhusu afya ya kijinsia yaweza kunitia wasiwasi.</td></tr><tr><td align=\"left\">TA2</td><td align=\"left\">Using the digital tool to seek sexual health information may make me feel uncomfortable.</td><td align=\"left\">Kutumia kifaa cha kidijitali kupata habari (ujumbe) kuhusu afya ya kijinsia yaweza kunikozesha amani.</td></tr><tr><td align=\"left\">TA3</td><td align=\"left\">Using the digital tool for sexual health information is a bad idea.</td><td align=\"left\">Kutumia kifaa cha kidijitali kupata habari (ujumbe) kuhusu afya ya kijinsia sio wazo zuri.</td></tr><tr><td align=\"left\">TA4</td><td align=\"left\">Using the digital tool for sexual health information is unpleasant.</td><td align=\"left\">Kutumia kifaa cha kidijitali kupata habari (ujumbe) kuhusu afya ya kijinsia haipendezi.</td></tr><tr><td align=\"left\" rowspan=\"4\">BI (Behavioral Intention to use a digital tool for sexual health information)</td><td align=\"left\">BI2</td><td align=\"left\">I expect that I would use the digital tool in the future to search for sexual-health-related information.</td><td align=\"left\">Natarajia kutumia kifaa cha kidijitali siku zijazo kutafuta habari (ujumbe) unaoambatana na afya ya kijinsia.</td></tr><tr><td align=\"left\">BI3</td><td align=\"left\">I would use the digital tool for information about contraceptive products.</td><td align=\"left\">Naweza kutumia kifaa cha kidijitali kutafuta habari (ujumbe) kuhusu mbinu za kupanga uzazi.</td></tr><tr><td align=\"left\">BI4</td><td align=\"left\">I would use the digital tool for information about my menstrual period.</td><td align=\"left\">Naweza kutumia kifaa cha kidijitali kutafuta habari (ujumbe) kuhusu hedhi yangu.</td></tr><tr><td align=\"left\">BI5</td><td align=\"left\">I plan to use a digital tool to get sexual health information.</td><td align=\"left\">Napanga kutumia kifaa cha kidijitali kupata habari (ujumbe) kuhusu afya ya kijinsia.</td></tr><tr><td align=\"left\" colspan=\"4\">\n<bold>Section C - SUS</bold>\n</td></tr><tr><td align=\"left\" colspan=\"3\">\n<bold>Question English</bold>\n</td><td align=\"left\">\n<bold>Question Swahili</bold>\n</td></tr><tr><td align=\"left\" colspan=\"3\">I think that I would like to use this system frequently.</td><td align=\"left\">Nadhani ningependa kutumia hii mbinu mara kwa mara.</td></tr><tr><td align=\"left\" colspan=\"3\">I found the system unnecessarily complex.</td><td align=\"left\">Nilipata mbinu hii kuwa ngumu kiasi.</td></tr><tr><td align=\"left\" colspan=\"3\">I thought the system was easy to use.</td><td align=\"left\">Nadhani mbinu hii ni rahisi kutumia.</td></tr><tr><td align=\"left\" colspan=\"3\">I think that I would need the support of a technical person to be able to use this system.</td><td align=\"left\">Nadhani nitahitaji msaada wa mtaalam mwenye ujuzi ili niweze kutumia kifaa hiki.</td></tr><tr><td align=\"left\" colspan=\"3\">I found the various functions in this system were well integrated.</td><td align=\"left\">Nilipata sehemu mbali mbali za mbinu hii zimeunganishwa vyema.</td></tr><tr><td align=\"left\" colspan=\"3\">I thought there was too much inconsistency in this system.</td><td align=\"left\">Nafikiri kuna sehemu ambazo sio sahihi kwa mbinu hii.</td></tr><tr><td align=\"left\" colspan=\"3\">I would imagine that most people would learn to use the system very quickly.</td><td align=\"left\">Nikidhani, watu wengi wanaweza kujifunza kutumia mbinu hii kwa wepesi.</td></tr><tr><td align=\"left\" colspan=\"3\">I found the system very cumbersome to use.</td><td align=\"left\">Nilipata mbinu hii kuwa ya kuchosha kutumia.</td></tr><tr><td align=\"left\" colspan=\"3\">I felt very confident using the system</td><td align=\"left\">Nalihisi kuwa na ujasiri kutumia mbinu hii.</td></tr><tr><td align=\"left\" colspan=\"3\">I needed to learn a lot of things before I could get going with this system.</td><td align=\"left\">Nahitaji kujifunza mambo mengi zaidi kabla ya kuanza kutumia mbinu hii.</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>CHERRIES for used web survey</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Item Category</th><th align=\"left\">Checklist Item</th><th align=\"left\">Explanation</th></tr></thead><tbody><tr><td align=\"left\">Design</td><td align=\"left\">Describe survey design</td><td align=\"left\">The target population are primarily young women in Kenya. Therefore we placed the study in community centers, girl schools and universities in Kenya. The sample is not a convenience sample.</td></tr><tr><td align=\"left\" rowspan=\"3\">IRB (Institutional Review Board) approval and informed consent process</td><td align=\"left\">IRB approval</td><td align=\"left\">The ethic committee of the university of Witten/Herdecke approved the survey on 06.08.2022 under the protocol code: S-119/2022.</td></tr><tr><td align=\"left\">Informed consent</td><td align=\"left\">Participants were informed on the welcome page of the survey. The Information and Consent for Participation in Research Study have been provided in English and Swahili. The participants in the study remain anonymous. By clicking the checkbox consent was confirmed.</td></tr><tr><td align=\"left\">Data protection</td><td align=\"left\">The survey was hosted and all data were stored on its own secure server. No personal information was linked to survey results in any way.</td></tr><tr><td align=\"left\">Development and pre-testing</td><td align=\"left\">Development and testing</td><td align=\"left\">The used methods (UTAUT &amp; SUS) consists of mainly standardised questions, which have been proved in various previous studies. Pre-testing was conducted using a sample of women in the community center in Kenya as well as women in Germany.</td></tr><tr><td align=\"left\" rowspan=\"3\">Recruitment process and description of the sample having access to the questionnaire</td><td align=\"left\">Open survey versus closed survey</td><td align=\"left\">The survey was an open survey.</td></tr><tr><td align=\"left\">Contact mode</td><td align=\"left\">The study was placed in community centres, girl schools and universities in Kenya. This was supported by the social initiative of Boehringer Ingelheim, Making More Health, (MHH). The participants were able to share the link to the study with friends e.g. via WhatsApp, Instagram and Facebook.</td></tr><tr><td align=\"left\">Advertising the survey</td><td align=\"left\">The study was announced by the contact persons in the community centres, girl schools and universities.</td></tr><tr><td align=\"left\" rowspan=\"11\">Survey administration</td><td align=\"left\">Web/E-mail</td><td align=\"left\">The survey was hosted on its own web server by the University Witten/Herdecke in Germany, using the software LimeSurvey.</td></tr><tr><td align=\"left\">Context</td><td align=\"left\">The landing page of the survey was publicly accessible and distributed through an URL. This ensured that participants were able to share the survey.</td></tr><tr><td align=\"left\">Mandatory/voluntary</td><td align=\"left\"><p>The survey was completely voluntary. Users could</p><p>access the landing page without completing the survey.</p></td></tr><tr><td align=\"left\">Incentives</td><td align=\"left\">No incentives were offered to participants.</td></tr><tr><td align=\"left\">Time/Date</td><td align=\"left\">The survey period was from 01.12.22 to 31.01.2023.</td></tr><tr><td align=\"left\">Randomization of items or questionnaires</td><td align=\"left\">Survey items were not randomized.</td></tr><tr><td align=\"left\">Adaptive questioning</td><td align=\"left\">No adaptive questioning was used.</td></tr><tr><td align=\"left\">Number of Items</td><td align=\"left\"><p>Section A: 9 questions</p><p>Section B: 6 questions with 4–8 statements to be assessed</p><p>Section C: 10 standardized questions</p><p>Section D: 3 open-ended questions</p></td></tr><tr><td align=\"left\">Number of screens (pages)</td><td align=\"left\">One welcome page and 9 pages with survey items</td></tr><tr><td align=\"left\">Completeness check</td><td align=\"left\">Most survey items were mandatory, and respondents were prompted to complete outstanding items before leaving the survey page.</td></tr><tr><td align=\"left\">Review step</td><td align=\"left\">Participants were able to review and change their answers by clicking the Back button.</td></tr><tr><td align=\"left\" rowspan=\"4\">Response rates</td><td align=\"left\">Unique site visitor</td><td align=\"left\">No cookies or IP controls were used to ensure that people could participate consecutively from the same device. Participation devices were brought to the survey.</td></tr><tr><td align=\"left\">View rate (Ratio of unique survey visitors/unique site visitors)</td><td align=\"left\">Not measured.</td></tr><tr><td align=\"left\">Participation rate (Ratio of unique visitors who agreed to participate/unique first survey page visitors)</td><td align=\"left\">Not measured.</td></tr><tr><td align=\"left\">Completion rate (Ratio of users who finished the survey/users who agreed to participate)</td><td align=\"left\"><p>Section A: 77/77 = 100%</p><p>Section B: 65/77 = ~ 84%</p><p>Section C: 64/77 = ~ 83%</p><p>Section D: 63/77 = ~ 82%</p></td></tr><tr><td align=\"left\" rowspan=\"4\">Preventing multiple entries from the same individual</td><td align=\"left\">Cookies used</td><td align=\"left\">No Cookies were used.</td></tr><tr><td align=\"left\">IP check</td><td align=\"left\">No cookies or IP controls were used.</td></tr><tr><td align=\"left\">Log file analysis</td><td align=\"left\">Indicate whether other techniques to analyze the log file for identification of multiple entries were used. If so, please describe.</td></tr><tr><td align=\"left\">Registration</td><td align=\"left\">Not necessary, since the survey was an open survey.</td></tr><tr><td align=\"left\" rowspan=\"3\">Analysis</td><td align=\"left\">Handling of incomplete questionnaires</td><td align=\"left\">Only completed questionnaires were included in the analysis.</td></tr><tr><td align=\"left\">Questionnaires submitted with an atypical timestamp</td><td align=\"left\">Not used.</td></tr><tr><td align=\"left\">Statistical correction</td><td align=\"left\">No statistical correction procedures or weightings were used in the analysis.</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Description Sample Characteristics</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Item</th><th align=\"left\" colspan=\"5\">Characteristics in Percentage (Frequency)</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"2\">Gender</td><td align=\"left\">\n<bold>Female</bold>\n</td><td align=\"left\">\n<bold>Male</bold>\n</td><td align=\"left\" colspan=\"3\">\n<bold>Diverse</bold>\n</td></tr><tr><td align=\"left\">89.6% (69)</td><td align=\"left\">9.1% (7)</td><td align=\"left\" colspan=\"3\">1.3% (1)</td></tr><tr><td align=\"left\" rowspan=\"2\">Location</td><td align=\"left\">\n<bold>Kenya</bold>\n</td><td align=\"left\">\n<bold>Germany</bold>\n</td><td align=\"left\" colspan=\"3\">\n<bold>Another African Country</bold>\n</td></tr><tr><td align=\"left\">93.5% (72)</td><td align=\"left\">3.9% (3)</td><td align=\"left\" colspan=\"3\">2.6% (2)</td></tr><tr><td align=\"left\" rowspan=\"2\">Age</td><td align=\"left\">\n<bold>Below 20 Years</bold>\n</td><td align=\"left\">\n<bold>20–30 Years</bold>\n</td><td align=\"left\" colspan=\"3\">\n<bold>Above 30</bold>\n</td></tr><tr><td align=\"left\">16.9% (13)</td><td align=\"left\">68.9% (53)</td><td align=\"left\" colspan=\"3\">14.3% (11)</td></tr><tr><td align=\"left\" rowspan=\"2\">Working Status</td><td align=\"left\">\n<bold>Student</bold>\n</td><td align=\"left\">\n<bold>Unemployed</bold>\n</td><td align=\"left\">\n<bold>Working</bold>\n</td><td align=\"left\" colspan=\"2\">\n<bold>Others</bold>\n</td></tr><tr><td align=\"left\">36.4% (28)</td><td align=\"left\">32.5% (25)</td><td align=\"left\">26% (20)</td><td align=\"left\" colspan=\"2\">5.2% (4)</td></tr><tr><td align=\"left\" rowspan=\"2\">Education</td><td align=\"left\">\n<bold>Primary Education</bold>\n</td><td align=\"left\">\n<bold>Secondary Education</bold>\n</td><td align=\"left\">\n<bold>Post-Secondary Education</bold>\n</td><td align=\"left\">\n<bold>Tertiary Education</bold>\n</td><td align=\"left\">\n<bold>Prefer not to answer</bold>\n</td></tr><tr><td align=\"left\">13% (10)</td><td align=\"left\">35.1% (27)</td><td align=\"left\">14.3% (11)</td><td align=\"left\">36.4% (28)</td><td align=\"left\">1.3% (1)</td></tr><tr><td align=\"left\" rowspan=\"2\">Relationship Status</td><td align=\"left\">\n<bold>Married</bold>\n</td><td align=\"left\">\n<bold>In a relationship, not married</bold>\n</td><td align=\"left\">\n<bold>Not in a relationship</bold>\n</td><td align=\"left\" colspan=\"2\">\n<bold>Others or prefer not to answer</bold>\n</td></tr><tr><td align=\"left\">32.5% (25)</td><td align=\"left\">29.9% (23)</td><td align=\"left\">27.3% (21)</td><td align=\"left\" colspan=\"2\">10.4% (8)</td></tr><tr><td align=\"left\" rowspan=\"2\">Digital Experience</td><td align=\"left\">\n<bold>Excellent</bold>\n</td><td align=\"left\">\n<bold>Good</bold>\n</td><td align=\"left\">\n<bold>Average</bold>\n</td><td align=\"left\" colspan=\"2\">\n<bold>Poor</bold>\n</td></tr><tr><td align=\"left\">20.8% (16)</td><td align=\"left\">35.1% (27)</td><td align=\"left\">36.4% (28)</td><td align=\"left\" colspan=\"2\">7.8% (6)</td></tr><tr><td align=\"left\">Online search behavior for ... information</td><td align=\"left\">\n<bold>Daily</bold>\n</td><td align=\"left\">\n<bold>Several days a week</bold>\n</td><td align=\"left\">\n<bold>About once a week</bold>\n</td><td align=\"left\">\n<bold>Less often</bold>\n</td><td align=\"left\">\n<bold>Never</bold>\n</td></tr><tr><td align=\"left\">general</td><td align=\"left\">49.4% (38)</td><td align=\"left\">11.7% (9)</td><td align=\"left\">6.5% (5)</td><td align=\"left\">15.6% (12)</td><td align=\"left\">16.9% (13)</td></tr><tr><td align=\"left\">health</td><td align=\"left\">7.8% (6)</td><td align=\"left\">20.8% (16)</td><td align=\"left\">20.8% (16)</td><td align=\"left\">36.4% (28)</td><td align=\"left\">14.3% (11)</td></tr><tr><td align=\"left\">sexual health</td><td align=\"left\">3.9% (3)</td><td align=\"left\">19.5% (15)</td><td align=\"left\">16.9% (13)</td><td align=\"left\">44.2% (34)</td><td align=\"left\">15.6% (12)</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Descriptive Statistics of Item Responses</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Item</th><th align=\"left\">Mean</th><th align=\"left\">Standard Deviation</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"3\">\n<bold>Performance Expectancy - Perceived usefulness (PU)</bold>\n</td></tr><tr><td align=\"left\">PePu1</td><td align=\"left\">5.8</td><td align=\"left\">1.6</td></tr><tr><td align=\"left\">PePu2</td><td align=\"left\">5.9</td><td align=\"left\">1.5</td></tr><tr><td align=\"left\">PePu3</td><td align=\"left\">6.1</td><td align=\"left\">1.2</td></tr><tr><td align=\"left\">PePu4</td><td align=\"left\">5.9</td><td align=\"left\">1.3</td></tr><tr><td align=\"left\" colspan=\"3\">\n<bold>Effort Expectancy - Perceived ease of use (EE)</bold>\n</td></tr><tr><td align=\"left\">EePeu1</td><td align=\"left\">6.1</td><td align=\"left\">1.1</td></tr><tr><td align=\"left\">EePeu2</td><td align=\"left\">6.0</td><td align=\"left\">1.1</td></tr><tr><td align=\"left\">EePeu3</td><td align=\"left\">6.1</td><td align=\"left\">0.8</td></tr><tr><td align=\"left\">EePeu4</td><td align=\"left\">6.0</td><td align=\"left\">1.3</td></tr><tr><td align=\"left\" colspan=\"3\">\n<bold>Social Influence (SI)</bold>\n</td></tr><tr><td align=\"left\">SI1</td><td align=\"left\">3.0</td><td align=\"left\">2.1</td></tr><tr><td align=\"left\">SI2</td><td align=\"left\">3.0</td><td align=\"left\">2.1</td></tr><tr><td align=\"left\">SI3</td><td align=\"left\">2.6</td><td align=\"left\">1.8</td></tr><tr><td align=\"left\">SI4</td><td align=\"left\">2.8</td><td align=\"left\">1.8</td></tr><tr><td align=\"left\">SI5</td><td align=\"left\">2.7</td><td align=\"left\">1.9</td></tr><tr><td align=\"left\" colspan=\"3\">\n<bold>Facilitating Conditions (FC)</bold>\n</td></tr><tr><td align=\"left\">FC1</td><td align=\"left\">4.9</td><td align=\"left\">2.2</td></tr><tr><td align=\"left\">FC2</td><td align=\"left\">6.0</td><td align=\"left\">1.3</td></tr><tr><td align=\"left\">FC3</td><td align=\"left\">5.7</td><td align=\"left\">1.5</td></tr><tr><td align=\"left\">FC4</td><td align=\"left\">4.4</td><td align=\"left\">2.3</td></tr><tr><td align=\"left\" colspan=\"3\">\n<bold>Attitude towards technology use (ATU)</bold>\n</td></tr><tr><td align=\"left\">ATU1</td><td align=\"left\">6.3</td><td align=\"left\">1.0</td></tr><tr><td align=\"left\">ATU2</td><td align=\"left\">6.0</td><td align=\"left\">1.5</td></tr><tr><td align=\"left\">ATU3</td><td align=\"left\">5.9</td><td align=\"left\">1.5</td></tr><tr><td align=\"left\">ATU4</td><td align=\"left\">6.2</td><td align=\"left\">1.1</td></tr><tr><td align=\"left\">TA1</td><td align=\"left\">4.0</td><td align=\"left\">2.1</td></tr><tr><td align=\"left\">TA2</td><td align=\"left\">2.5</td><td align=\"left\">1.6</td></tr><tr><td align=\"left\">TA3</td><td align=\"left\">2.0</td><td align=\"left\">1.3</td></tr><tr><td align=\"left\">TA4</td><td align=\"left\">2.0</td><td align=\"left\">1.6</td></tr><tr><td align=\"left\" colspan=\"3\">\n<bold>Behavioral intention to use the digital tool for sexual health information (BI)</bold>\n</td></tr><tr><td align=\"left\">BI1</td><td align=\"left\">6.1</td><td align=\"left\">0.9</td></tr><tr><td align=\"left\">BI2</td><td align=\"left\">6.2</td><td align=\"left\">1.1</td></tr><tr><td align=\"left\">BI3</td><td align=\"left\">6.2</td><td align=\"left\">1.2</td></tr><tr><td align=\"left\">BI4</td><td align=\"left\">6.2</td><td align=\"left\">1.0</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab5\"><label>Table 5</label><caption><p>CFA output concerning factor loadings and interactions</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Latent Variable</th><th align=\"left\">Item</th><th align=\"left\">Estimate</th><th align=\"left\">Std. Error</th><th align=\"left\">z-value</th><th align=\"left\">P(&gt;|z|)</th><th align=\"left\">Std.all</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"4\">PU</td><td align=\"left\">PePu1</td><td align=\"left\">0.86</td><td align=\"left\">0.17</td><td align=\"left\">5.13</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.59</td></tr><tr><td align=\"left\">PePu2</td><td align=\"left\">1.35</td><td align=\"left\">0.14</td><td align=\"left\">9.98</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.95</td></tr><tr><td align=\"left\">PePu3</td><td align=\"left\">1.06</td><td align=\"left\">0.12</td><td align=\"left\">8.73</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.87</td></tr><tr><td align=\"left\">PePu4</td><td align=\"left\">1.00</td><td align=\"left\">0.14</td><td align=\"left\">6.88</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.77</td></tr><tr><td align=\"left\" rowspan=\"5\">EE</td><td align=\"left\">EePeu1</td><td align=\"left\">0.97</td><td align=\"left\">0.11</td><td align=\"left\">8.96</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.90</td></tr><tr><td align=\"left\">EePeu2</td><td align=\"left\">0.88</td><td align=\"left\">0.11</td><td align=\"left\">7.89</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.83</td></tr><tr><td align=\"left\">EePeu3</td><td align=\"left\">0.43</td><td align=\"left\">0.09</td><td align=\"left\">4.90</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.53</td></tr><tr><td align=\"left\">EePeu4</td><td align=\"left\">1.00</td><td align=\"left\">0.13</td><td align=\"left\">7.61</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.81</td></tr><tr><td align=\"left\">PePu4</td><td align=\"left\">0.74</td><td align=\"left\">0.12</td><td align=\"left\">6.28</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.57</td></tr><tr><td align=\"left\" rowspan=\"5\">SI</td><td align=\"left\">SI1</td><td align=\"left\">1.49</td><td align=\"left\">0.23</td><td align=\"left\">6.50</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.72</td></tr><tr><td align=\"left\">SI2</td><td align=\"left\">1.48</td><td align=\"left\">0.23</td><td align=\"left\">6.52</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.72</td></tr><tr><td align=\"left\">SI3</td><td align=\"left\">1.66</td><td align=\"left\">0.18</td><td align=\"left\">9.43</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.93</td></tr><tr><td align=\"left\">SI4</td><td align=\"left\">1.32</td><td align=\"left\">0.20</td><td align=\"left\">6.50</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.72</td></tr><tr><td align=\"left\">SI5</td><td align=\"left\">1.22</td><td align=\"left\">0.22</td><td align=\"left\">5.67</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.65</td></tr><tr><td align=\"left\" rowspan=\"8\">ATU</td><td align=\"left\">ATU1</td><td align=\"left\">0.57</td><td align=\"left\">0.11</td><td align=\"left\">5.26</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.60</td></tr><tr><td align=\"left\">ATU2</td><td align=\"left\">0.38</td><td align=\"left\">0.18</td><td align=\"left\">2.06</td><td align=\"left\">0.04</td><td align=\"left\">0.25</td></tr><tr><td align=\"left\">ATU3</td><td align=\"left\">0.45</td><td align=\"left\">0.18</td><td align=\"left\">2.51</td><td align=\"left\">0.01</td><td align=\"left\">0.31</td></tr><tr><td align=\"left\">ATU4</td><td align=\"left\">0.31</td><td align=\"left\">0.12</td><td align=\"left\">2.58</td><td align=\"left\">0.01</td><td align=\"left\">0.30</td></tr><tr><td align=\"left\">TA1</td><td align=\"left\">-0.70</td><td align=\"left\">0.26</td><td align=\"left\">-2.70</td><td align=\"left\">0.01</td><td align=\"left\">− 0.33</td></tr><tr><td align=\"left\">TA2</td><td align=\"left\">-1.03</td><td align=\"left\">0.18</td><td align=\"left\">-5.65</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">− 0.63</td></tr><tr><td align=\"left\">TA3</td><td align=\"left\">-1.20</td><td align=\"left\">0.13</td><td align=\"left\">-9.39</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">− 0.91</td></tr><tr><td align=\"left\">TA4</td><td align=\"left\">-1.55</td><td align=\"left\">0.14</td><td align=\"left\">-10.93</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">− 0.99</td></tr><tr><td align=\"left\" rowspan=\"5\">BI</td><td align=\"left\">BI1</td><td align=\"left\">0.41</td><td align=\"left\">0.11</td><td align=\"left\">3.86</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.46</td></tr><tr><td align=\"left\">BI2</td><td align=\"left\">0.46</td><td align=\"left\">0.14</td><td align=\"left\">3.42</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.41</td></tr><tr><td align=\"left\">BI3</td><td align=\"left\">0.91</td><td align=\"left\">0.12</td><td align=\"left\">7.38</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.79</td></tr><tr><td align=\"left\">BI4</td><td align=\"left\">1.01</td><td align=\"left\">0.10</td><td align=\"left\">10.05</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.98</td></tr><tr><td align=\"left\">PePu4</td><td align=\"left\">-0.57</td><td align=\"left\">0.13</td><td align=\"left\">-4.31</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">− 0.44</td></tr><tr><td align=\"left\" colspan=\"7\">\n<bold>Interactions</bold>\n</td></tr><tr><td align=\"left\">\n<bold>Item1</bold>\n</td><td align=\"left\">\n<bold>Item2</bold>\n</td><td align=\"left\">\n<bold>Estimate</bold>\n</td><td align=\"left\">\n<bold>Std. Error</bold>\n</td><td align=\"left\">\n<bold>z-value</bold>\n</td><td align=\"left\">\n<bold>P(&gt;|z|)</bold>\n</td><td align=\"left\">\n<bold>Std.all</bold>\n</td></tr><tr><td align=\"left\">ATU3</td><td align=\"left\">ATU4</td><td align=\"left\">0.94</td><td align=\"left\">0.20</td><td align=\"left\">4.61</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.66</td></tr><tr><td align=\"left\">ATU2</td><td align=\"left\">ATU3</td><td align=\"left\">1.35</td><td align=\"left\">0.30</td><td align=\"left\">4.44</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.66</td></tr><tr><td align=\"left\">ATU2</td><td align=\"left\">ATU4</td><td align=\"left\">0.63</td><td align=\"left\">0.19</td><td align=\"left\">3.37</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.43</td></tr><tr><td align=\"left\">EePeu3</td><td align=\"left\">ATU4</td><td align=\"left\">0.25</td><td align=\"left\">0.07</td><td align=\"left\">3.39</td><td align=\"left\">&lt; 0.001</td><td align=\"left\">0.36</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab6\"><label>Table 6</label><caption><p>Cronbach α</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">PU</th><th align=\"left\">EE</th><th align=\"left\">SI</th><th align=\"left\">ATU</th><th align=\"left\">BI</th></tr></thead><tbody><tr><td align=\"left\">0.84</td><td align=\"left\">0.83</td><td align=\"left\">0.86</td><td align=\"left\">0.43</td><td align=\"left\">0.70</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab7\"><label>Table 7</label><caption><p>Correlation Between Latent Variables</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">EE</th><th align=\"left\">SI</th><th align=\"left\">ATU</th><th align=\"left\">BI</th></tr></thead><tbody><tr><td align=\"left\">PU</td><td align=\"left\">0.10</td><td align=\"left\">0.11</td><td align=\"left\">0.20</td><td align=\"left\">0.50</td></tr><tr><td align=\"left\">EE</td><td align=\"left\"/><td align=\"left\">− 0.21</td><td align=\"left\">0.16</td><td align=\"left\">0.23</td></tr><tr><td align=\"left\">SI</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">− 0.46</td><td align=\"left\">0.03</td></tr><tr><td align=\"left\">ATU</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\">0.32</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab8\"><label>Table 8</label><caption><p>Output regarding regression parameters</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Predictors</th><th align=\"left\">Estimated Parameter</th><th align=\"left\">95% CI</th><th align=\"left\">Std. Err</th><th align=\"left\">z-value</th><th align=\"left\"><italic>P</italic> (&gt;|z|)</th></tr></thead><tbody><tr><td align=\"left\">PU</td><td align=\"left\">0.65</td><td align=\"left\">[0.31, 0.99]</td><td align=\"left\">0.17</td><td align=\"left\">3.75</td><td align=\"left\">&lt; 0.001</td></tr><tr><td align=\"left\">EE</td><td align=\"left\">0.19</td><td align=\"left\">[-0.11, 0.49]</td><td align=\"left\">0.15</td><td align=\"left\">1.28</td><td align=\"left\">0.20</td></tr><tr><td align=\"left\">SI</td><td align=\"left\">0.26</td><td align=\"left\">[-0.08, 0.60]</td><td align=\"left\">0.17</td><td align=\"left\">1.55</td><td align=\"left\">0.12</td></tr><tr><td align=\"left\">ATU</td><td align=\"left\">0.37</td><td align=\"left\">[0.03, 0.71]</td><td align=\"left\">0.17</td><td align=\"left\">2.21</td><td align=\"left\">0.03</td></tr><tr><td align=\"left\">FC1</td><td align=\"left\">-0.15</td><td align=\"left\">[-0.27, -0.03]</td><td align=\"left\">0.06</td><td align=\"left\">-2.29</td><td align=\"left\">0.02</td></tr><tr><td align=\"left\">FC2</td><td align=\"left\">-0.25</td><td align=\"left\">[-0.47, -0.03]</td><td align=\"left\">0.11</td><td align=\"left\">-2.19</td><td align=\"left\">0.03</td></tr><tr><td align=\"left\">FC3</td><td align=\"left\">0.24</td><td align=\"left\">[0.04, 0.44]</td><td align=\"left\">0.10</td><td align=\"left\">2.39</td><td align=\"left\">0.02</td></tr><tr><td align=\"left\">FC4</td><td align=\"left\">0.01</td><td align=\"left\">[-0.11, 0.13]</td><td align=\"left\">0.06</td><td align=\"left\">0.24</td><td align=\"left\">0.81</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group><fn><p><bold>Publisher’s Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par5\">Trigeminal neuralgia (TN) is a clinical diagnosis defined as recurrent unilateral brief electric shock-like pain, with abrupt onset and termination, limited to the distribution of the trigeminal nerve, and triggered by innocuous stimuli [##REF##23771276##16##]. The classical notion of a “simple” neurovascular contact causing TN, introduced in 1932 by Dandy and popularized by Jannetta [##UREF##4##19##], has been challenged over the years, and possible additional factors have been discussed in its pathophysiology [##REF##2454303##4##, ##REF##31240582##26##, ##REF##11807642##35##].</p>", "<p id=\"Par6\">Microvascular decompression (MVD) is a first-choice surgical option for TN [##UREF##2##12##], being the only non-ablative procedure that can provide a cure [##REF##2454303##4##], with confirmed high success rates and minimal operative morbidity, especially in TN [##REF##8622156##3##, ##REF##10413149##27##, ##REF##18077952##36##, ##UREF##10##43##, ##REF##35436582##44##, ##REF##22547101##47##]. Pain recurrence may occur in 47% of patients at 8 years postoperatively [##REF##12164727##40##]. In the largest series with the longest follow-up of MVD for TN, the estimated annual risk of recurrence was about 2 percent over 5 years and less than 1 percent over 10 years [##REF##8622156##3##].</p>", "<p id=\"Par7\">Few studies have been aimed at identifying the risk factors and pathophysiological mechanisms leading to recurrence [##REF##18447706##1##, ##REF##25084464##2##, ##REF##31045760##9##–##UREF##1##11##, ##UREF##5##21##, ##REF##27419826##41##, ##REF##25759921##46##, ##REF##27763948##48##]. Possible causes for recurrence might include new vascular contact, scarring/adhesions, and Teflon granuloma, but in some patients, no definitive cause has been ascribed [##REF##20419329##5##, ##REF##31045760##9##, ##REF##29857210##17##, ##REF##34232394##18##, ##REF##34793993##39##, ##UREF##10##43##]. It is unclear which therapeutic options may be offered to a patient with recurrent TN—either after previous MVD or after ablative surgery. While several studies have suggested percutaneous trigeminal ganglion ablation (by thermocoagulation, balloon compression or glycerol injection), radiosurgery, or peripheral nerve field stimulation [##REF##22120364##7##, ##REF##30713521##23##, ##UREF##7##33##, ##UREF##8##34##], few other studies have advocated repeat microsurgical posterior fossa re-exploration.</p>", "<p id=\"Par8\">With regard to the limited data which has become available for surgical re-treatment after previous MVD for TN, we scrutinized the surgical findings and the clinical outcome in a consecutive series of 26 patients with recurrent TN who underwent microsurgical posterior fossa re-exploration according to a standard protocol over a period of 10 years.</p>" ]

[ "<title>Methods</title>", "<p id=\"Par9\">All patients with a diagnosis of TN who underwent microsurgical posterior fossa re-exploration over a 10-year period at the Department of Neurosurgery, Hannover Medical School, were screened for the present study. Inclusion criteria were a clinical diagnosis of classical drug-resistant TN and recurrence of pain after previous MVD. Patients with a diagnosis of multiple sclerosis were excluded, and data was published elsewhere [##UREF##3##15##]. Database screening according to the defined inclusion and exclusion criteria resulted in identification of 26 patients. Each patient had magnetic resonance imaging (MRI) before the microsurgical re-exploration.</p>", "<p id=\"Par10\">Departmental assessment methods and surgical techniques for MVD have been described in detail elsewhere [##REF##20419329##5##, ##REF##27676667##13##–##UREF##3##15##, ##REF##24438814##29##]. All surgeries were performed via a lateral retrosigmoidal approach. Patients were positioned in a modified “semi-concorde” prone position under general anesthesia. The head was flexed and rotated by 45° to the contralateral side. The site of the previous craniotomy was exposed. Then the cranioplasty material if present was removed, if necessary additional bone was drilled to visualize the edges of the sigmoid and the transverse sinus. After opening the dura under the microscope, the cerebellum was gently retracted. Local arachnoid membranes and scar tissue were dissected carefully and completely cut with microscissors to visualize the trigeminal nerve in the depth. The nerve was fully exposed from its entry site to the brainstem to its entrance to Meckel’s cave.</p>", "<p id=\"Par11\">The local topography of the trigeminal nerve was evaluated and assessed for the following findings: dislocation of the Teflon felt, presence of arachnoid scar tissue, evidence of Teflon granuloma, arterial compression, and venous contact. Furthermore, the site was inspected for deformation of the trigeminal nerve and evidence of pulsations transmitted to the trigeminal nerve (without direct arterial contact) through scar tissue or the previously inserted Teflon felt (the “piston effect”). The “piston effect” is defined as transmission of pulsations to the trigeminal nerve from systole to diastole from an arterial vessel via the interposed and hardened Teflon (Fig ##FIG##0##1##). All arachnoid scars were displayed and dissected meticulously to free the nerve and to resolve its deformity. Veins were dissected and coagulated in the case of venous compression. If Teflon granuloma was detected or the Teflon acted as a transmitter for the “piston effect,” the mass was removed gently. In order to eliminate the “piston effect,” that much Teflon felt was removed until there was no transmission of the pulsations from the artery to the trigeminal nerve. If the artery which had caused trigeminal nerve compression before the first surgery was fixed by scar tissue to the dura, such scars were not dissected. In some patients a new and smaller Teflon felt was inserted as a spacer between the trigeminal nerve and the artery. In no case, the trigeminal nerve was “pinched,” “combed,” or injured in any way. All procedures were performed by the senior author (JKK).</p>", "<p id=\"Par12\">The first clinical evaluation was obtained on the day after surgery. The first regular follow-up visit was scheduled at 3 months after surgery. Patients were followed in the further course either by periodic follow-up visits or by structured phone surveys. Patients were asked about the presence and nature of any facial pain and the severity of residual pain attacks in comparison with the preoperative state, the presence of sensory loss, and any medication for TN. Postoperative outcome was assessed according to the Barrow Neurological Institute (BNI) pain intensity score as adapted from Przybylowski et al.: (I) no pain, no pain medication; (II) occasional pain not requiring medication; (IIIa) no pain, but continued taking medication for fear of stopping; (IIIb) continued pain, adequately controlled with medication; (IV) some pain, not adequately controlled with medication; and (V) severe pain or no pain relief [##UREF##6##32##]. The 3-, 12-, and 24-month follow-up and long-term follow-up (mean 79.5 months, range 29–184 months) were analyzed. BNI I-III was interpreted as favorable outcome and BNI IV-V as poor.</p>", "<p id=\"Par13\">Statistic evaluation was based on logistic regression analysis. A non-parametric Friedman test was used to compare repeated measures of follow-up scores. Fisher’s exact test was conducted to evaluate differences in outcome with regard to age, gender, side, previous procedures other than MVD, the time between surgeries, symptom duration, and intraoperative identification of the “piston effect.” The JMP®, Version 16 (SAS Institute Inc., Cary, NC), statistical software was used for all analysis. <italic>p</italic>-values &lt;0.05 were considered to be statistically significant.</p>" ]

[ "<title>Results</title>", "<title>Patients and pain characteristics</title>", "<p id=\"Par14\">There were 13 women (50%) and 13 men (50%). Mean age at the time of surgery for recurrent TN was 59.1 years. The mean intervals between the initial surgery and microsurgical posterior fossa re-exploration are shown in Table ##TAB##0##1##. All patients had medically refractory classical TN pain attacks (BNI V), and 5 of them (19.2%) reported additionally a permanent pain component. The distribution of pain is presented in Table ##TAB##1##2##. Five of the 26 patients had mild hypaesthesia detected by the preoperative clinical examination (19.2%), two had complete or partial hearing loss (7.7%), and three presented with ataxia (11.5%).</p>", "<p id=\"Par15\">Four patients also had undergone percutaneous radiofrequency rhizotomy or percutaneous balloon compression of the trigeminal ganglion, and one patient had had gamma knife surgery. In 11 patients, the primary MVD surgery had been performed in another hospital (42.3%), while 15 patients had been operated primarily in the Department of Neurosurgery, Hannover Medical School (57.7%).</p>", "<title>Imaging studies</title>", "<p id=\"Par16\">MRI showed suspected Teflon granuloma in 12 patients (46.1%), suspected neurovascular contact in 3 patients (11.5%), and compression of the trigeminal nerve by a Meckel’s cave electrode for treatment of neuropathic pain in 1 patient (3.8%).</p>", "<title>Intraoperative findings</title>", "<p id=\"Par17\">Recurrent TN was associated with the several findings presented as shown in Table ##TAB##2##3##. Dislocation of the Teflon felt could not be identified in any instance. In one patient, the trigeminal nerve was compressed by an electrode in Meckel’s cave which had been inserted previously for treatment of neuropathic pain (1/26, 3.8%). Most patients (18/26, 69.2%) had a combination of findings. The patient in whom an arterial compression was identified at the trigeminal entry zone was a 52-year-old man (patient 20). He had had previous surgery in another hospital 70 months earlier. Deformation of the trigeminal nerve could not be assessed unambiguously in all patients during dissection of scar tissue but it was identified to some extent in almost all instances. Also, evidence of pulsations transmitted to the trigeminal nerve through the Teflon inserted previously/scar tissue was clearly identified in at least 15 patients (57.7%).</p>", "<p id=\"Par18\">Corresponding to the intraoperative findings several measures were taken (Table ##TAB##2##3##) tailored to the individual scenario.</p>", "<title>Postoperative outcome and follow-up</title>", "<p id=\"Par19\">There were no severe side effects after microsurgical posterior fossa re-exploration. Postoperative side effects occurred in 8 patients: new sensory deficit in 5 patients (19.2%), from which two cases were transient, transient diplopia secondary to trochlear palsy (3.8%), small asymptomatic cerebellar hemorrhage (3.8%), and delayed facial palsy (3.8%), in one instance, respectively. None of the patients with preoperative partial hearing loss or ataxia had symptom worsening postoperatively.</p>", "<p id=\"Par20\">Early postoperative pain relief was achieved in all instances. All patients were available for 24-month follow-up or longer. The individual BNI scores for the different follow-up assessments are shown in Table ##TAB##3##4##. The distribution of pain scores at the different follow-up periods is shown in Fig. ##FIG##1##2##. At 3-month follow-up, all patients benefited from pain relief as compared to preoperatively: BNI I 16/26 (62%), BNI II 1/26 (4%), and BNI IIIa 8/26 (31%), and only 1 patient had an unsatisfactory result with BNI IV (4%). At 12-month follow-up, BNI scores were distributed as follows: BNI I 17/26 (65%), BNI II 1/26 (4%), BNI IIIa 5/26 (19%), BNI IV 2/26 (8%), and BNI V 1/26 (4%). The distribution at 24-month follow-up was as follows: BNI I 16/26 (62%), BNI II 0/26, BNI IIIa 8/26 (31%), BNI IV 1/26 (4%), and BNI V 1/26 (4%) (Table ##TAB##3##4##).</p>", "<p id=\"Par21\">At long-term follow-up (mean 79.5 months, range 29–184 months), 10/26 patients (38.5%) had complete pain relief (BNI I), 11/26 patients (42.3%) had pain relief but continued taking medication (BNI IIIa), 3/26 patients (12%) had a BNI score of IV, and 2/26 patients (8%) had a BNI score of V. As compared to 24-month follow-up when only two patients had an unsatisfactory outcome (BNI scores IV and V), there were five such patients on long-term follow-up. The distribution of patients with BNI I-III versus BNI IV-V compared to the preoperative status was statistically significant at all follow-up evaluations (<italic>p</italic>&lt;0.0001). The five patients with poor outcome results on long-term follow-up were offered further surgical treatment, including percutaneous rhizotomy. The exact cause of re-recurrence could not be determined, since none of the patients underwent a new posterior fossa re-exploration. While three patients wanted to wait before making a decision, one patient was lost to follow-up, and another patient was severely disabled secondary to a stroke unrelated to posterior fossa surgery.</p>", "<title>Prognostic factors</title>", "<p id=\"Par22\">The intraoperative presence of arachnoid adhesions significantly predicted favorable outcome (<italic>p</italic>=0.037). Additionally, age was a significant predictor of favorable outcome (<italic>p</italic>=0.037). There was no statistically significant correlation between outcome and gender (<italic>p</italic>=1.00), side (<italic>p</italic>=0.187), affected branches (<italic>p</italic>=0.482), previous procedures other than MVD (<italic>p</italic>=0.920), the time between surgeries (<italic>p</italic>=0.965), symptom duration (<italic>p</italic>=0.757), and intraoperative identification of “the piston effect” (<italic>p</italic>=0.574).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par23\">There is no standard treatment for recurrent TN after a previously successful MVD procedure [##REF##31045760##9##]. There are few studies advocating repeat posterior fossa re-exploration in such a scenario; however, these studies are characterized by marked discrepancies regarding intraoperative findings, surgical maneuvers, and length of follow-up [##REF##25084464##2##, ##REF##20419329##5##, ##REF##31045760##9##, ##UREF##5##21##, ##REF##9802855##22##, ##REF##10898363##25##, ##UREF##10##43##, ##REF##1595390##45##, ##REF##25759921##46##]. While a recent systematic review has nicely summarized and compared the results of previous studies [##REF##34875391##20##], several questions remain open. Furthermore, there is a paucity of studies comparing posterior fossa re-exploration with other treatment options [##UREF##0##8##, ##UREF##10##43##, ##REF##27763948##48##]. Remarkably, according to a prospective study comparing “redo MVD” and percutaneous balloon compression, patients fared better after “redo MVD” considering complications, clinical outcome, and recurrence of TN over a 6-year period [##UREF##0##8##].</p>", "<p id=\"Par24\">Our study clearly demonstrates that posterior fossa re-exploration in patients with recurrent TN after previous MVD can achieve outcome which is similar to that of patients who underwent MVD for primary TN both with regard to intraoperative complications and to clinical outcome at long-term [##REF##18077952##36##]. Furthermore, we show that the development of recurrent TN may be due to a variety of findings which may act alone or in combination. Taking the individual findings into account will result in patient-specific measures such as removal of scar tissue and Teflon with or without adding new material for decompression. Such personalized approaches also obviate the possible need for additional intraoperative maneuvers to the trigeminal nerve such as combing or partial rhizotomy.</p>", "<p id=\"Par25\">There is quite a high variability in the description of intraoperative findings thought to underly recurrent TN, which may be due to both the surgical technique and the material used during the primary MVD but also to the eye of the beholder during the second surgery. When Jiao et al. tried to summarize the heterogeneous findings in their recent systematic review in 193 of 270 patients, who had repeat surgery for recurrent TN, they reported the following findings: granulomatous lesions or arachnoid adhesions in 31%, vascular conflict in 36%, slippage of Teflon felt in 9%, and no clearly identifiable cause in 29% [##REF##34875391##20##].</p>", "<p id=\"Par26\">Early studies, such as the study published by van Loveren et al. in 1982, predominantly had concentrated on vascular conflicts [##REF##6754883##42##]. In that study, 82% of patients were described to have an aberrant artery or vein in contact with the trigeminal nerve, but only 46% had a significant compression [##REF##6754883##42##]. These findings contrast with newer studies, for example with the study of Chen et al. reporting 50% of cases with Teflon granuloma, 30% cases with new arterial contact, 10% with venous compression, and 10% with negative findings [##REF##10773262##6##]. On the other hand, Cheng et al. reported severe arachnoid adhesions in all their 41 re-operated patients, from which 36.6% also had compression from an artery, 14.6% from a vein, and 19.5% from Teflon [##REF##31045760##9##]. Remarkably, in a recent study from China, incomplete or absent decompression or new vessels were noted in 50% of patients operated for recurrent TN, but all patients also had moderate to severe arachnoid adhesions [##UREF##10##43##].</p>", "<p id=\"Par27\">Over the past 20 years Teflon granuloma has been identified more frequently as a main cause for recurrent TN [##REF##10773262##6##]. Initially, Teflon had been introduced by Jannetta in 1985 after he had noted that several patients with recurrent TN had stiffened Ivalon implants during repeat surgery which allowed transmission of pulsations to the trigeminal nerve [##REF##9802855##22##]. Teflon was thought to be an inert material with a low rate of tissue reaction [##REF##10773262##6##, ##UREF##0##8##, ##REF##9802855##22##, ##REF##10898363##25##, ##REF##33189922##31##, ##REF##1595390##45##]. Some studies, however, have shown the migration of multinuclear giant cells in the Teflon material with abundant hyalinized scar tissue with blood supply to the granuloma deriving from the tentorium [##REF##20419329##5##, ##REF##10773262##6##]. While it was suspected that granuloma formation might be triggered by tissue glue that was used to fix the Teflon implant in several studies [##REF##20419329##5##], the findings of our study indicate that it is also the Teflon proper. We considered whether or not to use other materials, such as muscle or Ivalon to pad the vessel-nerve contact, or to use alternatively no-contact sling techniques. Interestingly, a recent multivariate analysis using a 2-center retrospective cohort study of TN patients with MVD using either Teflon or Ivalon found that the implant type had no impact on the final BNI score or the risk of relapse [##REF##33189922##31##].</p>", "<p id=\"Par28\">Arterial pulsations transmitted to the trigeminal nerve through the implanted material which was used for MVD during the primary MVD surgery have received only relatively little attention. We suggest to use the term “piston effect” for this phenomenon since we think that it denotates its mechanisms better than previous rather vague terms or the expression “secondary missile phenomenon” which has been applied by Jannetta to specifically refer to recurrence of TN or hemifacial spasm when using Ivalon [##REF##9802855##22##].</p>", "<p id=\"Par29\">The basis for the “piston effect” most likely is that arachnoid adhesions and granulomatous changes result in hardening of the Teflon, acting as a “glue” uniting the implant with the trigeminal nerve on one side, and the artery on the other. Thus, it appears that two key factors are essential for this phenomenon: hardening of the Teflon and a tight adhesion between nerve, graft and artery. Our study demonstrated that most patients (85%) had arachnoid adhesions or scar tissue at the site of the previous surgery. Indeed, after MVD, the surrounding arachnoid membranes tend to thicken and adhere, altering thus the anatomy of passing structures [##REF##11807642##35##], which may also contribute to compression along the trigeminal entry zone [##REF##31240582##26##], as well as transmission of pulsations from the decompressed vessels [##UREF##1##11##].</p>", "<p id=\"Par30\">It is quite possible that a high proportion of patients who were described to have unspecific findings upon re-exploration for recurrent TN in early studies had findings as described above or others which were not considered to be possibly causative [##REF##34232394##18##]. Other findings to be considered are distortion or indentation of the trigeminal nerve root [##REF##29857210##17##, ##REF##11807642##35##], which was also noted in our series.</p>", "<p id=\"Par31\">Complication rates for posterior fossa re-exploration after previous MVD vary greatly between different studies. The most commonly reported complication is facial numbness which, however, appears to be related to surgical technique [##UREF##5##21##, ##REF##10898363##25##, ##UREF##10##43##, ##REF##1595390##45##, ##REF##25759921##46##]. The frequency of other complications appears to be comparable to that of primary MVD surgery, being less than 5% [##REF##34875391##20##]. According to the systematic review of Jiao et al., the overall occurrence of hypaesthesia is 22% after posterior fossa re-exploration after previous MVD for TN. The frequency of hypaesthesia, however, is markedly higher in series from centers which use mechanical intraoperative measures to alter sensory function of the trigeminal nerve either routinely or in patients in whom no clear cause for recurrent TN has been detected [##UREF##10##43##]. Such measures include “combing” or “pinching” of the trigeminal nerve or “internal neurolysis,” as well as “partial transection of the trigeminal nerve,” “partial nerve section,” or “partial sensory rhizotomy” [##REF##18447706##1##, ##REF##7808609##10##, ##UREF##1##11##, ##REF##29857210##17##, ##REF##20920937##30##, ##UREF##10##43##, ##REF##34689474##49##]. Especially when using more extensive sectioning procedures, facial numbness may occur in more than 50% of patients after posterior fossa re-exploration [##REF##18447706##1##, ##REF##33189922##31##]. On the other hand, the rate of hypaesthesia is remarkably low when using sling methods to keep the offending vessel away from the trigeminal nerve in recurrent TN [##REF##34875391##20##, ##REF##25759921##46##].</p>", "<p id=\"Par32\">Since long-term results for pain relief appear to be similar for “repeat MVD” for recurrent TN regardless of intentional surgical microtrauma to the trigeminal nerve or not, but considering that there is a marked difference in the occurrence of facial numbness, we suggest that such measures may be avoided.</p>", "<p id=\"Par33\">No-contact techniques, such as “sling” or “clip” procedures, or gluing the offending artery to the dura have received more attention recently [##REF##25663308##24##, ##REF##10898363##25##, ##REF##34237450##28##, ##UREF##9##37##, ##REF##29676696##38##]. Such techniques certainly deserve further consideration both during primary and secondary MVD surgeries.</p>", "<p id=\"Par34\">Limitations of our study include retrospective nature, the lack of standardized preoperative MR protocols, and the lack of a categorical classification system for intraoperative findings, including the degree of trigeminal nerve compression. These limitations, however, are inherent to all studies on this subject. An advantage of our study is that there was no patient attrition and that all patients were available for follow-up of 2 years or longer. We are aware that the results of our study may not be adopted to other scenarios, since all surgeries have been performed by a dedicated senior surgeon. The latter fact, however, made it possible to tailor treatment specifically to intraoperative findings in a more consistent fashion.</p>" ]

[ "<title>Conclusions</title>", "<p id=\"Par35\">In the present study, we show that microsurgical posterior fossa re-exploration is a useful treatment option for recurrent TN. There were no serious side effects, and the frequency of postoperative hypaesthesia was relatively low. We conclude that microsurgical posterior fossa re-exploration avoiding any intentional damage to the trigeminal nerve is a very useful treatment option in this context. Any maneuvers such as dissecting or “combing” the trigeminal nerve may not be necessary. In the future, it might be worthwhile to explore the more frequent use of sling techniques in posterior fossa re-exploration for recurrent TN.</p>" ]

[ "<title>Objective</title>", "<p id=\"Par1\">Microvascular decompression (MVD) is a well-accepted treatment modality for trigeminal neuralgia (TN) with high initial success rates. The causes for recurrence of TN after previously successful MVD have not been fully clarified, and its treatment is still a matter of debate. Here, we present the surgical findings and the clinical outcome of patients with recurrent TN after MVD who underwent posterior fossa re-exploration.</p>", "<title>Methods</title>", "<p id=\"Par2\">Microsurgical posterior fossa re-exploration was performed in 26 patients with recurrent TN (mean age 59.1 years) who underwent MVD over a period of 10 years. The trigeminal nerve was exposed, and possible factors for recurrent TN were identified. Arachnoid scars and Teflon granulomas were dissected meticulously without manipulating the trigeminal nerve. Outcome of posterior fossa re-exploration was graded according to the Barrow Neurological Institute (BNI) pain intensity score. Follow-up was analyzed postoperatively at 3, 12, and 24 months and at the latest available time point for long-term outcome.</p>", "<title>Results</title>", "<p id=\"Par3\">The mean duration of recurrent TN after the first MVD was 20 months. Pain relief was achieved in all patients with recurrent TN on the first postoperative day. Intraoperative findings were as follows: arachnoid scar tissue in 22/26 (84.6%) patients, arterial compression in 1/26 (3.8%), venous contact in 8/26 (30.8%), Teflon granuloma in 14/26 (53.8%), compression by an electrode in Meckel’s cave used for treatment of neuropathic pain in 1/26 (3.8%), evidence of pulsations transmitted to the trigeminal nerve through the Teflon inserted previously/scar tissue (“piston effect”) in 15/26 (57.7%), and combination of findings in 18/26 (69.2%). At long-term follow-up (mean 79.5 months; range, 29–184 months), 21/26 (80.8%) patients had favorable outcome (BNI I-IIIa). New hypaesthesia secondary to microsurgical posterior fossa re-exploration occurred in 5/26 (19.2%) patients.</p>", "<title>Conclusions</title>", "<p id=\"Par4\">Posterior fossa re-exploration avoiding manipulation to the trigeminal nerve, such as pinching or combing, may be a useful treatment option for recurrent TN after previously successful MVD providing pain relief in the majority of patients with a low rate of new hypaesthesia.</p>", "<title>Keywords</title>", "<p>Open Access funding enabled and organized by Projekt DEAL.</p>" ]

[]

[ "<title>Author contributions</title>", "<p>Conceptualization: GHM, FWF, and JKK. Formal analysis and full investigation: GHM, FWF, HE, and SAA. Writing original draft preparation: GHM, FWF, and JKK. Writing—review and editing: FWF, GHM, HE, SAA, and JKK.</p>", "<title>Funding</title>", "<p>Open Access funding enabled and organized by Projekt DEAL.</p>", "<title>Data Availability</title>", "<p>The data is available from the corresponding author on reasonable request.</p>", "<title>Declarations</title>", "<title>Ethics approval</title>", "<p id=\"Par36\">This study was performed in line with the principles of the Declaration of Helsinki.</p>", "<title>Consent to participate</title>", "<p id=\"Par37\">Due to its retrospective nature, informed consent was not required by the ethics committee of Hannover Medical School in this study.</p>", "<title>Conflict of interest</title>", "<p id=\"Par38\">GHM, FWF, SAA, and HH have no conflict of interest to declare. JKK is a consultant for Medtronic, Boston Scientific, inomed, and Aleva.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Graphical demonstration of the “piston effect” defined as transmission of pulsations to the trigeminal nerve from systole to diastole from an arterial vessel via the interposed and hardened Teflon</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>Severity of pain according to the Barrow Neurological Institute (BNI) pain intensity score before and after posterior fossa re-exploration after previous microvascular decompression. Barrow Neurological Institute (BNI) pain intensity score (adapted from Przybylowski et al., 2018): I—no pain, no pain medication; II—occasional pain not requiring medication; IIIa—no pain, but continued taking medication for fear of stopping; IIIb—continued pain, adequately controlled with medication; IV—some pain, not adequately controlled with medication; and V—severe pain or no pain relief. FU: follow-up.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Demographic and clinical data of 26 patients with recurrent trigeminal neuralgia undergoing posterior fossa re-exploration after previous microvascular decompression</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Patient characteristics</th><th>Mean (range)</th></tr></thead><tbody><tr><td>Age (years)</td><td>59.1 (35–82)</td></tr><tr><td>Interval between surgeries (months)</td><td>59.0 (5–204)</td></tr><tr><td>Symptom duration (months)</td><td>20.2 (2–94)</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Distribution of pain of 26 patients with recurrent trigeminal neuralgia undergoing posterior fossa re-exploration after previous microvascular decompression</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Distribution of pain</th><th>No. of patients (%)</th></tr></thead><tbody><tr><td>Side</td><td/></tr><tr><td>Left</td><td>7 (26.9)</td></tr><tr><td>Right</td><td>19 (73.1)</td></tr><tr><td>Topography</td><td/></tr><tr><td>V1</td><td>0</td></tr><tr><td>V2</td><td>4 (15.4)</td></tr><tr><td>V3</td><td>5 (19.2)</td></tr><tr><td>V1 + V2</td><td>2 (7.7)</td></tr><tr><td>V2 + V3</td><td>2 (7.7)</td></tr><tr><td>V1 + V2 + V3</td><td>13 (50.0)</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Intraoperative findings and measures of 26 patients with recurrent trigeminal neuralgia undergoing posterior fossa re-exploration after previous microvascular decompression</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Intraoperative findings and measures (<italic>n</italic>=26)</th><th>No. of patients (%)</th></tr></thead><tbody><tr><td>Intraoperative findings</td><td/></tr><tr><td>Dislocation of Teflon</td><td>0 (0)</td></tr><tr><td>Arachnoid scar tissue</td><td>22 (84.6)</td></tr><tr><td>Arterial compression</td><td>1 (3.8)</td></tr><tr><td>Venous contact</td><td>8 (30.8)</td></tr><tr><td>Teflon granuloma</td><td>14 (53.8)</td></tr><tr><td>Cavum Meckeli electrode</td><td>1 (3.8)</td></tr><tr><td>Piston effect</td><td>15 (57.7)</td></tr><tr><td>Intraoperative measures</td><td/></tr><tr><td>Total removal of Teflon</td><td>9 (34.6)</td></tr><tr><td>Partial removal of Teflon</td><td>16 (61.5)</td></tr><tr><td>Removal of adhesions</td><td>22 (84.6)</td></tr><tr><td>Removal of electrode</td><td>1 (3.8)</td></tr><tr><td>Coagulation of vein</td><td>5 (19.2)</td></tr><tr><td>New Teflon inserted</td><td>10 (38.5)</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Longitudinal pain scores of 26 patients with recurrent trigeminal neuralgia undergoing posterior fossa re-exploration after previous microvascular decompression</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th rowspan=\"2\">Patient</th><th colspan=\"7\">Barrow Neurological Institute pain intensity score</th></tr><tr><th>Before PFRE</th><th>Early post operative</th><th>3-m FU</th><th>12-m FU</th><th>24-m FU</th><th>Last FU after PFRE</th><th>Length of FU after PFRE (months)</th></tr></thead><tbody><tr><td>1</td><td>V</td><td>IIIa</td><td>IIIa</td><td>IV</td><td>IIIa</td><td>IIIa</td><td>60</td></tr><tr><td>2</td><td>V</td><td>IIIa</td><td>IIIa</td><td>V</td><td>V</td><td>V</td><td>72</td></tr><tr><td>3</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>IIIa</td><td>54</td></tr><tr><td>4</td><td>V</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>44</td></tr><tr><td>5</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>I</td><td>64</td></tr><tr><td>6</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>IV</td><td>135</td></tr><tr><td>7</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>I</td><td>89</td></tr><tr><td>8</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>I</td><td>140</td></tr><tr><td>9</td><td>V</td><td>IIIa</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>93</td></tr><tr><td>10</td><td>V</td><td>IIIa</td><td>IV</td><td>IV</td><td>IV</td><td>IV</td><td>113</td></tr><tr><td>11</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>IIIa</td><td>IIIa</td><td>44</td></tr><tr><td>12</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>IIIa</td><td>IIIa</td><td>82</td></tr><tr><td>13</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>I</td><td>72</td></tr><tr><td>14</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>I</td><td>87</td></tr><tr><td>15</td><td>V</td><td>IIIa</td><td>II</td><td>II</td><td>I</td><td>V</td><td>78</td></tr><tr><td>16</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>IIIa</td><td>94</td></tr><tr><td>17</td><td>V</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>169</td></tr><tr><td>18</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>I</td><td>184</td></tr><tr><td>19</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>IIIa</td><td>48</td></tr><tr><td>20</td><td>V</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>92</td></tr><tr><td>21</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>IIIa</td><td>81</td></tr><tr><td>22</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>I</td><td>48</td></tr><tr><td>23</td><td>V</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>60</td></tr><tr><td>24</td><td>V</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>IIIa</td><td>IV</td><td>44</td></tr><tr><td>25</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>I</td><td>29</td></tr><tr><td>26</td><td>V</td><td>IIIa</td><td>I</td><td>I</td><td>I</td><td>I</td><td>34</td></tr></tbody></table></table-wrap>" 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[ "<table-wrap-foot><p>Comparison of the individual preoperative, early postoperative (first postoperative day), 3-month, 12-month, 24-month, and the last available follow-up BNI pain intensity scores. Follow-up periods are given as months. Barrow Neurological Institute (BNI) pain intensity score (adapted from Przybylowski et al., 2018): I—no pain, no pain medication; II—occasional pain not requiring medication; IIIa—no pain, but continued taking medication for fear of stopping; IIIb—continued pain, adequately controlled with medication; IV—some pain, not adequately controlled with medication; and V—severe pain or no pain relief</p><p><italic>FU</italic> follow-up, <italic>m</italic> months, <italic>PFRE</italic> posterior fossa re-exploration</p></table-wrap-foot>", "<fn-group><fn><p>Gökce Hatipoglu Majernik, Filipe Wolff Fernandes shared the first co-authorship.</p></fn><fn><p><bold>Publisher’s note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"701_2023_5877_Fig1_HTML\" id=\"MO1\"/>", "<graphic xlink:href=\"701_2023_5877_Fig2_HTML\" id=\"MO2\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"P2\">Targeted therapies have emerged as a promising therapeutic option in the management of metastatic colorectal cancer (mCRC) when used in combination with chemotherapy [##UREF##0##1##, ##UREF##1##2##]. However, the efficacy of targeted therapies can vary significantly among individuals due to their biological heterogeneity [##REF##29632055##3##]. Within the complex landscape of cancer biology, multiple genes and their interconnected pathways collaborate to influence tumor behavior and the response to therapy. incorporating gene interactions into gene expression profiling provides knowledge about cancer biology that can guide clinical care and improve treatment assignments for cancer patients [##UREF##2##4##–##UREF##6##8##].</p>", "<p id=\"P3\">Here, we constructed data-driven networks of gene transcripts and augmented them with germline genotype data. This augmentation, based on the principles of Mendelian randomization, enabled us to build the transcriptomic-causal network [##UREF##7##9##–##UREF##10##12##]. The causal network identification facilitates a data-driven approach for discovery of a group of genes that collectively constitute a gene signature for patient stratification. Exploring the network, we identified sub-networks and linked them to the OS to account for effect of confounding genes. We identified gene signatures corresponding to each sub-network, which includes genes impacting OS, and stratified patients under each treatment. We conducted enrichment analysis and explored the biological function of findings using immune signatures known as key inputs in characterizing the immune subtypes of cancer.</p>", "<p id=\"P4\">We performed this analysis using data from a randomized phase III trial (CALGB/SWOG 80405) comprising 1,165 patients with mCRC treated with cetuximab or bevacizumab combined with chemotherapy (FOLFOX or FOLFIRI). We replicated the findings using data from 103 patients initially excluded from the main analysis along with external cohorts, including the Cancer Genome Atlas (TCGA). We also assessed the biological relevance of interconnectivity among genes using STRING, a protein-protein interaction database. Integrating transcriptomic-causal network in the identification of gene signatures provided significant OS separation among patients. Integrating transcriptomic-causal networks in gene signature identification provided significant OS separation among patients. This approach has improved the biological interpretability and reproducibility of the findings while offering promising prognostic biomarkers to tailor treatment strategies for personalized medicine.</p>" ]

[ "<title>Materials and methods</title>", "<title>Data</title>", "<p id=\"P28\">Patients in this study were drawn from the Cancer and Leukemia Group B (CALGB; now a part of the Alliance for Clinical Trials in Oncology) and SWOG 80405 (Alliance) trial. The trial was initiated in September 2005 with a total of 2,326 patients randomized to the three treatment arms (bevacizumab, cetuximab, or their combination in addition to chemotherapy with FOLFIRI or FOLFOX).</p>", "<title>Genotyping.</title>", "<p id=\"P29\">DNA was extracted from peripheral blood of 2,334 patients. The first genotyping batch was performed on the Illumina HumanOmniExpress-12v1 platform at the Riken Institute (Tokyo, Japan) and included 731,412 genotyped variants. The second genotyping batch was performed on the Illumina HumanOmniExpress-8v1 and included 964,193 SNPs. A total of 719,461 SNPs from HapMap from batch 1 were also on the chip from batch 2. The quality control was performed to remove SNPs with mismatched annotation between the two platforms, genotyping call rates less than 99%, departures from Hardy–Weinberg equilibrium (<italic toggle=\"yes\">P</italic>&lt; 10⁻⁸), allele frequencies less than 0.05, and individuals with genotyping call rate &lt; 0.90. A total of 540,021 SNPs genotyped for 1,165 samples were remained [##REF##32958699##37##] after passing these filters.</p>", "<title>Tumor RNA sequencing.</title>", "<p id=\"P30\">Tumor RNA was extracted from pre-treatment formalin-fixed paraffin-embedded (FFPE) tumor blocks (96% primary, 2% metastatic, 2% unknown) from 584 CALGB/SWOG 80405 patients. TruSeq RNA access target enrichment and library preparation protocol were performed using 250 <italic toggle=\"yes\">ng</italic> of template RNA. Sequencing was done using synthesis chemistry targeting 50 M reads with a read length of 2×100 bp per sample on the HiSeq 2500. Data processing was conducted using standard procedures described by Kalari et al.[##UREF##22##38##].</p>", "<title>Clinical outcomes and covariates.</title>", "<p id=\"P31\">The primary endpoint of OS was calculated from the time of study entry to death or the last known follow-up for those without reported death. The median follow-up of 640 samples in bevacizumab and cetuximab arms was 65.7 months (95% confidence interval, 63.5–70). In addition, <italic toggle=\"yes\">BRAF</italic> and all <italic toggle=\"yes\">RAS</italic> mutation status were determined by BEAMing (beads, emulsion, amplification, magnetics; Hamburg, Germany) technology [##REF##31548349##39##] and included in the analysis as covariates in addition to age and gender.</p>" ]

[ "<title>Results</title>", "<p id=\"P5\">This study is conducted on a subset of Caucasian samples from CRC patients enrolled in a randomized phase III trial CALGB/SWOG 80405 due to the limited sample size from other ethnicities. Germline genotyped data was extracted from the peripheral blood of 1,165 patients (Figure S1). RNA-seq data were extracted from the primary tumor tissue of 469 patients, obtained from pre-treatment formalin-fixed paraffin-embedded blocks (Figure S2). The trial was designed to compare cetuximab, bevacizumab, or cetuximab + bevacizumab, each plus chemotherapy as first-line therapy in KRAS wild-type advanced or mCRC. The combined arm of the study (cetuximab + bevacizumab) reduced the efficacy of the treatment and therefore, was discontinued prematurely [##UREF##11##13##]. The clinical study did not show significant differences in OS of patients treated with bevacizumab versus cetuximab. However, these treatments differ at the molecular levels by targeting distinct biological pathways. Therefore, multiple genes and their interconnected pathways collaborate to influence the response to these therapies.</p>", "<p id=\"P6\">Here, we first constructed a data-driven network of genes and then augmented the network with genetic information to identify the transcriptomic-causal network based on Mendelian randomization technique [##UREF##7##9##, ##UREF##9##11##]. Toward this object, we conducted expression quantitative trait loci (eQTL) analysis and utilized the eQTLs as potential instrumental variables to estimate causal relationships among genes and construct the transcriptomic-causal network. We also included known gene regulatory pathways for identifying the causal network. We performed this analysis using the data from patients treated with either bevacizumab or cetuximab. We then assessed the stability of the network and replicated the edges using 103 samples from the bevacizumab + cetuximab arm of the study (Methods). Furthermore, we replicated the interconnectivity among genes using STRING database.</p>", "<p id=\"P7\">By mapping identified eGenes (genes with eQTLs) on the network, we identified sub-networks of genes regulated by eQTLs. We linked the network to OS and estimated the effects of genes on OS that were not attributed to the other genes in the study. Finally, we defined gene signatures corresponding to the sub-networks with genes associated with OS. For the analyses reviewed here, we accounted for the influence of the tumor microenvironment by adjusting for immune cell abundances estimated from RNA-seq data. In addition, all the analyses were adjusted for all RAS and BRAFv600E mutation status along with age, gender. The eQTL analysis is adjusted for batch effect correction. We replicated the findings using a subset of data from the cohort under study as well as two external cohorts. Additionally, we explored the relationship between our findings and immune signatures defined in the literature as key inputs to a description of the immune subtypes of cancer. ##FIG##0##Figure 1## represents the overall workflow of the study.</p>", "<title>Genetic regulatory variants of gene expression in CRC tumor tissue</title>", "<p id=\"P8\">To identify candidate susceptibility genes subject to regulation by genetic variants, we performed a <italic toggle=\"yes\">cis-</italic>eQTL analysis on 8,301 genes with adequate variation (standard deviation of normalized counts across samples &gt; 0.5) while adjusting for covariates. We assessed the relationship of 33,209,829 <italic toggle=\"yes\">cis</italic>-eQTL-gene pairs using 350 Caucasian samples with both RNA-seq and genotype data (##FIG##1##Fig. 2A##). We selected 352 top genes after applying the permutation test (<italic toggle=\"yes\">p</italic>-value &lt; 0.05) (Figures S4 and S5, Table S1). Enrichment analysis revealed a significant depletion of exons (##FIG##1##Fig. 2B##), and we observed a high enrichment of cis-eQTLs in bivalent chromatin states associated with enhancer sequences based on the Roadmap Epigenomics Consortium enhancer databases [##UREF##12##14##] (##FIG##1##Fig. 2C##). Bivalent enhancer refers to segments of DNA that have both repressing and activating epigenetic regulators in the same region.</p>", "<title>Transcriptomic-causal network</title>", "<p id=\"P9\">We combined the samples of bevacizumab and cetuximab arms to identify the transcriptomic-causal network since RNA seq data were recorded prior to the treatments (##FIG##1##Fig. 2A##). We identified the network on 8,301 genes and focused on 2,267 edges that passed the stability assessment. Using the 103 patients (treated with bevacizumab + cetuximab), who were initially excluded from the main analysis, we successfully replicated 71% of the interconnectivity among genes (Methods and Figures S7–9). By taking these steps, we reduced the likelihood of false positive discoveries and increased the chances of reproducibility for the identified edges. We then integrated <italic toggle=\"yes\">cis</italic>-eQTL analysis results for sub-networks identifications and the application of Mendelian randomization technique to identify causal relationships among genes (Methods).</p>", "<title>Gene signatures and patient stratification</title>", "<p id=\"P10\">We investigated the effect of the gene expression levels on OS by integrating the transcriptomic-causal network with Cox-proportional hazard models for the patients receiving bevacizumab (203 samples) and cetuximab (163 samples) separately. In this way, we controlled for the effects of confounding genes identified using the network. This analysis yielded the identification of three gene signatures corresponding to three sub-networks that comprised genes with significant effects (<italic toggle=\"yes\">p</italic>-value &lt; 0.1) on overall survival (OS) for either bevacizumab or cetuximab treatments (##FIG##2##Fig. 3##).</p>", "<p id=\"P11\">One of the sub-networks (##FIG##2##Fig. 3A##, sub-network 1) includes 4 genes <italic toggle=\"yes\">(UAP1L1, lL2RB, RELT,</italic> and <italic toggle=\"yes\">MYO1G).</italic> Among these genes, <italic toggle=\"yes\">MYO1G</italic> was identified as an eGene. To assess the effect of <italic toggle=\"yes\">MYO1G</italic> on OS, we controlled for the effect of <italic toggle=\"yes\">RELT</italic> as confounding gene and observed that <italic toggle=\"yes\">MYO1G</italic> prolonged, and <italic toggle=\"yes\">RELT</italic> shortened <italic toggle=\"yes\">OS</italic> under cetuximab treatment (HR: 0.65 and 1.37, <italic toggle=\"yes\">p</italic>-value: 0.05, 0.07, respectively) but not under bevacizumab treatment (HR: 0.86 and 1.14, <italic toggle=\"yes\">p</italic>-value: 0.51, 0.33, respectively), (##FIG##2##Fig. 3##).</p>", "<p id=\"P12\">The other sub-network (##FIG##2##Fig. 3A##, sub-network 2) involved 4 genes <italic toggle=\"yes\">(IDO1, GBP2, GBP4,</italic> and <italic toggle=\"yes\">GBP5),</italic> three of which belong to the guanylate-binding protein (GBP) family, including eGene <italic toggle=\"yes\">GBP5.</italic> We observed that the two directly connected genes <italic toggle=\"yes\">IDO1</italic> and <italic toggle=\"yes\">GBP4</italic> significantly prolonged OS of patients under bevacizumab treatment (HR: 0.79, 0.63, <italic toggle=\"yes\">p</italic>-value: 0.018, 0.015, respectively) and not under cetuximab (HR: 0.92, 0.93, <italic toggle=\"yes\">p</italic>-value: 0.455, 0.692, respectively), (##FIG##2##Fig. 3C##). In this analysis, we controlled for the confounding role of <italic toggle=\"yes\">IDO1</italic> on the <italic toggle=\"yes\">GBP4-OS</italic> relationship.</p>", "<p id=\"P13\">The third sub-network (##FIG##2##Fig. 3A##, sub-network 3) includes 6 genes <italic toggle=\"yes\">(BLM, FANCI, UBE2T, SNRPA1, PRC1,</italic> and <italic toggle=\"yes\">DTL),</italic> with <italic toggle=\"yes\">BLM</italic> being an eGene. We observed that <italic toggle=\"yes\">PRC1</italic> and <italic toggle=\"yes\">FANCI</italic> shortened the OS of patients treated with cetuximab (HR: 1.53, 1.43; <italic toggle=\"yes\">p</italic>-value: 0.03, 0.04 respectively) but not with bevacizumab (HR: 0.94, 1.31; <italic toggle=\"yes\">p</italic>-value: 0.71, 0.10), (##FIG##2##Fig. 3C##).</p>", "<p id=\"P14\">To facilitate the nomination of a scoring model for prospective testing as a gene signature and to support the visualization of survival plots, we dichotomized the expression level of each gene based on their third quartile and defined beneficial/non-beneficial expression levels with respect to OS specific to each treatment. We then classified patients based on having either beneficial or non-beneficial expression levels in each sub-network (Methods). We estimated the survival function based on Kaplan-Meier estimator using set of genes that collectively form a gene signature. We observed a notable decrease in the median OS from 43.5 to 16.1 months (<italic toggle=\"yes\">p</italic>-value: 0.0002) for patients with the beneficial vs. non-beneficial levels for both <italic toggle=\"yes\">RELT</italic> and <italic toggle=\"yes\">MYO1G</italic> in sub-network 1; from 38.1 months to 13.1 months (<italic toggle=\"yes\">p</italic>-value: 0.0059) for patients with beneficial vs. non-beneficial levels for both <italic toggle=\"yes\">FANCI</italic> and <italic toggle=\"yes\">PRC1</italic> in sub-network 3 (##FIG##3##Fig. 4##). For patients with beneficial expression for one gene and non-beneficial expression for other gene in the signatures see Figure S10. Due to the limitations associated with dichotomizing the data in Kaplan-Meier estimators, the statistical power of the analysis for sub-network 2 in the bevacizumab arm was insufficient to detect significant differences among patients.</p>", "<title>Immune feature enrichment</title>", "<p id=\"P15\">The biological functions of the gene signatures were assessed by clustering genes in the corresponding sub-networks (##FIG##4##Fig. 5##) based on 10 immune signatures known as key inputs to a description of the immune subtypes of cancer. These immune signatures including macrophages [##REF##19188147##15##], lymphocytes [##REF##18592372##16##], TGF-β [##UREF##13##17##], IFN-γ [##UREF##14##18##], wound healing [##UREF##14##18##], and CD8 + T cell [##REF##24916698##19##] measure a final common pathway of antitumor immune activity (cytotoxicity [##REF##24916698##19##]), characteriz the immune microenvironment (T follicular helper, Tfh, cells [##REF##24138885##20##]), and mediate the response to checkpoint inhibitors (B cells and T cells cooperation [##REF##31730857##21##]).</p>", "<p id=\"P16\">The median value of the genes (Tables S2 and S3) within an immune signature was assigned to each patient except for the cytotoxicity signature where the geometric mean was used, according to Rooney <italic toggle=\"yes\">et al.</italic> [##REF##25594174##22##]. In addition, we included immunoglobulin G[##UREF##15##23##] and single gene CD274[##UREF##16##24##] as prognosis biomarkers. Since in this study, gene expression levels are adjusted for enriched immune cell types, we first investigated the relationship of the immune signatures and the enriched immune cell types (Figure S11). We then clustered the genes in the sub-networks represented as a heat map plot in ##FIG##4##Fig. 5##.</p>", "<p id=\"P17\">As the heatmap in this figure shows, Sub-network 1 is divided into two clusters: one includes <italic toggle=\"yes\">IL2RB</italic> and <italic toggle=\"yes\">MYO1G</italic> and the other includes <italic toggle=\"yes\">UAP1L1</italic> and <italic toggle=\"yes\">RELT,</italic> consistent with what we observed in the OS analysis where <italic toggle=\"yes\">MYO1G</italic> prolonged OS and <italic toggle=\"yes\">RELT</italic> shortened OS. Both genes <italic toggle=\"yes\">(MYO1G</italic> and <italic toggle=\"yes\">RELT)</italic> in the corresponding signature, showed stronger correlation with macrophages.</p>", "<p id=\"P18\">All the gene in sub-network 2 are clustered together and represented strong correlation with cytotoxicity, CD8 + T cells, lymphocytes, and macrophages. The two genes <italic toggle=\"yes\">(IDO1</italic> and <italic toggle=\"yes\">GBP4)</italic> in the corresponding signature showed stronger correlation with cytotoxicity (##FIG##4##Fig. 5##).</p>", "<p id=\"P19\">On the other hand, sub-network 3 is only related to wound healing and all the genes in this pathway showed correlations with the wound healing (##FIG##4##Fig. 5##).</p>", "<title>Replications</title>", "<p id=\"P20\">In addition to using 103 samples for replication of edges in the network, we used three external replication cohorts. We replicated the edges in the network and investigated the functional relationships among genes within identified signatures using the STRING database, a biological database of protein–protein interactions [##REF##30476243##25##]. ##TAB##0##Table 1## represents confidence scores for protein interactions corresponding to the genes in the signatures. Scores above 0.5 are indicative of promising evidence for potential physical interactions.</p>", "<p id=\"P21\">We used the GSE146889 data from the Gene Expression Omnibus database as an external cohort, which includes 85 paired samples from normal and tumor tissue of colorectal cancer patients. For 21,983 genes measured in this cohort, we performed differential expression analysis. All the genes in sub-networks 1–3 except <italic toggle=\"yes\">IL2RB</italic> and <italic toggle=\"yes\">MYO1G</italic> were differentially expressed in this cohort after adjusting for multiple testing using the false discovery rate (FDR) method (FDR &lt; 0.05) (##TAB##1##Table 2##).</p>", "<p id=\"P22\">We also replicated the gene-OS relationships using data from the COAD project of The Cancer Genome Atlas (TCGA), consisting of 27 samples treated with bevacizumab and ten with cetuximab. Thus, we performed the replication analysis for the findings related to the bevacizumab arm and not for cetuximab due to the limited number of samples. We used the Wilcoxon rank-sum test to validate the findings for the follow-up time of 27 patients treated with bevacizumab who exhibited elevated expression levels of both <italic toggle=\"yes\">GBP4</italic> and <italic toggle=\"yes\">IDO1</italic> or the absence of high expression levels in both genes. The expression levels were dichotomized based on the third quartile, similar to the Kaplan-Meier analysis. We observed a significant difference between these two groups (<italic toggle=\"yes\">p</italic>-value = 0.038), confirming the association of both <italic toggle=\"yes\">GBP4</italic> and <italic toggle=\"yes\">IDO1</italic> in sub-network 2 with the overall survival of patients treated with bevacizumab.</p>" ]

[ "<title>Discussion</title>", "<p id=\"P23\">Prognostic gene expression signatures can help to improve treatment assignments and provide guidance for a personalized treatment-decision. To improve, the interpretability and reproducibility of the gene signatures for patient stratification, we integrated the interconnectivity of genes into analysis by identifying transcriptomic-causal networks. This allowed us to account for the impact of confounding genes on gene-OS relationship and to defined the set of genes that collectively stratifies patients. As a result, we replicated the findings using multiple external cohorts including an external protein-protein interaction database for replication of the functional relationships among genes within gene signatures.</p>", "<p id=\"P24\">One of the identified gene signatures includes <italic toggle=\"yes\">IDO1</italic> and <italic toggle=\"yes\">GBP4</italic> that are typically expressed at low-to-medium basal levels in the absence of acute activating signals [##REF##30755454##26##]. <italic toggle=\"yes\">IDO1,</italic> with an important role in regulating the innate and adaptive immune response, is overexpressed in many types of cancers, including CRC [##UREF##17##27##]. Currently, an increasing number of studies have demonstrated that <italic toggle=\"yes\">IDO1</italic> is associated with immune escape by suppressing T cell activity and enhancing regulatory T cells in different tumor types. However, our study revealed that an elevated level of <italic toggle=\"yes\">IDO1</italic> transcription is associated with longer OS in patients treated with bevacizumab. This disparity has been reported in the study of gynecologic and breast cancers [##REF##32227579##28##], which might be due to the activation of tissue-specific gene regulatory pathways in tumor cells. Our study connects the expression of <italic toggle=\"yes\">IDO1</italic> to the expression of 3 genes in the GBP family <italic toggle=\"yes\">(GBP5, GBP4,</italic> and <italic toggle=\"yes\">GBP2)</italic> that all shared the characteristic of high correlation with cytotoxicity signatures. This proposes a new mechanism involved in CRC tumor cytotoxicity, which is not through the IFN-γ pathway related to the negative effect of <italic toggle=\"yes\">IDO1.</italic> That may be potentially activated as a response to bevacizumab therapy.</p>", "<p id=\"P25\">The other signature includes <italic toggle=\"yes\">MYOIGthat</italic> constitutes the minor histocompatibility antigen HA-2 that binds to MHC class I molecules, makes the antigens recognizable by CD8 + T cells in tumor cells, and allows the destruction of harmful tumor cells [##REF##32451863##29##, ##REF##29224918##30##]. Interestingly, our OS analysis showed positive effects of high expression of <italic toggle=\"yes\">MYO1G</italic> on OS in the cetuximab arm. Its upstream gene in the identified sub-network, <italic toggle=\"yes\">RELT,</italic> is frequently overexpressed in colorectal cancer cell lines and primary colorectal carcinomas [##REF##24623448##31##], consistent with the negative effect of <italic toggle=\"yes\">RELT</italic> on OS shown in our study. <italic toggle=\"yes\">RELT</italic> activates NF-κB pathway and deregulates β-catenin activity in the majority of sporadic forms of colorectal cancer and colon cancer cell lines[##REF##24623448##31##]. β-catenin, on the other hand, has been associated with the expression of MHC class I in glioma stem cells[##REF##31591480##32##]. Given that MHC class I serves as the receptor of HA-2, the interaction between <italic toggle=\"yes\">RELT</italic> and <italic toggle=\"yes\">MYO1G</italic> identified in our study suggests a potential mechanism by which tumor cells can evade immune recognition by CD8 + T cells upon cetuximab therapy.</p>", "<p id=\"P26\">The protein regulator of cytokinesis 1 <italic toggle=\"yes\">(PRC1)</italic> in the signature corresponding to sub-network 3, reduced OS of patients treated with cetuximab, anti-EGFR therapy. <italic toggle=\"yes\">PRC1</italic> plays an important role in the pathogenesis of various cancers, including colon cancer [##UREF##18##33##]. <italic toggle=\"yes\">PRC1</italic> and all the genes in its sub-network showed relationships with wound healing signatures, most likely due to their common function in DNA damage and repair. DNA damage sensing and repair dysregulation causes genome instability and is a hallmark of many cancers [##UREF##19##34##]. In this study, we found that less than 1 <italic toggle=\"yes\">%</italic> of CMS4 tumors carry a beneficial expression level of <italic toggle=\"yes\">PRC1</italic> and <italic toggle=\"yes\">FANCI,</italic> when CMS4 is known for stromal infiltration and resistance to anti-EGFR therapy (Figure S12) [##UREF##20##35##, ##UREF##21##36##].</p>", "<p id=\"P27\">Collectively, our study highlights the significance of data-driven networks in gaining a deeper understanding of the functional mechanisms underlying treatment response. We have identified novel signatures that exhibited a higher potential for reproducibility and improved interpretability. Furthermore, our findings have generated novel hypotheses for experimental testing, shedding light on tumor progression and suppression. Our treatment-specific analysis has also unveiled promising biomarkers for personalized therapies and identified potential targets for the development of new anticancer drugs.</p>" ]

[]

[ "<p id=\"P1\">Predictive and prognostic gene signatures derived from interconnectivity among genes can tailor clinical care to patients in cancer treatment. We identified gene interconnectivity as the transcriptomic-causal network by integrating germline genotyping and tumor RNA-seq data from 1,165 patients with metastatic colorectal cancer (CRC). The patients were enrolled in a clinical trial with randomized treatment, either cetuximab or bevacizumab in combination with chemotherapy. We linked the network to overall survival (OS) and detected novel biomarkers by controlling for confounding genes. Our data-driven approach discerned sets of genes, each set collectively stratify patients based on OS. Two signatures under the cetuximab treatment were related to wound healing and macrophages. The signature under the bevacizumab treatment was related to cytotoxicity and we replicated its effect on OS using an external cohort. We also showed that the genes influencing OS within the signatures are downregulated in CRC tumor vs. normal tissue using another external cohort. Furthermore, the corresponding proteins encoded by the genes within the signatures interact each other and are functionally related. In conclusion, this study identified a group of genes that collectively stratified patients based on OS and uncovered promising novel prognostic biomarkers for personalized treatment of CRC using transcriptomic causal networks.</p>" ]

[ "<title>Methods</title>", "<title>Data preprocessing</title>", "<p id=\"P32\">Among the 584 samples with RNA-seq data, the majority (86%) were Caucasian, with 9% being African American and 5% from other ethnicities. Due to the small sample size of other ethnicities, our analysis focused specifically on primary tumor samples from Caucasian. To ensure data quality and reliability, we excluded genes with low expression variation (standard deviation &lt; 0.5) and low counts across the samples (&gt; 30% zeros). This resulted in a final set of 8301 genes for further analysis. we applied upper quartile normalization, which enabled comparability of gene expression values across different samples. We removed duplicated samples (n = 5) and tumors with low gene expression across the genome (&gt; 50% genes with zero counts; n = 1). Further details and visuals can be found in Figures S1 and S13. We then transformed the RNA-seq data into the log2 scale for the analysis.</p>", "<p id=\"P33\">To assess the presence of batch effects or hidden population stratification in the RNA-seq data, we conducted principal component analysis (PCA) (Figure S14). In order to validate the self-reported gender information, we utilized k-means clustering based on the expression patterns of genes on chromosome Y. This analysis revealed that 5 samples had a discrepancy between their recorded gender and their biological sex (Figure S15).</p>", "<p id=\"P34\">Given the influence of the tumor microenvironment on tumor biology, it is crucial to consider the heterogeneity of cell types within the tumor samples when analyzing RNA-seq data. The tumor microenvironment consists of various cell types, including tumor cells, immune cells, stromal cells, and others, which can have distinct gene expression profiles. Therefore, correcting for the abundance of different cell types becomes even more important in order to accurately capture the gene expression patterns specific to tumor cells and to account for any confounding effects introduced by non-tumor cell types. To this end, we estimated the abundance of immune cell types in our RNA-seq data using CIBERSORTx [##REF##31960376##40##] with the validated leukocyte gene signature matrix as a reference. We defined a cell phenotype to be enriched in our data if at most 30% of its estimated scores across samples are zero and its standard deviation is greater than 0.1. As a result, 9 hematopoietic cell phenotypes were enriched in our data: naive and memory B cells, plasma cells, CD8 + T cells, resting and activated memory CD4 + T cells, M0 and M2 macrophages, and activated mast cells [##REF##35176136##41##]. The relationships between the immune cell types and immune signatures are represented in Supplementary Figure S11.</p>", "<p id=\"P35\">We used 1,165 Caucasian samples with 540,021 SNPs for imputation and employed phased haplotypes from the Haplotype Reference Consortium (HRC) panel through the University of Michigan Imputation Server [##REF##27571263##42##] (<ext-link xlink:href=\"https://github.com/statgen/locuszoom-standalone/\" ext-link-type=\"uri\">https://github.com/statgen/locuszoom-standalone/</ext-link>). Phasing was done using Eagle v2.4 algorithm[##REF##27694958##43##]. The HRC panel combines sequence data across &gt; 32,000 individuals from &gt; 20 medical sequencing studies. The imputed genotype data with imputation score &gt; 0.7 and minor allele frequency (MAF) &gt;0.05 included 5,539,144 common SNPs. These SNPs were used in all the downstream analyses.</p>", "<p id=\"P36\">For analysis that includes pre-treatment or baseline data (samples with genotype and RNA-seq), we used samples in all arms. However, for the analysis that involved post-treatment data (samples with overall survival and events), we excluded Arm 3 that showed shorter overall survival with two other arms [##UREF##11##13##] (Figure S16). The comparisons between the population with the RNA-seq and the population without it are presented in Table S4.</p>", "<title>cis -eQTL analysis</title>", "<p id=\"P37\">To identify germline genetic variants associated with tumor gene expression, we focused on <italic toggle=\"yes\">cis</italic>-eQTL since gene expression is affected by nearby genetic variations [##UREF##9##11##, ##REF##29022597##44##]. Therefore, for all pairs of genes and SNPs within 1 Mb upstream and downstream of the genes’ transcription start sites (TSS), we applied a linear regression model. In our primary analysis, we estimated latent variables for the potential confounders using the probabilistic estimation of expression residuals (PEER) approach [##UREF##23##45##]. However, PEER factors did not explain the variation in RNA-seq data in our study (more details in Supplementary PEER Factors section and Figures S17–19).</p>", "<p id=\"P38\">To address the impact of heterogeneous cell types in our RNA-seq data and to mitigate potential batch effects, we applied several adjustments. We adjusted the expression level of the genes for the enriched cell types in our data estimated by CIBERSORTx [##REF##31960376##40##]. Additionally, we incorporated the first principal component of genotype data to remove the contribution of batch effects that may have arisen during sample processing and sequencing (Figure S3). Furthermore, we considered important covariates such as gender, age, and mutation status (including RAS and BRAFv600E) in our analysis to account for potential confounding factors. This analysis was performed using FastQTL [##REF##26708335##46##]. We applied the adaptive permutation mode of FastQTL while setting for 10,000 permutations and selected eGenes with at least one <italic toggle=\"yes\">cis</italic>-eQTL with an adjusted <italic toggle=\"yes\">p</italic>-value &lt; 0.05 at the gene level. These genes are the ones selected for the OS analysis.</p>", "<title>Enrichment analysis in genomic regions</title>", "<p id=\"P39\">We investigated the enrichment of identified <italic toggle=\"yes\">cis</italic>-eQTLs in the biological location in DNA, including genic, intron, exon, intergenic, distal intergenic, and upstream and downstream (&lt; = 300 bp) of a gene. To demonstrate that the number of <italic toggle=\"yes\">cis</italic>-eQTLs in any region is higher than expected by chance, we simulated the null distribution by permuting 1,000 random sets of SNPs with the size of <italic toggle=\"yes\">cis</italic>-eQTL matching <italic toggle=\"yes\">cis</italic>-eQTL in terms of allele frequency, gene density, distance from TSS, and density of tagSNPs arising from genomic variability of linkage disequilibrium [##REF##27668389##47##]. We then calculated the Z-score for the observed number of <italic toggle=\"yes\">cis</italic>-eQTL in each region based on the simulated null distribution.</p>", "<title>Overlap of cis-eQTL with enhancer databases</title>", "<p id=\"P40\">We looked for the overlap of <italic toggle=\"yes\">cis</italic>-eQTLs with enhancer database from the Roadmap Epigenomics Consortium [##UREF##24##48##]. In particular, we focused on active, genic, and bivalent chromatin states in colon sigmoid, mucosa, and smooth muscle. An active enhancer refers to the regulatory region of DNA that interacts with the promoter DNA region; a genic enhancer refers to regulatory regions in a gene; a bivalent enhancer refers to segments of DNA that have both repressing and activating epigenetic regulators in the same region. We counted the number of top <italic toggle=\"yes\">cis</italic>-eQTLs (the most significant associated SNP per gene) that lie within enhancer sequences in each tissue. We calculated the z-score for each tissue similar to the previous section (Enrichment in the genomic region) and tested the significant levels.</p>", "<title>Transcriptomic-causal networks</title>", "<p id=\"P41\">The transcriptomic-causal networks are data-driven networks augmented with the principles of Mendelian randomization (MR). The use of transcriptomic-causal networks enables us to uncover the intricate biological connections between genes and identify confounding genes in order to evaluate the direct impact of a gene on overall survival (OS). For the feasibility of constructing robust networks, we initially employed k-mean clustering and clustered genes in 4 classes. We then built a data-driven network for each cluster based on an order-independent implementation of the conditional independence properties, i.e., directed acyclic graph (DAG), learning PC-algorithm [##UREF##7##9##]. We also augmented the networks with eQTLs as instrumental variables (IVs) to identify causal networks established in the MR principles [##UREF##9##11##, ##REF##27501297##49##–##UREF##26##51##]. We identified stable networks by employing two different techniques and then focused on the edges commonly identified by both methods. One method was based on false discovery rate (FDR) where we built a dense network by retaining all the edges within each cluster. We then select significant edges between gene pairs with FDR ≤ 0.05.</p>", "<p id=\"P42\">The other method for the network stability determination was based on the Hamming distance metric (HD). In this method, we constructed the network for different values of , and calculated HD for each pair of networks where . Smaller leads to a sparser network, but the question is which yields the network corresponding to the actual sparsity level. To answer this question, we employed a piecewise regression model by regressing on as follows;\n</p>", "<p id=\"P43\">Here, is the indicator function of significant slope change for the breakpoint . The breakpoint represents a point that indicates a significant change in the slope of the regression model. By fitting this model for all possible breakpoints, , we identified the optimal corresponds to the maximum breakpoint. ##FIG##4##Figure 5## represents this procedure for one of the clusters in our analysis, whereas Figure S6 represents the illustration for all the clusters.</p>", "<title>Validating the edges in the network.</title>", "<p id=\"P44\">To validate the interconnectivity among genes, we used 103 additional patients for the combination arm of the study (bevacizumab + cetuximab plus chemotherapy) excluded from the main analysis. We considered this set as the test set and replicated the interconnectivity among genes using the predictive linear model as follows:\n</p>", "<p id=\"P45\">Here, is an matrix of the expression level of predictors for samples used for building the network. The predictors of each gene refer to its direct upstream and downstream genes in the network. is an matrix of predictors’ expression levels for samples (here, ) selected for the test set. We then calculated the correlation between observed and predicted values, and considered a link validated if the correlation was above 0.5.</p>", "<title>Identification of sub-networks and their regulatory genes.</title>", "<p id=\"P46\">We defined a sub-network as a set of genes directly connected to an eGene or after one step. We used the eQTLs to implement the Mendelian randomization (MR) technique in addition to the rule of v-structure (details in supplementary V-structure) for identification of causal relationship between genes [##UREF##9##11##, ##REF##27501297##49##, ##UREF##25##50##]. The causal relationships discovered genes with a high potential of having a regulatory role in the sub-networks. In addition, we included known gene regulatory pathways for identifying the causal network. For instance, it is known that <italic toggle=\"yes\">FANCI</italic> and <italic toggle=\"yes\">BLM</italic> are in the Fanconi anemia pathway, and <italic toggle=\"yes\">FANCI</italic> regulates <italic toggle=\"yes\">BLM</italic> [##UREF##27##52##]. Therefore, we used this knowledge to define causal relationship between these two genes in sub-network 3. There were some edges with unidentified directions, but we did not remove those from the study.</p>", "<title>Identifying gene signatures impacting OS.</title>", "<p id=\"P47\">We performed the Cox proportional hazard model for genes in each sub-network, considering the underlying relationship of the genes in the network by adjusting the analysis for the impact of confounding genes on the gene-OS relationship as\n\nwhere . Here, includes all the upstream genes of in the sub-network, represents a matrix of covariates, and represents a matrix of enriched cell types. This analysis estimated the effect of on OS since it included their adjusted expression level, , for the effect of their upstream genes in the sub-network.</p>", "<p id=\"P48\">In each sub-network, we defined gene signature as the set of genes with significant impacts on OS since these genes collectively impact OS. To facilitate the nomination of a scoring model for prospective testing as a biomarker and to support the visualization of survival plots, we dichotomized the expression level of each gene in each signature based on their third quartile in each arm and defined beneficial/non-beneficial with respect to OS. For instance, if the higher expression of a gene prolonged OS, a patient was stratified as being in a beneficial state if the expression level exceeded the third quartile or as a non-beneficial if it fell below the third quartile. After stratifying patients based on genes in each sub-network, we estimated the overall survival function using the Kaplan-Meier estimator for each treatment.</p>" ]

[ "<title>Acknowledgements.</title>", "<p id=\"P50\">Research reported in this publication was supported by the NCI of the NIH under Award Numbers U10CA180821, U10CA180882, and U24CA196171 (to the Alliance for Clinical Trials in Oncology). Also supported in part by funds from Bristol-Myers Squibb, Genentech, Pfizer, and Sanofi.</p>", "<title>Data availability.</title>", "<p id=\"P49\">The gene expression data generated in this study are publicly available in Gene Expression Omnibus at GSE196576.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><title>Overall workflow.</title><p id=\"P55\">Gene interactions were revealed as transcriptomic-causal-network and integrated into OS analysis to account for confounding genes and identify gene sets that each collectively impact OS as a gene signature. Gene signatures were used to stratify patients. The results were replicated across multiple replication sets. To elucidate the biological functions of the findings, we explored the association of the identified genes with immune signatures. We also investigated the functional relationships among genes based on the protein-protein interactions database.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><p id=\"P56\">Data and enrichment analysis for cis-eQTL annotation. <bold>A.</bold> The Venn diagrams represent the number of samples with either RNA-seq or germline genotype data in each arm of the study. <bold>B.</bold> Z-score of the enrichment analysis for genomic location. <bold>C.</bold> Z-score of the overlap of cis-eQTLs with Roadmap enhancers for colon and rectum tissue; an active enhancer refers to the regulatory region of DNA that interacts with the promoter DNA region; a genic enhancer refers to regulatory regions in a gene; a bivalent enhancer refers to segments of DNA that have both repressing and activating epigenetic regulators in the same region.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3</label><caption><p id=\"P57\">Transcriptomic-causal network linked to OS. <bold>A.</bold> Revealed transcriptomic-causal network and its sub-networks comprising genes with significant effects on OS. <bold>B.</bold> Confounder genes and mediators respect to OS. <bold>C.</bold> Treatment specific p-values for genes with influencing OS within the sub-networks, with significant level represented with dashed line.</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4</label><caption><p id=\"P58\">Kaplan-Meier plots based on the identified gene signatures coresponding to each sub-network. The genes are dichotomized into beneficial and non-beneficial based on their third quartile with respect to OS. The purple curve represents the OS for patients with beneficial levels for both genes in the signatures, and the orange curve for those with non-beneficial levels. The median OS exhibited a notable decrease from 43.5 to 16.1 months (<italic toggle=\"yes\">p</italic>-value: 0.0002) in sub-network 1, from 38.1 months to 13.1 months (<italic toggle=\"yes\">p</italic>-value: 0.0059) in sub-network 3, when comparing beneficial to non-beneficial signature level.</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5</label><caption><p id=\"P59\">Immune related biological function of genes in the sub-networks 1–3. The heatmap shows the relationship between immune signatures and genes in the sub-networks with impact on OS. The correlations above ± 0.4 are presented on the heatmap.</p></caption></fig>", "<fig position=\"float\" id=\"F6\"><label>Figure 6</label><caption><p id=\"P60\">Piecewise regression for assessing stability of the network based on the Hamming distance metric. Y-axis represents <italic toggle=\"yes\">p</italic>-values from model (1) for coefficients <italic toggle=\"yes\">β</italic>_{<italic toggle=\"yes\">3</italic>}. X-axis represents −log10 of α_i.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\"><label>Table 1</label><caption><p id=\"P61\">Replication of gene interactions in the identified signatures using the STRING database. The confidence scores greater than 0.5 indicate promising evidence for potential physical interactions of proteins encoded by the genes in the signatures.</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Protein</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Connected</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Score</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">FANCI → PRC1</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">directly</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.63</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">IDO1 → GBP4</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">directly</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.52</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">RELT → → → MYO1G</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">within two steps</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.65, 0.51, 0.58</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T2\"><label>Table 2</label><caption><p id=\"P62\">Summary result of differential expression analysis over 21,983 genes from the GSE146889 database as an external cohort. FC: fold change; SE: standard error; stat: the value of the test statistic for the gene. The adjusted p-value is based on FDR correction.</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Gene symbol</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">log2 (FC)</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">SE _log2(FC)</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">stat</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"><italic toggle=\"yes\">p</italic>-value</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Adjusted <italic toggle=\"yes\">p</italic>-value</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>BLM</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−1.75</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.16</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−10.98</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">4.60E-28</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9.49E-27</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>DTL</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−2.29</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.18</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−12.75</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3.16E-37</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.47E-35</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>FANCI</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−1.88</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.11</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−16.75</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5.54E-63</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">4.39E-60</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>GBP2</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.49</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.12</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">4.27</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.99E-05</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5.09E-05</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>GBP4</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−0.77</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.15</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−5.18</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.25E-07</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">7.15E-07</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>GBP5</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−0.97</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.20</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−4.88</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.05E-06</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3.11E-06</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>IDO1</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−2.35</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.27</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−8.62</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6.97E-18</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6.01E-17</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>PRC1</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−2.06</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.13</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−15.96</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.44E-57</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">8.49E-55</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>RELT</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−0.47</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.16</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−2.90</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3.68E-03</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6.91E-03</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>SNRPA1</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−0.77</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.06</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−12.75</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3.28E-37</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.52E-35</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>UAP1L1</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−0.40</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.11</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−3.62</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.91E-04</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6.41E-04</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>UBE2T</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−2.43</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.16</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">−15.30</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">7.35E-53</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.57E-50</td></tr></tbody></table></table-wrap>" ]

[ "<inline-formula><mml:math id=\"M1\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfenced open=\"{\" close=\"}\" separators=\"|\"><mml:mrow><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:msup><mml:mo>∣</mml:mo><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mn>22</mml:mn></mml:mrow></mml:mfenced></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M2\" display=\"inline\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M3\" display=\"inline\"><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mi>i</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M4\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M5\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M6\" display=\"inline\"><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M7\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"FD1\">\n<label>(1)</label>\n<mml:math id=\"M8\" display=\"block\"><mml:mi>H</mml:mi><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mfenced open=\"{\" close=\"}\" separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>&lt;</mml:mo><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"M9\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mfenced open=\"{\" close=\"}\" separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>&lt;</mml:mo><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M10\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M11\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mi mathvariant=\"normal\">s</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M12\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<disp-formula id=\"FD2\">\n<mml:math id=\"M13\" display=\"block\"><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mi>ˆ</mml:mi></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">test</mml:mi><mml:mspace width=\"0.25em\"/></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:msup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup><mml:mi>G</mml:mi></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup><mml:mi>g</mml:mi><mml:mo>.</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"M14\" display=\"inline\"><mml:mi>G</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M15\" display=\"inline\"><mml:mi>n</mml:mi><mml:mo>×</mml:mo><mml:mi>p</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M16\" display=\"inline\"><mml:mi>p</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M17\" display=\"inline\"><mml:mi>n</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M18\" display=\"inline\"><mml:mo>(</mml:mo><mml:mi>g</mml:mi><mml:mo>)</mml:mo></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M19\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">test</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M20\" display=\"inline\"><mml:mi>m</mml:mi><mml:mo>×</mml:mo><mml:mi>p</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M21\" display=\"inline\"><mml:mi>m</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M22\" display=\"inline\"><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>103</mml:mn></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M23\" display=\"inline\"><mml:mfenced separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">test</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M24\" display=\"inline\"><mml:mfenced separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mi>ˆ</mml:mi></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">test</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math></inline-formula>", "<disp-formula id=\"FD3\">\n<mml:math id=\"M25\" display=\"block\"><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:mfenced><mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">exp</mml:mi></mml:mrow><mml:mspace width=\"0.15em\"/><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mi>α</mml:mi><mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:mspace width=\"0.15em\"/><mml:mrow><mml:mtext>\\varvec</mml:mtext><mml:mi>γ</mml:mi><mml:mover accent=\"true\"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mo>˜</mml:mo></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:mspace width=\"0.15em\"/><mml:mrow><mml:mtext>\\varvec</mml:mtext><mml:mi>β</mml:mi><mml:mi>G</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:mspace width=\"0.15em\"/><mml:mrow><mml:mtext>\\varvec</mml:mtext><mml:mi>θ</mml:mi><mml:mi>Z</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:mspace width=\"0.15em\"/><mml:mrow><mml:mtext>\\varvec</mml:mtext><mml:mi>δ</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow><mml:mo>,</mml:mo></mml:math>\n</disp-formula>", "<inline-formula><mml:math id=\"M26\" display=\"inline\"><mml:mover accent=\"true\"><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mo>˜</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mo>˜</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mo>˜</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mspace width=\"0.25em\"/><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mo>˜</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mfenced separators=\"|\"><mml:mrow><mml:mi>I</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mspace width=\"0.25em\"/><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M27\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M28\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M29\" display=\"inline\"><mml:mi>Z</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M30\" display=\"inline\"><mml:mi>T</mml:mi></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M31\" display=\"inline\"><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>", "<inline-formula><mml:math id=\"M32\" display=\"inline\"><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mo>˜</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>" ]

[]

[]

[]

[]

[]

[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN1\"><p id=\"P51\">Competing interests.</p><p id=\"P52\">The authors declare no competing interests.</p></fn><fn id=\"FN2\"><p id=\"P53\">Declarations</p><p id=\"P54\">A statement of ethics approval: This study has been approved by the Alliance for Clinical Trials in Oncology.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"nihpp-rs3673588v1-f0001\" position=\"float\"/>", "<graphic xlink:href=\"nihpp-rs3673588v1-f0002\" position=\"float\"/>", "<graphic xlink:href=\"nihpp-rs3673588v1-f0003\" position=\"float\"/>", "<graphic xlink:href=\"nihpp-rs3673588v1-f0004\" position=\"float\"/>", "<graphic xlink:href=\"nihpp-rs3673588v1-f0005\" position=\"float\"/>", "<graphic xlink:href=\"nihpp-rs3673588v1-f0006\" position=\"float\"/>" ]

[]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"P5\">Inflammatory and glucolipotoxic (GLT) stress causing β-cell failure and destruction <italic toggle=\"yes\">in vitro</italic> differentially regulates hundreds of β-cell transcripts (##REF##32894309##1##, ##REF##32398822##2##). The upregulation of splicing factors and of proteins involved in pre-mRNA processing gives rise to alternative splicing (AS) events, which in turn deregulates the balance and turnover of transcript isoforms (##UREF##0##3##). Interestingly, most human mRNAs exhibit alternative splicing, but not all alternatively spliced transcripts are translated into functional proteins and are therefore targeted for degradation via the RNA decay pathways.</p>", "<p id=\"P6\">In addition to regulating the expression of normal transcripts, the human nonsense-mediated RNA decay (NMD) machinery functions to eliminate premature termination codon (PTC)-containing mRNAs, as reviewed extensively (##REF##23027648##4##). Alternatively spliced mRNA species and translation of dominant transcript isoforms vary in a cell-specific manner and depends on the capacity of cells to cope with damaged transcripts (##REF##15598820##5##, ####REF##22412385##6##, ##REF##33763030##7####33763030##7##). A substantial number (i.e., around 35%, but it depends on tissue and physiological conditions) of alternatively spliced variants contain a PTC (##REF##23027648##4##, ##REF##26773057##8##, ##REF##16418482##9##). Approximately 35% of the cytokine-regulated transcripts in human islets undergo alternative splicing (##REF##22412385##6##), and Cyt profoundly up-regulate NMD) in rat and human insulin-producing cell lines and primary β-cells, likely to handle the NMD-load inferred by PTC containing splice variants (##REF##23027648##4##, ##REF##17352659##10##, ##REF##30065031##11##).</p>", "<p id=\"P7\">However, in addition to canonical NMD in which all key NMD components function on target transcripts, a second branch of NMD is (in)dependently regulated in an autoregulatory feedback loop by its key factors including UPF2 and UPF3 in a cell type-specific manner as reviewed previously (##REF##23027648##4##, ##REF##17352659##10##).</p>", "<p id=\"P8\">In a previous study (##REF##30065031##11##) we profiled the <italic toggle=\"yes\">expressional level</italic> of NMD components and their regulation by cytokines and GLT in insulin-producing cells, but the NMD <italic toggle=\"yes\">activity</italic> and its consequences for the β-cell transcriptome remained to be investigated. Here, using a luciferase-based NMD activity reporter, gain-/loss-of function and RNA-sequencing analyses in rodent and human β-cell systems, we measured NMD activity and explored its consequences for function and viability of pancreatic β-cells under normal condition and inflammatory stress.</p>" ]

[ "<title>Materials and Methods</title>", "<title>Cell culture, human islet dispersion and treatment</title>", "<p id=\"P9\">INS1(832/13) (##REF##1370150##12##), EndoC-βH3 (##REF##21865645##13##) or dispersed human islet cells were cultured and manipulated according to the protocols and procedures described in ##SUPPL##0##Supplementary Methods##.</p>", "<title>Luciferase-based NMD activity assay</title>", "<p id=\"P10\">One million cells were co-transfected with 650 ng of plasmid encoding either human <italic toggle=\"yes\">Haemoglobin</italic>-<italic toggle=\"yes\">β</italic> (<italic toggle=\"yes\">HBB</italic>) wildtype (WT or PTC−) or with a PTC-containing mutation (NS39 or PTC+) fused with <italic toggle=\"yes\">Renilla (RLuc),</italic> in brief named PTC−and PTC+), respectively. <italic toggle=\"yes\">Firefly (FLuc)</italic> plasmid (##REF##16934750##14##) was used as transfection efficiency reference. <italic toggle=\"yes\">Renilla</italic> and <italic toggle=\"yes\">Firefly</italic> luminescence was measured by Dual-Luciferase Reporter Assay (Promega, Hampshire, England) (##SUPPL##0##Supplementary Methods##). <italic toggle=\"yes\">RLuc</italic> signals were normalized to the <italic toggle=\"yes\">FLuc</italic> control in both HBB(NS39) (i.,e., PTC+) and HBB(WT) (i.,e., PTC−), and NMD activity was calculated by dividing the <italic toggle=\"yes\">RLuc/FLuc</italic>-HBB(WT) by the <italic toggle=\"yes\">RLuc/FLuc</italic>-HBB(NS39) (##SUPPL##0##Supplementary Fig.1A##) (##REF##16934750##14##). Experiments where the control construct <italic toggle=\"yes\">RLuc/FLuc</italic>-HBB(WT) was affected by cytokines were excluded, so that the resulting NMD activity only denotes the PTC-containing HBB(NS39). The transfection efficiency was tested twice and resulted in an average of 80% in INS1 and EndoC-βH3 cells as measured by FACS analysis of cells transfected with a GFP expressing plasmid (##SUPPL##0##Supplementary Fig.1B##-##SUPPL##0##C##).</p>", "<title>Functional analysis of UPF3A/B overexpression</title>", "<p id=\"P11\">One million INS1(832/13) or EndoC-βH3 cells were transfected with 650 ng of plasmids encoding UPF3A, UPF3B or UPF3BΔ42 (##REF##28899899##15##), then simultaneously with NMD activity reporter plasmids ( as above, 650 ng/million cells), recounted and seeded for Western blotting, glucose-stimulated insulin secretion (GSIS), viability, apoptosis (detailed below) and NMD activity assays in relevant plates and pre-incubated for 48 h before treatment with cytokines as explained in the ##SUPPL##0##Supplementary Methods##.</p>", "<title>Lentiviral shRNA gene knock-down</title>", "<p id=\"P12\">GPIZ lentiviral shRNAs particles directed against <italic toggle=\"yes\">UPF2, Upf3A</italic> or <italic toggle=\"yes\">Upf3B</italic>, and a non-silencing shRNA (NS) as negative control, were produced using the Trans-Lentiviral shRNA Packaging System in HEK293 cells (Horizon, Cambridge, England) according to the manufacturer’s protocol (##SUPPL##0##Supplementary Methods##).</p>", "<title>Apoptosis and cell viability assays</title>", "<p id=\"P13\">Apoptosis assays were performed in duplicate by detection of caspase-3 activity using a fluorometric [μM AMC] (or/colorimetric [μM PNA/min/ml] unless stated) assay kit (Cat#APPA015-1KT/CASP3C-1KT, Sigma, London, England) according to the manufacturers’ protocols. Cell viability was measured by Alamarblue assay (Cat#DAL1025, LifeTechnologies, Renfrew, England) as previously described (##REF##30065031##11##).</p>", "<title>Library preparation, RNA-sequencing, and data analysis</title>", "<p id=\"P14\">Thirty-three independent biological replicates of total RNA from the NS control and or UPF2 KD EndoC-βH3 cells exposed to cytokines, GLT, or PBS (i.e., N=6 of each PBS-/or cytokine-exposed NS control and UPF2 KD, and N=4/N=5 of GLT-exposed NS control/UPF2 KD, respectively) was extracted using TRIZOL, treated with DNase, and precipitated with isopropanol (##SUPPL##0##Supplementary Methods##). One μg total RNA/per isolate was used as input for generation of sequencing libraries using NEBNext^®Ultra-TM RNA Library-Prep (Cat#E7770, NEB, Ipswich, USA) following manufacturer’s recommendations (##SUPPL##0##Supplementary Methods##). The RNA-seq raw data underwent quality control were mapped to human reference genome (##REF##34791404##16##) and analysed using the bioinformatic pipeline described in the ##SUPPL##0##Supplementary Methods##.</p>", "<title>cDNA synthesis and RT-qPCR</title>", "<p id=\"P15\">Purified total RNA (500 ng) was used for cDNA synthesis with the SuperScript^™ (Cat# 11904018, LifeTechnologies). Real-time Reverse Transcriptase-quantitative PCR (RT-qPCR) was performed on 12 ng cDNA with SybrGreen PCR mastermix (LifeTechnologies) and specific primers (##SUPPL##0##supplementary Table 2##) and run in an ABI? real-time PCR machine (Applied Biosystems, ThermoFisher Scientific). The raw data was analysed through −ΔCt as described in ##SUPPL##0##Supplementary Methods##.</p>", "<title>Western blotting</title>", "<p id=\"P16\">Western blotting was performed using antibodies against alpha-Tubulin (1:2000) (Cat# T5168, Sigma), UPF2 (1:1000) (Cat# PA5-77128, LifeTechnologies), Upf3A (1:1000) (Cat# PA5-41904, LifeTechnologies), UPF3B (1:1000) (Cat#PB9843, Boster-Bio, Pleasanton, USA) and α-1-antitrypsin (1:1000) (Cat#TA500375, LifeTechnologies) as described in ##SUPPL##0##Supplementary Methods## (##UREF##1##17##).</p>", "<title>Glucose-stimulated insulin secretion (GSIS)</title>", "<p id=\"P17\">Three hundred thousand INS1(832/13) or EndoC-βH3 cells were cultured in 12-well plates (Cat#150200, Nunc, Buckingham, England), and pre-incubated for two days. GSIS was carried out using Krebs-Ringer buffer containing 2 mM or 17 mM glucose as described (##REF##30065031##11##, ##REF##33932586##18##).</p>", "<title>Insulin assay</title>", "<p id=\"P18\">Insulin concentration (ng/ml or pM) was measured using rat insulin ultra-sensitive ELISA kit (Cat#62IN2PEG, Cisbio, Cambridge, England) or human insulin ELISA kit (Cat#90095, CrystalChem, IL, USA), respectively, according to manufacturer’s protocol.</p>", "<title>Statistical analysis</title>", "<p id=\"P19\">Data are presented as means ± SEM. Statistical analysis was carried out on raw data also in cases where figures give normalized data. Group comparisons were carried out by two- or one-way ANOVA as appropriate. Significant ANOVAs were followed by post-hoc paired Student’s t-test with Bonferroni-correction using GraphPad Prism 6.0 (La Jolla, USA). Paired t-test was chosen to normalize for inter-passage variability in outcome parameters. Since the experimental conditions did not allow sequential sampling from the same cell culture, parallel control and interventional plate wells were considered to be paired observations and analysed accordingly statistically. If the <italic toggle=\"yes\">post-hoc</italic> paired t-test did not reveal a carrying statistical difference by ANOVA, individual paired t-tests were performed and corrected for multiple comparisons. Bonferroni-corrected <italic toggle=\"yes\">P</italic>-values ≤0.05 were considered significant and ≤0.10 a trend.</p>" ]

[ "<title>Results</title>", "<title>Cytokines suppress, whereas glucolipotoxicity increases, NMD activity in β-cells.</title>", "<p id=\"P20\">We previously reported that cytokines and glucolipotoxicity differentially up or down-regulate NMD component transcripts in pancreatic β-cells (##REF##30065031##11##). However, whether this regulation leads to increased NMD <italic toggle=\"yes\">activity</italic> remained to be elucidated. Here, we used a luciferase-based NMD reporter (##SUPPL##0##Supplementary Fig.1A##) (##REF##16934750##14##) to examine NMD activity in rat INS1(832/13), human insulin-producing EndoC-βH3 cells and primary human islets. Luciferase activity analysis showed that cytokines (Cyt;150 pg/mL IL-1β +0.1 ng/mL IFNγ+0.1 ng/mL TNFα) significantly suppressed NMD activity by nearly 50% after 18 h, but not 6h, of exposure in INS1(832/13) cells (##SUPPL##0##Supplementary Fig.2A##).</p>", "<p id=\"P21\">We first tested the effects of cytokines on EndoC-βH3 cells and dispersed human islet cells. Cytokines (2.5 ng/ml IL-1β+10 ng/ml TNFα+10 ng/ml IFNγ, chosen from dose-response experiment shown in ##SUPPL##0##Supplementary Fig.2B##) attenuated NMD activity by 30% (<italic toggle=\"yes\">p</italic>=0.009, n=6) and 40% (<italic toggle=\"yes\">p</italic>=0.0006, n=6) after 18 h exposure of EndoC-βH3 cells (##FIG##0##Fig.1A##-left) and dispersed human islet cells (##FIG##0##Fig.1B##), respectively. Cyt increased the luciferase signal (<italic toggle=\"yes\">RLuc/FLuc</italic>) from the HBB(PTC+) (##SUPPL##0##Supplementary Fig.2B##-##SUPPL##0##C##) confirming that the NMD substrate HBB(PTC+) was restored due to NMD activity attenuation by Cyt.</p>", "<p id=\"P22\">We next examined whether cytokine-mediated suppression of NMD was consistent with an accumulation of HBB(PTC+) <italic toggle=\"yes\">transcripts</italic>. For this, we used a forward and reverse primer set to amplify the <italic toggle=\"yes\">Renilla</italic> gene and the junction of exons 1 and 2 (i.e., ensuring amplification of mature transcripts only), respectively. RT-qPCR analysis demonstrated that cytokines caused significant upregulation of HBB(PTC+), but not HBB(PTC−) mRNA levels, rendering a significant reduction of the relative PTC−/PTC+ mRNA levels in INS1(832/13) (<italic toggle=\"yes\">p</italic>=0.008) (##SUPPL##0##Supplementary Fig.2A##-right) and EndoC-βH3 (<italic toggle=\"yes\">p</italic>=0.001) cells (##FIG##0##Fig.1A##-right), which verified the suppressive effect of cytokines on NMD activity.</p>", "<p id=\"P23\">Since we previously noted differential regulation of NMD component expression between Cyt and GLT exposure in INS1 cells and human islets (##REF##30065031##11##), we wished to clarify if a similar selective action related to the nature of cellular stress pertained to NMD activity. In fact, in striking contrast to Cyt, high glucose (GL, 25 mM) and GLT increased the activity of the NMD pathway (##FIG##0##Fig.1C##, ##SUPPL##0##Supplementary Fig.2D##). Examination by RT-qPCR confirmed that GLT increased the relative HBB(PTC+), but not HBB(PTC−) mRNA level (##SUPPL##0##Supplementary Fig.2D##-right), yielding a significant increase in the relative PTC−/PTC+ mRNA levels in EndoC-βH3 cells (##FIG##1##Fig.2D##-right).</p>", "<p id=\"P24\">Taken together, these results show that cytokines suppress the activity of the NMD activity in a range of insulin secreting cell types. In contrast, glucotoxic and glucolipotoxic conditions increase the NMD activity EndoC-βH3 cells.</p>", "<title>Cytokines-induced suppression of NMD activity in β-cells is ER stress dependent</title>", "<p id=\"P25\">Whereas NMD degrades unfolded protein response (UPR)-induced transcripts in compensated ER stress, NMD is suppressed in response to pronounced endoplasmic reticulum (ER) stress to allow a full-blown UPR (##REF##25807986##19##, ##REF##28503708##20##). Cytokines induce a robust ER stress in pancreatic β-cells, largely via nuclear factor-κB (NF-κB) activation and production of nitroxidative species that inhibit the smooth endoplasmic reticulum Ca²⁺ ATPase (SERCA) 2B pump, leading to ER calcium depletion (##REF##30065031##11##, ##UREF##1##17##). We have previously shown that chemical inhibition of inducible nitric oxide synthase (iNOS) alleviated ER stress and normalised cytokine-mediated regulation of NMD components in INS1(832/13) cells (##REF##30065031##11##). Therefore, we asked, if cytokine-mediated reduction of NMD activity was dependent on an ER stress response in β-cells. We first demonstrated that thapsigargin (TG), a non-competitive inhibitor of SERCA (##REF##9874782##21##) and ER stress inducer (##REF##28972171##22##) inhibited NMD activity by 50% in EndoC-βH3 cells as measured by luciferase assay (##SUPPL##0##Supplementary Fig.3A##). Compared to untreated EndoC-βH3 cells as control (CTL), cytokines significantly augmented the increase in mRNA levels encoding the ER stress markers BiP, Xbp1 and Chop (FDR &lt;0.05) measured by RNA-sequencing analysis (##FIG##1##Fig.2A##-left), and later verified by RT-qPCR examination (##FIG##1##Fig.2A##-right). Finally, compared with CTL, cytokines significantly decreased the NMD activity by 30%, and this effect was counteracted by the protein kinase R-like endoplasmic reticulum kinase (PERK) phosphorylation inhibitor GSK157 (8 μM) and by the Inositol-Requiring Enzyme1 (IRE1α) endoribonuclease inhibitor 4μ8C (16 μM) in EndoC-βH3 cells (##FIG##1##Fig.2B##-left, ##SUPPL##0##Supplementary Fig.3B##-##SUPPL##0##C##). RT-qPCR analysis of the relative PTC−/PTC+ mRNA levels in EndoC-βH3 cells confirmed the NMD activity data (##FIG##1##Fig.2B##-right, ##SUPPL##0##Supplementary Fig.3C##).</p>", "<p id=\"P26\">Taken together, these results demonstrate that inhibition of UPR antagonises the cytokine-mediated reduction of NMD activity in EndoC-βH3, indicating that cytokine-mediated inhibition of NMD activity is UPR-dependent.</p>", "<title>Cytokine-induced suppression of NMD activity is associated with UPF3B downregulation and attenuated by UPF3 overexpression in β-cells</title>", "<p id=\"P27\">Since we observed in our previous study that cytokine-induced ER stress downregulated UPF3B expression in human and rodent β-cells, as recovering nitroxidative-driven ER stress using the inducible nitric oxide synthase (iNOS) inhibitor N-methyl-l-arginine (NMA) since (##REF##30065031##11##),transcripts encoding UPR components are NMD targets and have been shown to be stabilised by UPF3A/B depletion (##REF##25807986##19##) and since UPF3B is a NMD activator in mammalian cells (##REF##27040500##23##), which led to proposed Upf3-dependent and -independent branches of NMD pathway (##REF##23027648##4##, ##REF##17352659##10##, ##UREF##2##24##). we reasoned that UPF3 regulated NMD activity in β-cells We therefore first measured the UPF3A/B expression level and next investigated the functional impact of overexpressing UPF3A/B on cytokine-mediated suppression of NMD activity in β-cells. RT-qPCR examination showed that cytokines significantly downregulated UPF3B mRNA levels after 18 h in both EndoC-βH3 (##FIG##2##Fig.3A##) and INS1(832/13) (##SUPPL##0##Supplementary Fig.4A##) as previously reported (##REF##30065031##11##). Immunoblot analysis verified overexpression of UPF3A, UPF3B and the UPF3B dominant negative UPF3BΔ42 in both INS1(832/13) and EndoC-βH3 (##SUPPL##0##Supplementary Fig.4B##-##SUPPL##0##C##-left). Cytokines reduced NMD activity, overexpression of UPF3B significantly attenuated this reduction in EndoC-βH3 (##FIG##2##Fig.3B## and ##SUPPL##0##Supplementary Fig.4B##) and to a lesser extent in INS1(832/13) (##SUPPL##0##Supplementary Fig.4C##). Neither UPF3A nor UPF3BΔ42 overexpression counteracted cytokine-attenuated NMD activity.</p>", "<p id=\"P28\">This result suggests that cytokines reduce the NMD activity in β-cells through downregulation of UPF3B expression.</p>", "<title>UPF3 overexpression deteriorates cell viability and reduces insulin content, but not secretion in EndoC-βH3 cells</title>", "<p id=\"P29\">The above findings provide evidence that the UPF3-dependent branch of NMD is involved in cytokine-mediated suppression of NMD activity. Therefore, we next investigated the impact of UPF3A/B overexpression on cytokine-induced cell death and insulin secretion. While UPF3A or UPF3B over-expression increased basal cell death, it also exacerbated the cytokine-induced apoptosis in EndoC-βH3 cells (##FIG##3##Fig.4A##). In INS1(832/13) cells neither UPF3A nor UPF3B overexpression changed cell viability in the absence of Cyt exposure, but UPF3B over-expression significantly aggravated cytokine-induced cell death as measured by Alamarblue and caspase-3 activity assays (##SUPPL##0##Supplementary Fig.5A##). Therefore, we next explored the impact of UPF3A or UPF3B deficiency on β-cell viability. Lentiviral shRNA-mediated knockdown of UPF3A and or UPF3B (##SUPPL##0##Supplementary Fig.5B##) significantly reduced basal INS1(832/13) cell viability (##SUPPL##0##Supplementary Fig.5C##). Taken together, this data indicates that genetic manipulations of UPF3A/B could be possibly detrimental for the β-cell viability.</p>", "<p id=\"P30\">Given that GLT increases NMD activity (##FIG##0##Fig.1F##), we hypothesised that UPF3 deficiency might prevent glucolipotoxicity-induced cell death in β-cells. Measurements of Caspase-3 activity demonstrated that both UPF3A and UPF3B knockdown rendered a slight, but significant protection against 24 h glucolipotoxicity in INS1(832/13) cells in comparison with untreated cells (##SUPPL##0##Supplementary Fig.5D##). Further, treatment with the NMD activator Tranilast dose-dependently sensitised to glucolipotoxicity-, but not cytokines-induced, EndoC-βH3 cell death, measured by caspase-3 activity assay (##SUPPL##0##Supplementary Fig.4E##). Neither UPF3A nor UPF3B overexpression affected GSIS in EndoC-βH3 (##FIG##3##Fig.4B##) or INS1(832/13) (##SUPPL##0##Supplementary Fig.5F##) cells. Nonetheless, UPF3B overexpression profoundly lowered insulin content in EndoC-βH3 cells (##FIG##3##Fig.4B##). In contrast, knockdown of UPF3A or UPF3B significantly decreased the stimulatory index, as well as provoking a substantial increase in insulin content in control INS1(832/13) cells (##SUPPL##0##Supplementary Fig.5G##-##SUPPL##0##H##).</p>", "<p id=\"P31\">Taken together, these findings reveal that UPF3 overexpression induces basal cell death and exacerbates cytokine-mediated toxicity in β-cells.</p>", "<title>UPF2 knockdown potentiates cytokine suppression of NMD activity and slightly alleviates cytokine toxicity for cell viability and insulin content in EndoC-βH3 cells</title>", "<p id=\"P32\">Next, we investigated the effect of UPF2 deficiency on the viability and insulin secretion of β-cells because (i) Knowingly UPF3A and UPF3B are involved in regulating UPF2, a key core NMD activator in mammalian cells (##REF##28008922##25##, ##REF##15059251##26##) by sequestering away from and bridging the exon-junction complex (EJC) with UPF1 and UPF2, respectively, leading to the NMD activation (##REF##27040500##23##), (ii) genome-wide association data (GWAS) data reveal that the <italic toggle=\"yes\">UPF2</italic> variant rs145580445 is significantly associated with type 2 diabetes risk (##REF##33763030##7##). We therefore knocked down the <italic toggle=\"yes\">UPF2</italic> gene in EndoC-βH3 cells using RNA interference and chose the three cell lines in which UPF2 was most efficiently knocked down (KD) (##SUPPL##0##Supplementary Fig.6A##). Examination of NMD activity using the luciferase-based NMD reporter revealed that UPF2 KD profoundly reduced NMD activity in untreated and cytokine-treated EndoC-βH3 cells (##SUPPL##0##Supplementary Fig.6B##). Compared with NS control, UPF2 KD slightly, but significantly prevented cytokine-induced cell death (##FIG##4##Fig.5A##). UPF2 KD had no effect on the GSIS, but significantly increased insulin content (##FIG##4##Fig.5B##).</p>", "<p id=\"P33\">These data indicate that the UPF2 plays a crucial role in cytokine-induced β-cell apoptosis. In addition, the increase in insulin content in UPF2 deficient EndoCβH3 cells implies that insulin transcripts could possibly be targets of the UPF2-dependent NMD pathway branch.</p>", "<title>UPF2 knockdown differentially affects cytokine- and glucolipotoxicity-mediated deregulation of EndoC-βH3 transcripts</title>", "<p id=\"P34\">Consistent with our observations above (##FIG##4##Fig.5##), we previously reported (##REF##30065031##11##) that the deficiency of SMG6, an endoribonuclease and a key effector of NMD, rendered protection against cytokine-induced cell death and was associated with increased insulin content. Therefore, we aimed to identify potential NMD target transcripts and then, we used RNA-sequencing to assess the transcriptome of cytokine- or PBS-treated EndoC-βH3 cells stably transfected with a non-silencing shRNA (NS) or the specific shRNA (shRNA-1 named U1) against <italic toggle=\"yes\">UPF2</italic>. Since GLT exposure increased, whereas cytokine exposure decreased, NMD activation in β-cells (##FIG##0##Fig.1F##), we also performed RNA-sequencing after UPF2 KD vs. NS control EndoC-βH3 exposed to GLT compared with PBS-treated.</p>", "<p id=\"P35\">The RNA-seq datasets from either UPF2 KD or NS control EndoC-βH3 cell lines exposed to cytokines and or GLT was dimensionally reduced by principal component analysis (PCA) into into two main principal components, PC1 and PC2 (<italic toggle=\"yes\">p&lt;0.05</italic>). The PCA of the NS control EndoC βH3 cells demonstrated a high similarity between the biological replicates, a small within-group variance and a distinct clustering of the untreated, cytokines and GLT groups (##SUPPL##0##Supplementary Fig.7A##). Pearson correlation (<italic toggle=\"yes\">p&lt;0.05</italic>) between samples justified the clustering of biological replicates of cytokines, GLT and untreated conditions (##SUPPL##0##Supplementary Fig.7B##). PCA revealed that the UPF2 knockdown increased the majority of variance in transcript isoforms of the cell, hence patterns leading to visually dispersed biological replicates from the cytokine exposed isolates, whereas decreased the variances from the GLT and untreated biological replicates, hence clustered them together.</p>", "<p id=\"P36\">A Venn diagram of the RNA-seq datasets demonstrated around an approximate total of 14,000 commonly expressed transcripts (FDR&lt;0.05) and those differentially expressed transcripts regulated by PBS (i.e., untreated control), cytokines or GLT (##SUPPL##0##Supplementary Fig.7C##-##SUPPL##0##D##). Among them, cytokines regulated 54% (up) and 46% (down), whereas GLT impacted 48% (up) and 52% (down) of significantly expressed transcripts in NS control EndoC-βH3 cells (##FIG##5##Fig.6A##). UPF2 KD changed the cytokine-mediated regulation of significantly expressed transcripts by 59% (up) and 41% (down), whereas it did not change regulation of significantly expressed transcripts by GLT (up:47%, down:53%) in EndoC-βH3 cells (##FIG##5##Fig.6A##). This indicates UPF2 KD possibly alters the mRNA levels of identified transcript species.</p>", "<p id=\"P37\">To identify UPF2 KD-regulated transcripts possibly providing protection against cytokine-induced cytotoxicity, we interrogated cytokine-and/or GLT-regulated transcripts that were differentially expressed (<italic toggle=\"yes\">p&lt;0.05</italic>) in UPF2 KD versus NS control EndoC-βH3 cells using gene enrichment analysis (GEA). Ge Ontontolgy (GO) and Kyoto Encyclopaedia of Genes and Genomes (KEGG) gene enrichment analyses demonstrated that GLT significantly (<italic toggle=\"yes\">p&lt;0.05</italic>) regulated transcripts in the cellular functions of RNA splicing, mitochondrial inner membrane, and purine nucleoside metabolism (##SUPPL##0##Supplementary Fig.7E##), although these were not affected by UPF2 KD (##FIG##5##Fig.6B##). In contrast, Gene Enrichment Analysis (GEA) revealed that in both untreated and cytokine-treated EndoC-βH3 cells, UPF2 KD significantly regulated transcripts encoding proteins involved in synaptic transmission, extracellular matrix, basolateral plasma membrane, receptor complex, synaptic membrane, transcriptional repressor activity and enzyme inhibitor activity (##FIG##5##Fig.6B##-##FIG##5##C##; ##SUPPL##0##Supplementary Fig.7E##).</p>", "<p id=\"P38\">RT-qPCR confirmed the logarithmic fold change of UPF2 KD regulated transcripts in the cytokine-treated EndoC-βH3 cells <italic toggle=\"yes\">versus</italic> cytokines-treated NS control cells (##FIG##5##Fig.6D##). The role of many of these transcripts in cell viability or insulin secretion was previously identified in pancreatic β-cells. Among them, α-1-antitrypsin has been proposed as an antagonist against cytokine-induced pancreatic β-cell death (##REF##18852465##27##, ##REF##33667344##28##). Our expression profiling verified that cytokines upregulated α-1-antitrypsin, and this effect was further potentiated by UPF2 KD. To explore the potential importance of these changes, we knocked down α-1-antitrypsin using specific siRNAs in INS1(832/13) and EndoC-βH3 cells as confirmed by quantitative WB (##SUPPL##0##Supplementary Fig.8##). Both α-1-antitrypsin siRNAs aggravated cytokine-induced cell death in comparison with NS control (##FIG##6##Fig.7A##). Only one siRNA increased basal cell death. The effect of siRNA-mediated α-1-antitrypsin knockdown was inconclusive in EndoC-βH3 cells (##FIG##6##Fig.7B##). Compared with NS control, α-1-antitrypsin knockdown reduced GSIS index, but had no effect on insulin content in INS1(832/13) cells (##FIG##6##Fig.7B##), possibly due to the reduced cell number.</p>", "<p id=\"P39\">Cytokines reportedly upregulate &gt;30 splicing factors, affecting alternative splicing of 35% of genes in the human islet transcriptome (##REF##22412385##6##). We examined RNA-seq datasets for alternative splicing (AS) isoforms driven by cytokines or GLT versus untreated in the NS control and UPF2 KD EndoC-βH3 cells. Among 2123 and 2106 cytokine-driven AS isoforms, skipped exon (SE) isoforms constituted 70.89% (<italic toggle=\"yes\">p</italic>=0.1, n=6) and 72.5% (<italic toggle=\"yes\">p</italic>=0.1, n=6) in NS control and UPF2 KD cells, respectively. In contrast, 220 and 133 GLT-driven AS isoforms were identified in NS control and UPF2 KD, respectively (##SUPPL##0##Supplementary Fig.9##). This differential regulation could possibly provide a reliable measure for cytokines and GLT role in inducing AS isoforms in β-cells.</p>", "<p id=\"P40\">Taken together, the above transcriptome analysis of EndoC-βH3 cells indicates that cytokines increase 1–AT expression, and this is synergised by NMD attenuation.</p>" ]

[ "<title>Discussion</title>", "<p id=\"P41\">In this study, we demonstrate that cytokines decrease nonsense-mediated RNA decay (NMD) in INS1(832/13), EndoC-βH3 cells and dispersed human islets and. In contrast, glucolipotoxicity (GLT) increased the NMD activity in EndoC-βH3 cells. We also showed that the cytokine-mediated decrease of NMD activity was driven by ER stress and downregulation of UPF3B. Loss-/or gain-of function of NMD activity could be elicited by UPF3B over-expression or UPF2 knockdown, which led to increases in, or slight decreases in, cytokine-induced apoptosis associated with decreased and increased insulin contents, respectively, without affecting GSIS index in EndoC-βH3 cells. Transcriptome profiling indicated a potentiating effect of UPF2 knockdown on Cyt, but not GLT-mediated, NMD activity. Interestingly, this approach identified transcript targets encoding proteins belonging to the extracellular matrix such as α-1-antitrypsin. Importantly, the knockdown of this gene enhanced cytokine-induced cytotoxicity in β-cells.</p>", "<p id=\"P42\">To the best of our knowledge the present study represents the first demonstration of a functional effect of cytokines, in contrast to glucolipotoxicity, on NMD activity.</p>", "<p id=\"P43\">UPR activation is known to inhibit NMD via PERK activation and eIF2 phosphorylation to restore IRE1α accumulation and hence a robust UPR activation (##REF##25807986##19##, ##REF##28503708##20##, ##REF##27173476##29##); additional to the role of PERK activation, our findings suggest that IRE1α riboendonuclease activity (<italic toggle=\"yes\">p=0.1</italic>) was involved in cytokine-mediated NMD inhibition in EndoC-βH3 cells.</p>", "<p id=\"P44\">UPF3A and UPF3B act as a potent NMD inhibitor and activator, respectively, in HeLa cells and in mice (##REF##27040500##23##), consistent with our observations following UPF3B overexpression in β cells. However, the finding that forced UPF3A overexpression slightly increased NMD activity in β-cells seems inconsistent with previous findings. Recent studies (##UREF##3##30##, ##REF##35451084##31##) support our apparently discrepant finding regarding the effects of UPF3A overexpression by showing redundancy of UPF3A and UPF3B as modular activators of NMD (##UREF##2##24##). With these two earlier studies in mind, we cannot rule out the interference of endogenous UPF3A in the actions of UPF3B on NMD in β-cells.</p>", "<p id=\"P45\">We investigated the consequences of NMD activity for pancreatic β-cell function and viability. The increase of NMD activity by UPF3B overexpression induced basal and cytokine-induced cell death in EndoC-βH3 cells, highlighting the role of increased UPF3B level in β-cells. This appears to be relevant for β-cell viability in both normal and inflammatory stress conditions. Similarly, UPF3B knockdown also caused basal cell death in INS1 cells. Hence, basal UPF3A/B levels seem to play crucial roles in the cell viability of β-cells and perturbation of such a controlled level implicates in cell death. On the other hand, the slight protection against cytokine-induced cell death conferred by UPF2 knockdown in EndoC-βH3 cells (##FIG##4##Fig.5A##) and by SMG6 knockdown in INS1 cells (##REF##30065031##11##) implies a possibly protective mechanism against cytotoxicity of cytokines in β-cells, irrespective to the outcome cytokine-induced cell death. Moreover, our findings provide evidence that the reduced insulin content observed after UPF3B overexpression is related to overactivated NMD. In terms of NMD activation, though yet to explore the mechanism, glucolipotoxicity activated NMD pathway (##FIG##0##Fig.1G##). These findings suggest that the effect of GLT on NMD activation in T2D animal models needs to be explored further (##REF##33763030##7##).</p>", "<p id=\"P46\">Cytokine-induced perturbation of NMD, (potentiated by UPF2 silencing), might change the balance of anti-/pro-apoptotic transcripts. This, in turn, may contribute to cytotoxic damage. Consistent with this view, GEA revealed that cytokines deregulate transcripts encoding proteins that localise to and/or function in the extracellular matrix Thus, α-1-antitrypsin knockdown increased the detachment of MIN6 cells and exacerbated Thapsigargin-induced cell death as measured by Propidium-Iodide staining (##REF##33667344##28##) and in this study increased cytokine-induced cell death in INS1(832/13) cells associated with decreased GSIS index.</p>", "<p id=\"P47\">We speculate that the perturbation of NMD by cytokines leads to increased exon skipping and that this may be part of a feedback loop promoting β-cell plasticity and resilience against cytotoxic cytokines. Future studies will be needed to test this possibility. We note also that depletion of alternative splicing factors (reviewed in (##UREF##0##3##)) inhibits insulin secretion and induces basal apoptosis and after cytokine treatment in rodent and human β-cells (##UREF##0##3##, ##REF##33763030##7##, ##REF##25249621##32##, ##REF##28077579##33##). Moreover, antisense-mediated exon skipping of 48–50 exons of the dystrophin gene restores the open reading frame and allows the generation of partially to largely functional protein (##REF##17684229##34##).</p>", "<p id=\"P48\">In conclusion, we reveal that cytokines suppress NMD activity via ER stress signalling, possibly as a protective response against cytokines-induced NMD component expression. Our findings highlight the central importance of RNA turnover in β-cell responses to inflammatory stress.</p>", "<title>Limitations.</title>", "<p id=\"P49\">We used a luciferase-based NMD reporter based on two separate PTC− and PTC+ constructs whose labelled luciferase is separately measured. Thus, a yet-to develop NMD activity reporter by which transcripts RNA, protein, or their corresponding labelled luciferase activity of both PTC− and PTC+ transcripts could be examined in one cell rather than two separate cells will remove the limitation of current reporter based on transfection of the constructs into two separate cells. Furthermore, the constant overexpression of UPF3A and UPF3B may result in cell death as β-cells cannot cope with overwhelming level of these proteins above the basal level. This could explain that UPF3B overexpression reduced the basal cell viability.</p>", "<title>Translatability of the findings</title>", "<p id=\"P50\">The findings report NMD involvement in the development of islet autoimmunity and the destruction of pancreatic β-cells in type 1 diabetes as well as islet inflammation in type 2 diabetes. The identification of novel targets arisen from cytokines-driven NMD attenuation could possibly suggest new biomarkers to monitor disease progression and may also guide the development of protein-based vaccines or antisense mRNA therapeutics in individuals who are at risk of diabetes development and or other inflammatory and autoimmune disorders.</p>" ]

[]

[ "<p id=\"P1\">Lead investigator: <email>[email protected]</email>, <email>[email protected]</email></p>", "<p id=\"P2\">Author contributions</p>", "<p id=\"P3\">S.M.G. was lead investigator and grant holder (grant number: 9034-00001B) and principally designed and performed all experiments, analysed data, prepared figures, and wrote the first draft. G.A.R. co-designed experiments, co-analysed data and co-wrote the manuscript. T.M.P. discussed scientific data and edited drafts. J.H.N. mentored the DFF fellowship application and edited the first draft. P.M. and L.P. provided human islets and discussed the data. B.P. discussed the Upf2 data and edited the first draft. All authors approved the manuscript.</p>", "<p id=\"P4\">Proinflammatory cytokines are implicated in pancreatic β-cell failure in type 1 and type 2 diabetes and are known to stimulate alternative RNA splicing and the expression of Nonsense-Mediated RNA Decay (NMD) components. Here, we investigate whether cytokines regulate NMD activity and identify transcript isoforms targeted in β-cells. A luciferase-based NMD reporter transiently expressed in rat INS1(832/13), human-derived EndoC-βH3 or dispersed human islet cells was used to examine the effect of proinflammatory cytokines (Cyt) and/or glucolipotoxicity (GLT) on NMD activity. Gain-or loss-of function of two key NMD components UPF3B and UPF2 was used to reveal the effect of cytokines on cell viability and function. RNA-sequencing and siRNA-mediated silencing were deployed using standard techniques. Cyt, but not GLT, attenuated NMD activity in insulin-producing cell lines and primary human β-cells. These effects were found to involve ER stress and were associated with downregulation of UPF3B. Increases or decreases in NMD activity achieved by UPF3B overexpression (OE) or UPF2 silencing, raised or lowered Cyt-induced cell death, respectively, in EndoC-βH3 cells, and were associated with decreased or increased insulin content, respectively. No effects of these manipulations were observed on glucose-stimulated insulin secretion. Transcriptomic analysis revealed that, in contrast to GLT, Cyt increased alternative splicing (AS)-induced exon skipping in the transcript isoforms, and this was potentiated by UPF2 silencing. Gene enrichment analysis identified transcripts regulated by UPF2 silencing whose proteins are localized and/or functional in extracellular matrix (ECM) including the serine protease inhibitor SERPINA1/α-1-antitrypsin, whose silencing sensitised β-cells to Cyt cytotoxicity. Cyt suppress NMD activity via UPR signalling, potentially serving as a protective response against Cyt-induced NMD component expression. Our findings highlight the central importance of RNA turnover in β-cell responses to inflammatory stress.</p>" ]

[ "<title>Supplementary Material</title>" ]

[ "<title>Acknowledgments</title>", "<p id=\"P53\">Our special thanks to Dr. Gabriele Neu-Yilik, Professor Andreas E. Kulozik and Professor Matthias W. Hentze (Heidelberg University, Germany) for providing the plasmids <italic toggle=\"yes\">Renilla</italic> (<italic toggle=\"yes\">RLuc)</italic>-HBB (NS39), <italic toggle=\"yes\">RLuc</italic>-HBB (WT), <italic toggle=\"yes\">Firefly (FLuc)</italic>, Upf3A, Upf3B and Upf3BΔ42 and our gratitude to Dr. Gabriele Neu-Yilik for initially reviewing with her comments and guidance on the NMD activity data. We thank Novogene (Cambridge, United Kingdom) for providing RNA-sequencing and RNA-seq raw data analysis.</p>", "<title>Funding</title>", "<p id=\"P54\">This study was supported by grants to S.M.G. from the Medical Council for Independent Research Fund Denmark (Independent postdoctoral international mobility, grant number: 9034-00001B), and the Society for Endocrinology (SfE) (SEF/2021/ICL-SMG), London, United Kingdom. G.A.R. was supported by a Wellcome Trust Investigator (WT212625/Z/18/Z) Award, and an MRC (UKRI) Programme grant (MR/R022259/1) an NIH-NIDDK project grant (R01DK135268), a CIHR-JDRF Team grant (CIHR-IRSC TDP-186358 and JDRF 4-SRA-2023-1182-S-N), CRCHUM start-up funds, and an Innovation Canada John R. Evans Leader Award (CFI 42649). This project has received funding from the European Union’s Horizon 2020 research and innovation programme via the Innovative Medicines Initiative 2 Joint Undertaking under grant agreement No 115881 (RHAPSODY) to G.A.R. and P.M. Provision of human islets from Milan was supported by JDRF award 31-2008-416 (ECIT Islet for Basic Research program).</p>", "<title>Data accessibility</title>", "<p id=\"P51\">The RNA-seq data from the human insulin-producing cell line EndoC-βH3 that support the findings of ##FIG##5##Figure 6## of this study are deposited in the Sequence Read Archive (SRA) data repository (Accession numbers for 33 RNA-seq datasets: SRR22938756-SRR22938788) under the BioProject accession number (PRJNA916946) that are appreciated for further citations.</p>", "<p id=\"P52\">All other data generated or analysed during this study are included in this published article (and its supplementary information files).</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1.</label><caption><title>Cytokines suppress, whereas glucolipotoxicity increases, NMD activity in β-cells</title><p id=\"P57\">A-B. EndoC-βH3 cells (A) and dispersed human islet cells (B) were co-transfected with <italic toggle=\"yes\">Renilla</italic>-HBB(WT) and or <italic toggle=\"yes\">Renilla</italic>-HBB(NS39), and the <italic toggle=\"yes\">Firefly</italic> plasmids and exposed to cytokine combination and or PBS as control (CTL) simultaneously with or without Cycloheximide (CHX) as positive control for inhibited NMD activity. Luciferase activity was measured in the lysate of the transfected EndoC-βH3 cells (A-left) and dispersed human islet cells (B) exposed to cytokine combination (Cyt; 3 ng/mL IL-1β + 10 ng/mL IFNγ+ 10 ng/mL TNFα) for 18 hrs. A-right. mRNA level of <italic toggle=\"yes\">Renilla-HBB</italic> fused gene and <italic toggle=\"yes\">Firefly</italic> gene in the transfected EndoC-βH3 cells was quantified by RT-qPCR using specific primers extending the junction of exons 1 and 2 of the <italic toggle=\"yes\">HBB</italic> gene, and <italic toggle=\"yes\">Renilla</italic> gene, or only <italic toggle=\"yes\">Firefly</italic> gene and normalised to actin and tubulin, respectively.</p><p id=\"P58\">C. EndoC-βH3 cells were co-transfected with HBB(WT) and or HBB(NS39) and the <italic toggle=\"yes\">Firefly</italic> plasmid, and exposed to 5.6 mM glucose, 25 mM glucose (GL), glucolipotoxicity (GLT; 0.5 mM Palmitate+25 mM glucose), and GLT+ Cycloheximide (CHX) for 18 hrs. Luciferase activity was measured in the lysate of transfected EndoC-βH3 cells and represented as NMD activity calculated by dividing luciferase activity of HBB(WT or PTC−)/HBB(NS39 or PTC+) as explained in the methods. C-right. mRNA level of <italic toggle=\"yes\">Renilla-HBB</italic> fused gene and <italic toggle=\"yes\">Firefly</italic> gene in the transfected EndoC-βH3 cells was quantified by RT-qPCR using specific primers extending the junction of exons 1 and 2 of the <italic toggle=\"yes\">HBB</italic> gene, and <italic toggle=\"yes\">Renilla</italic> gene, or only <italic toggle=\"yes\">Firefly</italic> gene and normalised to tubulin. The data are means ± SEM of N=6. The symbol * indicates the Bonferroni-corrected paired t-test values of treatments versus untreated (Unt) (A-B), cytokines (Cyt) that is, otherwise, designated by a line on top of the bars (A-B). In ##FIG##0##Fig.1C##, the Bonferroni-corrected paired t-test values of treatments versus the control (CTL, containing 5.6 mM glucose). * ≤ 0.05, ** ≤ 0.01, *** ≤0.001, **** ≤ 0.0001. ns: non-significant.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2.</label><caption><title>Cytokines-induced suppression of NMD activity in β-cells is ER stress dependent</title><p id=\"P59\">A. mRNA levels of ER stress markers in EndoC-βH3 cells exposed to cytokine combination (Cyt; 3 ng/mL IL-1β + 10 ng/mL IFNγ+ 10 ng/mL TNFα) for 18 hrs was quantified by RNA-sequencing (left) with false discovery rate (FDR) &lt;0.05 presented as logarithmic fold change the cytokine (Cyt) treatment versus control (untreated), and RT-qPCR (right) which was normalised to tubulin mRNA.</p><p id=\"P60\">B. EndoC-βH3 cells were co-transfected with <italic toggle=\"yes\">Renilla</italic>-HBB (WT) and or <italic toggle=\"yes\">Renilla</italic>-HBB (NS39), and the <italic toggle=\"yes\">Firefly</italic> plasmid and exposed to PBS as control (CTL), cytokine combination (Cyt; 3 ng/mL IL-1β + 10 ng/mL IFNγ+ 10 ng/mL TNFα) alone, and or simultaneously with 16 μM of 48μC, an endoribonuclease inhibitor of IRE1α, and or 8 μM of GSK2656157 (GSK157), PERK inhibitor for 18 hrs. Luciferase activity was measured in the lysate of EndoC-βH3 cells transfected with <italic toggle=\"yes\">Renilla</italic>-HBB(WT) and or <italic toggle=\"yes\">Renilla</italic>-HBB(NS39), and the <italic toggle=\"yes\">Firefly</italic> plasmid exposed to PBS as control (CTL) or Thapsigargin (TG), an ER stress inducer, and represented as NMD activity calculated by dividing luciferase activity of HBB(WT or PTC−)/HBB(NS39 or PTC+) as explained in the methods. B-right. mRNA level of <italic toggle=\"yes\">Renilla-HBB</italic> fused gene and <italic toggle=\"yes\">Firefly</italic> gene in the EndoC-βH3 cells was quantified by RT-qPCR using specific primers extending the junction of exons 1 and 2 of the <italic toggle=\"yes\">HBB</italic> gene, and <italic toggle=\"yes\">Renilla</italic> gene, or only <italic toggle=\"yes\">Firefly</italic> gene and normalised to tubulin.</p><p id=\"P61\">The data are means ± SEM of N=6. The symbol * indicates the Bonferroni-corrected paired t-test values of treatments versus untreated (Unt) (A-B) or cytokines (Cyt) that is, otherwise, designated by a line on top of the bars (B). * ≤ 0.05, ** ≤ 0.01, *** ≤0.001, **** ≤ 0.0001.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3.</label><caption><title>Cytokines-induced suppression of NMD activity is associated with UPF3B downregulation and attenuated by UPF3 overexpression in β-cells</title><p id=\"P62\">EndoC-βH3 cells were co-transfected with empty vector (E.V.), UPF3A, UPF3B and or UPF3BΔ42 (dominant negative of UPF3B) plasmids, and then with <italic toggle=\"yes\">Renilla</italic>-HBB(WT) and or <italic toggle=\"yes\">Renilla</italic>-HBB(NS39), along with the <italic toggle=\"yes\">Firefly</italic> plasmid and exposed to cytokine combination (Cyt; 3 ng/mL IL-1β + 10 ng/mL IFN-γ+ 10 ng/mL TNFα) for 18 hrs.</p><p id=\"P63\">A. mRNA level of <italic toggle=\"yes\">Upf3A</italic> and <italic toggle=\"yes\">Upf3B</italic> genes in EndoC-βH3 cells was quantified by RT-qPCR and normalised to actin and tubulin mRNAs, respectively.</p><p id=\"P64\">B. Luciferase activity (lower) was measured in the lysate of the transfected cells and represented as NMD activity calculated by dividing luciferase activity of HBB(WT or PTC−)/HBB(NS39 or PTC+) as explained in the methods. The overexpression of UPF3A and UPF3B proteins was examined by Western blot analysis (top). The data are means ± SEM of N=6. The symbol * indicates the Bonferroni-corrected paired t-test values of treatments versus untreated E.V. (Unt) (A-B) or cytokines (Cyt)-treated E.V. that is, otherwise, designated by a line on top of the bars (B). * ≤ 0.05, ** ≤ 0.01. ns: non-significant.</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4.</label><caption><title>UPF3 overexpression deteriorates cell viability and reduces insulin content, but not secretion in EndoC-βH3 cells</title><p id=\"P65\">EndoC-βH3 cells were co-transfected with empty vector (E.V.), UPF3A and or UPF3B plasmids and exposed to cytokine combination (Cyt; 3 ng/mL IL-1β + 10 ng/mL IFNγ+ 10 ng/mL TNFα) for three days. A. Cell viability was measured by Alamarblue (left) and caspase-3 activity (right) assays (N=6). B. Glucose-stimulated insulin secretion (GSIS) (left) and insulin contents (right) were investigated in the transfected EndoC-βH3 cells. Insulin concentration (ng/ml) was measured by insulin ultra-sensitive assay (N=6). GSIS index (middle) was calculated by dividing insulin concentration measured in the treatments of 17 mM by 2 mM glucose. The data are means ± SEM of N=6. The symbol * indicates the Bonferroni-corrected paired t-test values of treatments versus untreated E.V. (Unt) or cytokines (Cyt)-treated E.V. that is, otherwise, designated by a line on top of the bars (A), or the Bonferroni-corrected paired t-test values of corresponding low versus high glucose, that is otherwise, designated by lines on the top of the bars (B). * ≤ 0.05, ** ≤ 0.01, **** ≤ 0.0001. ns: non-significant.</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5.</label><caption><title>UPF2 knockdown potentiates cytokine suppression of NMD activity and slightly alleviates cytokine toxicity for cell viability and insulin content in EndoC-βH3 cells</title><p id=\"P66\">EndoC-βH3 cell lines with the most efficient stable knock-down (KD) of Upf2 (three shRNAs) and <monospace>non-silencing shRNA</monospace> as nonsense control (NS) were co-transfected with <italic toggle=\"yes\">Renilla</italic>-HBB(WT) and or <italic toggle=\"yes\">Renilla</italic>-HBB(NS39), and the <italic toggle=\"yes\">Firefly</italic> plasmids and exposed to PBS as control (CTL) and or cytokine combination (Cyt; 3 ng/mL IL-1β + 10 ng/mL IFNγ+10 ng/mL TNFα).</p><p id=\"P67\">A. Cell viability was measured by Alamarblue (left) and caspase-3 activity (right) assays (N=6).</p><p id=\"P68\">B. Glucose-stimulated insulin secretion (GSIS) (left) and insulin contents (right) were investigated in the UPF2 KD EndoC-βH3 (D) cells. Insulin concentration (pM) was measured by human insulin ELISA (N=6). GSIS index (middle) was calculated by dividing insulin concentration measured in the treatments of 17 mM by 2 mM glucose. The data are means ± SEM. The symbol * indicates the Bonferroni-corrected paired t-test values of treatments versus untreated (Unt) NS control (A-B), otherwise, designated by a line on top of the bars or the Bonferroni-corrected paired t-test values of corresponding low versus high glucose, that is otherwise, designated by lines on the top of the bars (B). * ≤ 0.05, ** ≤ 0.01, *** ≤0.001.</p></caption></fig>", "<fig position=\"float\" id=\"F6\"><label>Figure 6.</label><caption><title>UPF2 knockdown differentially affects cytokine- and glucolipotoxicity-mediated deregulation of EndoC-βH3 transcripts</title><p id=\"P69\">EndoC-βH3 cell lines knocked-down for UPF2 (shRNA-1 named U1) or non-silencing shRNA as nonsense control (NS) were exposed to PBS as control or untreated (CTL or Unt), cytokine combination (Cyt; 3 ng/mL IL-1β + 10 ng/mL IFNγ+ 10 ng/mL TNFα) and or glucolipotoxicity (GLT; 0.5 mM Palmitate+25 mM glucose). Total RNA was extracted from the treated cells, cDNA library was made and sequenced using Hiseq platform as explained in methods. 33 RNA-seq datasets from NS/CTL (N=6), NS/Cyt (N=6), NS/GLT (N=4), U1/CTL (N=6), U1/Cyt (N=6) and U1/GLT (N=5) were analysed through the pipeline described in the ##SUPPL##0##supplementary methods##.</p><p id=\"P70\">A. Volcano plot of number of transcripts (FDR &lt;0.05) regulated by Cyt or GLT versus untreated in the NS or UPF2 KD (U1) cell lines.</p><p id=\"P71\">B. Top enriched pathways (left) and the number of transcripts (right) regulated by Cyt and/or GLT in the UPF2 KD (FDR &lt; 0.0.5). Enrichment is shown as log10 (adjusted <italic toggle=\"yes\">p</italic>-value &lt;0.05).</p><p id=\"P72\">C. Top GEA-identified transcripts regulated by Cyt compared to untreated in UPF2 KD vs. NS control EndoC-βH3 cells. The expression level is shown as log2 (adjusted <italic toggle=\"yes\">p</italic>-value &lt;0.05).</p><p id=\"P73\">D. RT-qPCR verification of the identified transcripts (D) regulated by Cyt in UPF2 KD vs. NS control EndoC-βH3 cells. The expression level is shown as log2 (adjusted <italic toggle=\"yes\">p</italic>-value &lt;0.05).</p></caption></fig>", "<fig position=\"float\" id=\"F7\"><label>Figure 7.</label><caption><title>SERPINA1 knockdown deteriorates cytokine cytotoxicity for viability and glucose-stimulated insulin secretion index in INS1(832/13) cells.</title><p id=\"P74\">EndoC-βH3 and INS1(832/13) cells were transfected with siRNAs against SERPINA1 (Si; two species-specific siRNAs for each cell type) and a non-silencing siRNA control (NS), incubated for 24 hours and exposed to PBS as control or untreated (CTL or Unt) and cytokine combination (Cyt for EndoC-βH3; 3 ng/mL IL-1β + 10 ng/mL IFN-γ+ 10 ng/mL TNFα) (Cyt for INS1(832/13); 150 pg/mL IL-1β + 0.1 ng/mL IFNγ+ 0.1 ng/mL TNFα) for 72 and 24 hrs, respectively (see ##SUPPL##0##Supplementary Methods##). The knockdown efficiency was checked using quantitative WB (##SUPPL##0##Supplementary Fig.7##). A. Cell viability was measured by Alamarblue and caspase-3 activity assays (N=6). B. Glucose-stimulated insulin secretion (GSIS) (left) and insulin contents (right) were investigated in the transfected EndoC-βH3 cells. Insulin concentration (ng/ml) was measured by insulin ultra-sensitive assay (N=6). GSIS index (middle) was calculated by dividing insulin concentration measured in the treatments of 17 mM by 2 mM glucose. The data are means ± SEM of N=6. The symbol * indicates the Bonferroni-corrected paired t-test values of treatments versus untreated (Unt) NS control or cytokines (Cyt)-treated NS that is, otherwise, designated by a line on top of the bars (A), or corresponding low versus high glucose that is, otherwise, designated by lines on the top of the bars (B). * ≤ 0.05, ** ≤ 0.01, *** ≤0.001, **** ≤ 0.0001. ns: non-significant.</p></caption></fig>" ]

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[ "<supplementary-material id=\"SD1\" position=\"float\" content-type=\"local-data\"><label>Supplement 1</label></supplementary-material>" ]

[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN3\"><p id=\"P55\">Conflict of interest</p><p id=\"P56\">G.A.R. is a consultant for, and has received grant funding from Sun Pharmaceuticals Inc.</p></fn></fn-group>" ]

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[ "<p id=\"P1\">Author Contributions</p>", "<p id=\"P2\">The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.</p>", "<p id=\"P3\">A modified protein fragment complementation assay has been designed and validated as a gain-of-signal biosensor for nucleic acid:nucleic acid interactions. The assay uses fragments of NanoBiT, the split luciferase reporter enzyme, that are esterified at their C-termini to steramers, sterol-modified oligodeoxynucleotides. The <italic toggle=\"yes\">Drosophila</italic> hedgehog autoprocessing domain, DHhC, served as a self-cleaving catalyst for these bioconjugations. In the presence of ssDNA or RNA with segments complementary to the steramers and adjacent to one another, the two NanoBiT fragments productively associate, reconstituting NanoBiT enzyme activity. NanoBiT luminescence in samples containing nM ssDNA or RNA template exceeded background by 30-fold and as high as 120-fold depending on assay conditions. A unique feature of this detection system is the absence of a self-labeling domain in the NanoBiT bioconjugates. Eliminating that extraneous bulk broadens the detection range from short oligos to full-length mRNA.</p>", "<title>Graphical Abstract</title>", "<p id=\"P4\">\n\n</p>" ]

[ "<p id=\"P5\">Proteins whose function can be turned on and off experimentally enable biosensing applications and provide valuable reagents to probe complex biological pathways. Temperature sensitive phenotypes, for example, are possible with destabilized mutant proteins that aggregate or fail to fold above some threshold temperature.[##REF##8252628##1##] Domain insertion into a protein with ligand binding modules,[##REF##15632292##2##] light responsive polypeptides, such as LOV2, or light responsive prosthetic groups, like azobenzene, provides an alternative and potentially reversible means for controlling protein function. [##REF##27980211##3##–##REF##21601095##6##]</p>", "<p id=\"P6\">Protein fragment complementation, the focus of the present work, is widely applied as an approach to achieve conditional activity.[##REF##11495741##7##] The protein of interest is “split” into non-functional, weakly interacting fragments. Under selected conditions, those fragments are driven to assemble into a functional complex, restoring function.[##REF##7937952##8##, ##REF##9237989##9##] Enzymatic and nonenzymatic proteins are amenable to this type of manipulation.[##REF##16426967##10##] Different triggers can be used to promote fragment complementation, such as noncovalent interactions of fused partner proteins, as in the two-hybrid assay,[##REF##2547163##11##] proteolytic cleavage,[##REF##20945451##12##] as well as by chemical dimerizers for concentration-responsive complementation.[##REF##10318894##13##, ##REF##12940738##14##]</p>", "<p id=\"P7\">Here we explore nucleic acid hybridization as a driver of enzyme fragment complementation with a view toward applications in RNA and DNA biosensing.[##REF##16461889##15##–##REF##20941727##19##] As depicted in ##FIG##4##Scheme 1##, we envisioned a split reporter enzyme where each polypeptide fragment was covalently linked to an oligodeoxynucleotide. The sequences of those oligos would be designed for hybridization at proximal sites on a DNA or RNA of interest. In this prototype biosensor, enzyme activity reports the presence of complementary nucleic acid.</p>", "<p id=\"P8\">For evaluation, we chose the split reporter enzyme, NanoBiT, derived by Wood and colleagues from the bioluminescent nanoluciferase. NanoBiT is comprised of a large (LrgBiT, 18 kDa) and small (SmBiT, 1.2 kDa) fragment.[##REF##26569370##20##] Intrinsic affinity of LrgBiT and SmBiT is weak (K_D ~ 100–200 µM), reducing spontaneous association. When LrgBiT and SmBiT are induced to associate productively at concentrations well below their KD value, the two fragments reconstitute NanoBiT as an engineered heterodimer. Enzymatic oxidation of coelenterazine or its synthetic analog furimazine by NanoBiT generates glow-type luminescence. Like nanoluciferase, NanoBiT activity is independent of NTP and cofactor/coenzymes, which simplifies experimental design. Other advantages of luciferase reporters include their sensitivity, the absence of background in most biological samples and the ease of signal detection, permitting qualitative yes/no analysis even with cellphone-based imaging devices.[##REF##37122471##18##, ##UREF##0##21##, ##REF##32124606##22##]</p>", "<p id=\"P9\">To attach oligodeoxynucleotides site-specifically to LrgBiT and SmBiT with 1:1 stoichiometry we exploited the enzymatic autoprocessing domain found in <italic toggle=\"yes\">Drosophila</italic> hedgehog protein, hereafter, DHhC. [##REF##31600061##23##] The native function of DHhC is to cleave off and covalently attach cholesterol to an adjacent N-terminal polypeptide within a precursor form of hedgehog. [##REF##8824192##24##, ##REF##9335337##25##] We and others have found that DHhC is substrate promiscuous, retaining cleavage/sterylation activity when fused to heterologous N-terminal polypeptides (e.g., nanoluciferase, maltose binding protein, lysozyme, cyan fluorescent protein, etc.) and accepting a broad range of cholesterol analogs as alternative substrates, including monosterylated oligonucleotides, or steramers.[##REF##31600061##23##, ##REF##9335337##25##, ##REF##26095399##26##] In a steramer, the sterol’s fused ring structure is maintained for substrate recognition by DHhC while the iso-octyl sterol side chain is modified to serve as a linker for covalent oligo attachment (<italic toggle=\"yes\">below</italic>). DHhC is active toward steramer substrates with oligos of varying sequence, length, chemical modification and secondary structure.[##REF##31600061##23##] We recently reported a cost-effective means for preparing steramers by solid phase synthesis using commercial reagents.[##REF##34890095##27##]</p>", "<p id=\"P10\">\n\n</p>", "<p id=\"P11\">The general method for protein-to-steramer bioconjugation using DHhC resembles other self-labeling proteins such as HaloTag or HUH tag.[##REF##18533659##28##, ##REF##28481515##29##] The key distinction is that DHhC self-cleaves from the product simultaneously with the coupling event whereas self-labeling domains remain attached. A chimeric gene construct is assembled first in which the protein of interest (POI) is fused at its C-terminus to DHhC. A single glycine residue represents the minimal spacer between the protein of interest and the first residue of DHhC. The alpha carboxyl group of that glycine spacer residue will be the site of sterol esterification. Following expression of the chimeric POI-DHhC gene under sterol-free conditions, such as in <italic toggle=\"yes\">E. coli</italic>, the protein precursor is activated by addition of steramer. No ATP or cofactors are involved. As mentioned above, protein-steramer bioconjugation displaces DHhC, furnishing an almost scar-free product (##FIG##0##Figure 1##). Note that, here, we incorporated (Gly-Ser) C-terminal spacer to allow for flexibility in split NanoBiT reassembly. [##REF##26569370##20##, ##REF##32587898##30##, ##REF##29203496##31##]</p>", "<p id=\"P12\">The chemi-enzymatic approach to generate the two NanoBiT steramer components of this prototype nucleic acid biosensor is summarized in ##FIG##1##Figure 2##. For the NanoBiT components, we designed constructs where LrgBiT and SmBiT were fused to DHhC, an N-terminal SUMO tag was incorporated as a solubility enhancer[##REF##2539593##32##] and a C-terminal His₆ tag was added to DHhC for precursor protein purification. The two constructs, SUMO-LrgBiT-DHhC-His₆ and SUMO-SmBiT-DHhC-His₆, were overexpressed in <italic toggle=\"yes\">E. coli</italic> in soluble form and purified under native conditions by Ni-NTA chromatography (##SUPPL##0##Supporting Figure 1##).</p>", "<p id=\"P13\">For proof of concept, oligo sequences for two steramers (##TAB##0##Table 1##) were designed for annealing at adjacent sites within the NSP10 gene from SARS-CoV-2. NSP10 encodes a protein cofactor of the viral 2’-O-RNA methyltransferase.[##REF##32728018##33##, ##REF##32709887##34##] Unlike variability reported in other SARS-CoV-2 coding sequences, NSP10 appears relatively stable,[##UREF##1##35##, ##REF##34297786##36##] making this region an attractive target for molecular detection. Steramers were prepared in two steps, first joining heptanediamine to 23,24-bisnor-5-cholenic acid-3β-ol to produce <bold>III</bold>, followed by amide coupling of <bold>III</bold> to resin-bound oligodeoxynucleotide equipped with 5’ NHS ester modification. After deprotection and resin cleavage, the two steramers, S1 and S2, were desalted and isolated by reverse phase chromatography.</p>", "<p id=\"P14\">Covalent steramer attachment by DHhC to the NanoBiT fragments was monitored by denaturing SDS-PAGE followed by staining with Gel-Red (nucleic acid specific) and Coomassie blue. In a typical experiment, reactions were initiated on the benchtop at 23 °C in Bis-Tris buffered solution (pH 7.1) containing 1.5 µM precursor protein and 50 µM steramer. Initial steramer concentration represents 2x the KM value with DHhC, determined separately using a FRET-based DHhC activity assay (##SUPPL##0##Supporting Figure 2##).[##REF##26095399##26##] Analysis of the reaction mixture by denaturing SDS-PAGE showed the consumption of precursor protein along with the accumulation of the respective bioconjugate. ##FIG##1##Figure 2##, right side, is a representative gel for the SUMO-LrgBiT coupling to S2. The calculated molecular weight is 36.6 kDa for the product, which agrees well with the extrapolated value of 37.1 kDa, based on comparison to the MW ladder. This species is visible with Coomassie staining and the nucleic acid stain Gel-Red. Only trace amounts of the hydrolytic unconjugated side-product, SUMO-LrgBiT, was observed (##FIG##1##Figure 2##, see gel, “h”). Reactions ranged from 50–80% conversion, with highest yields coming from precursor protein that was isolated by Ni-NTA chromatography then further purified by SEC. Agarose gel extraction offered a convenient and inexpensive means for the final isolation step (##SUPPL##0##Supporting Figure 3##).</p>", "<p id=\"P15\">Robust NanoBit luminescence signal was observed when the two bioconjugates, hereafter ^SUMOSm-S1 and ^SUMOLg-S2, were combined with a complementary single stranded NSP10 oligodeoxynucleotide, d-NSP10 (##FIG##3##Figure 3##A). NanoBiT complementation was assayed in solutions containing ^SUMOSm-S1 and ^SUMOLg-S2 and d-NSP10 added together in a ratio of 1:1:1 over a dilution series from 1 x 10⁻⁷ M to 48 x 10⁻¹² M. As a control, we measured spontaneous NanoBiT complementation from ^SUMOSm-S1 and ^SUMOLg-S2 over that same concentration range in the absence of d-NSP10 template (##SUPPL##0##Supporting Figure 4##). Additional negative control experiments were carried out using ^SUMOSm-S1 and ^SUMOLg-S2 mixed with ssDNA template of the same length as d-NSP10 but randomized sequence; samples with the random oligo generated NanoBiT signal equal to the “no template” readings (##FIG##3##Figure 3##A, “random”). We also attempted to measure NanoBiT reconstitution with a ssDNA template where the hybridization site for ^SUMOLg-S2 was mutated to a poly T track; luminescence again matched the background (no template) readings (##FIG##3##Figure 3##A, “d-NSP10 ΔS2”). Based on those results, we selected 25 x 10⁻⁹ M as the working concentrations of ^SUMOSm-S1 and ^SUMOLg-S2. This setup, which is similar to conditions used in related NanoBiT complementation assays (##SUPPL##0##Supporting Table 1##), conserved NanoBiT-steramer components while providing maximum signal / background of 30–60 fold and a lower limit for complementary template detection of 2 x10⁻⁹ M.</p>", "<p id=\"P16\">As shown in ##FIG##3##Figure 3##B, luminescence from reconstituted NanoBiT plotted as a function of increasing d-NSP10 template yielded a bell-shaped curve. “Hook-effect” behavior is consistent with non-cooperative tripartite interactions where one component of the assembly provides a scaffold to bring the remaining two components together.[##REF##37122471##18##, ##REF##23544844##37##] Here, d-NSP10 is the scaffolding element. Signal/background was highest when the three components were equimolar. To the right of the equivalence point, the excess template favors formation of enzymatically in-active binary complexes: d-NSP10 hybridized with ^SUMOSm-S1 only or with ^SUMOLg-S2 only.</p>", "<p id=\"P17\">We found that NanoBiT reconstitution remained durable with several “mutant” d-NSP10 templates (##FIG##3##Figure 3##C). We first tested templates where the ^SUMOLg-S2 site contained single or double nucleotide substitutions (see ##TAB##0##Table 1##: d-NSP10 (A to C); d-NSP10 (C to A); d-NSP10 (AC swap)). We observed only modest changes in luminescence, &lt; 1.5-fold, compared to the positive control, d-NSP10, ##FIG##3##Figure 3##C (yellow). Tolerance toward base-pair mismatches is not unexpected for a linear hybridization involving oligos of this length.[##REF##4022774##38##, ##REF##12732557##39##] In a related split enzyme system, using oligodeoxynucleotide-modified fragments of dihydrofolate reductase, split enzyme reassembly could be driven by hybridization to a ssDNA template carrying as many as five mismatches (##SUPPL##0##Supporting Table 1##).[##REF##20941727##19##]</p>", "<p id=\"P18\">Next, we tested templates in which the hybridization sites of ^SUMOSm-S1 and ^SUMOLg-S2 were perfectly complementary but spread farther apart by insertions of 2, 4, 6, or 8 deoxy thymidine nucleotides (##TAB##0##Table 1##. See d-NSP10(TT)_n). In samples containing the spacer templates d-NSP10(TT)₁, d-NSP10(TT)₂ and d-NSP10(TT)₃ we observed NanoBiT signal indistinguishable from samples mixed with d-NSP10 ##FIG##3##Figure 3##C (pink). Only d-NSP10(TT)₄ stood out in this comparison, and interestingly, the luminescence was enhanced, suggesting that there is room for further design optimization. In summary, these results indicate that there is tolerance toward single and double nucleotide substitutions in the nucleic acid target sequence and that there is flexibility in the distance separating the sites for hybridization.</p>", "<p id=\"P19\">With the modified NanoBiT complementation system working largely as intended on ssDNA, we turned our attention to the detection of target RNA. A synthetic SARS-CoV-2 NSP10 gene (460 bp) was cloned into the pGEM-3 vector and transcribed in vitro in the forward and reverse directions to generate sense (+) and anti-sense (−) NSP10 RNA transcripts. Samples containing ^SUMOSm-S1, ^SUMOLg-S2 and equivalent concentration (25 x 10⁻⁹ M) of the purified anti-sense transcript generated strong NanoBiT signal (Figure 4). RLU values were on par with those containing d-NSP10, used a positive control. A hook-effect in concentration-response plots was also apparent (##SUPPL##0##Supporting Figure 5##). By contrast, NanoBiT reconstitution was not supported by samples containing the NSP10 sense transcript, which harbors segments identical to the oligonucleotides in ^SUMOSm-S1 and ^SUMOLg-S2. In summary, the split NanoBiT-steramer conjugates provide hybridization-driven luminescence detection of ssDNA and RNA.</p>", "<p id=\"P20\">As this work was underway, a conceptually similar nucleic acid biosensor was reported that involves fusions of the NanoBiT fragments to a noncatalytic Cas9 protein (dCas9).[##REF##37122471##18##, ##REF##37122472##40##] In that system, the ~160 kDa dCas9 provided the linker to (non-covalently) join LrgBiT and SmBiT to an RNA oligonucleotide in a manner similar to the way the sterol molecule here covalently links ^SUMOLrgBiT and ^SUMOSmBiT to DNA oligonucleotides. Although the assay also suffered from a hook-effect, a practical advantage of using dCas9 is that the guide RNAs for dsDNA template recognition do not require chemical modification. One potential drawback is the restricted hybridization specificity of this large ribonucleoprotein complex: split NanoBiT reconstitution required that the target harbor a 5′-NGG-3′ PAM and its reverse complement 5′-CCN-3′ within 27–52 nt of each other. The shortest dsDNA template that supported NanoBiT reconstitution with the dCas9 linker was 450 bp. Nonetheless, once optimized, the dCas9/NanoBiT “LUNAS” method from Merkx and colleagues[##REF##37122471##18##] showed impressive sensitivity and appeared amenable to translation as a point-of-care diagnostic. Specificity of LUNAS to discriminate nucleotide mismatches was not disclosed.</p>", "<p id=\"P21\">In this study, we designed, prepared and validated a prototype split enzyme complementation assay driven by nucleic acid hybridization. We attached monosterylated oligodeoxynucleotides (steramers) site-specifically to the carboxy termini of SUMO-fused LrgBiT and SmBiT fragments using the self-cleaving <italic toggle=\"yes\">Drosophila</italic> hedgehog autoprocessing domain, DHhC. It seems reasonable to assume that direct covalent attachment of oligonucleotide to split enzyme fragments by a small molecule linker, rather than through a self-labeling macromolecule, would moderate steric interference to productive protein fragment complementation. Moreover, small molecule linkers may be beneficial for split protein assembly on shorter nucleic acids of interest including non-coding RNA. Although the solid phase synthesis of steramers could present a practical obstacle to the application of DHhC bioconjugation, an alternative solution phase steramer preparation has been reported involving straight-forward oxime chemistry.[##REF##31600061##23##]</p>", "<p id=\"P22\">Next generation nucleic acid biosensor assays of the general type described here will need to address limitations in mismatch discrimination as well as the hook-effect in concentration response plots, the latter causing diminished signal output with super stoichiometric template. We anticipate that using hairpin oligo components in place of the linear oligos, used here, will increase the sequence specificity of protein fragment complementation, as it does with molecular beacons.[##REF##9630890##41##, ##REF##21769413##42##] Protein-steramer bioconjugation by DHhC proceeds unimpeded with hairpin oligos, making this an attractive test system.[##REF##34890095##27##] The second challenge posed by the hook-effect is a design defect that threatens assay reliability through false negatives, where increasing nucleic acid template results in decreasing protein fragment complementation. Engineering cooperativity into the NanoBiT/template assembly offers one solution.[##REF##37816014##43##] A split enzyme-steramer complementation assay with cooperative 1:1:1 template hybridization would drive reporter enzyme reconstitution even with a template surplus.</p>", "<title>Supplementary Material</title>" ]

[]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1.</label><caption><title>Extraneous peptide sequences or self-labeling domains that remain on the selected bioconjugate product cover a wide range.</title><p id=\"P35\"><italic toggle=\"yes\">Drosophila</italic> Hedgehog HhC (DHhC) catalyzed protein-nucleic acid conjugation, used in the present study, leaves a small “scar” of a single glycine residue.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2.</label><caption><title>Convergent chemi-enzymatic synthesis of NanoBiT fragment-oligonucleotide conjugates.</title><p id=\"P36\">(Top) Design, <italic toggle=\"yes\">E. coli</italic> expression and purification of LrgBiT fusion with the bioconjugating domain, DHhC. Constructs contain an N-terminal SUMO (S) protein to enhance solubility and a C-terminal hexahistidine tag (His₆) for Ni-NTA chromatography. (Bottom) Steramer preparation. Monosterylation of oligodeoxynucleotides was carried out by amide coupling between (III) and NHS-ester modified oligodeoxynucleotides, followed by purification with reverse phase chromatography. (Side right) Bioconjugation. Precursor protein, ^SUMOLrgBiT-DHhC-His₆ is combined with steramer, leading to the displacement of DHhC-His₆ and covalent steramer attachment to ^SUMOLrgBiT, to yield ^SUMOLrgBiT-S2 (*). Competing side reaction where water replaces the steramer in the bioconjugation is minimal (h, product from hydrolysis). Bioconjugates were purified by agarose gel extraction, then concentrated by spin column.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 2.</label><caption><title>NanoBiT fragment-steramer bioconjugates enable nucleic acid hybridization driven protein fragment complementation.</title><p id=\"P37\">A. NanoBiT signal generated from samples containing the indicated ssDNA templates. Oligonucleotide templates, ^SUMOSm-S1, ^SUMOLg-S2 were (1:1:1), 25 x 10⁻⁹ M, final. Statistical analysis: one-way ANOVA followed by T-test (GraphPad). <bold>B</bold>. Hook-effect in concentration-response plot with increasing d-NSP10. <bold>C</bold>. Assessing NanoBiT luminescence with template variants. Conditions same as (A). Only (TT)₄ showed a significant difference from control, d-NSP10. **, P&lt;.0001</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 3.</label><caption><title>NanoBiT reconstitution reports the presence of SARS-CoV-2 NSP-10 RNA.</title><p id=\"P38\">In vitro transcribed anti-sense NSP10 RNA (squares) but not NSP10 sense RNA (triangles) triggers NanoBiT activity. Synthetic d-NSP10 was added as a positive control (circles). Nucleic acid and NanoBiT components were added (1:1:1) to 25 x10⁻⁹ M, final. Statistical analysis: one-way ANOVA followed by T-test (GraphPad).</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Scheme 1.</label><caption><title>Repurposing protein fragment complementation for DNA or RNA detection.</title><p id=\"P39\">Split NanoBiT luciferase fragments with covalently attached oligonucleotides assemble via hybridization on a suitable nucleic acid template.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\" orientation=\"landscape\"><label>Table 1.</label><caption><p id=\"P40\">Sequences of steramer 1 and 2 and nucleic acid targets evaluated here for potential hybridization-driven protein fragment complementation</p></caption><table frame=\"void\" rules=\"none\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">Name</th><th align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">Sequence</th></tr></thead><tbody><tr><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">Steramer-1 (S1)</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">sterol linker-5’-TTATGGCTGTAGTTGTGATC-3’</td></tr><tr><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">Steramer-2 (S2)</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">sterol linker-5’-CGTCTGCGGTATGTGGAA-3’</td></tr><tr><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">d-NSP10</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">5’-TTGATCACAACTACAGCCATAACCTTTCCACATACCGCAGACGGT-3’</td></tr><tr><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">random</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">5’-CTGATCCAGGAATACGTAAAAGATTATAATCGATATGACTGAGCG-3’</td></tr><tr><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">d-NSP10 ΔS2</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">5’-TTGATCACAACTACAGCCATAACCTAAAAAAAAAAAAAAAAAAT-3’</td></tr><tr><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">d-NSP10 (A to C)</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">5’-TTGATCACAACTACAGCCATAACCTTTCCACAT<bold>CC</bold>CGCAGACGGT-3’</td></tr><tr><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">d-NSP10 (C to A)</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">5’-TTGATCACAACTACAGCCATAACCTTTCCACAT<bold>AA</bold>CGCAGACGGT-3’</td></tr><tr><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">d-NSP10 (AC swap)</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">5’-TTGATCACAACTACAGCCATAACCTTTCCACAT<bold>CA</bold>CGCAGACGGT-3’</td></tr><tr><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">d-NSP10(TT)_n</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">5’-TTGATCACAACTACAGCCATAAC(TT)_nCTTTCCACATACCGCAGACGGT-3’</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material id=\"SD1\" position=\"float\" content-type=\"local-data\"><label>Supplement 1</label></supplementary-material>" ]

[ "<fn-group><fn id=\"FN2\"><p id=\"P23\">ASSOCIATED CONTENT</p><p id=\"P24\"><bold>Supporting Information</bold>. vector construction, protein expression, steramer kinetic assays, chemical synthesis, NMR and MALDI-TOF. This material is available free of charge via the Internet at <ext-link xlink:href=\"http://pubs.acs.org/\" ext-link-type=\"uri\">http://pubs.acs.org</ext-link>.</p></fn></fn-group>" ]

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{ "acronym": [ "DHhC", "SUMO", "MW", "ssDNA", "dsDNA", "NSP10", "Ni-NTA", "S1", "S2", "NSP" ], "definition": [ "Drosophila hedgehog C-terminal domain", "small ubiquitin modifier protein", "molecular weight", "single stranded DNA", "double stranded DNA", "nonstructural protein 10", "Nickel (2+)-Nitriloacetic acid", "steramer 1", "steramer 2", "nonstructural protein" ] }

[ "<title>Introduction</title>", "<p id=\"P4\">The circulating thyroxine (T4) and triiodothyronine (T3) levels are robustly regulated to stay within narrow (about 2-3 fold) physiological ranges (##UREF##5##Jain 2015##, ##REF##27737898##Welsh and Soldin 2016##). Deviations of the thyroid hormones (THs) from the normal ranges lead to a variety of adverse health outcomes (##REF##21460788##Combs et al. 2011##, ##REF##27811932##Jabbar et al. 2017##, ##REF##30157487##Prezioso et al. 2018##, ##REF##29767691##Silva et al. 2018##, ##REF##31286098##Yavuz et al. 2019##). By maintaining the setpoint levels of THs, the hypothalamic-pituitary-thyroid (HPT) axis is the primary mechanism for long-term TH homeostasis. To combat transient perturbations, it requires three major TH binding/distributor proteins (THBPs/THDPs) in the blood: thyroxine-binding globulin (TBG), transthyretin (TTR), and albumin (ALB).</p>", "<p id=\"P5\">The vast majority of circulating THs are bound to and transported by these THBPs (##REF##28257828##Janssen and Janssen 2017##, ##REF##28249735##McLean et al. 2017##), with a small fraction also bound to lipoproteins and members of the serine proteinase inhibitors (serpins) superfamily (##REF##3379137##Benvenga et al. 1988##, ##REF##11883864##Benvenga et al. 2002##, ##UREF##0##Benvenga 2013##). The molar abundances of the three THBPs in humans follow the order of ALB&gt;TTR&gt;TBG, with ALB more than a hundred times higher than TTR and TTR more than tens of times higher than TBG (##REF##103659##Attwood et al. 1978##, ##REF##7400337##Gardner and Scott 1980##, ##REF##6803653##Attwood and Atkin 1982##, ##REF##2044212##Vatassery et al. 1991##). However, the binding affinities of THs for the three THBPs follow the reverse order: TBG &gt; TTR &gt; ALB. The binding affinities of T4 for the three THBPs differ by more than two orders of magnitude, while in the case of T3, the binding affinities differ by at least one or two orders of magnitude (##REF##3928734##Murata et al. 1985##, ##REF##3584855##Yabu et al. 1987##, ##REF##9914537##Chang et al. 1999##, ##REF##17338921##Richardson 2007##, ##REF##28249735##McLean et al. 2017##). As a result of the vast differences in the THBP abundances and binding affinities, TBG accounts for ~75% of both plasma total T4 and total T3 despite its low abundance and having only one binding site, TTR accounts for ~15% of total T4, but less than 5% of total T3, and ALB accounts for less than 5% of total T4 and ~20% of total T3 (##REF##10718550##Schussler 2000##, ##REF##28257828##Janssen and Janssen 2017##, ##REF##28249735##McLean et al. 2017##). Overall, less than 0.03% of plasma T4 is free with the vast majority bound to THBPs, while about 0.3% of plasma T3 is free with the vast majority also bound to THBPs (##REF##2673754##Mendel 1989##, ##REF##17338921##Richardson 2007##, ##REF##28249735##McLean et al. 2017##). TBG is only about 20% saturated and TTR is less than 1% saturated by THs with the majority of their binding sites unoccupied by THs (##REF##10718550##Schussler 2000##, ##REF##28249735##McLean et al. 2017##). THBPs are believed to have several functions. (1) They produce a strong buffer in the plasma where a transient increase or decrease of free T4 and T3 can be quickly dampened. (2) THs stored in THBPs can act as a reservoir in the face of short-term TH deficiency. (3) THBPs are required to ensure uniform distribution of circulating THs through a perfused tissue; in the absence of THBPs, a steep gradient of tissue THs would result (##REF##3701605##Weisiger et al. 1986##, ##REF##3106010##Mendel et al. 1987##, ##REF##3407767##Mendel et al. 1988##).</p>", "<p id=\"P6\">While a lot has been learned about THBPs, many questions remain unanswered. THs are taken up by tissues in free form mainly via cell membrane transporters including MCT8, MCT10, LAT1, LAT2, OATP1c1 etc. (##REF##12351693##Pizzagalli et al. 2002##, ##REF##12871948##Friesema et al. 2003##, ##REF##25942657##Bernal et al. 2015##, ##REF##26305885##Zevenbergen et al. 2015##, ##UREF##4##Groeneweg et al. 2020##). However, the bulk of these free THs that move into the tissues must first be unloaded off of THBPs in the tissue capillary. The relative contributions of each THBP species in this regard are not completely established but are important for understanding TH tissue distribution as well as potential species differences in TH kinetics. It has been speculated that because of the faster dissociation rate constants for the binding between THs and ALB (i.e., shorter residence time of T4 and T3 on an ALB molecule), ALB contributes the most THs to the perfused tissues (##REF##2673754##Mendel 1989##, ##REF##10718550##Schussler 2000##). It is also argued that TTR may be the one that contributes the most T4 to tissues because of its “Goldilocks” (just-about-right) properties among the three THBPs, i.e., intermediate abundance and binding affinity (##REF##12553418##Robbins 2002##, ##REF##19725882##Richardson 2009##, ##REF##25737004##Alshehri et al. 2015##). Given that TBG is loaded with over two thirds of total T4 and total T3 in the blood, it is also possible that this high abundance compensates for its tight grip on THs such that it makes comparable or even greater contributions than ALB and TTR. In addition, the blood perfuses different tissues at different flow rates, thus THBPs carrying THs would clear through tissues in different amounts of time, during which free THs transport into and out of tissues at potentially different rates. It is unclear how all these factors – the residence time of THs on THBPs, THBP abundance, tissue blood transit time, as well as the rates of blood-tissue TH exchange – interplay to determine the amounts of THs that are unloaded from each THBP species in local tissue blood and eventually move into the tissue proper.</p>", "<p id=\"P7\">While ingenious experimentation can help clarify many of these questions, quantitative understanding of the thyroid system can also benefit greatly from mathematical modeling (##REF##4108034##DiStefano and Chang 1971##, ##REF##3965501##Oppenheimer and Schwartz 1985##, ##REF##2333963##Pilo et al. 1990##, ##REF##1538642##Curti and Fresco 1992##, ##REF##18844475##Eisenberg et al. 2008##, ##REF##19479014##McLanahan et al. 2009##, ##REF##23535361##Lumen et al. 2013##, ##UREF##1##Berberich et al. 2018##, ##REF##34117770##Handa et al. 2021##). A physiologically based kinetic (PBK) model of THs containing sufficient details of the THBP binding events with spatial configurations will help dissect the differential roles of THBPs and provide critical quantitative insights. Here we developed a spatial human PBK model of THs based on a nonspatial model we have recently published (##REF##37305053##Bagga et al. 2023##), and reported several novel findings on local TH tissue delivery and contributions by different THBP species.</p>" ]

[ "<title>Methods</title>", "<title>The nonspatial human PBK model of THs</title>", "<p id=\"P8\">The details of the nonspatial human PBK model of THs, including its construction and parameterization, were described in our recently published paper (##REF##37305053##Bagga et al. 2023##). The model was rigorously parameterized based on numerous published studies on THBP binding affinities, association/dissociation rate constants, TH and THBP concentrations, plasma/tissue partitioning, half-lives, etc. Briefly, the model contains four major compartments: <italic toggle=\"yes\">Body Blood</italic>, <italic toggle=\"yes\">Thyroid</italic>, <italic toggle=\"yes\">Liver</italic>, and <italic toggle=\"yes\">Rest-of-Body</italic> (<italic toggle=\"yes\">RB</italic>) (##FIG##0##Fig. 1A##). The <italic toggle=\"yes\">Liver</italic>, <italic toggle=\"yes\">RB</italic>, and <italic toggle=\"yes\">Thyroid</italic> compartments contain respective tissue blood (vasculature) and tissue proper (extra-vasculature) subcompartments. In each blood (sub)compartments, the binding events between THs and THBPs follow the law of mass action (##FIG##0##Fig. 1B##). As standard PBK practices, here the concentrations of molecular species in <italic toggle=\"yes\">Body Blood</italic> are treated as their arterial concentrations (CA), and the concentrations in tissue blood are treated as proxies for their venous concentrations (CVT, CVRB, and CVL) (##UREF##3##Fisher et al. 2020##). The model produced several novel findings, including fast and near-equilibrium blood-tissue TH exchanges as an intrinsic robust mechanism against local metabolic perturbation and tissue influx as a limiting step for transient tissue uptake of THs in the presence of THBPs (##REF##37305053##Bagga et al. 2023##). Since we were concerned with the steady-state behaviors of THs in local tissues, it was not necessarily to include the feedback regulation of TSH by THs in the model, which is a simplification that is also applied to the spatial model presented below.</p>", "<title>Construction of the spatial human PBK model of THs</title>", "<p id=\"P9\">In order to examine the gradients of TH distribution in tissues, we extended the nonspatial PBK model above into a spatial model. Specifically, the <italic toggle=\"yes\">RB</italic> and <italic toggle=\"yes\">Liver</italic> compartments are divided into 200 linear segments of equal size with each segment containing its own tissue blood and tissue proper subsegments (##FIG##0##Fig. 1C##). 200 segments provide a sufficient resolution to simulate the concentration gradients as higher numbers of segments do not appreciably improve the precision (simulation results not shown). The concentration of each molecular specie in the last (200^th) tissue blood subsegment is treated as its tissue venous blood concentration. Each subsegment is assumed to be well-mixed and has its own set of ordinary differential equations (ODEs) tracking the subsegment-specific rates of change of the variables. The method of lines is used to track the molecular species in each subsegment. The i^th and i+1^th blood subsegments are connected by the unidirectional blood flow of each molecular species. The kinetic constants <italic toggle=\"yes\">k</italic>₁ – <italic toggle=\"yes\">k</italic>₁₂ for TH and THBP binding in the blood (i.e., the association rate constants and dissociation rate constants) are assumed the same across all blood subsegments and equal to the values in the nonspatial PBK model. The bidirectional transport of free T4 and free T3 between the tissue blood and tissue proper subsegments as well as T4-to-T3 conversion metabolism and clearance metabolism within each tissue proper subsegment also occur as described in the nonspatial PBK model. The parameters describing the influx and efflux within each tissue segment (<italic toggle=\"yes\">k</italic>₂₁, <italic toggle=\"yes\">k</italic>₂₃, <italic toggle=\"yes\">k</italic>₂₅, <italic toggle=\"yes\">k</italic>₂₇, <italic toggle=\"yes\">k</italic>₂₈, <italic toggle=\"yes\">k</italic>₂₉, <italic toggle=\"yes\">k</italic>₃₀, and <italic toggle=\"yes\">k</italic>₃₁) are scaled from the values used in the nonspatial PBK model by dividing by 200, the total number of segments. The parameters describing the metabolism within each tissue segment (<italic toggle=\"yes\">k</italic>₂₄, <italic toggle=\"yes\">k</italic>₂₆, <italic toggle=\"yes\">k</italic>₃₂, <italic toggle=\"yes\">k</italic>₃₃, <italic toggle=\"yes\">k</italic>₃₄, and <italic toggle=\"yes\">k</italic>₃₅), which are first-order rate constants, remain the same as in the nonspatial PBK model. For homogenous simulations, the values of each of these parameters (<italic toggle=\"yes\">k</italic>₂₁, <italic toggle=\"yes\">k</italic>₂₃ – <italic toggle=\"yes\">k</italic>₃₅) are equal across all segments. For heterogeneous simulations where a parameter value may vary from the arterial to venous ends, a linear gradient was applied to each parameter across the segments such that the average is equal to the constant value in the homogenous case.</p>", "<p id=\"P10\">One additional consideration is the possible bidirectional diffusion of molecular species across the blood subsegments and across the tissue subsegments. The diffusion of a molecular species would be normally modeled as (<italic toggle=\"yes\">species</italic>_{<italic toggle=\"yes\">i</italic>–1} + <italic toggle=\"yes\">species</italic>_{<italic toggle=\"yes\">i</italic>+1} − 2 * <italic toggle=\"yes\">species_i</italic>)/<italic toggle=\"yes\">dx</italic>² where <italic toggle=\"yes\">i</italic> represents the current subsegment, <italic toggle=\"yes\">i</italic> − 1 the subsegment before, <italic toggle=\"yes\">i</italic> + 1 the subsegment after, and <italic toggle=\"yes\">d</italic>x is the length of the subsegment. Since the diffusion rate is much slower than the blood flow rate (##REF##3701605##Weisiger et al. 1986##), including cross-subsegment diffusions didn’t produce any difference in the model’s behavior based on our simulations (results not shown). Therefore, for all spatial PBK simulations we proceeded without considering cross-subsegment diffusions.</p>", "<title>Model parameters, equations, and simulation tools</title>", "<p id=\"P11\">Model parameter values and details of the source references and justifications, ordinary differential equations (ODEs), and algebraic equations were described in (##REF##37305053##Bagga et al. 2023##). Both the nonspatial and spatial models were constructed in MATLAB R2019a (MathWorks Natick, Massachusetts, USA), and <italic toggle=\"yes\">ode15s</italic> was used to numerically solve the ODEs. All MATLAB code is available at <ext-link xlink:href=\"https://github.com/pulsatility/2023-TH-PBK-Model.git\" ext-link-type=\"uri\">https://github.com/pulsatility/2023-TH-PBK-Model.git</ext-link>.</p>" ]

[ "<title>Results</title>", "<p id=\"P12\">Among the three THBPs, it has been argued that ALB and/or TTR play a larger role than TBG in supplying THs to tissues because of their lower binding affinity for THs and/or shorter residence time of T4 and T3 on these proteins (##REF##2673754##Mendel 1989##, ##REF##10718550##Schussler 2000##, ##REF##12553418##Robbins 2002##, ##REF##19725882##Richardson 2009##, ##REF##25737004##Alshehri et al. 2015##). We explored this issue with both the nonspatial and spatial versions of the PBK model below.</p>", "<title>Nonspatial PBK model</title>", "<title>TH unloading in <italic toggle=\"yes\">Liver blood</italic></title>", "<p id=\"P13\">Compared with the respective arterial concentrations, free T4 (<italic toggle=\"yes\">fT4</italic>), <italic toggle=\"yes\">T4TBG</italic>, <italic toggle=\"yes\">T4TTR</italic>, and <italic toggle=\"yes\">T4ALB</italic> in the venous blood leaving the <italic toggle=\"yes\">Liver</italic> compartment are only negligibly lower, dropping by 0.02%-0.18% (##TAB##0##Table 1##). <italic toggle=\"yes\">fT4</italic> drops by 0.18%, which is comparable to the percentage drop of <italic toggle=\"yes\">T4ALB</italic> (0.17%), however, <italic toggle=\"yes\">T4TTR</italic> drops only by 0.069% and <italic toggle=\"yes\">T4TBG</italic> by 0.022%. This result suggests that only <italic toggle=\"yes\">T4ALB</italic> is in near equilibrium with <italic toggle=\"yes\">fT4</italic> as the blood transits through the <italic toggle=\"yes\">Liver</italic>. Here, equilibrium refers to the state when the rate of association between T4 (or T3) and either <italic toggle=\"yes\">TBG</italic>, <italic toggle=\"yes\">TTR</italic> or <italic toggle=\"yes\">ALB</italic> equals the rate of dissociation of the respective TH-THBP complex. The difference in the percentage drops of these three T4-THBPs can be explained by the T4 residence time on these proteins relative to the blood transit time through the <italic toggle=\"yes\">Liver</italic>. Given the vascular volume and blood flow rate of the <italic toggle=\"yes\">Liver</italic> (##REF##37305053##Bagga et al. 2023##), the blood transit time is about 8.3 seconds. The average residence time of T4 on TBG, TTR and ALB, as determined by the inverse of the dissociation rate constants <italic toggle=\"yes\">k</italic>₂, <italic toggle=\"yes\">k</italic>₄, and <italic toggle=\"yes\">k</italic>₆, is 55.55, 12, and 0.77 seconds respectively. Therefore, it is expected that <italic toggle=\"yes\">T4ALB</italic> can unload quickly in response to the drop of <italic toggle=\"yes\">fT4</italic> as <italic toggle=\"yes\">fT4</italic> moves into the <italic toggle=\"yes\">Liver tissue</italic>, while <italic toggle=\"yes\">T4TTR</italic> unloads much more slowly, and <italic toggle=\"yes\">T4TBG</italic> unloads the slowest during this short transit time. Despite these differences, the absolute amounts of T4 unloaded from the three THBPs are not much different – the concentration of <italic toggle=\"yes\">T4TBG</italic> decreases the most, by 15.2 pM, followed by <italic toggle=\"yes\">T4ALB</italic>, 12.3 pM, and <italic toggle=\"yes\">T4TTR</italic>, 11.8 pM. This is because although the relative speed of T4 dissociation from TBG is much slower than from ALB, <italic toggle=\"yes\">T4TBG</italic> has a much higher (nearly 10 times) abundance than <italic toggle=\"yes\">T4ALB</italic>, resulting in a higher absolute amount of T4 unloaded in the <italic toggle=\"yes\">Liver blood</italic> of the nonspatial model.</p>", "<p id=\"P14\">For T3 in <italic toggle=\"yes\">Liver</italic>, the venous blood concentrations are negligibly lower than the arterial concentrations by 0.24%-0.42% (##TAB##0##Table 1##). Free T3 (<italic toggle=\"yes\">fT3</italic>) drops by 0.42%, comparable to the percentage drop of <italic toggle=\"yes\">T3ALB</italic> (0.4%), only slightly higher than <italic toggle=\"yes\">T3TTR</italic> (0.36%), but tangibly higher than <italic toggle=\"yes\">T3TBG</italic> (0.24%). The average residence time of T3 on TBG, TTR and ALB, as determined by the inverse of the dissociation rate constants <italic toggle=\"yes\">k</italic>₈, <italic toggle=\"yes\">k</italic>₁₀, and <italic toggle=\"yes\">k</italic>₁₂, is 6.06, 1.45, and 0.45 seconds respectively. Compared with the blood liver transit time of 8.3 seconds, it is clear that <italic toggle=\"yes\">T3ALB</italic> and <italic toggle=\"yes\">T3TTR</italic> can reach near equilibrium with <italic toggle=\"yes\">fT3</italic> during this time, while <italic toggle=\"yes\">T3TBG</italic> can only partially reach equilibrium. However, like the case of T4, <italic toggle=\"yes\">T3TBG</italic> contributes the most T3 unloaded in the <italic toggle=\"yes\">Liver blood</italic> with an absolute drop of 3.0 pM, followed by 1.3 pM by <italic toggle=\"yes\">T3ALB</italic>, and a much smaller 0.3 pM by <italic toggle=\"yes\">T3TTR</italic>.</p>", "<title>TH unloading in <italic toggle=\"yes\">RB blood</italic></title>", "<p id=\"P15\">For T4 in <italic toggle=\"yes\">RB</italic>, the venous blood concentrations are negligibly lower than the arterial concentrations by 0.017%-0.078% (##TAB##0##Table 1##). Similar to the case in <italic toggle=\"yes\">Liver</italic>, the percentage drop of <italic toggle=\"yes\">T4ALB</italic> is comparable to that of <italic toggle=\"yes\">fT4</italic>, while <italic toggle=\"yes\">T4TBG</italic> drops by a much smaller fraction. Given the vascular volume and blood flow rate of <italic toggle=\"yes\">RB</italic> (##REF##37305053##Bagga et al. 2023##), the blood <italic toggle=\"yes\">RB</italic> transit time is 20 seconds, which is much longer than the T4 residence time on ALB, nearly twice as long as on TTR, and less than half of the residence time on TBG. The absolute amounts of T4 unloaded from the three THBPs are quite different, where <italic toggle=\"yes\">T4TBG</italic> unloads the most by 11.8 pM, <italic toggle=\"yes\">T4TTR</italic> the second by 7.5 pM, and <italic toggle=\"yes\">T4ALB</italic> the least by 5.6 pM.</p>", "<p id=\"P16\">For T3 in <italic toggle=\"yes\">RB</italic>, the venous blood concentrations are negligibly lower than the arterial concentrations by 0.062%-0.088% (##TAB##0##Table 1##). <italic toggle=\"yes\">fT3</italic> drops by 0.088%, similar to that of <italic toggle=\"yes\">T3ALB</italic> and <italic toggle=\"yes\">T3TTR</italic>, and only slightly higher than that of <italic toggle=\"yes\">T3TBG</italic> (0.062%), which can be explained by the much longer blood <italic toggle=\"yes\">RB</italic> transit time than the residence times of T3 on these THBPs (20 vs. 6.06, 1.45, and 0.45 seconds). Similar to T3 in <italic toggle=\"yes\">Liver</italic>, <italic toggle=\"yes\">T3TBG</italic> contributes nearly 70% of T3 unloaded in <italic toggle=\"yes\">RB blood</italic> with an absolute decrease of 0.78 pM, followed by 0.29 pM by <italic toggle=\"yes\">T3ALB</italic>, and a much negligible 0.07 pM by <italic toggle=\"yes\">T3TTR</italic>.</p>", "<title>Spatial PBK model</title>", "<p id=\"P17\">The nonspatial PBK model provides some preliminary insights into the relative contributions of THBPs to local THs. A caveat of the nonspatial model is that the <italic toggle=\"yes\">Liver</italic> and <italic toggle=\"yes\">RB</italic> are treated as well-mixed compartments, so it is unable to predict the concentration gradients of THs in the tissues. To this end, we utilized the spatial PBK model, where both the <italic toggle=\"yes\">Liver</italic> and <italic toggle=\"yes\">RB</italic> compartments are simulated as 200 interconnected, consecutive segments as detailed in <xref rid=\"S2\" ref-type=\"sec\">Methods</xref> (##FIG##0##Fig. 1C##). The half-lives of plasma T4 and T3 and the decay profiles in simulated T4 or T3 tracer experiments, in either the presence or absence of THBPs, are nearly identical to the nonspatial model (simulation results not shown).</p>", "<title>Overall contributions of THBPs to TH unloading in tissue blood</title>", "<p id=\"P18\">The spatial PBK model predicts that the arterial and venous plasma concentrations of both T4 and T3 are marginally lower than the corresponding concentrations in the nonspatial PBK model (##TAB##1##Tables 2## vs. ##TAB##0##1##). For a given TH variable, the differences between the arterial and venous concentrations are similar between the two models, and the relative contributions of TBG, TTR and ALB to providing THs to the tissues are by and large in the same order as predicted by the nonspatial model. One exception is <italic toggle=\"yes\">Liver</italic> T4, where although the contributions of the three THBPs are still comparable, ALB unloads (14.4 pM) 10% more than TBG (13.1 pM) does, while in the nonspatial model TBG unloads 20% more than ALB does. The percentage contributions by each THBP species in the spatial model are summarized in ##FIG##1##Fig. 2##.</p>", "<title>TH concentration gradients</title>", "<p id=\"P19\">The spatial plasma concentrations of <italic toggle=\"yes\">fT4</italic> and the three T4-THBPs in tissue blood exhibit very different gradient profiles (##FIG##2##Fig. 3A##-##FIG##2##3E##). <italic toggle=\"yes\">fT4</italic> (##FIG##2##Fig. 3A##) and <italic toggle=\"yes\">T4ALB</italic> (##FIG##2##Fig. 3D##) in both <italic toggle=\"yes\">Liver blood</italic> and <italic toggle=\"yes\">RB blood</italic> show similar, exponential-like drops (with the exception that <italic toggle=\"yes\">fT4</italic> exhibits a discrete drop in the first segment). When examined on logarithmic scale of the Y axis, the decays do not follow a straight line (results not shown), indicating that they are not truly exponential. In contrast, <italic toggle=\"yes\">T4TBG</italic> exhibits a decreasing trend that is concave downward (##FIG##2##Fig. 3B##), while <italic toggle=\"yes\">T4TTR</italic> is concave downward initially but then transitions to a nearly linear decay phase, especially in <italic toggle=\"yes\">RB blood</italic> (##FIG##2##Fig. 3C##). Interestingly, <italic toggle=\"yes\">Total T4</italic> decays in a basically linear fashion (##FIG##2##Fig. 3E##). The concentration gradients of T4 in <italic toggle=\"yes\">Liver tissue</italic> and <italic toggle=\"yes\">RB tissue</italic> follow the same trend and profile as <italic toggle=\"yes\">fT4</italic> in the respective tissue blood, and the percentage drops from the first segment to the last segment, which are negligibly small, are also the same as the drop of <italic toggle=\"yes\">fT4</italic> (results not shown). These gradient profiles are qualitatively similar in both the <italic toggle=\"yes\">Liver</italic> and <italic toggle=\"yes\">RB</italic> compartments, but the concentration drops are generally steeper in <italic toggle=\"yes\">Liver</italic> than in <italic toggle=\"yes\">RB</italic>. The gradient profiles of T3 are similar to those of T4 (##FIG##2##Fig. 3F##-##FIG##2##3J##), except that <italic toggle=\"yes\">T3TTR</italic> exhibits a concave-upward profile for the most part (##FIG##2##Fig. 3H##) whereas <italic toggle=\"yes\">T4TTR</italic> is more concave-downward (##FIG##2##Fig. 3C##).</p>", "<title>Nonlinear, spatially dependent THBP unloading of THs in tissue blood</title>", "<p id=\"P20\">We next calculated the difference in T4-THBP concentrations between two consecutive tissue blood segments, which represents the amount of T4 that is unloaded by the THBPs in the downstream segment of the two. It is interesting to note that the concentration drop of each THBP species varies depending on the location of the segments (##FIG##3##Fig. 4A## and ##FIG##3##4B##). ALB clearly contributes the most T4 in the upstream <italic toggle=\"yes\">Liver</italic> segments, but as the blood perfuses downstream, the contributions by TBG and TTR start to catch up halfway through. Toward the venous end of the <italic toggle=\"yes\">Liver</italic> compartment, TBG becomes the dominant contributor, TTR the intermediate contributor, and ALB the smallest contributor (##FIG##3##Fig. 4A##). A similar change of order in contribution between ALB, TBG and TTR also occurs in the <italic toggle=\"yes\">RB</italic> compartment, although more upstream, nearly quarter-way through (##FIG##3##Fig. 4B##). Interestingly, the <italic toggle=\"yes\">T4TTR</italic> drop exhibits a nonmonotonic profile in <italic toggle=\"yes\">RB</italic>. The contributions of TBG and ALB to T3 unloaded in <italic toggle=\"yes\">RB blood</italic> exhibit similar profiles to the case of T4, with changes in the order of contribution occurring at further upstream locations (##FIG##3##Fig. 4C## and ##FIG##3##4D##). <italic toggle=\"yes\">T3TTR</italic> has the smallest contribution also showing nonmonotonicity in the <italic toggle=\"yes\">Liver blood</italic> and <italic toggle=\"yes\">RB blood</italic> compartments. At downstream locations in <italic toggle=\"yes\">RB</italic>, the amount of T3 unloaded from each of the three THBPs seems to reach a constant.</p>", "<p id=\"P21\">This nonlinear, spatially dependent, differential TH unloading by the three THBPs is due, at least in part, to the differences in the residence time of T4 and T3 on the THBP molecules. In the upstream segments, for instance, as <italic toggle=\"yes\">fT4</italic> moves into the tissue such that its local concentration drops, <italic toggle=\"yes\">T4ALB</italic> releases T4 much more quickly than <italic toggle=\"yes\">T4TTR</italic> and <italic toggle=\"yes\">T4TBG</italic>. The T4 released by <italic toggle=\"yes\">T4ALB</italic> likely further slows down the release of T4 from <italic toggle=\"yes\">T4TTR</italic> and <italic toggle=\"yes\">T4TBG</italic>, resulting in the concave-downward gradients of the two species (##FIG##2##Fig. 3B## and ##FIG##2##3C##). As the blood flows through the remaining segments downstream where <italic toggle=\"yes\">fT4</italic> drops to lower levels, <italic toggle=\"yes\">T4TTR</italic> and <italic toggle=\"yes\">T4TBG</italic> are further removed from their equilibrium with <italic toggle=\"yes\">fT4</italic> (##FIG##4##Fig. 5A##-##FIG##4##5B## and ##FIG##4##5D##-##FIG##4##5E##). This built-up “tension” or disequilibrium begins to force <italic toggle=\"yes\">T4TTR</italic> and <italic toggle=\"yes\">T4TBG</italic> to release more T4 than in the upstream segments. While in contrast, <italic toggle=\"yes\">T4ALB</italic> moves further toward equilibrium, therefore releasing less T4 (##FIG##4##Fig. 5C## and ##FIG##4##5F##). The evolution of the binding disequilibrium between T3 and THBPs through tissue blood shows similar trends to the case of T4 (##FIG##5##Fig. 6##). <italic toggle=\"yes\">T3TTR</italic> binding disequilibrium exhibits nonmonotonic changes in both <italic toggle=\"yes\">Liver blood</italic> (##FIG##5##Fig. 6B##) and <italic toggle=\"yes\">RB blood</italic> (##FIG##5##Fig. 6E##), similar to <italic toggle=\"yes\">T4TTR</italic> in <italic toggle=\"yes\">RB blood</italic> (##FIG##4##Fig. 5E##), which explains the bell-shaped contributions to T3 (##FIG##3##Fig. 4C##-##FIG##3##4D##) and T4 (##FIG##3##Fig. 4B##) by TTR.</p>", "<p id=\"P22\">The differential contribution to the amounts of THs unloaded in tissue blood by different THBPs cannot be explained, however, solely by the difference in the residence time of T4 or T3 on THBP molecules. There is a quantitative mismatch. For instance, the residence time of T4 on ALB is 70 times shorter than on TBG and nearly 16 times shorter than on TTR. However, in the first segment of the <italic toggle=\"yes\">Liver</italic>, <italic toggle=\"yes\">T4ALB</italic> contributes only about 7 times higher T4 than <italic toggle=\"yes\">T4TBG</italic>, and the <italic toggle=\"yes\">T4TBG</italic> contribution is on par with <italic toggle=\"yes\">T4TTR</italic> contribution (##FIG##3##Fig. 4A##). An examination of the absolute association rates and dissociation rates of all TH/THBP pairs reveals that the relative contributions to THs unloaded by the three THBPs in the most upstream segments are basically proportional to the absolute dissociation rates in the arterial blood (##FIG##4##Figs. 5## and ##FIG##5##6## and ##SUPPL##1##Table S1##). These dissociation rates are determined not only by the dissociation rate constant parameters (which determine the residence time), but also by the concentrations of TH-THBP complexes. In the case of contributions to the amount of T4 unloaded in local tissues, <italic toggle=\"yes\">T4TBG</italic> and <italic toggle=\"yes\">T4TTR</italic> have higher concentrations than <italic toggle=\"yes\">T4ALB</italic>, which compensate to a great extent their longer residence time.</p>", "<p id=\"P23\">Despite the nonlinear, varying contribution by each of the three THBPs across tissue segments, the total amounts of T4 and T3 unloaded by all three THBPs combined is highly constant across tissue segments (##FIG##3##Fig. 4##, green lines). The amount of T4 unloaded and thus delivered in the last segment is only 0.16% and 0.07% lower than that in the 1^st segment in the <italic toggle=\"yes\">Liver</italic> and <italic toggle=\"yes\">RB</italic> compartments respectively, and in the case of T3, the difference is 0.82% and 0.12% respectively. These results support the notion that THBPs play a crucial role in maintaining a constant concentration and uniform delivery of THs through the tissues. This is further demonstrated when all three THBPs are removed in the spatial model (##FIG##6##Fig. 7##). The arterial concentration of <italic toggle=\"yes\">fT4</italic> increases dramatically to 34 pM, compared with 15 pM when all three THBPs are present, and the venous concentrations leaving the <italic toggle=\"yes\">Liver</italic> and <italic toggle=\"yes\">RB</italic> compartments drop to 2.5 and 6.5 pM respectively, forming steep exponential gradients (##FIG##6##Fig. 7A## and ##FIG##6##7B##, solid line). T4 concentrations in the <italic toggle=\"yes\">Liver tissue</italic> and <italic toggle=\"yes\">RB tissue</italic> follow a similar descending gradient with nearly identical percentage drops as <italic toggle=\"yes\">fT4</italic> (##FIG##6##Fig. 7C## and ##FIG##6##7D##, solid line). The arterial concentration of <italic toggle=\"yes\">fT3</italic> increases to 5.81 pM, compared with 5 pM when all three THBPs are present. Surprisingly, unlike T4 which exhibits a monotonically descending gradient, plasma <italic toggle=\"yes\">fT3</italic> exhibits a bell-shaped gradient in both <italic toggle=\"yes\">Liver</italic> and <italic toggle=\"yes\">RB</italic> when all three THBPs are absent (##FIG##6##Fig. 7E## and ##FIG##6##7F##, solid line). <italic toggle=\"yes\">fT3</italic> first increases slightly less than 1/10^th way through the <italic toggle=\"yes\">Liver blood</italic>, and then decreases in a nearly linear fashion followed by a slightly concave-upward decay down to 1.76 pM (##FIG##6##Fig. 7E##). The bell shape of the plasma <italic toggle=\"yes\">fT3</italic> gradient in <italic toggle=\"yes\">RB</italic> is more dramatic, where <italic toggle=\"yes\">fT3</italic> first rises to 6.16 pM more than 1/5^th way through the tissue, and then decreases to 4.46 pM as it exits the tissue (##FIG##6##Fig. 7F##). T3 in <italic toggle=\"yes\">Liver tissue</italic> follows a similar nonmonotonic gradient profile as <italic toggle=\"yes\">fT3</italic> in <italic toggle=\"yes\">Liver blood</italic> (##FIG##6##Fig. 7G##, solid line) but T3 in <italic toggle=\"yes\">RB tissue</italic> only exhibits a monotonic decay (##FIG##6##Fig. 7H##, solid line). The nonmonotonicity of <italic toggle=\"yes\">fT3</italic> gradients in the tissue blood in the absence of all three THBPs is explained in ##SUPPL##0##Fig. S1##.</p>", "<p id=\"P24\">The above results clearly demonstrate that THBPs are essential to maintain constant concentrations and uniform delivery of THs through the tissues. We next examined if any one of the three THBPs is sufficient to fulfill this function. When only either TBG, TTR, or ALB remains in the model, the concentrations of T4 and T3 across the blood and tissue segments exhibit negligible drops and the arterial concentrations are nearly identical to the concentrations when all three THBPs are present (##FIG##7##Figs. 8##-##FIG##9##10##). The only notable changes are <italic toggle=\"yes\">fT3</italic> concentration when only TTR is present, where it drops from 5.2 pM in the arterial blood to 4.9 pM in <italic toggle=\"yes\">Liver</italic> venous blood (##FIG##8##Fig. 9C##). This is because <italic toggle=\"yes\">T3TTR</italic> is the least abundant among the three T3-THBPs, thus a large percentage drop is needed to meet the demand by <italic toggle=\"yes\">Liver tissue</italic>. Interestingly, just like the concentration gradients of <italic toggle=\"yes\">Total T4</italic> and <italic toggle=\"yes\">Total T3</italic> when all three THBPs are present, the concentration gradients of <italic toggle=\"yes\">T4TBG</italic> and <italic toggle=\"yes\">T3TBG</italic>, <italic toggle=\"yes\">T4TTR</italic> and <italic toggle=\"yes\">T3TTR</italic>, or <italic toggle=\"yes\">T4ALB</italic> and <italic toggle=\"yes\">T3ALB</italic> (which are essentially the total T4 and T3 respectively since only a single THBP species is present in each case) are also basically linear and exhibit the same absolute drop (e.g., about 40 pM for T4 and 4.5 pM for T3 from segment 0 to segment 200 in <italic toggle=\"yes\">Liver blood</italic>). These results indicate that any of the three THBPs is sufficient to ensure uniform delivery of THs in tissues.</p>", "<p id=\"P25\">Lastly, we examined the situation when only one THBP is absent. With a combination of any two THBPs present, the concentration gradients of <italic toggle=\"yes\">Total T4</italic> and <italic toggle=\"yes\">Total T3</italic> are also basically linear and exhibit the same absolute drops as when all three THBPs are present, indicative of uniform delivery of THs in tissues (##SUPPL##0##Figs. S2##-##SUPPL##0##S4##). <italic toggle=\"yes\">fT4</italic> and the T4-THBP species that has the shorter residence time for T4 always exhibit concave-upward decay profiles, while the T4-THBP species that has the longer residence time exhibits a concave-downward decay profile. The combination of ALB and TBG produces the most contrasted profiles (##SUPPL##0##Fig. S3##).</p>", "<title>Heterogeneous transport and metabolism across tissue segments</title>", "<p id=\"P26\">The above spatial analysis has the underlying assumption that each of the cross-membrane transport and metabolism parameters remains constant across the tissue segments from the arterial to venous ends. However, this assumption may not be true for all tissues or organs. For instance, the liver is known for its zonal heterogeneity in gene expression (##REF##25024605##Gebhardt and Matz-Soja 2014##). Thus, transporters such as MCT10 as well as TH-metabolizing enzymes such as DIO1, UGT and SULT might have possible gradients from the periportal to central vein layers within the liver lobule (##REF##28166538##Halpern et al. 2017##). In this section, we explored how such intra-tissue heterogeneity alters THBP contribution to TH tissue delivery and TH tissue concentrations. For simplicity, a 10-fold linear gradient (either increasing or decreasing) was applied to one parameter (<italic toggle=\"yes\">k</italic>₂₁, <italic toggle=\"yes\">k</italic>₂₃ – <italic toggle=\"yes\">k</italic>₃₅) at a time with the average equal to the default value used in the homogenous case above. Below we reported the results for a select few parameters for the <italic toggle=\"yes\">Liver</italic> and the results for the remaining parameters are provided in ##SUPPL##0##Supplemental Material##.</p>", "<p id=\"P27\">When <italic toggle=\"yes\">k</italic>₂₅, the rate constant for T4 liver influx, increases from segment 1 through 200 (##FIG##10##Fig. 11A##), <italic toggle=\"yes\">fT4</italic> in <italic toggle=\"yes\">Liver blood</italic> barely drops percentage-wise, exhibiting a small concave-down gradient (##FIG##10##Fig. 11B##, red line) as compared with the concave-up gradient when <italic toggle=\"yes\">k</italic>₂₅ is constant (##FIG##10##Fig. 11B##, gray line). <italic toggle=\"yes\">Total T4</italic> in <italic toggle=\"yes\">Liver blood</italic> decreases with a concave-down gradient vs. a linear gradient with constant <italic toggle=\"yes\">k</italic>₂₅, but by the same amount in both cases (##FIG##10##Fig. 11C##). As far as T4 contributions by THBPs are concerned, they are all spatially dependent, but their nonlinearities are modulated by <italic toggle=\"yes\">k</italic>₂₅ variation as compared with constant <italic toggle=\"yes\">k</italic>₂₅ (##FIG##10##Fig. 11E## vs. ##FIG##3##Fig. 4A##). ALB remains as the highest contributor throughout most segments, while TBG and TTR have similar contributions. The total amount of T4 unloaded by all three THBPs combined increases linearly across the segments. Correspondingly, T4 in <italic toggle=\"yes\">Liver tissue</italic> increases linearly as well, by nearly 10-fold (##FIG##10##Fig. 11D##). The kinetic changes of T4 caused by the <italic toggle=\"yes\">k</italic>₂₅ gradient also affect T3. <italic toggle=\"yes\">fT3</italic> in <italic toggle=\"yes\">Liver blood</italic> exhibits a small nonmonotonic profile (##FIG##10##Fig. 11G##) and <italic toggle=\"yes\">Total T3</italic> also has a U-shaped profile vs. a linear one with constant <italic toggle=\"yes\">k</italic>₂₅, but it drops by the same amount in both cases (##FIG##10##Fig. 11H##). TBG still contributes to T3 the highest throughout most segments particularly downstream ones followed by ALB, and interestingly both revert to net loading of T3 toward the venous end (##FIG##10##Fig. 11J##). The total amount of T3 unloaded by all three THBPs combined decreases linearly across the segments. Despite that T4 in <italic toggle=\"yes\">Liver tissue</italic> increases by 10-fold across the segments, T3 in <italic toggle=\"yes\">Liver tissue</italic> is insensitive to the change, increasing only by 3% (##FIG##10##Fig. 11I##). A decreasing <italic toggle=\"yes\">k</italic>₂₅ gradient has opposite effects compared with an increasing <italic toggle=\"yes\">k</italic>₂₅ gradient. When <italic toggle=\"yes\">k</italic>₃₀, the rate constant for T4 liver efflux, varies from segment 1 through 200, it has an opposite effect to varying <italic toggle=\"yes\">k</italic>₂₅ (##SUPPL##0##Fig. S5##). It is worth noting that T4 in <italic toggle=\"yes\">Liver tissue</italic> is sensitive but T3 in <italic toggle=\"yes\">Liver tissue</italic> is insensitive to <italic toggle=\"yes\">k</italic>₃₀ changes, with highly hyperbolic-like gradients despite <italic toggle=\"yes\">k</italic>₃₀ varying linearly (##SUPPL##0##Fig. S5D## and ##SUPPL##0##S5I##). This nonlinearity is explained in ##SUPPL##0##Fig. S5## legend.</p>", "<p id=\"P28\">When <italic toggle=\"yes\">k</italic>₂₇, the rate constant for T3 liver influx, increases or decreases from segment 1 through 200 (##FIG##11##Fig. 12A##), the gradients of all T4-related variables are barely affected (##SUPPL##0##Fig. S6##). With an increasing <italic toggle=\"yes\">k</italic>₂₇ gradient, T3 in <italic toggle=\"yes\">Liver blood</italic> is loaded onto as opposed to unloaded from the THBPs in the upstream quarter of the segments (##FIG##11##Fig. 12B##). Moving downstream, ALB unloads more T3 than TBG does, but ultimately TBG contribution surpasses ALB contribution. With a decreasing <italic toggle=\"yes\">k</italic>₂₇ gradient, ALB initially contributes more T3, while TBG contributes the most T3 throughout the majority of the downstream segments, and all THBPs revert to net loading of T3 toward the venous end (##FIG##11##Fig. 12C##). The total amount of T3 unloaded by all three THBPs combined changes linearly across the segments (##FIG##11##Fig. 12 B## and ##FIG##11##12C##). As <italic toggle=\"yes\">k</italic>₂₇ varies, <italic toggle=\"yes\">fT3</italic> and <italic toggle=\"yes\">Total T3</italic> in <italic toggle=\"yes\">Liver blood</italic> exhibit nonmonotonic profiles but the percentage changes are negligibly small (##FIG##11##Fig. 12D##-##FIG##11##12E##). T3 in <italic toggle=\"yes\">Liver tissue</italic> is sensitive to <italic toggle=\"yes\">k</italic>₂₇ – i.e., as <italic toggle=\"yes\">k</italic>₂₇ increases or decreases linearly 10-fold, T3 changes by nearly 10-fold in the same direction (##FIG##11##Fig. 12F##). When <italic toggle=\"yes\">k</italic>₃₁, the rate constant for T3 liver efflux, varies from segment 1 through 200, it has an opposite effect to varying <italic toggle=\"yes\">k</italic>₂₇, and produces a sensitive hyperbolic gradient of T3 in <italic toggle=\"yes\">Liver tissue</italic> without affecting T4-related variables (##SUPPL##0##Fig. S7##).</p>", "<p id=\"P29\">When <italic toggle=\"yes\">k</italic>₂₆, the rate constant for T4-to-T3 conversion in <italic toggle=\"yes\">Liver tissue</italic>, varies from segment 1 through 200 by 10-fold (##FIG##12##Fig. 13A##), the gradients of <italic toggle=\"yes\">fT4</italic> and <italic toggle=\"yes\">Total T4</italic> in <italic toggle=\"yes\">Liver blood</italic> deviate slightly from the case when <italic toggle=\"yes\">k</italic>₂₆ is constant (##FIG##12##Fig. 13B##-##FIG##12##13C##). When <italic toggle=\"yes\">k</italic>₂₆ increases, the gradient of T4 in <italic toggle=\"yes\">Liver tissue</italic> is steeper than when <italic toggle=\"yes\">k</italic>₂₆ is constant, but the percentage drop remains negligible, only about −0.5% (##FIG##12##Fig. 13D##, red line). When <italic toggle=\"yes\">k</italic>₂₆ decreases, the gradient of T4 in <italic toggle=\"yes\">Liver tissue</italic> is reversed, which increases by about 2% (##FIG##12##Fig. 13D##, green line). The relative contributions to T4 in <italic toggle=\"yes\">Liver blood</italic> by THBPs are similar to the case when <italic toggle=\"yes\">k</italic>₂₆ is constant (##FIG##12##Fig. 13E## and ##FIG##12##13F## vs. ##FIG##3##Fig. 4A##). <italic toggle=\"yes\">fT3</italic> and <italic toggle=\"yes\">Total T3</italic> in <italic toggle=\"yes\">Liver blood</italic> exhibit U-shaped profiles (##FIG##12##Fig. 13G##-##FIG##12##13H##). Despite that the liver T4-to-T3 conversion rate constant <italic toggle=\"yes\">k</italic>₂₆ varies by 10-fold across the segments, T3 in <italic toggle=\"yes\">Liver tissue</italic> is insensitive to the changes, increasing or decreasing only by about 3% (##FIG##12##Fig. 13I##). The changes in the relative contributions of THBPs to T3 unloaded in <italic toggle=\"yes\">Liver blood</italic> are similar to the case of varying <italic toggle=\"yes\">k</italic>₂₅ (##FIG##12##Fig. 13J## and ##FIG##12##13K##). Varying <italic toggle=\"yes\">k</italic>₃₄, the rate constant for T4 metabolism in <italic toggle=\"yes\">Liver tissue</italic>, has a small effect on T4 gradient in <italic toggle=\"yes\">Liver tissue</italic>, which only changes by slightly &gt; 1% from the arterial to venous ends, and the effect on T3 in <italic toggle=\"yes\">Liver tissue</italic> is negligible (##SUPPL##0##Fig. S8##). Varying <italic toggle=\"yes\">k</italic>₃₅, the rate constant for T3 metabolism in <italic toggle=\"yes\">Liver tissue</italic>, has no effect on T4-related variables, and only causes a T3 gradient in <italic toggle=\"yes\">Liver tissue</italic> that varies by &lt; 6% from the arterial to venous ends (##SUPPL##0##Fig. S9##).</p>", "<p id=\"P30\">Simulations were also conducted for <italic toggle=\"yes\">RB</italic>-related parameters <italic toggle=\"yes\">k</italic>₂₁, <italic toggle=\"yes\">k</italic>₂₈, <italic toggle=\"yes\">k</italic>₂₃, <italic toggle=\"yes\">k</italic>₂₉, <italic toggle=\"yes\">k</italic>₂₄, <italic toggle=\"yes\">k</italic>₃₂, and <italic toggle=\"yes\">k</italic>₃₃ respectively (##SUPPL##0##Figs. S10##-##SUPPL##0##S16##). In general, when compared to their counterparts in <italic toggle=\"yes\">Liver</italic>, applying gradients to these parameters modulates the THBP contributions to T4 and T3 tissue delivery in a qualitatively similar fashion. Compared to the corresponding parameters in <italic toggle=\"yes\">Liver</italic>, however, T3 in <italic toggle=\"yes\">RB tissue</italic> is much more sensitive to the gradients of <italic toggle=\"yes\">k</italic>₂₁, <italic toggle=\"yes\">k</italic>₂₄, <italic toggle=\"yes\">k</italic>₂₈, <italic toggle=\"yes\">k</italic>₃₂, and <italic toggle=\"yes\">k</italic>₃₃, but less sensitive to the gradients of <italic toggle=\"yes\">k</italic>₂₃ and <italic toggle=\"yes\">k</italic>₂₉. T4 in <italic toggle=\"yes\">RB tissue</italic> is still largely insensitive to the gradients of most parameters except <italic toggle=\"yes\">k</italic>₂₁ and <italic toggle=\"yes\">k</italic>₂₈.</p>" ]

[ "<title>Discussion</title>", "<title>Contribution of THBPs to TH tissue delivery</title>", "<title>Uniform TH tissue delivery</title>", "<p id=\"P31\">One of the most prominent functions of THBPs is to transport THs from the thyroid gland through the blood circulation to various target tissues and ensure uniform delivery of THs within the tissue where necessary as the blood perfuses. Our PBK models show minimal drops between the arterial and venous concentrations of free THs (##TAB##0##Tables 1## and ##TAB##1##2##), suggesting that the cells at the vascular entrance of the tissues are exposed to nearly identical TH concentrations as those at the exit. The spatial model further demonstrates that nearly constant amounts of THs are unloaded in the first through the last tissue blood segments when no parameter gradients exist. Removing all three THBPs results in markedly steep concentration gradients of <italic toggle=\"yes\">fT4</italic> and <italic toggle=\"yes\">fT3</italic> as well as tissue T4 and T3, which is consistent with what was observed experimentally using <italic toggle=\"yes\">ex vivo</italic> liver perfusion (##REF##3701605##Weisiger et al. 1986##, ##REF##3106010##Mendel et al. 1987##, ##REF##3407767##Mendel et al. 1988##), and confirms the obligatory role of THBPs in maintaining uniform TH tissue delivery. However, the three THBPs are redundant and thus cross-compensatory in this role, as either one alone appears to be sufficient to maintain relatively constant TH concentrations through tissues. The sufficiency of ALB alone was previously demonstrated experimentally by using <italic toggle=\"yes\">ex vivo</italic> liver perfusion (##REF##3106010##Mendel et al. 1987##). It was further shown that T4 uptake in the liver, kidney and brain of Nagase analbuminemic rats was not different from normal rats (##REF##2910905##Mendel et al. 1989##). TTR knockout mice are euthyroid with normal circulating free T4, T3, and TSH concentrations although total T4 and T3 are lower as expected (##REF##7806543##Palha et al. 1994##). TBG-deficient patients have similar TH profiles and are also euthyroid (##REF##31352644##Mimoto and Refetoff 2020##). Our results are consistent with these studies in that not all three THBPs specie are needed to maintain normal free TH concentrations and tissue uptake.</p>", "<title>Contribution of individual THBP species</title>", "<p id=\"P32\">It has been argued that ALB supplies the most THs to tissues due to the shortest residence time of THs on ALB; in contrast a smaller fraction of THs is supplied by TBG since THs have much longer residence time on TBG (##REF##2673754##Mendel 1989##, ##REF##10718550##Schussler 2000##). It has also been proposed that TTR makes the most contribution to immediate T4 tissue delivery because of its Goldilocks properties – it has the intermediate binding affinity for T4 and the abundance of T4TTR is intermediate among the three T4-THBPs (##REF##12553418##Robbins 2002##, ##REF##19725882##Richardson 2009##, ##REF##25737004##Alshehri et al. 2015##). The roles of TTR and TBG were also compared analogously to bank accounts, where TTR is the checking and TBG the saving accounts such that T4TTR is tapped first for T4 tissue delivery and then it is replenished by T4TBG later on after they leave the tissue (##REF##12553418##Robbins 2002##, ##REF##19725882##Richardson 2009##, ##REF##32003587##Hamers et al. 2020##). However, our models suggest neither of the two hypotheses on the roles of ALB and TTR relative to TBG is valid.</p>", "<p id=\"P33\">Simulations of both the nonspatial and spatial models indicate that TBG, TTR and ALB contribute comparable amounts of T4 in <italic toggle=\"yes\">Liver</italic>, but TBG plays a dominant role in <italic toggle=\"yes\">RB</italic> (##FIG##1##Fig. 2## and ##TAB##0##Tables 1## and ##TAB##1##2##). This is because the amounts of T4 unloaded by T4-THBPs are determined by both their abundances and T4 residence time relative to the blood tissue transit time. TBG binds the most T4 (75%) out of the three THBPs, which compensates for the longer residence time and thus slower release of T4. The longer blood transit time in <italic toggle=\"yes\">RB</italic> provides more time for TBG to unload T4, which explains the much higher contribution of TBG to tissue T4 in <italic toggle=\"yes\">RB</italic> than in <italic toggle=\"yes\">Liver</italic>. Therefore, for slowly perfused tissues, the contribution to tissue T4 follows the order of the abundances of T4-THBPs. The residence time of T3 on THBPs are generally much shorter than T4, especially for T3TBG and T3TTR, therefore the unloaded amounts are more aligned with their relative abundance rather than the residence time. This explains the consistent order of contribution to T3 by THBPs in both tissue compartments (##FIG##1##Fig. 2##), where TBG contributes about 3 times more than ALB, and ALB nearly 4 times more than TTR.</p>", "<p id=\"P34\">Generally, for tissue uptake of a substance bound to plasma proteins, the dissociation rate, rate of transportation or diffusion into the tissues, and blood perfusion rate can all play some limiting roles depending on the parameter conditions (##REF##3856281##Weisiger 1985##). The dissociation of THs from a THBP species could be rate-limiting in providing THs for tissue uptake, the significance of which depends on the tissue blood perfusion rate. For very slowly perfused tissues, the THBP contribution to tissue TH follows the order and proportion of the abundances of the TH-THBPs. In contrast, for rapidly perfused tissues, the percentage contribution by ALB will become higher while that by TBG will become lower since the instantaneous T4 release matters more. Ultimately it is the absolute dissociation rate, which is determined by both the TH residence time on THBP and the abundance of TH-THBP, that dictates the amount of TH released in the most rapidly perfused tissues.</p>", "<p id=\"P35\">In extremely rapidly perfused extrathyroidal tissues, it is conceivable that ALB is the most liquid checking account, from which the vast majority of T4 and T3 are withdrawn and delivered to the tissue. As the blood goes through the thyroid, ALB is quickly loaded with nearly half of the newly synthesized T4, while TBG and TTR are loaded with the remaining half (##FIG##1##Fig. 2A##, ##TAB##0##Tables 1## and ##TAB##1##2##). Then as the thyroid venous blood mixes with the venous blood coming back from other tissues, T4 will be transferred from ALB to TBG and TTR, as demonstrated by the positive or negative sign of the difference between the association and dissociation rates for each TH/THBP pair in <italic toggle=\"yes\">Body Blood</italic> (##SUPPL##1##Table S1##). For T3, because of the relatively faster binding kinetics and lower binding affinities between T3 and THBPs compared with T4, TBG seems to be the THBP that is loaded with the most T3 in <italic toggle=\"yes\">Thyroid blood</italic> (##FIG##1##Fig. 2B##, ##TAB##0##Tables 1## and ##TAB##1##2##). Interestingly, after mixing with the venous blood coming back from other tissues in <italic toggle=\"yes\">Body Blood</italic>, both ALB and TTR transfer T3 to TBG (##SUPPL##1##Table S1##).</p>", "<p id=\"P36\">If T4TBG is too slow to release enough T4, it begs the question why TBG alone is sufficient to unload a similar amount of T4 in <italic toggle=\"yes\">Liver</italic> and <italic toggle=\"yes\">RB</italic> as when all three THBPs are present (##FIG##7##Figs. 8## vs. ##FIG##2##3##). This is because with TBG alone, if <italic toggle=\"yes\">T4TBG</italic> does not unload enough T4 in the tissue, <italic toggle=\"yes\">T4TBG</italic> in the venous blood would leave the tissue at a higher concentration than it would be when T4 is sufficiently unloaded. After perfusing through the thyroid, <italic toggle=\"yes\">T4TBG</italic> will be recharged by the secreted T4 to an even higher level. With <italic toggle=\"yes\">T4TBG</italic> in the arterial blood coming back at a higher level to the tissue, the dissociation rate will be higher, thus more T4 is unloaded. The slightly higher arterial <italic toggle=\"yes\">T4TBG</italic> concentration is evidenced by the levels shown for segment 0 in ##FIG##7##Fig. 8## compared with ##FIG##2##Fig. 3##. In addition, because of the difference in residence time, a larger drop of free THs in the first segment occurs when TBG is the only THBP (##FIG##7##Fig. 8A## and ##FIG##7##8C##), which contrasts with the smaller drop of free THs when ALB is the only THBP (##FIG##9##Fig. 10A## and ##FIG##9##10C##). The initial larger drop in free THs moves TH-TBG further away from equilibrium, reducing the association rate and allowing sufficient amounts of THs to be released from TH-TBG.</p>", "<title>Spatially dependent variation of THBP contribution</title>", "<p id=\"P37\">An interesting and novel finding of the present study is that the three THBPs unload THs in a spatially dependent manner, where ALB generally contributes the most at the beginning whereas TBG contributes the most at the end of the perfused tissue. This is due to different residence times among the THBPs relative to the tissue blood transit time and different abundances of TH-THBPs. At the beginning of tissue perfusion, there is not enough time for THs to dissociate from TBG or TTR due to their longer residence time. As a result, ALB, which has a shorter residence time, will unload more. The relative contribution of the THBPs to tissue THs in the most upstream segments thus follows the order of the absolute dissociation rates of each TH-THBP (##FIG##4##Figs. 5## and ##FIG##5##6##). As the blood continues to move through the tissue with free THs dropping, TBG and TTG will become further removed from their equilibrium, resulting in a higher differential between the dissociation rate and association rate, thus more THs are unloaded from TBG and TTR. Therefore, with longer perfusion time, the TH contribution by TBG and TTR will catch up, while that by ALB will lessen. If the perfusion time is long enough, like in <italic toggle=\"yes\">RB</italic> for T3, the distances from equilibrium for the three THBPs will approach some constants (##FIG##5##Fig. 6D##-##FIG##5##6F##), and the relative contribution by these three TH-THBPs will reach a fixed ratio proportional to their abundances (##FIG##3##Fig. 4D##). At such a state, TBG is the one that is mostly removed from equilibrium, while ALB is closest to equilibrium, with TTR in the middle. Being the furthest removed from equilibrium allows TBG to deliver the most TH despite its tight grip on T4 and T3. The spatially dependent, nonlinear kinetics of THBPs may explain some of the discrepancies in the overall amounts of unloaded THs predicted by the spatial vs. nonspatial models. For instance, the spatial model predicts that T4ALB unloads a slightly higher amount of T4 than T4TBG does in <italic toggle=\"yes\">Liver</italic> (##FIG##1##Fig. 2A## and ##TAB##1##Table 2##), while the nonspatial model predicts the opposite (##TAB##0##Table 1##). Such discrepancy underscores the importance of modeling tissue blood flow in greater details for more accurate predictions.</p>", "<p id=\"P38\">The above conclusions are based on simulations with the assumption that the influx, efflux, and metabolism parameter values are constant across the tissue segments. Our simulations with spatial parameter heterogeneities indicate that they tend to modulate the contributions of different THBP species to TH tissue delivery and may ultimately impact the intra-tissue concentrations of T4 and T3 in some cases. Because of the near-equilibrium kinetics of TH influx and efflux between the blood and tissues, especially for T4 in <italic toggle=\"yes\">Liver</italic> and <italic toggle=\"yes\">RB</italic> and for T3 in <italic toggle=\"yes\">Liver</italic> (##REF##37305053##Bagga et al. 2023##), the intra-tissue T4 and T3 concentrations are mostly sensitive to gradients of the respective influx and efflux parameters. In contrast, they are almost insensitive to gradients of metabolism parameters except for T3 in <italic toggle=\"yes\">RB</italic>. In the latter case, 10-fold spatial variations of the metabolism parameters, such as T4-to-T3 conversion and elimination of T4 and T3, can lead up to 1.66-fold changes in T3 in <italic toggle=\"yes\">RB tissue</italic>. An implication of these results is that for regulating local T3, varying DIO2 and DIO3 may not be sufficient in some cases, and the membrane transporters may also need to be regulated. However, cautions should be exercised when extrapolating to specific tissues as the <italic toggle=\"yes\">RB</italic> here represents the average of all extra-hepatic tissues. In tissues that do not operate in equilibrium mode, i.e., the influx and efflux rates are considerably different, it is expected that local intracellular T3 would be more sensitively regulated by variations in DIO2 and DIO3 than we predicted here. It is also well known that multiple local feedback and feedforward mechanisms mediated by DIOs operate to regulate tissue T3 homeostasis (##REF##7651427##Toyoda et al. 1995##, ##REF##10487701##Hosoi et al. 1999##, ##REF##11075806##Gereben et al. 2000##, ##REF##17452445##Sagar et al. 2007##, ##REF##30366665##Rastogi et al. 2018##). Therefore, the extent to which a tissue operates in the equilibrium mode and the strength of local feedback/feedforward regulation may coordinately control local T3 concentrations, depending on the developmental stages and tissues.</p>", "<p id=\"P39\">One example is how T3 determines the retina pattern formation during development. In mice, T3 concentration forms a roughly 2.5-fold dorsal-to-ventral decreasing gradient in the nascent retina on postnatal day 10 (##REF##16606843##Roberts et al. 2006##). This gradient drives the formation of a pattern where the medium-wavelength opsin expressing cone cells dominate in the dorsal retina while the short-wavelength opsin expressing cone cells dominate in the ventral retina. The T3 gradient was not due to variations of DIO expression because DIO1 and 2 levels were undetectable while DIO3 displayed a uniform distribution in the developing retina. It was argued that graded TH transporter expression may create the T3 gradient in the retina, which is consistent with our findings that intra-tissue T3 distribution is sensitive to influx and efflux parameter gradients. In comparison, during metamorphosis of the frog, there is a DIO3 gradient concentrated in the dorsal half of the developing retina, which leads to T3 accumulation in the ventral half that drives the proliferation of ventral ciliary marginal zone (CMZ) cells and the formation of the ipsilateral projection (##REF##10624950##Marsh-Armstrong et al. 1999##).</p>", "<p id=\"P40\">The liver lobule is known for its zonal heterogeneity in gene expression (##REF##25024605##Gebhardt and Matz-Soja 2014##, ##REF##28166538##Halpern et al. 2017##). To meet different functional demands for THs in different zones, there may exist spatial gradients of transporters and enzymes in their abundance and affinity such that the rates of local influx, efflux, and metabolism of THs, and consequently, hepatic TH concentrations may vary across zones. Based on our analysis, even if there are gradients in the abundance and/or activities of the metabolic enzymes, but no gradients for the transporters, T4 and T3 are not expected to form tangible gradients within the liver lobule. By perfusing the rat liver through the portal vein with human serum containing ¹²⁵I-labeled T4, it was shown that the hepatic T4 uptakes were similar in the different layers of the rat liver lobule (##REF##3106010##Mendel et al. 1987##), suggesting that there is no heterogeneous distribution of T4 transporters. Antegrade and retrograde liver perfusions in the absence of THBPs further corroborated that the periportal-to-central capacities for T4 uptake are relatively constant, at least in rat livers (##REF##3701605##Weisiger et al. 1986##, ##REF##3407767##Mendel et al. 1988##).</p>", "<title>Evolutionary perspective on THBPs and their biological functions</title>", "<p id=\"P41\">Power et al., reviewed a wide array of studies on primitive vertebrates including fish, amphibians, reptiles, and birds, suggesting that ALB emerged earlier than TTR evolutionarily (##REF##11017772##Power et al. 2000##). Since ALB binds many compounds weakly, it is not surprising that more TH-specific THBPs arose as the organisms evolved. TBG appeared the latest with mammals being the only animal class that have all three THBPs (##REF##28249735##McLean et al. 2017##). Interestingly, these THBPs share some homologous sequences in the domains involved in TH binding, especially a 5-residue TH-binding motif, with TH-binding apolipoproteins, cell membrane TH transporters, and TH receptors, suggesting the origin of a common ancestor gene (##REF##29293436##Benvenga and Guarneri 2018##).</p>", "<p id=\"P42\">Our PBK model indicates that efficient, uniform TH unloading in tissue blood can be ensured even when only one of the three THBP species is present. It thus follows that the THBPs that emerged later may have additional functions besides supplying THs to tissues. TTR can be synthesized in the choroid plexus in the brain and placenta where it plays a unique role in transporting THs against the local concentration gradients (##REF##23664144##Landers et al. 2013##, ##REF##25784853##Richardson et al. 2015##, ##REF##31440205##Rabah et al. 2019##). The late-emerging TBG appears to provide several additional functions. T4 and T3 are predominantly associated with TBG in the plasma, and moreover, among the three THBPs, TBG has the highest binding affinities for T4 and T3. Therefore, the buffering function of THBPs against transient TH perturbations is mainly carried out by TBG. Furthermore, the total amount of T4 in the plasma is equivalent to about 2-days’ worth of T4 produced by the thyroid; therefore, THBPs, especially TBG, may function as an extrathyroidal reservoir of THs, in addition to the intrathyroidal store in thyroglobulin, in the event of disruption of TH production. TBG-deficient patients present with low total T4 and total T3 but have normal concentrations of free THs and TSH and thus are euthyroid (##REF##31352644##Mimoto and Refetoff 2020##). Our simulations showed that in the absence of TBG the uniformity of TH unloading in tissues blood is not affected, which is consistent with the lack of clinical manifestations in TBG-deficient patients. The relative contribution of THBPs to tissue THs does not appear to be fixed and may vary under different physiological and pathological conditions. For instance, TBG can be regulated during acute inflammation, where locally released serine proteases can cleave TBG to a low-binding-affinity form that can unload more THs to the inflammatory sites (##REF##3143075##Pemberton et al. 1988##, ##REF##11095421##Jirasakuldech et al. 2000##). During pregnancy, the maternal TBG level steadily doubles through the mid-term because of the rising estrogen and reduced plasma clearance, which may function to titrate the increasingly produced maternal THs and prevent them from rising too high (##REF##24449667##Moleti et al. 2014##).</p>", "<title>Limitations, potential applications, and future directions of the PBK models</title>", "<p id=\"P43\">Our current model does not include the HPT negative feedback loop. This is because we are studying the local intra-tissue TH and THBP kinetics at steady state with constant inputs of these species from the arterial blood. Therefore, it is not necessary to consider any actions of THs on TSH and vice versa, the inclusion of which in the model is not expected to change the results and conclusions in the present study. The uncertainty in the amounts of THs unloaded in the tissue blood by each THBP species predicted by our models lies in the values reported in the literature for the relative abundance of TH-THBPs and dissociation rate constants, which can vary depending on individuals and their physiological conditions as well as the experimental methodologies. The fraction of T4TBG may be overstated when measured with zone electrophoresis by as much as more than 10% because of the comigration of T4-binding high-density lipoprotein (HDL) and some Serpin proteins with TBG (##REF##11883864##Benvenga et al. 2002##). Despite such overestimation, the abundance of T4TBG is still expected to be much higher than T4TTR and T4ALB; therefore, for slowly perfused tissues, the order of contribution to T4 by each THBP species would remain largely unchanged because it primarily scales with the abundance of each TH-THBP species. In contrast, the predicted relative contribution by each THBP species for liver, a rapidly perfused tissue, is more uncertain because both the abundances and dissociation rate constants of THBPs play a role here, but fewer experiments have been done to determine the latter parameters.</p>", "<p id=\"P44\">While the three THBPs modeled here play major roles in regulating the amounts of THs unloaded and thus available for tissue uptake, previous research indicated that lipoproteins including HDL and low-density lipoprotein (LDL) also affect TH influx into and efflux out of cells (##REF##2105208##Benvenga and Robbins 1990##, ##REF##9751514##Benvenga and Robbins 1998##). In static cultures of skin fibroblasts or hepatocytes, the net efflux of T4 out of the cultured cells was facilitated by the lipoproteins as well as by THBPs (##REF##9751514##Benvenga and Robbins 1998##). The facilitation can be partially explained by these proteins acting as a sink to lower free T4 in the medium to reduce re-entry. As a matter of fact, as long as the free fraction of T4 was titrated to the same level, regardless of using either TBG, TTR, or ALB in the cell culture, T4 influx or uptake by the cells was reduced to a similar magnitude (##REF##2105208##Benvenga and Robbins 1990##). On an equal molar basis, the order of efflux facilitation was TBG &gt; LDL &gt; TTR &gt; HDL &gt;&gt; ALB, which cannot be explained by the relative binding affinities of these proteins for T4 because lipoproteins are generally weaker T4 binders. It was postulated that LDL and HDL may engage some nonspecific physical contact with the plasma membrane and their hydrophobic nature helps to desorb T4 off of the plasma membrane. LDL is better than HDL in facilitating the efflux because of its higher lipophilic content. Counterintuitively, it was also shown that LDL may facilitate T4 uptake (##REF##2105208##Benvenga and Robbins 1990##). This occurs in cells expressing LDL receptor (LDLR) (##REF##2105208##Benvenga and Robbins 1990##), where after T4-carrying LDL docks to LDLR, T4 can be released and internalized by cells. Such LDLR-mediated T4 entry may play a role in regulating lipid metabolism in local tissues. Future iterations of our model may consider including LDL and HDL-mediated TH tissue uptake.</p>", "<p id=\"P45\">The liver is an organ that can take up and secrete fatty acids, which can compete with THs for ALB binding. Serum fatty acid concentrations can fluctuate as the blood transits through different segments of the liver tissue when they are taken up from or secreted into the blood. Thus, theoretically it is possible that such fluctuations may impact the local TH binding to ALB and affect the spatial model accuracy. But the impact is unlikely to be significant, because at physiological concentrations, THs only occupy a tiny fraction (about 0.0016%) of the binding sites on ALB (##REF##10718550##Schussler 2000##), so there are likely plenty sites available for fatty acids and other molecules such as steroid hormones without tangibly displacing THs from ALB. Moreover, experimental studies demonstrated that when myristate, a fatty acid that can bind to ALB with high affinity (##REF##17827235##Fujiwara and Amisaki 2008##), is present at concentrations 30-fold greater than physiological levels, human serum ALB still retains a high-affinity binding site for T4 (the affinity was only reduced by half), due to some conformational changes of ALB (##REF##12743361##Petitpas et al. 2003##). The intracellular distribution of THs can be heterogeneous as well. For instance, the nucleus tends to accumulate much higher free T3 than the cytosol (##REF##3965501##Oppenheimer and Schwartz 1985##). Future iterations of the model may consider further compartmentalizing a tissue into cytosolic and nuclear spaces if needed. Future updates to the model could also include splitting the <italic toggle=\"yes\">RB</italic> into separate tissue compartments as sufficient tissue-specific data such as TH tissue concentrations, transporter and TH-metabolizing enzyme related kinetic parameters become available.</p>", "<p id=\"P46\">For simplicity, we used a linear gradient and varied one parameter at a time to examine their impact on THBP contributions to TH tissue delivery and tissue TH concentrations. In reality, the gradients of transporters and metabolic enzymes can vary in shape which could be exponential, nonmonotonic, or others. Moreover, the gradients of THs may not be controlled by a single protein or enzyme. Rather, their gradients may multiplex to collectively regulate local TH distributions within a tissue. As more tissue-specific spatial information becomes available, they can be included in future models.</p>", "<p id=\"P47\">Our models and their future iterations can have several potential applications. It is well known that the thyroid system is subjected to circadian regulation. In humans there are strong daily rhythms of TSH and fT3 accompanied with less pronounced rhythm of fT4 (##REF##3940263##Nimalasuriya et al. 1986##, ##REF##18364382##Russell et al. 2008##). A strong temporal relationship between TSH and fT3 has been suggested to associate with familial longevity (##REF##26230295##Jansen et al. 2015##). Since THBPs carry THs in blood circulations, they may affect the relative timing of TH tissue distribution. Applying control theory, we are currently using the PBK model to study the amplitude and phase relationships between TSH, fT3, fT4 and tissue THs and the potential mechanisms underpinning the circadian patterns. Our model may also help with clinical chronotherapy to guide the timing of dosing of thyroid drugs to ensure that the fluctuations of plasma THs are more closely aligned with the natural circadian rhythm. Many environmental chemicals can interfere with TH binding to THBPs because of their similar structures to THs. These endocrine-disrupting chemicals (EDCs) include polychlorinated biphenyls (PCBs), dibenzo-p-dioxins, dibenzofurans, and brominated flame retardants (##UREF##2##Cheek et al. 1999##, ##REF##16601080##Hamers et al. 2006##). EDCs, especially those with high binding affinities, can displace THs from THBPs to cause a potential increase in fT4 or fT3 levels (##REF##9460170##Brouwer et al. 1998##, ##REF##31487205##Noyes et al. 2019##). Using the nonspatial version of the PBK model, we have recently predicted that for persistent exposure to these EDCs the impact on free THs is minimal, whereas daily intermittent exposure to EDCs with short half-lives may cause concern as free THs can fluctuate to an extent that interferes with the circadian rhythm (##REF##37305053##Bagga et al. 2023##). The spatial model can be used to further investigate the impacts of thyroid EDCs on tissue TH delivery by THBPs.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"P48\">In summary, we have constructed a spatial PBK model with explicit considerations of TH and THBP interactions and explored their differential contributions in supplying TH to tissues. While it remains to be experimentally validated, the model provides novel insights into the debated roles of THBPs in this regard and local TH regulations.</p>" ]

[ "<p id=\"P1\">Author Contributions</p>", "<p id=\"P2\">QZ conceived the model structure, ADB constructed and simulated the model in MATLAB, ADB and QZ conducted the parameter justification and estimation, and formal analysis of simulation results, ADB and QZ wrote the initial draft and revised the manuscript. BPJ critically reviewed and revised the manuscript.</p>", "<p id=\"P3\">Plasma thyroid hormone (TH) binding proteins (THBPs), including thyroxine-binding globulin (TBG), transthyretin (TTR), and albumin (ALB), are important in regulating thyroxine (T4) and triiodothyronine (T3). THBPs carry THs to extrathyroidal sites, where THs are unloaded locally and then taken up via membrane transporters into the tissue proper. The respective roles of THBPs in supplying THs for tissue uptake are not completely understood. To investigate this question, we developed a spatial human physiologically based kinetic (PBK) model of THs, which produces several novel findings. <bold>(1)</bold> Contrary to postulations that TTR and/or ALB are the major local TH contributors, the three THBPs may unload comparable amounts of T4 in <italic toggle=\"yes\">Liver</italic>, a rapidly perfused organ; however, their contribution in <italic toggle=\"yes\">Rest-of-Body (RB)</italic>, which is more slowly perfused, follows the order of abundance of T4TBG, T4TTR, and T4ALB. The T3 amounts unloaded in both compartments follow the order of abundance of T3TBG, T3ALB, and T3TTR. <bold>(2)</bold> Any THBP alone is sufficient to maintain spatially uniform TH tissue distributions. <bold>(3)</bold> The TH amounts unloaded by each THBP species are spatially dependent and nonlinear in a tissue, with ALB being the dominant contributor near the arterial end but conceding to TBG near the venous end. <bold>(4)</bold> Spatial gradients of TH transporters and metabolic enzymes may modulate these contributions, producing spatially invariant or heterogeneous TH tissue concentrations depending on whether the blood-tissue TH fluxes operate in near-equilibrium mode. In summary, our modeling provides novel insights into the differential roles of THBPs in local TH tissue distribution and homeostasis.</p>" ]

[ "<title>Supplementary Material</title>" ]

[ "<title>Funding Acknowledgements</title>", "<p id=\"P49\">This research was supported in part by NIEHS Superfund Research grant P42ES04911 and NIEHS HERCULES grant P30ES019776.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1.</label><caption><title>Schematic illustrations of the human PBK model for T4 and T3.</title><p id=\"P53\"><bold>(A)</bold> The overall structure of the PBK model (##REF##37305053##Bagga et al. 2023##). <italic toggle=\"yes\">RB</italic>: rest-of-body. QC: cardiac output (plasma portion); QT, QRB, and QL: rate of blood (plasma) flow to <italic toggle=\"yes\">Thyroid</italic>, <italic toggle=\"yes\">RB</italic>, and <italic toggle=\"yes\">Liver</italic> respectively; CA: arterial concentration of a TH species; CVT, CVRB, and CVL: venous concentration of a TH species in <italic toggle=\"yes\">Thyroid</italic>, <italic toggle=\"yes\">RB</italic>, and <italic toggle=\"yes\">Liver</italic> respectively; <italic toggle=\"yes\">k</italic>₂₀ – <italic toggle=\"yes\">k</italic>₃₅: rate constants for production, transport, or metabolism processes as indicated. <bold>(B)</bold> Reversible binding of T4 and T3 with TBG, TTR and ALB. These binding events occur in the plasma of all blood (sub)compartments/subsegments. <italic toggle=\"yes\">k</italic>₁ – <italic toggle=\"yes\">k</italic>₁₂: rate constants for association and dissociation between THs and THBPs as indicated. Refer to (##REF##37305053##Bagga et al. 2023##) for definitions and values of these parameters in (A) and (B). <bold>(C)</bold> Illustration of tissue segmentation in the spatial PBK model. The tissue influx, efflux, T4-to-T3 conversion, and metabolism occur in all segments, but are only illustrated in the 1^st segment.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2.</label><caption><title>Percentage contributions of THBPs to loading and unloading of T4 (A) and T3 (B) in tissue blood in the spatial PBK model.</title><p id=\"P54\">Percentage contributions by ALB, TTR, and TBG in <italic toggle=\"yes\">RB</italic>, <italic toggle=\"yes\">Liver</italic>, and <italic toggle=\"yes\">Thyroid</italic> compartments are indicated. Negative values denote TH unloading from THBPs and positive values denote TH loading to THBPs.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3.</label><caption><title>Concentration gradients of free THs, TH-THBPs, and total THs in tissue blood in the spatial PBK model.</title><p id=\"P55\"><bold>(A-E)</bold> Plasma concentrations of <italic toggle=\"yes\">fT4</italic>, <italic toggle=\"yes\">T4TBG</italic>, <italic toggle=\"yes\">T4TTR</italic>, <italic toggle=\"yes\">T4ALB</italic>, and <italic toggle=\"yes\">Total T4</italic> in <italic toggle=\"yes\">Liver blood</italic> (orange) and <italic toggle=\"yes\">RB blood</italic> (green) respectively. <bold>(F-J)</bold> Plasma concentrations of <italic toggle=\"yes\">fT3</italic>, <italic toggle=\"yes\">T3TBG</italic>, <italic toggle=\"yes\">T3TTR</italic>, <italic toggle=\"yes\">T3ALB</italic>, and <italic toggle=\"yes\">Total T3</italic> in <italic toggle=\"yes\">Liver blood</italic> (orange) and <italic toggle=\"yes\">RB blood</italic> (green) respectively. Concentrations in segment 0 represent the arterial concentrations and the concentrations in segment 200 represent the venous concentrations.</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4.</label><caption><title>Segment-to-segment net differences in concentrations of free THs, TH-THBPs, and total THs in tissue blood in the spatial PBK model.</title><p id=\"P56\">The value of concentration drop for each segment is calculated by subtracting the concentration in the current segment from that in the immediate upstream segment. For the drop in the first segment, the arterial blood concentration (segment 0) is used as the upstream concentration. <bold>(A-B)</bold> Segment-to-segment drops in plasma concentrations of <italic toggle=\"yes\">fT4</italic>, <italic toggle=\"yes\">T4TBG</italic>, <italic toggle=\"yes\">T4TTR</italic>, <italic toggle=\"yes\">T4ALB</italic>, and <italic toggle=\"yes\">Total T4</italic>, as indicated in (A), in <italic toggle=\"yes\">Liver blood</italic> and <italic toggle=\"yes\">RB blood</italic> respectively. <bold>(C-D)</bold> Segment-to-segment drops in plasma concentrations of <italic toggle=\"yes\">fT3</italic>, <italic toggle=\"yes\">T3TBG</italic>, <italic toggle=\"yes\">T3TTR</italic>, <italic toggle=\"yes\">T3ALB</italic>, and <italic toggle=\"yes\">Total T3</italic>, as indicated in (C), in <italic toggle=\"yes\">Liver blood</italic> and <italic toggle=\"yes\">RB blood</italic> respectively.</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5.</label><caption><title>Evolution of the binding kinetics between T4 and THBPs through tissue blood in the spatial PBK model.</title><p id=\"P57\"><italic toggle=\"yes\">Association rate</italic> (solid blue), <italic toggle=\"yes\">Dissociation rate</italic> (solid green), and <italic toggle=\"yes\">Distance from equilibrium</italic> (dashed orange) for <italic toggle=\"yes\">fT4</italic> binding to <italic toggle=\"yes\">TBG</italic>\n<bold>(A)</bold>, <italic toggle=\"yes\">TTR</italic>\n<bold>(B)</bold>, and <italic toggle=\"yes\">ALB</italic>\n<bold>(C)</bold> in <italic toggle=\"yes\">Liver blood</italic>. <italic toggle=\"yes\">Association rate</italic>, <italic toggle=\"yes\">Dissociation rate</italic>, and <italic toggle=\"yes\">Distance from equilibrium</italic> for <italic toggle=\"yes\">fT4</italic> binding to <italic toggle=\"yes\">TBG</italic>\n<bold>(D)</bold>, <italic toggle=\"yes\">TTR</italic>\n<bold>(E)</bold>, and <italic toggle=\"yes\">ALB</italic>\n<bold>(F)</bold> in <italic toggle=\"yes\">RB blood</italic>. <italic toggle=\"yes\">Distance from equilibrium</italic> = (<italic toggle=\"yes\">Dissociation rate</italic> - <italic toggle=\"yes\">Association rate</italic>) / <italic toggle=\"yes\">Association rate</italic> × 100%. Segment 0 represents the arterial blood. The same color scheme is used for all panels as indicated in (A).</p></caption></fig>", "<fig position=\"float\" id=\"F6\"><label>Figure 6.</label><caption><title>Evolution of the binding kinetics between T3 and THBPs through tissue blood in the spatial PBK model.</title><p id=\"P58\"><italic toggle=\"yes\">Association rate</italic> (solid blue), <italic toggle=\"yes\">Dissociation rate</italic> (solid green), and <italic toggle=\"yes\">Distance from equilibrium</italic> (dashed orange) for <italic toggle=\"yes\">fT3</italic> binding to <italic toggle=\"yes\">TBG</italic>\n<bold>(A)</bold>, <italic toggle=\"yes\">TTR</italic>\n<bold>(B)</bold>, and <italic toggle=\"yes\">ALB</italic>\n<bold>(C)</bold> in <italic toggle=\"yes\">Liver blood</italic>. <italic toggle=\"yes\">Association rate</italic>, <italic toggle=\"yes\">Dissociation rate</italic>, and <italic toggle=\"yes\">Distance from equilibrium</italic> for <italic toggle=\"yes\">fT3</italic> binding to <italic toggle=\"yes\">TBG</italic>\n<bold>(D)</bold>, <italic toggle=\"yes\">TTR</italic>\n<bold>(E)</bold>, and <italic toggle=\"yes\">ALB</italic>\n<bold>(F)</bold> in <italic toggle=\"yes\">RB blood</italic>. <italic toggle=\"yes\">Distance from equilibrium</italic> = (<italic toggle=\"yes\">Dissociation rate</italic> - <italic toggle=\"yes\">Association rate</italic>) / <italic toggle=\"yes\">Association rate</italic> × 100%. Segment 0 represents the arterial blood. The same color scheme is used for all panels as indicated in (A).</p></caption></fig>", "<fig position=\"float\" id=\"F7\"><label>Figure 7.</label><caption><title>Concentration gradients of THs in blood and tissues in the absence all THBPs in the spatial PBK model.</title><p id=\"P59\"><bold>(A-B)</bold> Plasma concentrations of <italic toggle=\"yes\">fT4</italic> in <italic toggle=\"yes\">Liver blood</italic> and <italic toggle=\"yes\">RB blood</italic> respectively. <bold>(C-D)</bold> Concentrations of T4 in <italic toggle=\"yes\">Liver tissue</italic> and <italic toggle=\"yes\">RB tissue</italic> respectively. <bold>(E-F)</bold> Concentrations of <italic toggle=\"yes\">fT3</italic> in <italic toggle=\"yes\">Liver blood</italic> and <italic toggle=\"yes\">RB blood</italic> respectively. <bold>(G-H)</bold> Concentrations of T3 in <italic toggle=\"yes\">Liver tissue</italic> and <italic toggle=\"yes\">RB tissue</italic> respectively. Solid line: all THBPs absent, dashed line: all THBPs present. Concentrations in segment 0 represent the plasma concentrations in arterial blood.</p></caption></fig>", "<fig position=\"float\" id=\"F8\"><label>Figure 8.</label><caption><title>Plasma concentration gradients of free THs and TH-TBGs in tissue blood with TBG as the only THBP present in the spatial PBK model.</title><p id=\"P60\"><bold>(A-B)</bold> Plasma concentrations of <italic toggle=\"yes\">fT4</italic> and <italic toggle=\"yes\">T4TBG</italic> in <italic toggle=\"yes\">Liver blood</italic> (orange) and <italic toggle=\"yes\">RB blood</italic> (green) respectively. <bold>(C-D)</bold> Plasma concentrations of <italic toggle=\"yes\">fT3</italic> and <italic toggle=\"yes\">T3TBG</italic> in <italic toggle=\"yes\">Liver blood</italic> (orange) and <italic toggle=\"yes\">RB blood</italic> (green) respectively. Concentrations in segment 0 represent the plasma concentrations in arterial blood.</p></caption></fig>", "<fig position=\"float\" id=\"F9\"><label>Figure 9.</label><caption><title>Plasma concentration gradients of free THs and TH-TTRs in tissue blood with TTR as the only THBP present in the spatial PBK model.</title><p id=\"P61\"><bold>(A-B)</bold> Plasma concentrations of <italic toggle=\"yes\">fT4</italic> and <italic toggle=\"yes\">T4TTR</italic> in <italic toggle=\"yes\">Liver blood</italic> (orange) and <italic toggle=\"yes\">RB blood</italic> (green) respectively. <bold>(C-D)</bold> Plasma concentrations of <italic toggle=\"yes\">fT3</italic> and <italic toggle=\"yes\">T3TTR</italic> in in <italic toggle=\"yes\">Liver blood</italic> (orange) and <italic toggle=\"yes\">RB blood</italic> (green) respectively. Concentrations in segment 0 represent the plasma concentrations in arterial blood.</p></caption></fig>", "<fig position=\"float\" id=\"F10\"><label>Figure 10.</label><caption><title>Plasma concentration gradients of free THs and TH-ALBs in tissue blood with ALB as the only THBP present in the spatial PBK model.</title><p id=\"P62\"><bold>(A-B)</bold> Plasma concentrations of <italic toggle=\"yes\">fT4</italic> and <italic toggle=\"yes\">T4ALB</italic> in <italic toggle=\"yes\">Liver Blood</italic> (orange) and <italic toggle=\"yes\">RB Blood</italic> (green) respectively. <bold>(C-D)</bold> Plasma concentrations of <italic toggle=\"yes\">fT3</italic> and <italic toggle=\"yes\">T3ALB</italic> in <italic toggle=\"yes\">Liver blood</italic> (orange) and <italic toggle=\"yes\">RB blood</italic> (green) respectively. Concentrations in segment 0 represent the plasma concentrations in arterial blood.</p></caption></fig>", "<fig position=\"float\" id=\"F11\"><label>Figure 11.</label><caption><title>Effects of <italic toggle=\"yes\">k</italic>₂₅ gradients across <italic toggle=\"yes\">Liver</italic> segments on TH concentrations in <italic toggle=\"yes\">Liver</italic> and contributions of THBPs to TH loading and unloading in the spatial PBK model.</title><p id=\"P63\"><bold>(A)</bold> Linearly increasing, decreasing, or constant <italic toggle=\"yes\">k</italic>₂₅ gradients implemented as indicated. <bold>(B-D)</bold> T4 concentrations in <italic toggle=\"yes\">Liver blood</italic> and <italic toggle=\"yes\">Liver tissue</italic> as indicated. <bold>(E-F)</bold> Segment-to-segment net differences in plasma concentrations of <italic toggle=\"yes\">fT4</italic>, <italic toggle=\"yes\">T4TBG</italic>, <italic toggle=\"yes\">T4TTR</italic>, <italic toggle=\"yes\">T4ALB</italic>, and <italic toggle=\"yes\">Total T4</italic> as indicated in <italic toggle=\"yes\">Liver blood</italic> for increasing and decreasing <italic toggle=\"yes\">k</italic>₂₅ gradients respectively. <bold>(G-I)</bold> T3 concentrations in <italic toggle=\"yes\">Liver blood</italic> and <italic toggle=\"yes\">Liver tissue</italic> as indicated. <bold>(J-K)</bold> Segment-to-segment net differences in plasma concentrations of <italic toggle=\"yes\">fT3</italic>, <italic toggle=\"yes\">T3TBG</italic>, <italic toggle=\"yes\">T3TTR</italic>, <italic toggle=\"yes\">T3ALB</italic>, and <italic toggle=\"yes\">Total T3</italic> as indicated in <italic toggle=\"yes\">Liver blood</italic> for increasing and decreasing <italic toggle=\"yes\">k</italic>₂₅ gradients respectively. The same color scheme is used for panels (B-D, G-I) as indicated in (A), for panel (F) as in (E), and for panel (K) as in (J).</p></caption></fig>", "<fig position=\"float\" id=\"F12\"><label>Figure 12.</label><caption><title>Effects of <italic toggle=\"yes\">k</italic>₂₇ gradients across <italic toggle=\"yes\">Liver</italic> segments on T3 concentrations in <italic toggle=\"yes\">Liver</italic> and contributions of THBPs to T3 loading and unloading in the spatial PBK model.</title><p id=\"P64\"><bold>(A)</bold> Linearly increasing, decreasing, or constant <italic toggle=\"yes\">k</italic>₂₇ gradients implemented as indicated. <bold>(B-C)</bold> Segment-to-segment net differences in plasma concentrations of <italic toggle=\"yes\">fT3</italic>, <italic toggle=\"yes\">T3TBG</italic>, <italic toggle=\"yes\">T3TTR</italic>, <italic toggle=\"yes\">T3ALB</italic>, and <italic toggle=\"yes\">Total T3</italic> as indicated in <italic toggle=\"yes\">Liver blood</italic> for increasing and decreasing <italic toggle=\"yes\">k</italic>₂₇ gradients respectively. <bold>(D-F)</bold> T3 concentrations in <italic toggle=\"yes\">Liver blood</italic> and <italic toggle=\"yes\">Liver tissue</italic> as indicated. The same color scheme is used for panels (D-F) as indicated in (A), and for panel (C) as in (B).</p></caption></fig>", "<fig position=\"float\" id=\"F13\"><label>Figure 13.</label><caption><title>Effects of <italic toggle=\"yes\">k</italic>₂₆ gradients across <italic toggle=\"yes\">Liver</italic> segments on TH concentrations in <italic toggle=\"yes\">Liver</italic> and contributions of THBPs to TH loading and unloading in the spatial PBK model.</title><p id=\"P65\"><bold>(A)</bold> Linearly increasing, decreasing, or constant <italic toggle=\"yes\">k</italic>₂₆ gradients implemented as indicated. <bold>(B-D)</bold> T4 concentrations in <italic toggle=\"yes\">Liver blood</italic> and <italic toggle=\"yes\">Liver tissue</italic> as indicated. <bold>(E-F)</bold> Segment-to-segment net differences in plasma concentrations of <italic toggle=\"yes\">fT4</italic>, <italic toggle=\"yes\">T4TBG</italic>, <italic toggle=\"yes\">T4TTR</italic>, <italic toggle=\"yes\">T4ALB</italic>, and <italic toggle=\"yes\">Total T4</italic> as indicated in <italic toggle=\"yes\">Liver blood</italic> for increasing and decreasing <italic toggle=\"yes\">k</italic>₂₆ gradients respectively. <bold>(G-I)</bold> T3 concentrations in <italic toggle=\"yes\">Liver blood</italic> and <italic toggle=\"yes\">Liver tissue</italic> as indicated. <bold>(J-K)</bold> Segment-to-segment net differences in plasma concentrations of <italic toggle=\"yes\">fT3</italic>, <italic toggle=\"yes\">T3TBG</italic>, <italic toggle=\"yes\">T3TTR</italic>, <italic toggle=\"yes\">T3ALB</italic>, and <italic toggle=\"yes\">Total T3</italic> as indicated in <italic toggle=\"yes\">Liver blood</italic> for increasing and decreasing <italic toggle=\"yes\">k</italic>₂₆ gradients respectively. The same color scheme is used for panels (B-D, G-I) as indicated in (A), for panel (F) as in (E), and for panel (K) as in (J).</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\"><label>Table 1.</label><caption><p id=\"P66\">Loading and unloading of THs in tissues by THBPs in nonspatial PBK model</p></caption><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th rowspan=\"3\" align=\"center\" valign=\"middle\" colspan=\"1\"/><th rowspan=\"3\" align=\"center\" valign=\"middle\" colspan=\"1\"/><th colspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\">Free TH</th><th colspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\">TH-TBG</th><th colspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\">TH-TTR</th><th colspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\">TH-ALB</th></tr><tr><th rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">Conc<break/> (pM)</th><th align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">CV-CA (pM)</th><th rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">Conc<break/> (pM)</th><th align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">CV-CA (pM)</th><th rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">Conc<break/> (pM)</th><th align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">CV-CA (pM)</th><th rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">Conc<break/> (pM)</th><th align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">CV-CA (pM)</th></tr><tr><th align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">% change</th><th align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">% change</th><th align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">% change</th><th align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">% change</th></tr></thead><tbody><tr><td rowspan=\"7\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">T4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">CA</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">15.0019</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">7.0046E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">1.6003E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">7.2754E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td></tr><tr><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">CV_Thyroid</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">17.4036</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+2.4017</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">7.055E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+504</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">1.6467E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+463</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">8.1783E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+903</td></tr><tr><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+16.01%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+0.7198%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+2.895%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+12.41%</td></tr><tr><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">CV_Liver</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">14.9745</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0274</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">7.0031E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−15.175</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">1.5992E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−11.753</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">7.2631E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−12.264</td></tr><tr><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.1824%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0217%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0734%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.169%</td></tr><tr><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">CV_RB</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">14.9901</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0117</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">7.0034E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−11.782</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">1.5996E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−7.496</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">7.2698E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−5.592</td></tr><tr><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0781%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0168%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0468%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.077%</td></tr><tr><td rowspan=\"7\" align=\"center\" valign=\"middle\" colspan=\"1\">T3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">CA</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">5.0089</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">1.2758E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">82.2077</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">331.36</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td></tr><tr><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">CV_Thyroid</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">5.9337</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+0.9247</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">1.3466E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+70.8</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">92.0188</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+9.811</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">383.59</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+52.23</td></tr><tr><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+18.46%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+5.55%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+11.93%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+15.76%</td></tr><tr><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">CV_Liver</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">4.9879</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.021</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">1.2727E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−3.0267</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">81.9134</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.2943</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">330.04</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−1.3206</td></tr><tr><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.4194%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.237%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.358%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.3985%</td></tr><tr><td rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">CV_RB</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">5.0045</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0044</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">1.275E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.7841</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">82.1397</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0681</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">331.07</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.2876</td></tr><tr><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">−0.0879%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">−0.0615%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">−0.0828%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">−0.08679%</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T2\"><label>Table 2.</label><caption><p id=\"P68\">Loading and unloading of THs in tissues by THBPs in spatial PBK model</p></caption><table frame=\"hsides\" rules=\"groups\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th rowspan=\"3\" align=\"center\" valign=\"middle\" colspan=\"1\"/><th rowspan=\"3\" align=\"center\" valign=\"middle\" colspan=\"1\"/><th colspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\">Free TH</th><th colspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\">TH-TBG</th><th colspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\">TH-TTR</th><th colspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\">TH-ALB</th></tr><tr><th rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">Conc<break/> (pM)</th><th align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">CV-CA (pM)</th><th rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">Conc<break/> (pM)</th><th align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">CV-CA<break/> (pM)</th><th rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">Conc<break/> (pM)</th><th align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">CV-CA (pM)</th><th rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">Conc<break/> (pM)</th><th align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">CV-CA (pM)</th></tr><tr><th align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">% change</th><th align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">% change</th><th align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">% change</th><th align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">% change</th></tr></thead><tbody><tr><td rowspan=\"7\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">T4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">CA</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">14.9978</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">7.0033E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">1.5999E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">7.2734E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td></tr><tr><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">CV_Thyroid</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">17.3995</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+2.4017</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">7.0537E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+504</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">1.6462E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+463</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">8.1764E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+903</td></tr><tr><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+16.01%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+0.72%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+2.90%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+12.41%</td></tr><tr><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">CV_Liver</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">14.9671</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0307</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">7.0020E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−13.0744</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">1.5987E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−11.7385</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">7.2590E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−14.3806</td></tr><tr><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.20%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0187%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0734%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.1977%</td></tr><tr><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">CV_RB</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">14.9853</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0125</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">7.0022E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−10.74</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">1.5991E4</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−8.0595</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">7.2673E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−6.0681</td></tr><tr><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.084%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0153%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0504%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0834%</td></tr><tr><td rowspan=\"7\" align=\"center\" valign=\"middle\" colspan=\"1\">T3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">CA</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">5.0047</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">1.2748E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">82.1376</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">331.077</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">na</td></tr><tr><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">CV_Thyroid</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">5.9294</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+0.9247</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">1.3456E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+70.8145</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">91.9485</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+9.8109</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">383.303</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+52.226</td></tr><tr><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+18.48%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+5.56%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+11.94%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">+15.77%</td></tr><tr><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">CV_Liver</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">4.9829</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0218</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">1.2718E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−2.9384</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">81.8216</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.3161</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" colspan=\"1\">329.684</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−1.3933</td></tr><tr><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.44%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.2305%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.3848%</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.4208%</td></tr><tr><td rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">CV_RB</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">5.0003</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0044</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">1.2740E3</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.7791</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">82.0688</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.0688</td><td rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">330.787</td><td align=\"center\" valign=\"middle\" style=\"border-bottom: solid 1px\" rowspan=\"1\" colspan=\"1\">−0.2896</td></tr><tr><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">−0.088%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">−0.0611%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">−0.0838%</td><td align=\"center\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">−0.0875%</td></tr></tbody></table></table-wrap>" ]

[]

[]

[]

[]

[]

[ "<supplementary-material id=\"SD1\" position=\"float\" content-type=\"local-data\"><label>Supplement 1</label></supplementary-material>", "<supplementary-material id=\"SD2\" position=\"float\" content-type=\"local-data\"><label>Supplement 2</label></supplementary-material>" ]

[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN2\"><p id=\"P50\">Conflict of Interest</p><p id=\"P51\">The authors declare no conflict of interest.</p></fn></fn-group>", "<table-wrap-foot><fn id=\"TFN1\"><p id=\"P67\">Note: na, not applicable. CA: arterial concentration, CV: venous concentration, % change = (CV-CA)/CA.</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"TFN2\"><p id=\"P69\">Note: na, not applicable. CA: arterial concentration, CV: venous concentration, % change = (CV-CA)/CA.</p></fn></table-wrap-foot>" ]

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[ "<title>Introduction</title>", "<p id=\"P2\">The microgravity environment induces a plethora of pathophysiological changes that resemble accelerated aging including wasting of skeletal muscle,^{##REF##35171699##1##} bone demineralization^{##REF##35383182##2##}, and metabolic and cardiovascular dysregulation ^{##REF##19851058##3##, ##REF##31527661##4##,##UREF##0##5##}. In the case of bone mineral homeostasis, the kidneys control the excretion and retention of calcium, phosphate and other essential ions^{##REF##25287933##6##}. The kidneys are also responsible for generation of the active form of vitamin D, 1α,25-(OH)₂ vitamin D₃, which plays a critical role in a multitude of biological functions including bone health^{##UREF##1##7##}.</p>", "<p id=\"P3\">Directly evaluating the impact of microgravity on kidney function at the molecular and cellular level is obviously not feasible in astronauts due to the invasiveness and inherent risk of performing renal biopsies^{##REF##33060161##8##}. While studies can be conducted in rodents, the results may not truly reflect changes occurring in humans. To address the question of how microgravity affects human physiology, the National Center for Advancing Translational Sciences (NCATS) at the National Institutes of Health formed the “Tissue Chips in Space” program (Tissue Chips in Space | National Center for Advancing Translational Sciences (<ext-link xlink:href=\"http://nih.gov\" ext-link-type=\"uri\">nih.gov</ext-link>)) that leveraged novel tissue engineering platforms to recapitulate human physiology in the environment of space. Selected research teams were allowed to each send two projects to the International Space Station National Lab (ISSNL) during the four-year funding period.</p>", "<p id=\"P4\">Microfluidic-based microphysiologic systems (MPS) represent an advancement in cell culture techniques aimed at better replicating the tissue-specific <italic toggle=\"yes\">in vivo</italic> environment. We have previously reported our development of a MPS-based model of the kidney proximal tubule (PT-MPS) utilizing a commercially available platform developed by Nortis Inc.^{##REF##27521113##9##} The Nortis^™ system is designed for use with a tubing-free pneumatic-driven pump system but requires a substantial footprint, presents logistical challenges within the lab and is not suitable in all research contexts. Therefore, the Kidney Chip Perfusion Platform (KCPP), a piston-based device, was developed by BioServe Space Technologies to support an MPS-based kidney proximal tubule model.</p>", "<p id=\"P5\">The PT-MPS has been used to study a variety of disease states (e.g., aristolochic acid nephropathy and proteinuria^{##UREF##2##10##,##REF##35197326##11##}) and the responses to drug/xenobiotic-induced kidney injury^{##REF##26260164##12##–##UREF##3##14##}. In addition, the robustness of this system was independently tested in collaboration with the NCATS-funded Tissue Chips Testing Centers^{##REF##30869201##15##,##REF##30291268##16##}. To test the premise that microgravity is an accelerated environment for aging/disease progression, we evaluated proteinuria, kidney vitamin D metabolism, and nephrolithiasis (kidney stone disease)^{##REF##29999169##17##}. To adapt the system to the infrastructure aboard the ISSNL, we created a completely novel hardware support system with BioServe Space Technologies, our Implementation Partner and Payload Developer. BioServe Space Technologies is a research center within the University of Colorado, Boulder and has a proven track record designing life sciences hardware for microgravity experiments with their hardware having flown on over 85 space flight missions. Herein we report the development of the KCPP in support of two missions to the ISSNL.</p>" ]

[ "<title>Methods</title>", "<title>Tissue Acquisition &amp; Cell Culture:</title>", "<p id=\"P20\">Whole human kidneys that were not suited for human transplantation were obtained from Novabiosis, Inc. (Research Triangle Park, NC) with all patient identifiers removed in accordance with a biospecimens procurement agreement. Primary human proximal tubule epithelial cells (PTECs) were isolated by mechanical and enzymatic dissociation and cultured as previously reported^{##UREF##4##18##}. PTEC cultures were maintained serum-free in DMEM/F12 (Gibco, Grand Island, NY, Cat. # 11330-032) supplemented with 1× insulin-transferrin-selenium-sodium pyruvate (ITS-A, Gibco, Cat. # 51300044), 50 nM hydrocortisone (Sigma, St. Louis, MO, Cat. # H6909), and 1× Antibiotic-Antimycotic (Gibco, Cat. # 15240062). Upon reaching 75–80% confluence, PTECs were passaged by enzymatic digestion with 0.05% trypsin EDTA (Gibco, Cat. # 25200056) and manual cell scraping to obtain a single-cell suspension which was subsequently neutralized with defined trypsin inhibitor/DTI (Gibco, Cat. # R007100) at a volume:volume ratio of 2:1 DTI:trypsin, then the cells were pelleted by centrifugation at 200 × g for 7 minutes, resuspended in maintenance media, and plated in cell culture treated flasks at &gt; 25% confluency (referred to as passage 1 or P1). For both EVTs and CRS-17/22 missions, media cassettes were loaded and then stored for 1 week at 4° C before being warmed to 37° C immediately prior to integration with the KCPP. At the end of the treatment duration, sample effluents were frozen at −80° C.</p>", "<title>Preparation of Nortis Kidney MPS</title>", "<p id=\"P21\">Kidney MPS devices were purchased from Nortis, Inc (Woodinville, WA). Device preparation and PTEC injections were performed by the investigators as previously reported^{##REF##27521113##9##}. PTEC MPS cultures were maintained serum-free in DMEM/F12 (Gibco, 11330-032) supplemented with 1× insulin-transferrin-selenium-sodium pyruvate (ITS-A, Gibco, 51300044), 50 nM hydrocortisone (Sigma, H6909), and 1x Antibiotic-Antimycotic (Gibco, 15240062). In brief, for all the experiments run for experimental validation testing (EVT) as well as for Commercial Resupply Services (CRS) missions CRS-17 and CRS-22, PTECs of passage 2 or lower were used from each individual donor kidney. In the cases of experiments run for CRS-17 &amp; CRS-22, PTECs were shipped on dry ice to a lab at Kennedy Space Center. Following recovery from cryopreservation and expansion in 2D culture, MPS were seeded and allowed to culture as detailed in ##FIG##5##Figs. 6## and ##FIG##7##8##. As part of the EVT experimental design, a media cassette change was performed eight days after initiating KCPP flow.</p>", "<title>Quantification of organ-specific injury biomarker KIM-1</title>", "<p id=\"P22\">DuoSet^© ELISA kits were used to quantify human KIM-1 (R&amp;D Systems, Minneapolis, MN, Cat. # DY1780B) in PT-MPS effluents following the manufacturer’s protocol. In brief 50–100 μL of effluent were tested in duplicates and concentrations determined based on the standard curves generated from manufacturer-supplied controls.</p>", "<title>RNAseq data generation and analysis</title>", "<p id=\"P23\">To collect RNA samples from PTEC tubules, the PT-MPS devices were flushed with a volume of 1 mL RLT buffer (Qiagen, #79216) delivered through the abluminal inlet using a 1 mL slip-tip syringe (BD, 309659) equipped with a 22-gauge needle (BD, 305142) and collected at the outlet port. The RNA samples in RLT buffer were stored at −80° C until extraction which was performed as described by Lidberg et. al ^{##UREF##5##19##}.</p>" ]

[ "<title>Results</title>", "<title>Kidney Chip Perfusion Platform System Overview</title>", "<p id=\"P6\">In partnership with BioServe Space Technologies, we developed the KCPP hardware, addressed NASA safety and regulatory requirements, and facilitated the transition to a spaceflight certified and capable system. The KCPP is a precision, syringe pump-based platform designed to perfuse up to six Nortis^™ Triplex (each unit has three independently perfused tubules) PT-MPS built to support the NIH/NCATS Kidney Cell experiments. The platform is composed of five components, the kidney MPS, the MPS housing and valve block, media cassettes, fixative cassettes, and the programable precision syringe pump. Each KCPP as shown in ##FIG##0##Fig. 1## is comprised of over 2500 custom-designed and machined components. In the lab, preparing and assembling these components for experiments is a lengthy process and requires sustained, active engagement. The astronauts aboard the ISS have a set number of hours to operate scientific experiments and operate on a strict schedule. The innovation of the KCPP over the in-lab process is a dramatic reduction in complexity and time commitment. For example, in the lab, switching between maintenance and experimental media can be a multi-hour effort. This process was simplified with a pump interfacing to the MPS housing and valve block which can accept pre-loaded media or fixative cassettes. The pump provides a continuous flow of media or fixative while maintaining temperature control at 37° C. The pump uses a stepper motor to provide translation of a carriage which simultaneously depresses 18 syringe plungers. Preloading the media and fixative cassettes on the ground during the final pre-launch preparation phase streamlines the on-orbit protocol followed by the assigned astronaut on board the ISSNL. Additionally, the software for the pump only requires 5 operating modes for the experiment: the “Purge” command initiates the pump to engage the syringe pistons an prime the channels connecting the cassettes and the MPS, the “Run” command initiates perfusion with media at 0.5 μL/min, the “Fix” command perfuses fixative at 10 μL/min, the “Retract” command resets the pump plunger positions for sample housing and valve block and media/fixative cassette removal, and “Halt” stops all piston movement. The perfusion rate for media and fixation is programmable. Thus, while the KCPP is a complex work of engineering, the interface for users on the ISSNL is intuitive and user-friendly.</p>", "<p id=\"P7\">Media is loaded into nine channels separated by effluent bag cavities within one media cassette. The media is contained in the channels between an O-ring piston and a septum. A cannula from the valve block pierces the septa when installed and allows the piston to push media into the PT-MPS. The media flows through the PT-MPS and is collected in the effluent bags that are sealed with septa that are also pierced by cannula. The effluent bag cavities have containment plugs with O-rings on a retention plate. The waste media fills the effluent bags and is contained for post-flight analysis.</p>", "<title>Chip Housing &amp; Valve Block</title>", "<p id=\"P8\">The MPS housing and valve block system is a protective sealed enclosure, designed with functions for purging bubbles during media or fixative cassette installation (##FIG##1##Fig. 2##). Considerable effort is taken in the lab to mitigate the risk of bubbles entering the MPS since this will lead to disruption of media flow and compromise the integrity of the PTEC lumen. Because of the unpredictable nature of air bubbles in microgravity, they may not be subject to the same effects of buoyancy as on earth. Thus, it is possible that bubbles may bypass the traps in the MPS that are designed to utilize that buoyancy to trap bubbles above the path of the media. The enclosure interfaces to the media cassette and fixative cassette via four alignment pins and 18 cannulas. The housing vents are sealed with two adhesive covers during launch operations to maintain a 5% CO₂ and 100% humidity environment within the MPS housing. The valve block is designed with a valve bar system to direct flow through the valve block. Purging is performed when the valve bar is in the upper position as shown in ##FIG##1##Fig. 2D##. When the valve bar is in purge mode, flow is diverted from the PT-MPS directly into the effluent bags. When the valve bar is in flow mode, flow is directed into the PT-MPS.</p>", "<title>Media and Fixative Cassettes</title>", "<p id=\"P9\">The media cassette was designed to integrate directly with the chip housing and valve block and the KCPP to provide sufficient media to perfuse the PT-MPS for 10 days at a rate of 750 μL/day (##FIG##2##Fig. 3A##). The cassette consists of nine individual channels machined into an Ultem thermoplastic resin block. Each channel has a usable volume of 7.75 mL. The fluid is dispensed by mechanical plunger translation via the syringe pump. Once the fluid has passed out of the cassette and through the PT-MPS it returns to the housing and is stored in individually sealed bags in an adjacent chamber to the media channels. The fluid interfaces with the valve block via 18 cannulas piercing the corresponding septa in the bottom of the media cassette.</p>", "<p id=\"P10\">The sample effluent collection volume is sealed at the top of the cassette which provides an additional level of containment. When two levels of containment are required during cassette change out operations, the KCPP system can be operated within the Microgravity Science Glovebox (MSG) or Life Science Glovebox (LSG) which provide an additional level of containment during astronaut manipulations of the media or fixative cassettes or the KCPP in general.</p>", "<p id=\"P11\">The fixative cassette is a modified version of the media cassette (##FIG##2##Fig. 3B##). The cassette provides fixative for the final stage of the experiment to preserve the cells in the PT-MPS. The cassette has nine individual channels machined into a block of Ultem. Each channel has a maximum volume of 3.8 ml. The fluid is dispensed via mechanical plunger translation via the syringe pump. Once the fluid has passed out of the cassette and through the PT-MPS it returns to the housing and is absorbed into layers of absorbent material in adjacent chambers to the fixative channels.</p>", "<p id=\"P12\">The fluid interface to the valve block is via 18 cannulas piercing the corresponding septa in the bottom of the fixative cassette. The fixative cassette provides two levels of containment using O-rings on the pistons. Additional containment can be provided via outer bags, if needed.</p>", "<title>KCPP Integration</title>", "<p id=\"P13\">The integration and assembly of the individual components of the KCPP are shown in ##FIG##3##Fig. 4##. In brief, ##FIG##3##Figs. 4A##–##FIG##3##C## depicts a valve block, PT-MPS and integrated assembly, respectively. A media cassette is shown in and all the assembled components are seen in ##FIG##3##Fig. 4E##.</p>", "<title>KCPP Space Reduction Advancements</title>", "<p id=\"P14\">Although the overall footprint of an individual PT-MPS in the lab is small, the specialized equipment required to perfuse the devices is relatively large. As shown in ##FIG##4##Fig. 5A##/##FIG##4##B##, the individual components required to run experiments in our conventional fashion require an entire tissue culture incubator. The availability of space on ISSNL is limited but the KCPP reduces that required footprint 8-fold (1100 L to 136 L) allowing 24 PT-MPS to be housed and perfused within the locker space allocated to our group on board the ISSNL (##FIG##4##Figs. 5C##–##FIG##4##E##). As previously stated, the Nortis^™ pneumatic system does not require the use of tubing but the BioServe platform is syringe pump-based. We have previously used commercially available syringe pumps to run PT-MPS experiments and 24 PT-MPS require eight of these pumps to independently perfuse each of the 72 PT-MPS tubules. As shown in <bold>Supplementary Fig. 1,</bold>, the system accommodates two pumps per tissue culture incubator, necessitating four separate incubators for 24 PT-MPS. In addition to the significant space reductions from the KCPP, we have also eliminated the use of tubing, as the PT-MPS directly interface with the media blocks in the valve assembly. With syringe pumps, each individual PT-MPS tubule requires approximately 1 meter of tubing to connect media syringes outside of the incubators with PT-MPS within the incubator (<bold>Suppl. Figure 1</bold>). Thus, in addition to creating a simplistic user-interface for operation on the ISSNL, the KCPP exponentially shrinks the footprint requirements compared to conventional terrestrial PT-MPS experiments.</p>", "<title>Testing and Validation of the System</title>", "<p id=\"P15\">An experiment validation test (EVT) was performed prior to launch to assess the ability of the perfusion platform to maintain kidney PT-MPS cultures over the duration of the proposed experiments (##FIG##5##Fig. 6##). Kidney PT-MPS were loaded into the MPS housing and then integrated with the valve block and then into the perfusion platform. The devices were then cultured for six days in maintenance media to simulate a period of acclimation to microgravity. At day six, maintenance media cassettes were exchanged for treatment media cassettes and perfusion was continued for a 48-h treatment phase. At day eight, treatment media cassettes were removed and exchanged for fixative cassettes containing either RNAlater^® or formalin. The effluent from both the maintenance and treatment media were stored at −80°C for later analysis. Once the fixative cassette was integrated with the system, fixative/preservative was perfused for 1 hour after which the platform components were deintegrated and the PT-MPS were stored at −80° C or 4° C for later analysis.</p>", "<p id=\"P16\">Kidney Injury Molecule-1 (KIM-1) is a protein secreted into the urinary filtrate by proximal tubule health of our PT-MPS during the EVT, we measured the secretion of KIM-1 in effluents. We have previously shown that basal secretion of KIM-1 by PT-MPS is low but is markedly increased in response to nephrotoxic insults^{##UREF##3##14##–##REF##30291268##16##}. As shown in ##FIG##6##Fig. 7##, we observed low levels of KIM-1 from multiple PT-MPS evaluated in the EVT. For reference, a sample of 2D PTEC culture supernatant was included, but it should be noted that higher KIM-1 levels are expected in 2D cultures due to the cells being in a proliferative state while PTECs cultured in MPS devices are not proliferating^{##REF##30869201##15##}.</p>", "<title>KCPP System Performance</title>", "<p id=\"P17\">To date, we have completed two launches of the KCPP system to the ISSNL. The first launched on board SpaceX Commercial Resupply Services mission 17 (CRS-17) and the second on SpaceX CRS-22. On the first launch we evaluated vitamin D metabolism and proteinuric responses and the second launch on CRS-22 studied a calcium oxalate microcrystal model of nephrolithiasis. To assess overall performance of the KCPP hardware, we evaluated the ability to recover PT-MPS effluents for biomarker analyses as well as successful perfusion of RNAlater^™ for gene expression studies. The basic study design and timelines for CRS-17 and CRS-22 are shown in ##FIG##7##Figs. 8A##/##FIG##7##B##, respectively. Each launch consisted of 24 PT-MPS in-flight (microgravity) with a matched cohort of 24 ground-based PT-MPS. The CRS-17 launch consisted of 4 different PTEC donors (two males &amp; two females) while CRS-22 included 6 different donors (three males &amp; three females). The number of samples obtained for RNAseq analysis are shown in Tables 1 and 2 for CRS-17 and -22, respectively while Tables 3 and 4 show a similar breakdown for effluent retrievals for CRS-17 and -22, respectively.</p>", "<p id=\"P18\">The criteria for determining a “usable” sample for RNAseq was based on the ability to retrieve RNA from the PT-MPS tubules with a detergent solution, and subsequent total RNA isolation. Quality controls included Bioanalyzer^™ RNA integrity determination, RNA concentrations as well as subsequent RNAseq analysis (data not shown) and reported in <bold>Table 5.</bold> The criterium for “usable” sample for effluent analysis was based on retrieval of media in individual effluent bags after thawing of the KCPP media cassette blocks. It is worth noting the differences in the rates of “usable samples” between CRS-17 and CRS-22. In CRS-17, approximately 30% of the samples (RNAseq and effluents) were unusable for both flight and ground due to mold contamination of the PT-MPS. In contrast, nearly 100% of the samples were usable in CRS-22. The mold contamination observed in CRS-17 was not related to KCPP performance. Instead, it was likely driven by a combination of multiple launch delays that necessitated greater handling/transport of the PT-MPS from standard cell culture incubators to launch lockers and small amounts of residual media on the cell injection port on the PT-MPS. Approximately one week into the launch delay, an additional media cassette exchange procedure was carried out to ensure a fresh supply of media to the PT-MPS devices. To mitigate these issues for CRS-22, we employed PT-MPS cleaning protocols as well as applied a medical-grade silicone-based sealant (Silastic A^®) over the PT-MPS cell injection ports. It is also worth noting that the issues with launch delays in CRS-17 did not occur with CRS-22.</p>" ]

[ "<title>Discussion</title>", "<p id=\"P19\">KCPP is an integrated, automated, piston-based perfusion platform and enclosed cell culture environment designed to support MPS-based life sciences experimentation on board the ISSNL. Its compact design enables a significant reduction in the logistical challenges and spatial footprint required to implement these experiments aboard the confined space of the ISSNL. KCPP has been verified and space flight certified and is compliant with current NASA safety and interface requirements for space flight and use aboard the ISSNL. The system has been successfully utilized to support two space-based experimental studies designed to test the impact of microgravity on the function and pathophysiology of PTECs cultured in MPS aboard the ISSNL. In each instance, KCPP performed nominally and facilitated the execution of experiments otherwise impossible to be conducted terrestrially in simulated microgravity. The improvement in of samples recovery between missions emphasizes the importance of developing countermeasures against factors responsible for tubule or MPS device attrition. Looking to the future, extended studies using the KCPP system will facilitate the understanding of the long-term effects of spaceflight on renal physiology. Future development of autonomous MPS-based platforms can be used to predict human health concerns caused by spaceflight and long-term residence in microgravity that will occur during long term human space exploration.</p>" ]

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[ "<p id=\"P1\">Study of the physiological effects of microgravity on humans is limited to non-invasive testing of astronauts. Microphysiological models of human organs recapitulate many functions and disease states. Here we describe the development of an advanced, semi-autonomous hardware platform to support kidney microphysiological model experiments in microgravity.</p>" ]

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[ "<title>Acknowledgements:</title>", "<p id=\"P24\">This work was supported by the National Center for Advancing Translational Sciences (UH3TR000504, UG3TR002158 and UH3TR002178), jointly by the National Center for Advancing Translational Sciences and the Center for the Advancement of Science in Space (UG3TR002178), the National Institute of Environmental Health Sciences (P30ES00703) and an unrestricted gift from Northwest Kidney Centers to the Kidney Research Institute. BioServe’s work was supported in part by NASA contracts 80JSC020F0019 and 80JSC017F0129. We would like to thank the Life Science and Research Support Staff at Kennedy Space Center, in particular John Catechis and Anne Currin. In addition, we would like to express our gratitude to SpaceX and NASA for supporting our studies on CRS-17 and -22, in particular the astronauts Christina Koch &amp; Anne McClain (CRS-17) and Megan McArthur &amp; Mark Vande Hei (CRS-22). Finally, we would like to acknowledge the members of the BioServe Space Technologies team who enabled the successful design, development and launch of the KCPP.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><p id=\"P28\">Schematic (A) and real life (B) view of the novel KCPP programmable perfusion platform designed by BioServe Space Technologies showing six Triplex chips (1) situated within a housing unit after integration into the adapter unit (2) and media cassette (3).</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><p id=\"P29\">Nortis TCC, Triplex Housing &amp; Valve Block Assembly <bold><italic toggle=\"yes\">(A)</italic></bold> Schematic depicting a Nortis Triplex Chip including; (1) port valve, (2) injection port, (3) matrix plug holes, (4) bubble traps, <bold><italic toggle=\"yes\">(B)</italic></bold> An integrated view of a Chip Housing and Valve Block Assembly, <bold><italic toggle=\"yes\">(C)</italic></bold> An exploded view of a Chip Housing &amp; Valve Block Assembly including; (1) Triplex Housing, (2) Vent Cover, (3) Absorbent Padding, (4) Nortis Triplex Chips, (5) Valve Block, (6) Valve Bar, (7) Cannula, (8) Alignment Pins, <bold><italic toggle=\"yes\">(D)</italic></bold> Diagram of the valve block assembly oriented such that media flows from left to right depicting the internal mechanism set to the purge configuration.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3</label><caption><p id=\"P30\">Media &amp; Fixative Cassettes, <bold><italic toggle=\"yes\">(A)</italic></bold> Integrated view of a media cassette assembly, <bold><italic toggle=\"yes\">(B)</italic></bold> Exploded view of a media cassette assembly, <bold><italic toggle=\"yes\">(C)</italic></bold> Integrated view of a fixative cassette assembly, <bold><italic toggle=\"yes\">(D)</italic></bold> Exploded view of a media cassette assembly (1) septa, (2) polyetherimide block, (3) piston, (4) Piston o-ring, (5) FEP bags (media cassette only), (6) absorbent padding (fixative cassette only), <bold>(E)</bold> Diagram depicting the self-sealing septa mechanism in separated <bold>(L)</bold> and connected <bold>(R)</bold> configurations.</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4</label><caption><p id=\"P31\">Stepwise integration of the chip environment, <bold><italic toggle=\"yes\">(A)</italic></bold> Valve block, <bold><italic toggle=\"yes\">(B)</italic></bold> Three triplex chips integrated onto the valve block, <bold><italic toggle=\"yes\">(C)</italic></bold> Valve block &amp; triplex housing integrated assembly, <bold><italic toggle=\"yes\">(D)</italic></bold> Launch media cassette, <bold><italic toggle=\"yes\">(E)</italic></bold> Media cassette, valve block &amp; triplex housing.</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5</label><caption><p id=\"P32\">Reduction of the chip experimental footprint. <bold><italic toggle=\"yes\">(A)</italic></bold> Image depicting the amount of space 24 Triplex Chips occupy when the platforms and shelving is placed with the docking stations within tissue culture incubators, with the pneumatic pumps mounted on the top of the incubator, <bold><italic toggle=\"yes\">(B)</italic></bold> Image depicting the amount of laboratory space 44 Nortis Triplex Chips occupy when attached to platforms and shelves in a six foot biosafety cabinet, <bold><italic toggle=\"yes\">(C)</italic></bold> Image of KCPP programmable perfusion platforms designed by BioServe Space Technologies Four platforms are capable of perfusing 24 Nortis Triplex Chips, <bold><italic toggle=\"yes\">(D)</italic></bold> The powered locker used in the Kidney Chips experiment supplies power to four KCPP pumps, <bold><italic toggle=\"yes\">(E)</italic></bold> The KCPP platform reduces the space required to perfuse 48 Nortis Triplex Chips by 8-fold enabling astronauts to work within the limited space of the Life Sciences Glovebox. [2X incubators + 1X CO₂ Tank = 1100 L Each Locker measures ~68 L (67.3L)]</p></caption></fig>", "<fig position=\"float\" id=\"F6\"><label>Figure 6</label><caption><p id=\"P33\">Experimental timeline for engineering validation test</p></caption></fig>", "<fig position=\"float\" id=\"F7\"><label>Figure 7</label><caption><p id=\"P34\">KIM-1 levels in effluents of kidney MPS cultured for 11 days in the KCPP and in 2D cultured PTECs at passage 0 after 7 days.</p></caption></fig>", "<fig position=\"float\" id=\"F8\"><label>Figure 8</label><caption><p id=\"P35\">Experimental timelines for: <bold><italic toggle=\"yes\">(A)</italic></bold> CRS-17 &amp; <bold><italic toggle=\"yes\">(B)</italic></bold> CRS-22</p></caption></fig>" ]

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[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN1\"><p id=\"P25\"><bold>Competing interests statement:</bold> All authors declare no financial or non-financial competing interests.</p></fn><fn id=\"FN2\"><p id=\"P26\">Tables</p><p id=\"P27\">Tables 1 to 5 are available in the Supplementary Files section.</p></fn></fn-group>" ]

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[ "<title>INTRODUCTION</title>", "<p id=\"P4\">The blood vasculature is topologically organized as branched trees or a hybrid combination of trees and connecting vessels between branches, which effectively create loops^{##REF##15681697##1##,##REF##10820900##3##}. Most vessel networks, including the skeletal^{##REF##15604133##4##–##REF##4021836##6##} and cardiac muscle^{##REF##7247913##7##–##REF##14340824##11##} and skin vasculature^{##UREF##2##12##} exhibit the hybrid topological organization which provides functional advantages while minimizing flow resistance. The vessels that interconnect adjacent branches of the same vascular tree are known as collaterals or arcades, and these provide redundancy and alternate pathways for flow through the network.</p>", "<p id=\"P5\">Blood vessels respond to alterations in flow in normal and pathological conditions^{##UREF##0##2##,##REF##15604133##4##,##REF##15681697##13##} by changing both the inner vascular diameter and the vascular wall thickness^{##UREF##0##2##,##REF##15604133##4##,##REF##15681697##13##}. Vasomotion and changes in arterial tone control the overall flow of blood to specific tissues and can also redirect flow to specific regions of the network. Depending on regional demand, the flow in collateral vessels can stagnate or reverse, thus controlling flow laterally across a network rather than down the branching hierarchy. Collateral vessels are generally redundant in baseline tissue function, but are very important during microvascular alterations such as reestablishing blood supply during ischemic revascularization^{##REF##15604133##4##,##UREF##3##14##–##UREF##5##17##}, or temperature control^{##UREF##6##18##} and brain function^{##REF##33893567##19##,##REF##32703056##20##}.</p>", "<p id=\"P6\">Selective manipulation of vascular networks can be used in the clinic to enhance microvascular function or reduce vessel density. Diabetic retinopathy (abnormal vascular growth of retinal vasculature) is treated locally by selective laser photocoagulation to reduce vascular density^{##REF##24852439##21##,##REF##25420029##22##}. Preclinical studies in other tissue beds (cremaster muscle, mesentery, saphenous and femoral artery and vein) showed significant vascular remodeling following laser ablation^{##REF##32297542##23##}, but an integrative and predictive network model which can serve as a planning tool for clinical interventions is still lacking.</p>", "<p id=\"P7\">In the present study, we examined vascular redundancy and the ability of vascular networks to adjust their diameter and redirect flow following focal ablation of blood flow while the remaining network was undamaged (##FIG##0##Figure 1##). Our hypothesis was that alterations in blood flow to major vessel segments in a microvascular network should result in predictable remodeling of the remaining vessels, collateral pathway development and redistribution of blood flow to areas with decreased supply. To test this hypothesis, we selectively laser-ablated arterial and venous vessels in the dorsal skin vasculature of five mice and measured the topological changes.</p>", "<p id=\"P8\">We then analyzed one representative network in more detail to demonstrate how a computational model of microvascular flow distribution might be used to predict which collateral vessels are redundant in a microvascular network and are susceptible to compensatory flow reversal in response to adjacent vessel blockage. The model was used for two related purposes. First, using the available data, we demonstrate simulated annealing as a tool for estimating unknown relative pressures at the edges of the field of view based only on the directions of flow within the observed network. While independent measures of these pressures are currently not feasible, our simulated annealing method provides information about the uncertainty in the predictions. The second use of the model is to demonstrate the concept that the redundant vessels that permit rerouting of flow around a disruption can often be identified as those with flow that is most sensitive to changes in pressure distal from the ablation. A tool for making such predictions is likely to be valuable when planning selective ablations in settings such as the retina or coronary circulation. Similar computational models can be developed to test clinical hypotheses in other microvascular networks.</p>" ]

[ "<title>METHODS</title>", "<title>Experimental model</title>", "<title>Animal models.</title>", "<p id=\"P39\">The protocol for the animal experiments was reviewed and approved by the Institutional Animal Care and Use Committee of the Massachusetts General Hospital. The procedures were performed in accordance with the approved guidelines. The reporting in the manuscript follows the recommendations in the ARRIVE guidelines. We have generated αSMA⁺-DsRed/Tie2⁺-GFP/FVB double transgenic mice line by crossing the Tie2⁺-GFP/FVB mice^{##UREF##7##24##} with the αSMA DsRed mice^{##REF##22065738##25##}. Once established, this transgenic line has been backcrossed to FVB mice at least 10 generations^{##REF##22065738##25##–##REF##1547488##27##}.</p>", "<title>Dorsal skin fold chamber (DSFC).</title>", "<p id=\"P40\">A DSFC was implanted in ten mice as described before^{##REF##9033284##28##} (##FIG##0##Fig. 1##). Briefly the animals were anesthetized via intraperitoneal K&amp;X injection. The back fur was removed over a 2.5×2.5cm area with an electric shaver and topical hair removal lotion. The skin is lifted, and the chamber is fixed to the skin with metal screws. A circular skin area of 1cm in diameter is removed to install the glass slide. The chamber was maintained throughout the duration of the experiment. The glass coverslip was not removed during the laser ablation and imaging procedures. The initial set of five mice was employed to refine the experimental and imaging techniques (data not presented). The subsequent group of five mice, identified as Mice A-E (refer to sFig. 1 and sTables 1–6), was included in the present study. These mice were assessed at different time intervals following diverse levels of laser ablation to determine general remodeling patterns. Mouse E, selected as a representative case, is thoroughly discussed to exemplify consistent changes observed in all mice and to furnish anatomical data for the mathematical model. The diameter data presented in sTables 1–5 are expressed as mean values along with their respective standard deviations, for three equidistant radial regions: proximal, medial, and distal. To define these regions, we calculated the distance from each arterial/venous vessel fragment to the nearest corresponding ablation site. Subsequently, all these distances were sorted in ascending order, and the maximum distance was divided into three equal parts, with segments falling into each third reported accordingly. sTable 6 was constructed based on the observations presented in sFig. 1, where “Y” indicates the occurrence of a specific remodeling pattern, and “N” indicates the absence of such a pattern.</p>", "<title>Multiphoton laser imaging and ablation.</title>", "<p id=\"P41\">A modified in-house multi-photon laser^{##UREF##8##29##} was used for the laser ablation and imaging. The laser beam was programmed to scan a single line oriented perpendicular to the vessel longitudinal axis, with the length of approximately 1.5x the vessel diameter. The laser power was increased to 1W. Several line scans were run side to side across the vessel until the vessel wall constricted and then disrupted, leading to blood stasis (evaluated via light microscopy). Subsequent imaging of the vessel was performed with the same MPLSM after returning to normal laser power. A Nikon SMZ 2500 stereomicroscope (Nikon Instruments Inc., Malville, NY) was used to image the microvascular structure <italic toggle=\"yes\">in vivo</italic> throughout the remodeling process.</p>", "<title>OCT imaging.</title>", "<p id=\"P42\">A custom-build OCT system^{##REF##19749772##30##} was used for angiographic and quantitative flow imaging. Angiographic methods follow those described before^{##REF##19749772##30##}. Quantitative flow information was derived from the OCT amplitude-decorrelation rate^{##UREF##9##31##–##REF##21633451##33##}. Briefly, the temporal rate of the OCT amplitude signal decorrelation was measured at each voxel and used as an indication of relative flow speed. The OCT data were not used to draw conclusions on absolute flow speed. Due to gradient effects^{##REF##32541887##32##,##REF##27627357##34##}, measurements in vessels with diameters comparable to or slightly larger than the imaging resolution (~15 μm) were not used in the analysis.</p>", "<title>Computational vascular network model</title>", "<title>Microvascular network fluid dynamics.</title>", "<p id=\"P43\">The vascular network data (number of vessels, connectivity matrix, vessel diameters and lengths) were measured from digital images of the microvascular network in all visible segments using the NIH ImageJ software. Arteries and veins were identified by morphology, location and flow direction from live observation and movies. Arteries and veins are usually paired, and the artery has a smaller diameter. Smaller arterioles and venules were identified by tracing them back to the larger arteries and veins. The map of the vascular network was extracted from low magnification images presented in ##FIG##0##Figs. 1A## and ##FIG##0##B## and ##FIG##1##Fig. 2##. The diameter measurements were performed at a higher resolution (e.g. ##FIG##2##Fig. 3A## and sFig. 2) which allowed clear visualization of the inner vessel diameter of viable blood vessels as marked by the presence of the blood in their lumen. The measurements were performed at the same magnification at all time points. The statistical analysis including the number of total vessels of a given diameter and % diameter changes as well as the graphical representation of the data were performed in Microsoft Excel and GraphPad Prism version 9.5.0 for macOS, GraphPad Software, San Diego, California USA, <ext-link xlink:href=\"http://www.graphpad.com/\" ext-link-type=\"uri\">www.graphpad.com</ext-link>. The mathematical model for calculating the hemodynamic parameters (blood flow, intravascular pressure, resistance) was described before in detail^{##REF##15681697##1##,##REF##15604133##4##}. Whereas calculations of flow and pressure were previously used to predict adaptation in the vessel diameters^{##REF##15681697##1##,##REF##15604133##4##}, here we assume fixed diameters, but seek the sensitivity of flow in each vessel to distant changes in the network topology. The present model aims to provide insight into the sensitivity of flow to distant changes in the network topology rather than to predict adaptation due to shear stress and pressure as was done before^{##REF##15681697##1##,##REF##15604133##4##}.</p>", "<p id=\"P44\">Briefly, the vessel segments were represented as cylinders with bifurcation nodes between them. Assuming steady state laminar flow of a Newtonian fluid, the flow rate for the segment connecting nodes and is given by , where is the flow resistance of segment , and , and and are its length and diameter, is viscosity. Conservation of mass at node requires that:\n\nwhere is the resistance of the terminal arteries relative to the venous network which is assumed to be at a constant pressure . The resulting system of linear equations can be solved for the ’s and the flows in each segment provided that all of the pressures at the boundaries of the field of view (FOV) are known. Here we assume a known pressure for the largest vessel entering the FOV. The remaining smaller vessels that cross the boundary have pressures that are unknown but can be estimated from a simulated annealing method (SAM) that seeks the optimal fit between the experimentally observed flow directions in all segments within the FOV and the directions obtained from our estimated boundary pressures.</p>", "<title>Simulated annealing method.</title>", "<p id=\"P45\">The theoretical basis of the simulated annealing method was described before^{##UREF##10##35##}. To implement this method, we make a random assumption/guess of the unknown boundary pressures, and then calculate the flow directions throughout the network based on this guess. By iteratively adjusting the guessed boundary pressures, the SAM seeks a global minimum for an error function (E) equal to the total number of segments with calculated flow directions that differ from those experimentally observed. The error function can be unweighted, weighted by the estimated flow or diameter, or both. The SAM usually accepts a better guess but avoids getting trapped in local minima by occasionally accepting a less good estimate with a probability proportional to , where n is the annealing iteration number and T is a gradually decaying pseudo temperature. The boundary pressures are updated 5,000 times for each set of initial guesses until the error function reaches a stable minimum.</p>", "<p id=\"P46\">The entire SAM optimization process is repeated 100 times with a new, random set of initial pressures for each run yielding a set of boundary pressures that minimizes the number of incorrectly predicted flow directions. Because the optimization process is stochastic, the variability in segment flow rate between optimization trials serves as a measure of how sensitive flow in that segment is to changes elsewhere in the network such as those imposed by vessel ablation.</p>" ]

[ "<title>RESULTS</title>", "<title>Laser ablation of the dorsal skinfold chamber (DSFC) microcirculation</title>", "<p id=\"P9\">The general patterns of skin microvascular remodeling were similar in all five mice studied (Supplementary Figure 1, sFig. 1 and Supplementary Tables 1–6, sTables 1–6). sFig.1 provides an extensive depiction of the time course and patterns of microvascular network remodeling observed in five distinct DSFC experiments, denoted as Rows A-E, each row representing a separate animal experiment. For the in-depth analysis, we focused on Mouse E as a representative case. For this network, we performed a comprehensive anatomical data analysis and mathematical modeling. In sFig. 1, the locations of laser ablations are denoted by the circles (red, arterial; blue, venous). White crosses signify collateral outward remodeling from previously very small vessels, and blue crosses represent outward or inward remodeling of existing arterial/venous segments. Red and blue brackets indicate arterial and venous ablated segment reopening, respectively, and red and blue cross-brackets denote interruption of perfusion in arterial and venous segments, respectively. One to three ablations (except for mouse B D1+) were performed at select locations in the middle of the microvascular network in the largest visible arteries and veins. Remarkably, all specimens exhibit substantial remodeling at different time points from as early as days 1–3 (sFig.1, Row B d1 and mouse C d3) and up to day 20 (mouse E, d20) and later (see below). In sFig.2, mouse A, proximal venous ablation was bypassed through the development of an existing transverse venule, which underwent outward remodeling to match the initial vein diameter. The distal venous ablation revascularized by day 12, while the main vein initially underwent inward remodeling until day 12 and subsequently returned to its pre-ablation diameter by day 17. Arterial ablations and one venous ablation reopened by day 12 in mouse A. In mouse B, by day 5, venous ablations either led to bypass through outward remodeling of transverse veins (mouse C, d5, upper half) or caused inward remodeling of the main venous branch (mouse C, d5, lower half). Mouse C illustrates venous ablations bypassed by pronounced collateral development, while arterial ablation successfully revascularized. In mouse D, initial arterial and venous ablations reopened as early as day 2, while other ablations targeted the main artery and vein and two of their branches to induce more permanent flow changes. Mouse E showcases a combination of all remodeling patterns, albeit with varying time courses. Venous ablations revascularize through collateral growth, and arterial occlusions reopen. The majority of vessels display visible remodeling, and diameter data is further described and modeled in subsequent sections of the study.</p>", "<p id=\"P10\">The detailed diameter values are reported in sTable 1 for intact pre-ablation vessels and sTables 2–5 for remodeling time points reported in sFig.1 for the proximal, medial, and distal regions from the closest ablation.</p>", "<p id=\"P11\">The primary observed remodeling patterns, which encompass outward/inward remodeling of existing arteries and veins, collateral growth of previously small vascular segments, segment reopening, and permanent segment occlusions, are summarized in sTable 6, with accompanying diameter data provided in sTables 2–5, and illustrated in sFig. 1. One of the notable findings was the presence of both outward and inward remodeling phenomena in both arterial and venous segments, a dynamic process that persisted throughout the observation period. From the onset, the immediate aftermath of laser ablation at day 0 there were significant diameter changes as shown in sTables 2–5 although these changes are difficult to observe in sFig. 1. Starting at day1, there was visible collateral remodeling in most specimens.</p>", "<p id=\"P12\">Furthermore, sTable 6 also highlights the segment occlusion which was the goal of each initial laser ablation. While certain vessels maintained their occluded state throughout the observation period, a subset of vessels displayed the ability to gradually reopen over time. This observation indicates the dynamic nature of microvascular responses and their potential for adaptive adjustments over extended timeframes.</p>", "<p id=\"P13\">Due to variations in time course remodeling among specimens, a representative mouse (mouse E in sFig. 1 and sTables 1–6) was chosen to show the observed remodeling process for the remainder of the study. The typical mouse microcirculation within the DSFC contains a main artery and vein pair (##FIG##0##Fig. 1A##, ##FIG##0##B##, solid green arrowhead, and sFig. 1) and smaller artery - vein pairs (open green arrowheads). There are multiple arcade/collateral vessels that connect arteries to other arteries on separate branches of the arterial or tree or veins to veins between venous branches. A few arterial collaterals are indicated by red and venous collaterals by blue arrowheads, respectively in (A). These arcading vessels provide vascular redundancy by allowing redistribution of blood flow. Arteries have significantly smaller diameters than the paired veins and have tighter concentric layers of smooth muscle cells (red and yellow in ##FIG##0##Fig. 1##, Pre-ablation 1–3 and Post-ablation 1–3).</p>", "<p id=\"P14\">The laser ablation was performed at three major locations (##FIG##0##Fig. 1A## and ##FIG##0##B##, two artery/vein pairs in regions 1 and 3, and an artery in region 2) in the center of the window to maximize blood flow redistribution and to allow long term observation of the developing vascular changes (as some drifting of the tissue occurs within the DSFC over two weeks). The ablated vessels experienced rapid vasoconstriction upstream and downstream from the ablation site (##FIG##0##Fig. 1##, Post-ablation 1–3). There was complete blood flow interruption in segments just distal and proximal from the ablations (sVideos 1A, 2A and 3A). The laser ablation procedure was focused only on the target vessels, effectively cauterizing them while having little effect on the surrounding tissue. The brown scar tissue located in the muscle fascia subsides at later time points (sVideos 1B, 2B and 3B). Note that in region 2, the ablation of the artery had no effect on the diameter of the adjacent large vein or the blood flow in that vessel (##FIG##0##Fig. 1##, Post-ablation 2; sVideo 2A).</p>", "<title>Time-course of vascular network remodeling</title>", "<p id=\"P15\">By day 6 after ablation, there was clear evidence of vascular remodeling throughout the network (compare ##FIG##1##Figs. 2##D0−/+ and 2D6). Vessel segments associated with the ablated vessels had reduced diameter at day 6, while there was increased diameter in a number of collateral vessels (regions 4, 5 in ##FIG##1##Fig. 2D6##). By day 13, vessel diameters had qualitatively returned to pre-ablation values for much of the network (##FIG##1##Fig. 2D13##). This was due to remodeling of collateral vessels, which allowed an increase in compensatory flow entering tissue regions previously supplied by the ablated vessels. There were also large increases in diameter in a few small vessels that restored flow through the veins by bypassing the ablation sites (arrowheads in ##FIG##1##Fig. 2##, D13, and sVideo 1A-D, 3A-D).</p>", "<p id=\"P16\">These structures formed from sequences of smaller microvessels that were part of the original vascular bed. The increased flow through these small bypass channels likely caused the expansion of vessel diameter which eventually matched that of the original vein, similar to previous observations in the mouse gracilis muscle^{##UREF##0##2##}.</p>", "<p id=\"P17\">Some branches from the two small networks in regions 1 and 3 associated with the new vein segments appeared to be pruned or regressed as the new segments became part of the large veins. Albeit observed at low/medium resolution in transmitted light images, in these veins, there was no visible evidence of extensive angiogenesis or new vessel growth contributing to the regeneration of the network or restoration of flow. Rather, the rerouting occurred through remodeling of existing vessel segments, most of which could be visualized even before the ablations were performed (see ##FIG##4##Fig 5a##, ##FIG##4##b##).</p>", "<p id=\"P18\">However, we did observe reconnection of venous segments through the ablation site via endothelial migration in other networks (sFig.1, rows A-C and E). The response to injury appears to be related to the effective blood pressure difference across the ablation. In ##FIG##1##Fig. 2##, regions 1 and 3, the ablations are situated such that there is a large pressure drop across the ablation sites. This forces the blood to reroute through the smaller vessels early after the injury. However, in sFig.1 row A, there were two ablations performed on the same large vein. In this case, the upstream ablation has little pressure drop because the downstream ablation is preventing outflow. For this reason, very little flow re-routing or vessel remodeling occur at the upstream ablation, and this region was instead reperfused by direct reconnection of the vein via angiogenesis (sFig. 1, row A, d12 and d17).</p>", "<p id=\"P19\">On the arterial side, in region 2 we did not observe re-routing locally through pre-existing microvessels, and their subsequent enlargement, as in the veins of regions 1 and 3, ##FIG##1##Fig. 2##. Instead, flow was redistributed through the preexisting arterial arcades to circumvent the ablation and compensate for the lowered flow distal to the ablation sites (##FIG##1##Fig. 2##. D6 and D13, areas 4–6, and sFig. 1C, d3 and d18). Compared with the venous rerouting in regions 1 and 3 in ##FIG##1##Fig. 2##, which occurred over very short distances (~1mm) around the ablations, rerouting on the arterial side extended over much larger distances (~5–10mm) through the arcade vessels. In the ablated arteries, we did observe reconnection of the vessel through the ablation site via angiogenesis. On days 20, 23, 28 and 30, there was evidence of regeneration on the arterial side, as the artery ablated in Region 1 (##FIG##1##Fig. 2##) reconnected (for example, see the arterial ablation in region 1 (##FIG##1##Fig. 2##, D6–30, green arrowheads, and sVideo 1A-D). As this new vessel segment grew, original flow through the artery was restored, and the diameters of the major compensating collaterals decreased (##FIG##1##Fig. 2##, D28, region 8). The artery in region 2 (##FIG##1##Fig. 2##, D6–30, yellow arrowheads) did not achieve reconnection by the 30-day time point although some small flow pathways can be traced (sVideos 2C and 3C). The arterial flow in region 3 was re-established by day 30 but via smaller vessels than the original artery (##FIG##1##Fig. 2##, D30 blue arrowhead), with blood flow evident via doppler OCT at day 14 (##FIG##4##Fig. 5##, D14b) and intravital BF imaging at later time points (sVideo 3D).</p>", "<title>Angiogenesis at the ablation sites</title>", "<p id=\"P20\">Because of the endogenous reporters expressed by the mice, we were able to visualize endothelial cells (TIE2-GFP - green) and smooth muscle cells (aSMA-dsRed - red) longitudinally at the ablation sites. <italic toggle=\"yes\">In vivo</italic> laser confocal imaging of the regions 2 and 3 in ##FIG##0##Figure 1## revealed migration of the endothelial and smooth muscle cells through the ablation sites (##FIG##2##Fig. 3##). In region 3, the vascular pathway was re-established, and blood flow was observed (##FIG##2##Fig 3D##). Both endothelial and smooth muscle cells migrated into the damaged region and appeared to establish a connection by day 30, based on doppler OCT imaging (see ##FIG##4##Figure 5##). A similar process was observed for the other artery, which was ablated at location 2 in ##FIG##0##Figure 1## (##FIG##2##Fig. 3A##, ##FIG##2##B##), although this vessel did not reconnect by the end of our observation period. Angiogenesis was not observed in the large vein that remodeled in region 3, but the remodeled region acquired a covering of smooth muscle cells (##FIG##2##Fig. 3C##). After day 30, the relevant vessels had shifted out of the window chamber and were no longer observable.</p>", "<title>Time course of diameter remodeling</title>", "<p id=\"P21\">Overall, both arteries and veins changed their diameters collectively over time (##FIG##3##Fig. 4## and sFig. 1 and sTables 2–6). Because of resolution limitations, we restricted the quantitative analysis to the main arteries and veins and their transverse branches with inner diameters larger than 11 μm; therefore, the histograms do not include smaller vessels and capillaries. The smallest arteries (30 μm centered bin) stayed almost constant during the time points studied. A small dip at day 6 was recovered and slightly increased at the later time points. Combined with changes at other time points this could mean that smaller vessels became larger and therefore visible in this diameter range. The largest change in diameter distribution was observed in the 60 μm bin which was increased at days 6–20 and went back to normal values by day 30 which suggests a transient increase in vessel diameters to accommodate the early changes in blood flow as we noticed before in the gracilis artery remodeling^{##UREF##0##2##,##REF##15604133##4##}. Some larger vessels also constricted, moving from the 90–150 μm to the 60 μm range. At day 16, this trend reversed temporarily while between days 20–28 a lot of the larger arteries were still constricted. By day 30 diameter distribution of all arteries was close to post-ablation and pre-ablation values even in with the absence of the ablated large artery suggesting that blood redistribution can be accomplished through the contribution of the network of smaller arterioles even in the absence of the large artery.</p>", "<p id=\"P22\">The vein diameter distribution is more spread over a larger range of diameters suggesting a larger adaptation of the veins to accommodate flow changes. The largest variation in diameter distribution was observed in the 30 μm bin although slight transient tendency is also observed between days 6 and 28 with a decrease to normal values at day 30. During the transient increase period, an interesting second transient decrease was observed at day 16. Veins in the 80 μm range exhibited a gradual increase starting from post-ablation and peaching at day 30. The veins with diameters in 130–180 μm range showed the larges increased in density at early and medium time points (days 6 and 16). The largest veins stayed open immediately following the ablation, at day 6 they were reduced in diameter, at days 16 and 20 they were close to normal values but by day 30, the number of larger veins was drastically reduced suggesting again that on the venous side like the arterial side, flow redistribution could also be accomplished via a larger network of smaller venules.</p>", "<p id=\"P23\">We next focused on individual vessels to determine how specific vessels contributed to the flow redistribution. Using quantitative flowmetry OCT methods based on amplitude-decorrelation which can be used to estimate flow rate as well as lumen diameters ^{##REF##22475930##36##,##UREF##11##37##}, we analyzed a number of segments distal and proximal to the ablations sites before and following the ablations (##FIG##4##Fig. 5##). We also used intravital BF microscopy to determine flow directions (see Supplementary Videos 1–3). In the intact network, the blood flows from left to right from the large artery (#2, ##FIG##4##Fig. 5##) to its branches (#4, 6 and 9). The blood flows from the venous branches (#3, 5, 7, 8 and 10) towards the main vein (#1). Following ablation, the blood flow stopped in the ablated segments, but both upstream and downstream arteries continued to be perfused by arcading vessels from adjacent vascular trees (#2,4,6 and 9). Immediately after and at day 2 post-ablation, the segments near the ablations were not perfused. Nonetheless, at day 14, there is a signal of blood flow (##FIG##4##Fig. 5##, D14 green arrowheads) confirming the data from bright field microscopy (green arrowheads in ##FIG##1##Fig. 2##, D6–30). The arteries upstream from the ablation (#2 and 4) have a decreased diameter and flow velocity during the first few days post-ablation while the more peripheral arteries (#6 and 9 with reversed flow as observed experimentally) increased their diameters from day 2 post-ablation and through day 14, suggesting that they are largely responsible for the compensatory flow being rerouted from the parallel arteries (which are outside of the field of the window).</p>", "<p id=\"P24\">The main vein (#1 and 3) significantly decreased its diameter at day 2 but by day 14 the main vein and its small branch (#10) as well as a contiguous series of microvessels became enlarged to match the size of the vein (##FIG##4##Fig 5a##, ##FIG##4##b##). Venous branch #5 maintained its diameter throughout the 14-day time-course, as its flow was not directly affected by the ablations, and exit flow proceeded through the main vein through this pathway. After the ablation, flow through vein #7 was rerouted through vein #8, causing flow reversal in this vessel (Supplemental video 3A). Once the connection between these segments and the main vein was reestablished, the flow direction in vein #8 returned to normal (Supplementary video 3B). These changes in flow direction and topology resulted in large changes in diameter and flow rate in this region (##FIG##4##Fig 5##, D14). A side branch, venule #10 was affected little by the ablations, and maintained exit flow through the main vein. The ablation completely stopped exit flow in vein #11 by day 14, the connection is rerouted, and flow and diameter are returning to pre-ablation levels.</p>", "<p id=\"P25\">Diameter measurements at later time points show that main artery segments #2 and 4 recover after the initial diameter decrease probably due to vasoconstriction caused by the ablation. They continue to remodel outwards from day 16–28 with a transient dip at day 14 (##FIG##5##Fig. 6## top histograms). The transverse arteriole #6 diameter increased throughout the time course although the flow direction changed (Supplementary videos 2A-C). Despite interruption from the main artery 52, its distal arteriole branch #9 had undergone outward remodeling (with a transient lower rate at day 13) due to collateral and reversed flow from adjacent arterioles.</p>", "<p id=\"P26\">The main vein segments #1, 3 and 8 remodeled inward at early time points and then outward from day 14 on. The transverse venules #5 and 7 remodeled outward, likely to compensate for the main vein interruption. Interestingly, the distal part of the small venule #10 remodeled outward rapidly to match diameter and re-route flow to the main vein. Its diameter increased by 40% at day 6 to 221% at day 13, 229% at day 14, 306% at day 16 and 343% at day 20. Vessel #10’s outward diameter remodeling peaked at day 23 at 379% increase from normal (close to 400%) and decreased by the end of the observation period at day 28 to 282% of the original diameter at day 23, suggesting a possible transient remodeling (##FIG##4##Figure 5b## and ##FIG##5##Fig. 6##, venous segment #10).</p>", "<title>Computational model simulation</title>", "<p id=\"P27\">We next investigated flow patterns in the network before and after the ablations. To do this, we used a computational approach to estimate flow in each segment. The first step in computational modeling is extraction of the network topology and characterization from bright field images taken with the stereo microscope (##FIG##6##Fig. 7##). The venous network roughly parallels the arterial network with visibly larger diameter vessels. The direction of the flow for each segment was observed from the live BF microscopy recordings and marked on the network map (##FIG##6##Fig. 7a## and ##FIG##6##b##).</p>", "<p id=\"P28\">We then used a simulated annealing method to estimate flow rates and pressures throughout the network (see <xref rid=\"S9\" ref-type=\"sec\">Methods</xref>). Guesses are made for the terminal segment pressures, and the flows are calculated based on topology and measured vessel diameters. The predicted flow direction in each segment is compared to the observed direction, and an error function is calculated based on the number of incorrect directions. The error is used to scale a set of new guesses for the pressures, which is also subjected to a random function (this is the basis for the simulated annealing method). The process is then repeated to minimize the number of incorrect flow directions in individual segments. Using this method, we find that most large vessels have flow that varies little between trials (blue in ##FIG##7##Fig. 8##), but that flow direction in a few vessels (red in ##FIG##7##Fig. 8##) is relatively uncertain - showing a high sensitivity to distant changes in pressure. This suggests that these vessels can readily serve as collaterals that are available to redirect flow in either direction if necessary.</p>", "<p id=\"P29\">First, the flow distribution of individual vessels was optimized based on network topology and flow directions in the normal non-ablated state for vessels with different levels of uncertainty/flow levels (##FIG##7##Fig. 8##). Before ablation, the larger arteries have low uncertainty, suggesting that they rarely change flow direction (##FIG##7##Fig. 8##, blue and yellow color vessels). For example, the vessel fragment in Fig, ##FIG##7##8##, panel C has a low level of uncertainty (indicated by blue color on the vessel map) and the relative values of the volumetric flow rate are mostly around 20% of that in the largest vessel (which is assumed at a value of 1000). The segments with the highest uncertainty mostly carry lower flow and are located near the center of the network (##FIG##7##Fig. 8##, red and orange color vessels). To illustrate this point, vessel fragments in ##FIG##7##Fig. 8##, panels A, B and D have a higher uncertainty (orange and red on the vessel map) and therefore a wider range of possible values. Note that the segments in panels A and B stabilize at zero or close to zero values which reflects a low priority for these collateral vessels prior to ablation.</p>", "<p id=\"P30\">Using this method, we estimated flow through the network before (##FIG##8##Fig. 9 A## and ##FIG##8##C##) and after ablation (##FIG##8##Fig. 9##, ##FIG##8##B## and ##FIG##8##D##) for arteries and veins, respectively. The venous network had more segments with higher flow rate pre-ablation (##FIG##8##Fig. 9##, C vs. A). In arteries, after ablation, flow tends to be reversed in vessels with a high uncertainty index in the pre-ablation model close to the site of ablation (##FIG##8##Fig. 9##). There was no flow reversal in the vein network although the flow magnitude was slightly changed in many vessel fragments.</p>" ]

[ "<title>DISCUSSION</title>", "<p id=\"P31\">In this study, we analyzed flow redistribution and remodeling in vessel networks after laser ablations of vascular segments. Laser ablation cauterizes vessels and causes acute temporary vasoconstriction lasting only a few minutes after ablation. This was observed in the ablated vessels as well as segments downstream and upstream from the ablation location. The diameters of the vasoconstricted vessels returned nearly to control values within a few minutes of ablation. This is direct evidence that the vasoactivity of the ablated vessels and their immediate branches was not altered by the laser ablation. This new approach can be further used to investigate shear-based remodeling in injured or developing networks.</p>", "<p id=\"P32\">The specific diameter changes and patterns of remodeling were observed in detail in five specimens. The present data demonstrated that microvascular remodeling patterns are similar and reproducible but differ in detail from mouse to mouse (sFig. 1 ad Tables 2–6). The comprehensive data presented in sTable 6 not only underscores the diversity of remodeling patterns but also the intricate and adaptive nature of microvascular networks in response to laser ablation, offering valuable insights into their behavior and potential clinical relevance.</p>", "<p id=\"P33\">sFig. 1, in conjunction with sTables 2–6, provides a comprehensive insight into the dynamic behavior of microvascular networks in response to laser ablation. The figures and data within sFig. 1 offer a detailed visual representation of the time course and various remodeling patterns observed across five distinct animal experiments (Mice A-E). These patterns include collateral outward remodeling, reopening of arterial and venous segments, and instances of permanent segment occlusion. The selection of Mouse E as a representative case for in-depth analysis in sFig. 1 serves to illustrate consistent changes seen across all mice while supplying essential anatomical data for subsequent biological and mathematical modeling endeavors. sTable 6 complements this by summarizing the observed remodeling patterns at different time points, highlighting the persistence of both outward and inward remodeling in arterial and venous segments throughout the observation period. Additionally, the findings emphasize the network’s remarkable adaptability, with the ability to achieve permanent occlusion in some vessels while also demonstrating the capacity for gradual reopening over time in others. Together, sFig. 1 and sTable 6 could offer critical insights that have relevance for both experimental investigations and potential clinical applications.</p>", "<p id=\"P34\">The mouse shown on row E in sFig. 1 was found to be representative of the remodeling processes and was used for more detailed analysis. Extensive flow redirection and vascular remodeling occurred after ablation in the arterial side of the vasculature (##FIG##1##Fig. 2##). The pattern of upstream and downstream remodeling of the network vessels is consistent with previous results in the mouse gracilis artery^{##REF##15604133##4##}. Interestingly, veins remodeled more than arteries, but with no observable change in flow direction.</p>", "<p id=\"P35\">The process of vessel regeneration via angiogenesis was consistently observed on the arterial side of the vasculature. There was directed, collective migration of endothelial and smooth muscle cells from the damaged segments that eventually spanned the ablation sites and reconnected the damaged arteries. In one case, the ablated artery segment was in the process of reconnection at the termination of our observations (##FIG##2##Fig 3A##).</p>", "<p id=\"P36\">On the venous side, flow was often rerouted locally around the ablation sites through small postcapillary venules that were not readily observable prior to the ablation, which subsequently increased dramatically in diameter to re-establish the original vein diameter. This rerouting depended on the pressure drop across the ablation site: if the ablation interrupted flow in a large vein, rerouting and subsequent vessel remodeling occurred to compensate for the ablation (sVideos 1A-D, 3A-D). Although some of the small vessels that constituted these new, remodeled pathways could be identified in the pre-ablation images, it is not clear whether angiogenesis was involved in making some of the rerouting connections to the vein. Interestingly, if there was little pressure drop across the ablation site, the re-routing was not apparent. Veins that were ablated at two locations along the length resulted in a downstream interruption with high trans-ablation pressure drop, but also an upstream ablation with low pressure drop (because flow in the vessel had already been blocked by the downstream ablation). In this case, there was re-routing around the downstream ablation, but not around the upstream ablation (sFig. 1A). Conversely, there was no reconnection of the downstream vessel via angiogenesis, but there was for the upstream segment.</p>", "<p id=\"P37\">The study has several notable limitations. First, we applied multiple laser ablations in each network rather than single ablations. Systematic analysis of the effects of individual ablations is challenging because the network in each DSFC is unique, so identifying equivalent vessels in different mice is difficult. In addition, in our experience, ablating only one vessel results in little perturbation in flow over the entire network. For this reason, multiple ablations were more appropriate to analyze flow redistribution. In addition, some aspects of this study focused on one specific case study (Mouse E), and the findings may not be fully generalizable. However, we observed qualitatively similar adaptation processes in Mice A-D (sFig. 1). Such common remodeling patterns included venous and arterial collateral development and segment reopening, and both outward and inward remodeling, with variations in the time frames for each mouse. We should also reiterate that our imaging was not, in general, able to resolve all the vessels in the DSFC, and there were many capillaries and microvessels that were not visible or quantified.</p>", "<p id=\"P38\">Network connectivity plays a major role in flow patterns and vascular remodeling. Flow reversals are unequivocal evidence of redundancy, indicating that a region of tissue can be reached by more than one pathway. Because they can arise from variable demand, they can maintain distal vessels with flow reversal and probably shear stress adaptation (not addressed in the present work). The vessels showing the highest degree of uncertain flow direction (yellow and red segments in ##FIG##7##Figs. 8## and ##FIG##8##9##) seem to be most likely to undergo flow reversal after others are ablated in the arterial network. These segments were also more likely to increase in diameter to accommodate the change in flow. Nonetheless, due probably to the large density of vessels in the network, the vessels with reversed flow did not remodel significantly as observed previously in the gracilis artery of the mouse^{##UREF##0##2##}. Several vessels at the periphery changed direction, but they did not experience a large, visible increase in diameter, which could mean they have changed direction of flow before or that change in the direction in flow by itself does not increase the shear stress and diameter significantly. Regardless, a change in direction of flow was necessary to restore tissue perfusion and maintain pressure in peripheral vessels, which were observed to maintain flow and had little changes in diameter. While full validation of the model awaits more rigorous testing with a larger data set, our present study demonstrates concepts of vascular redundancy and remodeling that are relevant to other microvascular networks with collateral vessels such as the brain, retina, and coronary circulation. Furthermore, the experimental and modeling approaches described here could be used to estimate the remodeling potential of any vascular network under changing blood flow conditions.</p>" ]

[]

[ "<p id=\"P1\">AUTHOR’S CONTRIBUTIONS</p>", "<p id=\"P2\">G.G., J.B., B.J.V., T.P.P. and L.L.M. conceived the experiment(s), G.G., S.M., D.B., L.G.G. and M.O.L. conducted the experiment(s), G.G., J.B., B.J.V., T.P.P. and L.L.M. analyzed the results, L.G.G, B.J.V and L.L.M. secured the funding. All authors reviewed the manuscript.</p>", "<p id=\"P3\">Overly dense microvascular networks are treated by selective reduction of vascular elements. Inappropriate manipulation of microvessels could result in loss of host tissue function or a worsening of the clinical problem. Here, experimental, and computational models were developed to induce blood flow changes via selective artery and vein laser ablation and study the compensatory collateral flow redistribution and vessel diameter remodeling. The microvasculature was imaged non-invasively by bright-field and multi-photon laser microscopy, and Optical Coherence Tomography pre-ablation and up to 30 days post-ablation. A theoretical model of network remodeling was developed to compute blood flow and intravascular pressure and identify vessels most susceptible to changes in flow direction. The skin microvascular remodeling patterns were consistent among the five specimens studied. Significant remodeling occurred at various time points, beginning as early as days 1–3 and continuing beyond day 20. The remodeling patterns included collateral development, venous and arterial reopening, and both outward and inward remodeling, with variations in the time frames for each mouse. In a representative specimen, immediately post-ablation, the average artery and vein diameters increased by 14% and 23%, respectively. At day 20 post-ablation, the maximum increases in arterial and venous diameters were 2.5x and 3.3x, respectively. By day 30, the average artery diameter remained 11% increased whereas the vein diameters returned to near pre-ablation values. Some arteries regenerated across the ablation sites via endothelial cell migration, while veins either reconnected or rerouted flow around the ablation site, likely depending on local pressure driving forces. In the intact network, the theoretical model predicts that the vessels that act as collaterals after flow disruption are those most sensitive to distant changes in pressure. The model results match the post-ablation microvascular remodeling patterns.</p>" ]

[]

[ "<title>ACKNOWLEDGEMENTS</title>", "<p id=\"P47\">Research reported in this publication was supported by the Center for Biomedical OCT Research and Translation through Grant Number P41EB015903, awarded by the National Institute of Biomedical Imaging and Bioengineering of the National Institutes of Health, and the Norwegian Financial Mechanism 2014–2021 under the project RO-NO-2019–0138, 19/2020 “Improving Cancer Diagnostics in Flexible Endoscopy using Artificial Intelligence and Medical Robotics” IDEAR, Contract No. 19/2020.</p>", "<title>DATA AVAILABILITY</title>", "<p id=\"P48\">The datasets and computer code generated during and/or analyzed during the current study are available from the corresponding authors on reasonable request.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1.</label><caption><p id=\"P51\">Vascular laser ablation. A, B: Bright field (BF) microscopy images of the skin vasculature before (A) and at four hours after ablation (B) of specific large vessels. The major and minor artery-vein pairs are indicated by solid and open green arrowheads, respectively. The three regions of ablation are shown with the numbers. Red and blue arrowheads indicate some of the major arcading/collateral vessels on the arterial and venous sides, respectively. The lower panels show multiphoton laser microscopy (MPLM) images of microvasculature before and immediately after laser ablation of each region indicated. Red: smooth muscle cells (aSMA-DsRed); green: endothelial cells (TIE-2-GFP); orange: overlapped SMCs and ECs; bright green: autofluorescence of scar tissue resulting from the ablation procedure. White scale bars are 1mm.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2.</label><caption><p id=\"P52\">Time course of vascular remodeling post-ablation. D0⁻ and D0⁺ indicate pre-ablation and post-ablation on Day 0, respectively. Regions 1–3 indicate the ablation regions and site (yellow line). Shown are images through Day 30 (D30). Initially, at Days 6 vessel redundancy and remodeling in areas compensate for the ablation-induced ischemia. By day 13, the venous connections were reestablished (clear and black arrowheads). From day 20, the artery in Region 1 has reconnected (green arrowhead), increasing flow to the downstream network. There was no angiogenic regeneration of the ablated veins; instead, flow quickly rerouted through small pre-existing venules that appeared to pre-existing connections at either side of the damage site (clear and black arrowheads, D13–30). The white scale bars are 1mm.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3.</label><caption><p id=\"P53\">Vessel regeneration at Day 30. At top is a brightfield image of regions 2 and 3 from ##FIG##0##Figure 1##. Four regions are shown in detail with multiphoton imaging of the endogenous TIE2-GFP (endothelial cells) and aSMA-dsRed (smooth muscle cells). The ablated regions are shown by the circles. In these regions, there was evidence of angiogenesis in the arterial network as endothelial cells (solid arrowheads) and smooth muscle cells (open arrowhead) migrated into the ablated regions. At this time point, the remodeled vein segment in region 3, ##FIG##0##Fig. 1## has matured, with a covering of smooth muscle cells (arrow, C). The scale bar is 1mm.</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4.</label><caption><p id=\"P54\">The frequency distribution of vessel diameter for arteries (top) and veins (bottom) pre and up to 30 days post-ablation. Post-ablation, the distribution of artery diameters is skewed towards more smaller diameter vessels suggesting the blood is redirected from large arteries to smaller alternative pathways. This trend is reversed towards a more normal distribution (more larger vessels) past day 16. On the venous side, the distribution of diameters is more stable reflecting a larger capacity of the venous side to accommodate blood flow redistribution without major diameter changes in most of the vessels.</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5.</label><caption><p id=\"P55\">Blood flow visualized by decorrelation-based quantitative flowmetry OCT before ablation (D0⁻), just after ablation (D0⁺) and on days 2 (D2) and 14 (D14). The three ablation sites are marked with blue circles at D0⁻ (see also ##FIG##1##Figs. 2## D0⁻ and D0⁺). Areas in the blue boxes at D0− and D14 (a, b) appear at bottom at higher magnification. Immediately post-ablation, flow is completely interrupted in the segments just downstream from the ablations and diverted to alternative pathways. The venous connection in left side ablation site (circle 1 in ##FIG##1##Figs. 2## D0⁻ and D0⁺) is reconnected by day 14 while the arterial segment is not reconstructed. The flow is reversed in artery 6 which received blood from the bottom vascular network from day 0 to day 30 when the direction of flow is restored to pre-ablation direction from the large artery segments 2 and 4 towards segment 6 (Supplementary Videos 1–3). Venous segment 10 remodels close to 400% from a venule to a major vein. Smaller post-capillary venules also appear to be involved in this rerouting of flow (arrowheads). By day 14, angiogenesis has partially reconnected the artery in this region, and some flow is evident (arrow, b).</p></caption></fig>", "<fig position=\"float\" id=\"F6\"><label>Figure 6.</label><caption><p id=\"P56\">Time course of diameter changes for the representative vessel segments imaged by OCT (see ##FIG##4##Fig. 5##). The venous connection in area 1 is re-established by day 14 while the arterial segment #2 is not reconstructed. The flow is reversed in artery 6 which received blood from the vessels of the distal network at the bottom region of the ##FIG##0##Figs. 1## and ##FIG##1##2##. Venous segment 10 remodels close to 400% from a precapillary venule to a major vein.</p></caption></fig>", "<fig position=\"float\" id=\"F7\"><label>Figure 7.</label><caption><p id=\"P57\">Vascular network topology and flow patterns. The arterial (<bold>a</bold>) and venous networks (<bold>b</bold>) are traced separately based on intravital images, and digitized versions are extracted. The observed flow directions are indicated by arrows.</p></caption></fig>", "<fig position=\"float\" id=\"F8\"><label>Figure 8.</label><caption><p id=\"P58\">Computational model results of the pre-ablation network. Flow rates have been normalized relative to a value of 1000 assigned to largest vessel segment located on the left side. A-D) The histograms show the frequency of flow rates in representative vessel segments obtained from 100 runs of the simulated annealing algorithm. Numbers in the network map show the average flow rate for each segment, calculated over the 100 runs. The network map is color coded to show the relative uncertainty (standard deviation/mean) of the flow rates in each segment.</p></caption></fig>", "<fig position=\"float\" id=\"F9\"><label>Figure 9.</label><caption><p id=\"P59\">Computational model results of the pre- (a and c) and post-ablation (b and d) networks. Numbers in the network maps indicate flow rate. The arterial network (a and b) had fewer fragments with high flow rate uncertainty than the venous network (c and d). However, flow reversal was common in the arteries but not the veins.</p></caption></fig>" ]

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[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN2\"><p id=\"P49\">COMPETING INTERESTS</p><p id=\"P50\">The author(s) declare no competing interests.</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p id=\"P4\">There has been a long appreciation of the relationship between aging and a loss of mitochondrial function, including bioenergetic, biophysical, and replicative properties ^{##REF##23746838##1##}. In addition to being considered a “hallmark of aging” ^{##REF##23746838##1##}, deficits in mitochondrial homeostasis also directly affect other processes associated with aging including cellular senescence, inflammation, and energetics ^{##REF##35879417##2##–##REF##35048548##4##}. Many longevity interventions have been reported to drive beneficial change to mitochondrial function ^{##REF##27741510##5##–##UREF##0##7##}. Methionine restriction (MR) has been shown to extend longevity and directly impact mitochondrial function ^{##REF##25773352##8##–##REF##16770005##18##}. In rodent models, MR has been established to decrease mitochondrial reactive oxygen species (ROS) generation in different tissues in conjunction with decrease mitochondrial DNA oxidation ^{##REF##25773352##8##,##REF##25765145##14##}, despite increased oxygen consumption ^{##REF##25773352##8##,##REF##26278039##13##–##REF##16770005##18##}. While the beneficial effects of MR, including improved metabolic function and longevity, may be driven partially by its effects on mitochondria, the mechanisms by which these are regulated have yet to be determined and could be developed as targets for novel interventions to improve healthy aging.</p>", "<p id=\"P5\">Our group has explored the potential role that methionine sulfoxide reductase A (MsrA), a regulator of methionine oxidation/reduction (redox), may play in the molecular and physiological effects of MR ^{##REF##35332198##19##,##UREF##3##20##}. The chemical structure of methionine is easily oxidized to methionine sulfoxide which has been shown to modify protein structure and function ^{##REF##9923602##21##–##REF##9512014##25##}. MsrA and other methionine sulfoxide reductases have evolved to repair this oxidative damage ^{##REF##21670260##26##–##REF##14699060##29##} and provide cellular protection from oxidative stress ^{##REF##19487311##30##–##REF##11867705##32##}. There is growing evidence that MsrA may regulate some functional properties of mitochondria ^{##REF##26448611##33##}, though because MsrA is localized in both cytosol and mitochondria it is not clear if these are direct or indirect actions of the enzyme ^{##REF##12693988##34##}. In yeast, lack of MsrA reduces mitochondrial efficiency and increases ROS production suggesting that MsrA is required for normal mitochondrial function ^{##REF##20799725##35##}. Similarly, mitochondria from the muscle of mice lacking MsrA (MsrA KO) have been shown to have bioenergetic deficits under normal conditions ^{##REF##33829213##36##} while studies from an Alzheimer’s Disease mouse model lacking MsrA showed decreased Complex 4 activity and oxygen consumption in brain-derived samples ^{##REF##26786779##37##}. In cell based models, knockdown of MsrA was shown to reduce mitochondrial ATP content and reduce Complex 4 activity while increasing expression of MsrA had the reverse effect ^{##REF##24120970##38##}. Taken together these studies indicate a regulatory or functional role between MsrA and mitochondrial function.</p>", "<p id=\"P6\">Previously we showed that MsrA was not required to mediate the beneficial physiological effects of MR on glucose metabolism and body weight/composition. In these studies, we found that the lack of MsrA in mice manifested greater loss in lean and fat mass with MR, as well as larger improvements to glucose metabolism relative to normal diet compared to that in control mice ^{##REF##35332198##19##}. These results suggest MsrA has an uncharacterized role in mediating or modulating MR. In this study we investigated if MsrA was directly involved in the regulation of mitochondrial function in response to MR. We specifically tested the response of mitochondria in bioenergetics, physiology, and complex composition in multiple tissues from mice under MR and in the presence/absence of MsrA. Overall, our reports point to tissue- and sex-dependent effects of MR on mitochondrial respiration and generation of oxygen radicals, and hints at interactions with MsrA directly. Overall, our studies add to the growing evidence for MsrA in maintaining normal mitochondrial homeostasis and a potential role for this enzyme in mediating the molecular effects of MR in the liver and kidney.</p>" ]

[ "<title>Methods</title>", "<title>Animal Usage and Ethical Procedures</title>", "<p id=\"P7\">All animal experiments were approved by the Institutional Animal Care and Use Committees and UTHSA (Animal Protocol 20170190AR), and have been reported following ARRIVE guidelines. All methods were conducted in accordance with international ethical standards and guidelines.</p>", "<title>Animal use and diets</title>", "<p id=\"P8\">We have described all husbandry conditions, the genetic mutant MsrA KO mouse, and diets used in this study in our previous publications ^{##REF##35332198##19##,##UREF##3##20##,##REF##27821326##39##}. All mice were maintained in the C57Bl/6J background and MR diets used were control diet (CD) (0.86% Met, TestDiet 578F w/0.86% MET – 5SFD) or MR (0.15% Met, 0% Cys, Test Diet 96D2, modified TestDiet 58B0). 7–10 mice were assigned randomly to each group and the age range at time of enrollment was 11.4 to 14.3 months old with average age similar for all groups. Mice were fed each diet for approximately 2 months before the end of the study and were euthanized via CO₂ asphyxiation at which time tissues were collected for isolated mitochondrial studies. Tissue samples collected from an independent group of animals reported previously ^{##REF##35332198##19##} were used for frozen mitochondrial assays and immunoblots.</p>", "<title>Mitochondrial Isolation</title>", "<p id=\"P9\">Fresh liver and kidney were collected and rinsed twice with ice cold Mitochondrial Isolation Buffer (MSHE) (70 mM Sucrose, 210 mM Mannitol, 5 mM HEPES, 1 mM EGTA, pH 7.2 with KOH) to remove as much blood as possible from the samples. The buffer was decanted and the tissue samples were minced in a petri dish on ice with a razor blade until the tissue pieces reached approximately a few millimeters per dimension. The minced tissues were then disrupted by hand with a Dounce homogenizer on ice in 10 ml of MSHE. The disrupted tissue was transferred to a centrifuge tube and spun at 800 g for 10 min at 4°C. The top layer containing fat was aspirated with a pipette, the supernatant was removed with a pipette and strained through synthetic cheese cloth into a new tube. Samples were spun at 8000 g for 10 min at 4°C. The supernatant was decanted and the pellet was carefully resuspended in 25 ml of ice cold MSHE. This sample was centrifuged at 8000 g for 10 min at 4°C. The supernatant was decanted and the pellet of isolated mitochondria was resuspended in 60–200 μl MSHE. Concentration was measured by the Bradford method using a spectrophotometer. Isolated mitochondria were kept on ice and immediately used for oxygen consumption assessment and H₂O₂ generation assay.</p>", "<title>Oxygen Consumption Rate of Isolated Mitochondria</title>", "<p id=\"P10\">The Oxytherm (Hansatech Instruments) reaction chamber containing a Clark electrode was pre-filled with 0.5 ml of assay buffer (MHPM) (120 mM KCl, 10 mM KH₂PO₄, 2 mM MgCl₂, 5 mM HEPES, 1 mM EGTA, pH 7.2 at 37°C) containing 0.3% defatted BSA warmed to 37°C minus the volume of the isolated mitochondrial sample. The isolated mitochondria were diluted 1:1 with the MHPM buffer containing 0.3% defatted BSA to a final concentration of 5–10 μg/μl. The Oxytherm chamber was allowed to equilibrate with the MHPM buffer and 0.5 μM Rotenone with the chamber closed. Then 75 μg of the diluted isolated mitochondria were added resulting in a final chamber volume of 500 μl which was then allowed to equilibrate for 2–5 min. After equilibration, succinate was added to 5 mM (5 μl of 500 mM stock) to drive State 2 respiration and once curve linearity was established (2–3 min) ADP was added to 0.3 mM (10 μl of 15 mM stock) to drive State 3 respiration (2–3 min). Results were graphed and maximum slope was determined for State 2 and State 3 by best-fit over a 20 second time period. RCR was calculated as the ratio of State 3 to State 2. Samples in which no response was shown to ADP were interpreted as being hyper-permeabilized and were censored from the dataset.</p>", "<title>Hydrogen Peroxide (H₂O₂) Production of isolated mitochondria</title>", "<p id=\"P11\">Reaction buffer master mix was prepared in MHPM with 0.3% defatted BSA containing saturating amounts of superoxide dismutase (SOD; 35–50 U/ml), 5 U/ml horseradish peroxidase (HRP), and 50 μM Amplex UltraRed (A36006, Invitrogen). Isolated mitochondria were added to a final concentration of 0.15 μg/ml and distributed in technical replicate 200 μl reactions in a clear, flat-bottom, black-walled 96-well plate (30 μg of isolated mitochondria per well). 2 μl of fresh 10 mM PMSF in 200 proof ethanol was added to each well (0.1 mM final concentration) and fluorescence was measured at 540 nm excitation/590 nm emission every minute in a 37°C plate reader (Molecular Devices SpectraMax M2) for 10 min to establish baseline. The plate was removed and 2 μl of 500 mM succinate was added (5 mM final concentration). Wells were mixed by pipetting and the plate was returned to the plate reader for an additional 10–11 min of reads at 1 min intervals. The plate was again removed and 2 μl of 400 uM rotenone was added (4 μM final concentration). Wells were mixed by pipetting and the plate was returned to the plate reader for an additional 10–11 min of reads at 1 min intervals. Results were graphed and instantaneous slopes measured for the first 10 data points (10 min) of each segment of each run and corrected for baseline H₂O₂ production rate. Instantaneous rate was the average of the slopes for pairs of neighboring points which was more resistant to skewing by aberrant data points with tended to result in significant changes to best-fit calculations given the small number of data points.</p>", "<title>Mitochondrial Oxygen Consumption from Frozen Tissue</title>", "<p id=\"P12\">Sample prep and assays were performed using the methods of by Osto, et al.^{##UREF##4##40##} Briefly, ~ 50 mg of frozen tissues were homogenized at 10:1 ratio (buffer to tissue) in 1x MAS buffer (70 mM Sucrose, 220 mM Mannitol, 5 mM KH₂PO₄, 5 mM MgCl₂, 1 mM EGTA, 2 mM HEPES, pH 7.4 using KOH) using a TissueLyser Lite (Quiagen) at 35 Hz for 70 sec. The homogenate was then centrifuged at 200 g for 5 min at 4 C and supernatant was collected. Total protein concentration was measured via BCA assay (Pierce). Samples were diluted in 1x MAS buffer to 1.0 μg/μl for use in assays. XF96 Seahorse Extracellular Flux plates (Seahorse XFe96/XF Pro FluxPac, 103792–100, Agilent) were prepared and calibrated as per manufacturer’s instructions. The plate was equilibrated and measured at 28°C. 20 μl of the 1.0 μg/μl liver homogenate were added to the tissue plate (20 μg total protein) and allowed to incubate on ice for 52 min while the injection ports were calibrated. The tissue plate loaded with the mitochondria was then centrifuged at 2000 × g for 5 min at 4 C without centrifuge breaking enabled (25 min total cycle time). 130 μl of ice cold 1x MAS buffer with 10 μg/ml Cytochrome C was added to each well. The tissue plate was then tested with the protocol indicated by Osto, et al.^{##UREF##4##40##}</p>", "<title>Western Blot</title>", "<p id=\"P13\">Approximately 50 mg of frozen tissue were homogenized in ~ 500 μl RIPA (10 μl RIPA: 1 mg tissue) with protease and phosphatase inhibitors (Pierce). Tissue samples were homogenized using a TissueLyserII (Quiagen) for 1 min at 30 Hz for liver. Supernatant after centrifugation was collected and protein concentration measured using Pierce BCA assay kit (Bio-Rad). Blotting was performed using Criterion TGX gels (Bio-Rad) followed by transfer to PVDF membranes (Bio-Rad). Total protein was measured with Ponceau S (Sigma) staining imaged with a Perfection V39 flatbed scanner (Epson). Membranes were blocked with 10% non-fat dry milk in TBST. Membranes were incubated with primary antibody overnight at 4°C with agitation. The primary antibody was prepared in TBST with 2%BSA and 0.01% sodium azide. Antibody used: Total OxPhos/MitoProfile (MitoSciences, MS604/AbCam, ab110413). Membranes were washed with TBST and incubated with HRP secondary (Santa Cruz Biotechnology) for 1hr at room temperature. Membranes were then washed with TBST and developed with Pierce ECL Plus Western Blotting Substrate (ThermoFisher). Membranes were imaged on a Typhoon FLA 7000 (Amersham). All quantifications were performed in ImageStudioLite (Li-Cor). Quantifications were normalized to total protein. A standard sample was used for normalization and comparison between membranes.</p>", "<title>Statistics</title>", "<p id=\"P14\">Area Under the Curve was calculated by the trapezoid method in Excel. Best-fit (Oxytherm) and Instantaneous Rate (H₂O₂ Production) were calculated in Excel. Analyses were performed in Prism8. Results were analyzed by 2-Way ANOVA with Sidac multiple comparison corrections within each sex. Post-hoc testing was done to assess diet effect. 3-Way ANOVAs were completed to assess sex-effects without post-hoc testing.</p>" ]

[ "<title>Results</title>", "<p id=\"P15\">For isolated mitochondria studies, liver and kidney were processed immediately following collection from MR-treated mice and controls. Oxygen consumption rate (OCR) measured in isolated kidney mitochondria indicated that State 2 respiration (respiration in the presence of substrate but not ADP) was significantly reduced by MR in females, but there was no significant effect in males. The lack of MsrA was associated with higher respiration compared to controls in males regardless of diet while this effect was not present in females (##FIG##0##Fig. 1A##). In contrast, State 3 respiration (respiration in the presence of substrate and ADP) was unaffected by MR, but was significantly affected by lack of MsrA in disparate ways in males and females. State 3 OCR was decreased in females but increased in males (##FIG##0##Fig. 1B##). Respiratory control ratio (RCR), the ratio of State 3 to State 2 respiration, was increased by MR in females only. Additionally, lack of MsrA was associated with decreased RCR in females. RCR in males was unaffected by MR or MsrA status (##FIG##0##Fig. 1C##).</p>", "<p id=\"P16\">We assessed these kidney mitochondria for H₂O₂ production since previous studies have shown significant reduction of mitochondrial reactive oxygen species (ROS) production with MR in various tissues ^{##REF##25773352##8##,##REF##25765145##14##–##REF##16770005##18##}. Surprisingly, in our kidney-derived samples, MR was associated with increased H₂O₂ production in the presence of succinate as Complex 2 substrate in both sexes with 3-Way ANOVA indicating a sex-effect in which females had greater production than males (##FIG##1##Fig. 2A##). The addition of rotenone to prevent H₂O₂ generation via electron back-feeding from Complex 2 to Complex 1 showed a large decrease in H₂O₂ production rate in both sexes and loss of the MR main effect (##FIG##1##Fig. 2B##). Despite this drop in H₂O₂ production, females still had higher H₂O₂ production as indicated by the 3-Way ANOVA results. In females the wild type mice showed a decrease in H₂O₂ production in response to MR while the MsrA KO did not receive any effect.</p>", "<p id=\"P17\">Contrary to those from the kidney, mitochondria isolated from the liver show a significant decrease in succinate-stimulated H₂O₂ production in MR-treated females, though MR had no effect in males. For both sexes, the lack of MsrA is associated with increased H₂O₂ production relative to wild type mice. 3-Way ANOVA indicated a sex-effect with females having higher H₂O₂ production than males (##FIG##2##Fig. 3A##). The addition of rotenone decreased H₂O₂ production, however females still had higher H₂O₂ production as indicated by the 3-Way ANOVA. Under these conditions, MR was associated with decreased H₂O₂ production in the females while there was no effect in the males. Additionally, MsrA status has no effect in females or males (##FIG##2##Fig. 3B##).</p>", "<p id=\"P18\">To begin to address potential molecular mechanisms for these differences in mitochondrial physiology among sex, diet, and genotype, we first addressed whether there was evidence for global changes in the expression of the electron transport chain complexes. In tissue lysates from liver and kidney, we found no significant effect of sex, genotype, or MR on expression of Complex 1, 3, and 5, with the exception of increased Complex 3 expression in the kidney of male wild-type mice on MR (##FIG##3##Fig. 4##, ##FIG##4##Fig. 5##).</p>", "<p id=\"P19\">Using a method to measure mitochondrial respiration from whole tissue homogenates generated from frozen samples ^{##UREF##4##40##}, we addressed specific mitochondrial complex activity using samples generated from frozen liver homogenates (##FIG##5##Fig. 6##, ##FIG##6##Fig. 7##). In samples from males, there was largely no effect of MsrA or MR on Complex 1/2 or Complex 2 activity aligning largely with our results from isolated mitochondria. OCR of Complex 1/2 in females showed minimal effects of MR or MsrA with the exception of block of Complex 3 with Antimycin A (AA) which resulted in an interaction effect (##FIG##5##Fig. 6##). In females, we found that MR increased OCR of Complex 2 substrates with MsrA having no effect no outcome. Blocking of Complex 3 with Antimycin A (AA) largely eliminated these differences, suggesting that the differences in OCR with substrate was driven by the level of coupling between the complexes. Testing of Complex 4 with TMPD/Ascorbic Acid (TMPD/AscA) showed a substrate dependent effect in the females in which MR and MsrA KO both caused increased OCR, suggesting either greater Complex 4 expression or more efficient function. Complex 5 inhibition with azide showed substrate dependent effects, but was highly variable. Within-sex groups were generally similar with the exception of the MsrA KO group on MR (##FIG##6##Fig. 7##). The exact mechanism for this is unclear. It is also unclear why the female OCR was significantly higher than the males in the liver. ECAR was also determined for all runs and with the exception of some diet effects in the wild-type control males no other effects were observed (Supplemental Fig. 1, Supplemental Fig. 2).</p>" ]

[ "<title>Discussion</title>", "<p id=\"P20\">Previous studies have shown MR to have a variety of impacts on mitochondrial function ^{##REF##25773352##8##,##REF##26278039##13##–##REF##16770005##18##,##REF##25273919##41##,##REF##31472257##42##}. Here we expand on these findings and point to sex- and tissue-specific differences in mitochondrial response to MR as well as potential roles for methionine oxidation repair via MsrA in this process. Moreover, we demonstrate that different tissues within the same donor respond differently to MR. In particular, we show that MR increased H₂O₂ production from mitochondria isolated from the kidney and decreased this output in mitochondria isolated from the liver (##FIG##1##Fig. 2##, ##FIG##2##Fig. 3##). The mechanisms and consequences for this disparity are still unknown, but important to consider when considering the potential for interventions like MR in benefiting health.</p>", "<p id=\"P21\">In this study, we found that males tended to have blunted response to MR than females in terms of mitochondrial function. This is interesting in light of our previous studies on physiological changes in response to MR which showed that males tended to benefit more in terms of physiological measures and glucose homeostasis ^{##REF##35332198##19##}. In this study the mice also responded similar to treatment in terms of body weight and food consumption (Supplemental Fig. 3). These findings raise interesting questions regarding the mechanistic link between improved metabolism at the organismal level with MR and mitochondrial function. However, further exploration of other tissues important for glucose regulation, including muscle, adipose, and pancreatic β-cells would likely be required to better address this question. It is also important to note that many of these experiments were performed with succinate as a substrate and that different substrate/inhibitor combinations may change these outcomes.</p>", "<p id=\"P22\">Overall our data show that MR tends to have mixed effects on mitochondrial H₂O₂ production that are dependent on both sex and, to some degree, the presence of MsrA. MR had a pronounced effect on H₂O₂ production in the kidney with it being increased in the presence of succinate, but this increase was lost in the presence of rotenone (##FIG##1##Fig. 2##). This is unexpected since MR generally decreases or has no effect on H₂O₂ production although this has been demonstrated in tissues other than the kidney ^{##REF##25773352##8##,##REF##25765145##14##–##REF##18283555##16##,##REF##16770005##18##,##REF##31472257##42##}. The liver results were more in line with other published results with MR decreasing H₂O₂ production however this was only significantly changed in the females (##FIG##2##Fig. 3##). While MsrA largely had little effect on mitochondrial function overall, we did show some discreet effects of this enzyme on both mitochondrial function and the effect of MR. Interestingly, in kidney the absence of MsrA blocked the effect of MR on H₂O₂ production. With MsrA localized in the mitochondria endogenously ^{##REF##12693988##34##}, the enzyme could play an important role in the maintaining the mitochondrial proteome homeostasis ^{##REF##18946048##43##}. While more work is required, this raises an intriguing possibility that at least some mitochondrial physiology regulated by MR requires either reduction of methionine oxidative damage or regulation of methionine redox properties, both of which are driven by MsrA.</p>", "<p id=\"P23\">Comparisons between sexes showed some interesting dichotomies that warrant further exploration. In both kidney and liver, mainly females showed changes in respiration in response to MR as shown by increased OCR in the presence of complex 2 substrate for the liver, but decreased OCR for the kidney (##FIG##5##Fig. 6##, ##FIG##6##Fig. 7##). The genotype effect present in the kidney OCR results was not as prevalent in the liver. It is also important to note that the female liver had significantly higher OCR than the males, regardless of the substrate tested – Complex 1/2 versus Complex 2. The reason for this large difference in OCR between the sexes is unclear. Data in the literature has shown that OCR can be different between sexes in various tissues, including the liver, however not to this degree ^{##REF##17118599##44##,##REF##28325789##45##}. Mitochondrial complex expression was shown to be largely unchanged in either liver or kidney (##FIG##3##Fig. 4##, ##FIG##4##Fig. 5##). These results are in line with existing literature indicating generally no change, although there are some exceptions ^{##REF##26278039##13##–##REF##16770005##18##,##REF##25273919##41##,##REF##20871132##46##}. It may be that there are differences in supercomplex/respirasome formation that could account for this difference but would require additional investigation.</p>" ]

[]

[ "<p id=\"P1\">Author Contributions</p>", "<p id=\"P2\">Study was formulated by ABS and KMT. Data was collected and analyzed by KMT and RC. Figures and manuscript drafts were prepared by KMT. ABS, RC, and KMT reviewed, edited, and approved the manuscript.</p>", "<p id=\"P3\">Methionine restriction (MR) has been shown to affect mitochondrial function including altering oxygen consumption, reactive oxygen species (ROS) generation, Complex expression, and oxidative damage. The sulfur-containing amino acid methionine can become oxidized forming methionine sulfoxide which can lead to changes in protein function and signaling. Methionine sulfoxide reductases are endogenous enzymes capable of reducing methionine sulfoxide, with Methionine sulfoxide reductase A (MsrA) being ubiquitously expressed in mammals. Here we investigated if the effects of MR on mitochondrial function required functional MsrA in the liver and kidney which are the major tissues involved in sulfur biochemistry and both highly express MsrA. Moreover, MsrA is endogenously found in the mitochondria thereby providing potential mechanisms linking diet to mitochondrial phenotype. We found sex-specific changes in oxygen consumption of isolated mitochondria and females showed changes with MR in a tissue-dependent manner – increased in liver and decreased in kidney. Loss of MsrA increased or decreased oxygen consumption depending on the tissue and which portion of the electron transport chain was being tested. In general, males had few changes in either tissue regardless of MR or MsrA status. Hydrogen peroxide production was increased in the kidney with MR regardless of sex or MsrA status. However, in the liver, production was increased by MR in females and only slightly higher with loss of MsrA in both sexes. Mitochondrial Complex expression was found to be largely unchanged in either tissue suggesting these effects are driven by regulatory mechanisms and not by changes in expression. Together these results suggest that sex and MsrA status do impact the mitochondrial effects of MR in a tissue-specific manner.</p>" ]

[]

[ "<title>Acknowledgements</title>", "<p id=\"P24\">We would also like to acknowledge the experimental assistance of Yuhong Liu throughout this study, and Dr. Jonathan Dorigatti and Dr. Kennedy Mdaki for experimental assistance and helpful discussions. This research was funded in part by R01 AG050797, R01 AG057431, T32 AG021890 and the San Antonio Area Foundation. ABS is partially supported by the Geriatric Research, Education and Clinical Center of the South Texas Veterans Health Care System. The funding sources played no role in the study design, data collection, analysis or interpretation of the data, or the writing of the manuscript. This material is the result of work supported with resources and the use of facilities at South Texas Veterans Health Care System, San Antonio, Texas. The contents do not represent the views of the U.S. Department of Veterans Affairs or the United States Government.</p>", "<title>Data Availability</title>", "<p id=\"P25\">The data presented in the work are available from the corresponding author upon request.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><title>Kidney Isolated Mitochondria Oxygen Consumption Rate</title><p id=\"P29\">Oxygen Consumption Rate (OCR) was measured via Oxytherm Clark electrode. Isolated mitochondria were allowed to stabilize oxygen consumption in the presence of rotenone. Succinate was used as substrate for State 2 respiration (<bold>A</bold>), followed by the addition of ADP for State 3 respiration (<bold>B</bold>). Respiratory Control Ratio (RCR) was calculated as the ratio of State 3 to State 2. Analysis was completed within each sex via 2-Way ANOVA for main effects with post-hoc analysis performed with Sidac multiple comparisons correction to assess the diet effect within each genotype. 3-Way ANOVA was performed to assess sex-effect. Graphs represent mean ± SD. Group sizes were 4–7. (* p &lt; 0.05; ** p &lt; 0.01; *** p &lt;0.001)</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><title>Kidney Isolated Mitochondria H₂O₂ Production</title><p id=\"P30\">H₂O₂ production of kidney isolated mitochondria measured by AmplexRed Ultra fluorescence assay with succinate as substrate (<bold>A</bold>) followed by the addition of rotenone (<bold>B</bold>). Analysis was completed within each sex via 2-Way ANOVA for main effects with post-hoc analysis performed with Sidac multiple comparisons correction to assess the diet effect within each genotype. 3-Way ANOVA was performed to assess sex-effect. Graphs represent mean ± SD. Group sizes were 7–10. (* p &lt; 0.05; ** p &lt; 0.01)</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3</label><caption><title>Liver Isolated Mitochondria H₂O₂ Production</title><p id=\"P31\">H₂O₂ production of kidney isolated mitochondria measured by AmplexRed Ultra fluorescence assay with succinate as substrate (<bold>A</bold>) followed by the addition of rotenone (<bold>B</bold>). Analysis was completed within each sex via 2-Way ANOVA for main effects with post-hoc analysis performed with Sidac multiple comparisons correction to assess the diet effect within each genotype. 3-Way ANOVA was performed to assess sex-effect. Graphs represent mean ± SD. Group sizes were 7–10. (* p &lt; 0.05; ** p &lt; 0.01; *** p &lt; 0.001)</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4</label><caption><title>Kidney Mitochondrial Complex Expression</title><p id=\"P32\">Samples from cohort 2 were used to measure the expression of Mitochondrial Complexes in liver tissue homogenates via Western blot. Complex 1 (<bold>A</bold>), Complex 3 (<bold>B</bold>), and Complex 5 (<bold>C</bold>) were measured. Sexes were analyzed separately via 2-Way ANOVA. A 3-Way ANOVA was performed to compare between sexes for main effects. Graphs represent mean ± SD, group sizes were 7–10 mice.</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5</label><caption><title>Liver Mitochondrial Complex Expression</title><p id=\"P33\">Samples from cohort 2 were used to measure the expression of Mitochondrial Complexes in kidney tissue homogenates via Western blot. Complex 1 (<bold>A</bold>), Complex 3 (<bold>B</bold>), and Complex 5 (<bold>C</bold>) were measured. Sexes were analyzed separately via 2-Way ANOVA. A 3-Way ANOVA was performed to compare between sexes for main effects. Graphs represent mean ± SD, group sizes were 7–10 mice.</p></caption></fig>", "<fig position=\"float\" id=\"F6\"><label>Figure 6</label><caption><title>Liver Isolated Mitochondria Oxygen Consumption Rate, Complex 1/Complex 2</title><p id=\"P34\">Liver isolated mitochondria from frozen tissue were measured for Oxygen Consumption Rate (OCR) via 96-well Seahorse. 10μg of isolated mitochondria were used in MAS buffer in the presence of 10μg/ml Cytochrome C. Injections were as follows: Complex 1/Complex 2 OCR using NADH as substrate (C1/C2), Complex 3 inhibition with Antimycin A (AA), Complex 4 OCR with N,N,N’,N’-tetramethyl-<italic toggle=\"yes\">p</italic>-phenylenediamine and Ascorbic Acid (TMPD/AscA), Complex 4 Inhibition with Sodium Azide (Azide). Area Under the Curve (AUC) was calculated via the trapezoid method for each segment. Analysis was completed within each sex via 2-Way ANOVA for main effects with post-hoc analysis performed with Sidac multiple comparisons correction to assess the diet effect within each genotype. Graphs represent mean ± SD. Group sizes were 5. (* p &lt; 0.05; ** p &lt; 0.01)</p></caption></fig>", "<fig position=\"float\" id=\"F7\"><label>Figure 7</label><caption><title>Liver Isolated Mitochondria Oxygen Consumption Rate, Complex 2</title><p id=\"P35\">Liver isolated mitochondria from frozen tissue were measured for Oxygen Consumption Rate (OCR) via 96-well Seahorse. 10μg of isolated mitochondria were used in MAS buffer in the presence of 10μg/ml Cytochrome C. Injections were as follows: Complex 2 OCR using Succinate and Rotenone as substrate and Complex 1 inhibitor (C2), Complex 3 inhibition with Antimycin A (AA), Complex 4 OCR with N,N,N’,N’-tetramethyl-<italic toggle=\"yes\">p</italic>-phenylenediamine and Ascorbic Acid (TMPD/AscA), Complex 4 Inhibition with Sodium Azide (Azide). Area Under the Curve (AUC) was calculated via the trapezoid method for each segment. Analysis was completed within each sex via 2-Way ANOVA for main effects with post-hoc analysis performed with Sidac multiple comparisons correction to assess the diet effect within each genotype. Graphs represent mean ± SD. Group sizes were 5. (* p &lt; 0.05; ** p &lt; 0.01)</p></caption></fig>" ]

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[ "<fn-group><fn id=\"FN2\"><p id=\"P26\">Declarations</p><p id=\"P27\">Competing Interests</p><p id=\"P28\">The authors declare no competing interests.</p></fn></fn-group>" ]

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[ "<title>Background</title>", "<p id=\"P7\">Promoting sustained engagement in antiretroviral therapy (ART) services is a major focus of HIV programs throughout sub-Saharan Africa [##UREF##0##1##]. New or returning ART clients (defined as those on ART &lt; 3 months) are at increased risk of treatment interruption [##REF##33505567##2##–##REF##24648320##6##]. The first few months after (re)engagement represent a critical period. With targeted interventions, access-related barriers can be reduced, and clients can develop the problem-solving skills and external support needed to maintain ongoing engagement in care [##REF##33125587##7##, ##REF##34048443##8##].</p>", "<p id=\"P8\">Differentiated service delivery models (DSDs) are a leading strategy for easing access to lifelong services for stable clients (defined as virally suppressed and/or 6 + months post-ART initiation) [##UREF##0##1##]. Examples include multimonth dispensing, community-based ART delivery, and tailored ongoing counseling and peer support. DSD models have a range of benefits, including increased access to care [##REF##34234023##9##], greater acceptability among clients [##REF##34234023##9##, ##REF##32015007##10##], and improved or noninferior retention and viral suppression [##REF##36443853##11##–##UREF##2##14##]. However, new and returning ART clients have largely been excluded from DSD models [##REF##33247517##15##], even though they may benefit most from convenient, private, and lower-cost services [##REF##28770588##16##]. Several DSD models, including incentives [##REF##32891234##17##, ##UREF##3##18##], peer support [##UREF##4##19##, ##REF##32780187##20##], community-based ART initiation [##REF##36827327##13##, ##UREF##2##14##], and e-health interventions [##REF##27118443##21##–##REF##32314120##23##], have been tested among this population in study settings but are only now being considered for scale and national policy [##UREF##0##1##].</p>", "<p id=\"P9\">Stakeholder input can improve intervention design, translation, adoption, and scale-up of successful strategies. However, stakeholder perceptions are not systematically explored, and multilevel stakeholders are often not consulted [##UREF##6##24##–##UREF##7##27##]. A deeper understanding of stakeholders’ priorities and constraints can also help researchers offer decision-relevant information, such as data on expected costs and outcomes for different implementation and scale-up scenarios [##UREF##8##28##, ##REF##30635001##29##].</p>", "<p id=\"P10\">To our knowledge, no published literature on stakeholder perceptions of DSD models for new or returning ART clients in sub-Saharan Africa exists. Despite its highly resource-constrained health system, Malawi has been the vanguard of innovative public health strategies in the region (such as Option B + and HIV self-testing), making it an ideal study setting. [##UREF##9##30##–##REF##31981557##33##].</p>", "<p id=\"P11\">We qualitatively explored stakeholder perceptions of DSD models for new or returning ART clients and decision-making criteria for scaling DSD models and compared the views of internationally based and Malawi-based stakeholders.</p>" ]

[ "<title>Methods</title>", "<title>Study Design</title>", "<p id=\"P12\">We conducted in-depth interviews with internationally based stakeholders (from foundations, multilateral organizations, and NGOs) and Malawi-based stakeholders (from the Ministry of Health and local implementing partners) to explore perspectives and priorities regarding interventions targeting new or returning ART clients. The study was embedded in the Identifying Efficient Linkage Strategies for Men (IDEaL) randomized controlled trial, which aimed to develop and test the impact of male-tailored differentiated models of care on men’s ART initiation, reinitiation, and early retention in Malawi. The trial is described elsewhere [##UREF##10##34##].</p>", "<title>Theoretical Framework</title>", "<p id=\"P13\">We used the Assessing Cost-Effectiveness (ACE) approach as our theoretical framework for data collection and analysis [##UREF##11##35##]. The ACE is a structured method for conducting health policy priority-setting studies. It combines rigorous economic evaluation with a qualitative assessment of other implementation factors that influence policy adoption. Stakeholders provide guidance and feedback at every stage of the study process [##UREF##11##35##].</p>", "<title>Population and Recruitment</title>", "<p id=\"P14\">We aimed to recruit policy and programmatic stakeholders who were experts on DSDs and/or who were willing to provide care for new or returning ART clients. We sampled purposively to represent different organizations, including international foundations, multilateral organizations, nongovernmental organizations (NGOs), the Malawi Ministry of Health (MoH), and local implementing partners within Malawi. Although clients and community-based organizations are important stakeholders, they were not included in the sample, as this study focused on policy and guideline decision-making.</p>", "<p id=\"P15\">We identified twenty-nine potential participants through internet searches and the research team’s professional networks. We also included eight additional contacts through snowball sampling. We made two direct outreach attempts to each stakeholder by email. We continued recruitment until we reached saturation of the themes were saturated within in-depth interviews. We intentionally attempted to sample similar numbers of individuals from the international and national stakeholder categories to facilitate understanding of the similarities and differences between the two groups.</p>", "<title>Data collection and analysis</title>", "<p id=\"P16\">We developed an interview guide based on the ACE framework as well as the literature on interventions to improve engagement in care among new or returning ART clients in sub-Saharan Africa [##REF##33865471##36##, ##REF##37132119##37##]. The interview guide included open-ended questions about stakeholders’ HIV-related priorities, perceptions of factors influencing noninitiation or attrition immediately following initiation, challenges and solutions for financing and implementing relevant interventions, and desired data to inform decision-making.</p>", "<p id=\"P17\">Stakeholders also completed two interactive tasks. First, they rated five scale-up decision-making criteria commonly used in the Assessing Cost-Effectiveness (ACE) approach (cost, effectiveness, equity, feasibility, acceptability) as “lower,” “moderate” or “high” priority. We encouraged them to rate no more than three criteria as high priority and at least one as lower priority. We asked stakeholders to verbalize their thoughts as they completed the task, following the “think-aloud” method,” for qualitative research [##UREF##12##38##]. After the task was complete, we offered participants the option to rate any additional decision-making criteria and engaged them in retrospective reflection about the activity and the reasons for their ratings.</p>", "<p id=\"P18\">In accordance with the same procedure as in the previous task, the stakeholders rated the relative priority of seven ART initiation and early retention interventions. Options included monetary incentives, nonmonetary incentives, community-based care, ongoing peer/mentor support and counseling, eHealth, facility changes, multimonth dispensing, and an “other” category. To construct an average rating for each intervention, we assigned each high-priority rating ten points, each moderate-priority rating 5 points, and each lower-priority rating 0 points [##REF##22339981##39##].</p>", "<p id=\"P19\">We conducted two pilot interviews to refine and finalize the data collection tool. Two researchers conducted each interview in English via video conference. We recorded the interviews and transcribed them verbatim.</p>", "<p id=\"P20\">We developed a codebook using a priori codes informed by the literature and the theoretical framework [##UREF##11##35##], as well as inductive codes based on emergent themes [##UREF##13##40##]. Using Atlas.ti v9 [##UREF##14##41##], two researchers (KP and KH) piloted the codebook with six interviews. Four researchers (KD, SH, KH, and KP) reviewed the coded transcripts, discussed discrepancies, and refined the codebook. One researcher (KH) coded the remaining data, and other researchers performed spot checks on half of the transcripts to ensure consistency. We extracted coded texts and performed thematic content analysis [##UREF##15##42##]. In the present analysis, we compared and explored differences in themes between international and national stakeholders. We prioritized themes mentioned by many participants and explored divergent views.</p>" ]

[ "<title>Results</title>", "<title>Participant characteristics</title>", "<p id=\"P21\">We conducted 22 interviews between October 2021 and March 2022—thirteen with internationally based stakeholders and nine with nationally based stakeholders (##TAB##0##Table 1##). Participants had varied levels of experience with HIV service policies, guidelines, and practice, ranging from 2–21 years of working in the HIV field.</p>", "<title>Decision-making criteria</title>", "<p id=\"P22\">Overall, intervention effectiveness and acceptability were the highest-priority decision-making criteria (##FIG##0##Fig. 1##). Approximately half of the stakeholders explained that the decision-making criteria are interrelated, making it difficult to fully distinguish them. For example, a feasible intervention would likely also be low-cost, and an equitable intervention may have greater long-term effectiveness.</p>", "<p id=\"P23\">Effectiveness was a high priority for nearly all international stakeholders. International stakeholders frequently described effectiveness as their first consideration: “There’s no use in implementing something that will not help us reach the expected outcome.” Fewer than half of international stakeholders rated feasibility as a high priority. They explained that implementation challenges can be overcome through innovation, change management, and quality improvement: “Even if an intervention doesn’t seem feasible initially, if you can continually create evidence that... it’s effective, then systems can be built to figure out how that [can] work.”</p>", "<p id=\"P24\">International stakeholders were less concerned about the costs of interventions. They believed that long-term costs are minimal after initial investments and noted that interventions and programs can become more efficient over time: “Just because [something] is expensive today, we might figure out a way to do it more cheaply later”. Some noted that interventions can even be cost-saving in the long run: “In the long-term, [care] might be less costly if you do not need to be... spending that much time with people who are coming with advanced disease.”</p>", "<p id=\"P25\">In contrast, national stakeholders acknowledged the importance of effectiveness but believed feasibility, cost, and sustainability were more important. National stakeholders emphasized the fit of interventions within existing systems and human resource and equipment constraints:</p>", "<p id=\"P26\">If [an intervention] is not feasible, it will not come close to success...it needs to be straightforward and fit well into the rest of the programs and activities…and healthcare workers’ capacities and capabilities. (National Stakeholder)</p>", "<p id=\"P27\">For a long time, as you know, the whole HIV program like in Malawi is donor dependent, donor-driven… after the project mode has been phased out, it’s very difficult for us as a country to maintain it. So…[it] is necessary to make sure that we have sustainable interventions that can be implemented without donor support. (National Stakeholder)</p>", "<p id=\"P28\">Nearly all the national stakeholders were deeply concerned about the ongoing costs of interventions, citing the expectation that reductions in donor funding are imminent. As one national stakeholder summarized,</p>", "<p id=\"P29\">It is easy to find funding for innovations, but it is very difficult to find funding to sustain services. We know that every year the funding for HIV programming goes down... If we’re talking about introducing a certain innovation, are we able to sustain this? When [a donor] is no longer there, this intervention...will die a natural death. (National Stakeholder)</p>", "<p id=\"P30\">Both international and national stakeholders considered acceptability, especially client acceptability, to be a high priority. As one international stakeholder summarized, “If clients do not like it, then it is just going to fail, so it does not matter how cheap and easy it is to do.” Healthcare worker acceptability was also considered important due to the pervasiveness of healthcare worker burnout and the difficulty of implementing programs without their buy-in:</p>", "<p id=\"P31\">I think some things might not be acceptable to healthcare workers but be super acceptable to people living with HIV or vice versa…but I would advocate that maybe if health care workers don’t love [an intervention] but it is great for communities and people living with HIV, then we’ve got to try [it]. (International Stakeholder)</p>", "<p id=\"P32\">Some stakeholders discussed the tradeoff between equity and efficiency, explaining that reaching key populations and serving rural areas may be more expensive but still worthwhile. A couple of national stakeholders prioritized efficiency over equity:</p>", "<p id=\"P33\">At the end of the day [it comes down to] the number of people it impacts because that will make it more cost-effective. Yes, I may want to ensure equity by having those extended clinic hours, but I’m only seeing eight men. And will eight men improve overall retention for [the] country? (National Stakeholder)</p>", "<title>Perspectives on Interventions for New or Returning ART Clients</title>", "<p id=\"P34\">Overall, international and national stakeholders had similar priorities for DSD interventions (##FIG##1##Fig. 2##). They noted that multiple models are needed to address the full range of factors causing treatment interruption among new or returning ART clients. International stakeholders were more interested in ongoing peer support/counseling than were national stakeholders. The highest-rated interventions were facility changes followed by multimonth dispensing (MMD) (##TAB##1##Table 2##).</p>", "<title>Facility Efficiencies</title>", "<p id=\"P35\">Most international and national stakeholders prioritized facility changes (such as extending service hours and reducing wait times) because they saw these changes as feasible within existing systems and infrastructure. They saw healthcare facilities as the foundation of HIV service delivery and central to client experiences of care: “We already have the facilities in place, but what we have not done is try to look at things like the client flow, opening hours. This could be easily done.” (National Stakeholder)</p>", "<p id=\"P36\">A couple of stakeholders also supported the idea of differentiating facility-based care so that more intensive services are offered to clients with the greatest needs:</p>", "<p id=\"P37\">We already have workflows that are stripped to the bare bones. At most sites, patient consultations take only a few minutes and are not conducted by a trained health worker. There is no room to simplify any further, so the question is what we can add for patients who need it rather than what we can simplify. (National Stakeholder)</p>", "<title>Multimonth Dispensing (MMD)</title>", "<p id=\"P38\">MMD was also prioritized by both international and national stakeholders because it is feasible, acceptable, low-cost, and frees up facility resources to support higher-risk clients. Several stakeholders emphasized that new or returning ART clients may require several visits before being given MMD to ensure that they are adequately prepared for lifelong treatment; however, many international stakeholders believe that MMD should be offered as early as possible:</p>", "<p id=\"P39\">I assume that people look at the first six months of ART and dread the amount of work it involves…the less they have to do, the less onerous it will seem, and we hope that will make [HIV treatment] feel like something they want to do and want to stick with. (International stakeholder)</p>", "<title>Peer Support and Counseling</title>", "<p id=\"P40\">International stakeholders prioritized peer support highly, emphasizing its effectiveness. In contrast, national stakeholders focused on the high cost and human resource requirements of these programs, though they did support a new MoH program in which a facility-based counselor provides support to clients living with depression, anxiety, and substance use disorders.</p>", "<p id=\"P41\">Stakeholders believed that peer support must be provided in addition to interventions addressing service accessibility: “We use peers a lot… However, if you just make it easier to access services, then you will not need all of this additional support” (International stakeholder)</p>", "<title>Community-Based Care</title>", "<p id=\"P42\">Both groups rated community-delivered ART as a moderate priority. Perceived benefits included reduced client time and cost for accessing services and a reduced risk of unwanted status disclosure at facilities. Despite these benefits, both international and national stakeholders were concerned about the human resources needed to implement community services. For this reason, several believed that community services should be limited to key populations or those who are especially ill:</p>", "<p id=\"P43\">It is very difficult to imagine how this will be done at scale in the national program. We have to remember that this takes health workers away from the facilities [and] it takes many more nurses to see the same number of clients if you send them out in the community. So I think in a health system that is overall understaffed, it is a great luxury to send nurses to meet people in the community. (National Stakeholder)</p>", "<title>E-Health Strategies</title>", "<p id=\"P44\">The majority of stakeholders rated e-Health interventions as a moderate priority. Most stakeholders described e-Health as a new, “untapped” platform that could increase the reach of existing interventions, such as peer support, health education, appointment reminders, SMS check-ins between visits, and tracing clients with missed visits. However, several believed that the technology and evidence were not yet sufficient for widespread implementation in Malawi.</p>", "<p id=\"P45\">Stakeholders discussed both equity concerns and benefits related to e-health interventions. More than half of the stakeholders discussed uneven access to phones and the internet as critical barriers to effective and equitable e-health interventions, particularly in resource-constrained settings such as rural Malawi. National stakeholders noted that individuals often share phones with relatives, creating a risk of unwanted disclosure. However, several stakeholders noted e-Health’s unique potential to connect with harder-to-reach populations, such as youth, men, and mobile and rural populations.</p>", "<title>Incentives</title>", "<p id=\"P46\">Monetary and nonmonetary incentives were the lowest-rated interventions. Stakeholders perceived incentives as highly effective but dismissed the possibility of implementing them because of their cost and complexity. Stakeholders also believed that incentives may reduce clients’ intrinsic motivation for lifelong treatment:</p>", "<p id=\"P47\">If you tell them, ‘we can give you transport money,’ and then we don’t have it; the next time, they do not show up. However, if at the beginning you did not tell them that [they would] receive cash, they would find their own means to come to the clinic. [Because of incentives], they develop a dependency syndrome. (National Stakeholder)</p>", "<p id=\"P48\">Stakeholders noted that targeted incentives can improve equity by supporting poorer clients in meeting daily needs and overcoming socioeconomic barriers to care but were concerned that other clients would find targeted incentives unfair. Some have suggested that free or low-cost nonmonetary incentives could instead be used to reward or appreciate clients for their engagement in care.</p>", "<title>Overarching Service-Delivery Priority: Person-Centered Care</title>", "<p id=\"P49\">Person-centered care (PCC) organically emerged as a high priority for both national and international stakeholders. Stakeholders described PCC as flexible and tailored services that respond to clients’ holistic needs. The three components of PPC discussed most frequently were 1) segmented care, 2) integrated care, and 3) positive and empowering interactions with healthcare workers.</p>", "<title>Segmented Care</title>", "<p id=\"P50\">The majority of both international and national stakeholders expressed the view that services should be adapted to meet the needs of diverse groups, such as clients returning to care vs. stable clients, men, key populations, clients who prefer private services, and clients with psychosocial needs: “...it’s not a one-size-fits-all approach... what works for female sex workers may not necessarily work for MSM” (international stakeholder).</p>", "<p id=\"P51\">A few international stakeholders believed that segmented services may reduce health system costs by limiting costlier interventions to the clients who would benefit from them the most. However, one national stakeholder emphasized that tailoring services for different populations is an “extra project” that is not government funded and requires additional training and staff time.</p>", "<title>Integrated Services</title>", "<p id=\"P52\">Integrating HIV services with other healthcare was a priority for both international and national stakeholders, although several national stakeholders believed that integration would be costly and infeasible. Stakeholders described the holistic benefits of integrated care, including reduced costs for clients (because visits are combined), reduced risk of unwanted disclosure of one’s HIV status, and improved overall health outcomes:</p>", "<p id=\"P53\">They come to the clinic, they get their ART refilled, and tomorrow they’re supposed to go to the clinic to get their diabetes medication or hypertension medication, that to me is a bad idea. [If we had a] one-stop center where you get your ART refills and your other medications, I think that would help. (National Stakeholder)</p>", "<title>Positive and Empowering Interactions with Healthcare Workers</title>", "<p id=\"P54\">Positive client-healthcare worker interactions were considered key to retaining clients in care and reducing HIV-related stigma. As one national stakeholder stated, interactions with healthcare workers should be “empowering and supportive” rather than “coercive and threatening.” Stakeholders believe that counseling sessions should not be “generic” but rather tailored to clients’ concerns and designed to foster trust so that clients can feel comfortable discussing their barriers to treatment:</p>", "<p id=\"P55\">What I’ve seen from my own experience is that most providers do not understand that we are all human beings. We have other things that we do apart from coming to the clinic to get medication. So sometimes when a client misses an appointment and comes for a refill, he is treated like he is being punished. [If we] change the attitudes of our providers, I do not think we are going to struggle with retention or even initiation. (National Stakeholder)</p>", "<p id=\"P56\">Approximately one-quarter of international and national stakeholders noted that the lack of support and education clients receive when initiating ART leads to treatment interruptions:</p>", "<p id=\"P57\">If there is no time to discuss [treatment barriers] and to encourage disclosing such issues they will of course drop out because they had problems that haven’t been addressed. And I think that can only be addressed if there is … more time for [clients] to fully understand [their diagnosis] and [problem-solve] with treatment supporters and family members. (National Stakeholder)</p>", "<p id=\"P58\">Nearly all international stakeholders and a couple of national stakeholders emphasized the importance of empowering clients by giving them choices in how they receive care, such as community or facility-based care, peer support, and the choice of different facilities. A few stakeholders noted that client needs change over time, so explicitly asking about client preferences on an ongoing basis can help identify and meet evolving needs:</p>", "<p id=\"P59\">[If] as a client I make a choice…whether it is home-based care or whether it is multimonth dispensing, that’s the one that is acceptable for me and therefore I’m likely to adhere to that intervention and have improved linkage and early retention. (National Stakeholder)</p>" ]

[ "<title>Discussion</title>", "<p id=\"P60\">In resource-constrained settings, stakeholders make difficult trade-offs across multiple criteria (such as effectiveness, equity, budgetary and practical constraints, and political considerations) when deciding which interventions to implement and scale up. In this study, we explored how stakeholders make these decisions in the context of interventions for new or returning ART clients in Malawi. Our study suggests that both national and international stakeholders prioritize client acceptability but diverge in other areas: program effectiveness was a higher priority for international stakeholders, while ongoing costs, feasibility, and sustainability were higher priorities for national stakeholders. Despite these differences, international and national stakeholders had similar intervention preferences; they prioritized simple, low-cost, facility-based interventions that remove barriers to care, such as multimonth dispensing and extended facility hours. Most stakeholders attributed their interest in various interventions to PCC, whereby clients are provided tailored services with positive and empowering healthcare worker interactions.</p>", "<p id=\"P61\">Our analysis suggests notable differences in how international and national stakeholders perceive the long-term costs of interventions. International stakeholders viewed interventions as having low ongoing costs after initial investments. In contrast, national stakeholders were deeply concerned about the long-term costs and resource requirements of interventions. National stakeholders described their experiences watching promising new programs end after donors left. International stakeholders believed that factors such as cost and feasibility could be addressed through implementation strategies, a view that was not expressed by national stakeholders.</p>", "<p id=\"P62\">These findings highlight differences in experiences and perceptions of both global and local history and context. There is growing consensus that external funding should be aligned with national priorities [##UREF##16##43##, ##REF##35067229##44##]. However, transitioning from preferred narratives to actual practice may be slow. Historically, key donor funding institutions have driven the process of intervention selection and initiatives have been managed by numerous NGOs (some locally and some internationally based) rather than by national governments [##UREF##17##45##]. International donor funds predominantly support vertical programs rather than infrastructure or health system strengthening efforts. [##UREF##18##46##] A locally driven decision-making process may start with a range of intervention options selected by the Ministry of Health and community stakeholders that are then decided upon jointly. This approach would help ensure that international and national stakeholders are working together to fund programs that are country owned, sustainable, and coordinated.</p>", "<p id=\"P63\">Despite differences in their decision-making criteria, international and national stakeholders had similar intervention priorities, perhaps because of their ongoing discussions. Interestingly, cost and feasibility seemed to have the greatest influence on intervention preferences, although effectiveness and acceptability received the highest ratings in the think-aloud priority-setting tasks in this study. Like in a previous qualitative study [##REF##35190943##47##], high-level stakeholders favored simple interventions with minimal costs that removed structural barriers to care (e.g., extended hours, MMD) over those that were perceived as highly effective but needed additional systems and human resources (e.g., peer support and community-based care).</p>", "<p id=\"P64\">Stakeholders universally agreed on the importance of PCC, despite their concerns about resource constraints and sustainability challenges. In line with findings from previous PCC studies [##REF##31795741##48##], stakeholders were more interested in the impact of PCC on health system goals (e.g., retention in care, reducing costs) than on client goals (e.g., living a full life). They viewed PCC as a strategy to reduce healthcare costs by improving the effectiveness and efficiency of programs and recognized the critical importance of client choice and tailored, respectful, holistic care in improving retention and adherence. Additional evidence on the impact of PCC and best practices in sub-Saharan Africa is urgently needed. PCC practices were developed in high-income countries, but little quantitative evidence on PCC in LMICs exists [##UREF##19##49##]. Several aspects of PCC (such as segmented and integrated care) may require additional resources to be successfully implemented and sustained in historically vertical, disease-specific programs with scarce resources. In such contexts, innovative, low-cost strategies for offering tailored and/or integrated services may be needed. However, recent qualitative research in Malawi suggests that returning male ART clients value positive and empowering relationships with healthcare workers more than where and how ART is delivered, suggesting that key components of PCC could be taken to scale at low cost [##UREF##20##50##].</p>", "<p id=\"P65\">Our study has several limitations. First, the think-aloud priority-setting tasks did not fully mimic real-world situations in which stakeholders consider many nuances of a particular context. Second, there may have been social desirability bias or differences between expressed and revealed preferences. We believe this was minimal, as interviewers expressed neutrality and encouragement toward all comments and conversations were frank and casual. Third, not all stakeholders were represented in this study; critical stakeholders such as clients and community advocacy groups were not included due to the study’s focus on policy and guideline decision-making. Fourth, the quantitative results of the two ranking tasks should be interpreted with caution due to the small sample size. Fifth, some findings may not be generalizable beyond Malawi. Despite these limitations, we believe that the priorities and preferences expressed by stakeholders in this study are reflected in real-world settings.</p>" ]

[ "<title>Conclusions</title>", "<p id=\"P66\">We found that the top priorities of international and Malawi national stakeholders regarding DSD interventions for new or returning ART clients are effectiveness, feasibility, sustainability, and client acceptability. International stakeholders should recognize and act upon the greater priority that national stakeholders place on feasibility and sustainability. Person-centered care was emphasized by all stakeholders and should be incorporated into any intervention for new or returning ART clients. The findings can inform HIV treatment intervention development and research. Further research is needed to understand how differing priorities affect public health discussions, decision-making, and impact and how to ensure that national and local needs are prioritized.</p>" ]

[ "<p id=\"P1\">Authors’ contributions</p>", "<p id=\"P2\">KD and SH conceptualized and designed the study. SH, KP, and MT conducted qualitative data collection. KP coded the qualitative data. KD, SH, KH, and KP reviewed the coded transcripts, discussed discrepancies, and refined the codebook. SH conducted the preliminary data analysis and drafted the initial manuscript. All the authors reviewed, edited, and approved the final manuscript.</p>", "<title>Background</title>", "<p id=\"P3\">New or returning ART clients are often ineligible for differentiated service delivery (DSD) models, though they are at increased risk of treatment interruption and may benefit greatly from flexible care models. Stakeholder support may limit progress on development and scale-up of interventions for this population. We qualitatively explored stakeholder perceptions of and decision-making criteria regarding DSD models for new or returning ART clients in Malawi.</p>", "<title>Methods</title>", "<p id=\"P4\">We conducted in-depth interviews with internationally based stakeholders (from foundations, multilateral organizations, and NGOs) and Malawi-based stakeholders (from the Malawi Ministry of Health and PEPFAR implementing partners). The interviews included two think-aloud scenarios in which participants rated and described their perceptions of 1) the relative importance of five criteria (cost, effectiveness, acceptability, feasibility, and equity) in determining which interventions to implement for new or returning ART clients and 2) their relative interest in seven potential interventions (monetary incentives, nonmonetary incentives, community-based care, ongoing peer/mentor support and counseling, eHealth, facility-based interventions, and multimonth dispensing) for the same population. The interviews were completed in English via video conference and were audio-recorded. Transcriptions were coded using ATLAS.ti version 9. We examined the data using thematic content analysis and explored differences between international and national stakeholders.</p>", "<title>Results</title>", "<p id=\"P5\">We interviewed twenty-two stakeholders between October 2021 and March 2022. Thirteen were based internationally, and nine were based in Malawi. Both groups prioritized client acceptability but diverged on other criteria: international stakeholders prioritized effectiveness, and Malawi-based stakeholders prioritized cost, feasibility, and sustainability. Both stakeholder groups were most interested in facility-based DSD models, such as multimonth dispensing and extended facility hours. Nearly all the stakeholders described person-centered care as a critical focus for any DSD model implemented.</p>", "<title>Conclusions</title>", "<p id=\"P6\">National and international stakeholders support DSD models for new or returning ART clients. Client acceptability and long-term sustainability should be prioritized to address the concerns of nationally based stakeholders. Future studies should explore the reasons for differences in national and international stakeholders’ priorities and how to ensure that local perspectives are incorporated into funding and programmatic decisions.</p>" ]

[]

[ "<title>Acknowledgements</title>", "<p id=\"P67\">We acknowledge and thank the individuals who took the time to participate in the study. Their honest and generous engagement made this study possible.</p>", "<title>Funding</title>", "<p id=\"P68\">This work was supported by the Bill and Melinda Gates Foundation (INV-001423) and the National Institute of Mental Health (R01-MH122308). KD and MT were supported by the Fogarty International Center (K01-TW011484–01). MT was also supported by NIMH (T32MH080634) and the UC Global Health Institute (D43TW009343).</p>", "<title>Data Availability Statement</title>", "<p id=\"P69\">The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><p id=\"P78\">Ratings of decision-making criteria by national and international stakeholders</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><p id=\"P79\">DSD intervention ratings by national and international stakeholders</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\"><label>Table 1</label><caption><p id=\"P80\">Participant characteristics</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"top\" span=\"1\"/><col align=\"left\" valign=\"top\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Characteristic</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">N (%) N = 22</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Location</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">International</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">13 (59%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">National</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9 (41%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Gender</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Male</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">15 (68%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Female</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">7 (32%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"># of years in current role (Range)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5 (1–20)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mean # of years working in the HIV field (Range)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">14 (2–21)</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T2\"><label>Table 2</label><caption><p id=\"P81\">Stakeholder perceptions of interventions</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"top\" span=\"1\"/><col align=\"left\" valign=\"top\" span=\"1\"/><col align=\"left\" valign=\"top\" span=\"1\"/><col align=\"left\" valign=\"top\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Intervention</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Average Rating^{<xref rid=\"TFN1\" ref-type=\"table-fn\">†</xref>}</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Positive Perceptions</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Negative Perceptions</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Facility efficiencies (e.g., extended hours, improving workflows)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9.0</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Acceptable (clients)<break/>Feasible<break/>Low-cost</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Multimonth dispensing</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">7.9</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Acceptable (clients, policy-makers)<break/>Feasible<break/>Low-cost</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Community-based care</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">7.5</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Acceptable (clients) Equitable</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">High-cost<break/>Infeasible (complexity + HR requirements)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Ongoing peer support</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">7.4</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Acceptable (clients) Effective</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">High-cost<break/>Infeasible (complexity + HR requirements)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">e-Health</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6.6</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Effective (high future potential)<break/>Low-cost</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Inequitable (access to phones and the Internet)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Incentives</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3.3<break/>(nonmonetary)<break/>0.2 (monetary)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Acceptable (clients)<break/>Effective (short- term)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">High-cost<break/>Ineffective (long-term)<break/>Infeasible (determining and tracking eligibility)</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN2\"><p id=\"P70\">Competing interests</p><p id=\"P71\">The authors declare that they have no competing interests.</p></fn><fn fn-type=\"COI-statement\" id=\"FN3\"><p id=\"P72\"><bold>Additional Declarations:</bold> No competing interests reported.</p></fn><fn id=\"FN4\"><p id=\"P73\">Declarations</p><p id=\"P74\">Ethical approval and consent to participate</p><p id=\"P75\">The study was approved by the National Health Science Research Committee (NHSRC) in Malawi and by the University of California–Los Angeles Institutional Review Board. All participants gave oral consent before completing an interview. No identifiable information was collected.</p></fn><fn id=\"FN5\"><p id=\"P76\">Consent for publication</p><p id=\"P77\">Not Applicable</p></fn></fn-group>", "<table-wrap-foot><fn id=\"TFN1\"><label>†</label><p id=\"P82\">Rating scale: High Priority = 10 pts, Moderate Priority = 5 pts, Low Priority = 0 pts</p></fn></table-wrap-foot>" ]

[ "<graphic xlink:href=\"nihpp-rs3725505v1-f0001\" position=\"float\"/>", "<graphic xlink:href=\"nihpp-rs3725505v1-f0002\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"P3\">Adipose tissue is comprised of two major cell types- white and brown adipocytes-that have opposing energetic functions. White adipocytes primarily function to store excess lipid and an over-abundance of white adipose tissue (WAT) leads to unfavorable outcomes due to disruption in many metabolic and hormonal activities. In contrast, brown adipocytes consume glucose and lipid and brown adipose tissue (BAT) dissipates energy through uncoupled oxidative phosphorylation-induced heat generation, a process necessary for thermoregulation in small animals and infants. Interestingly, in response to various environmental cues, including following cold exposure, a subset of adipocytes within WAT undergo browning and these so-called “beige” fat cells exhibit many of the beneficial metabolic features of classical brown adipocytes including dissipation of energy through thermogenesis and the secretion of endocrine factors that promote whole body metabolism. As obesity results from a disparity between energy intake and expenditure, the identification of new molecules and signals that can drive fat browning may lead to attractive targets for the treatment of obesity and related cardiometabolic disorders.</p>", "<p id=\"P4\">WAT and BAT are located in discrete depots that are distributed throughout the body and are derived from a diverse group of progenitor cells. While we do not yet know the precise origins of cells within each of these tissues, current lineage tracing studies indicate that Myf5-expressing progenitor cells that originate in the dermomyotome give rise to greater than 90% of inter- and sub-scapular BAT (iBAT and sBAT) which are the largest BAT depots in mice^{##REF##24942009##1##, ##REF##24898859##2##}. Other adipose depots likely arise from a more heterogeneous population of progenitors. For example, the contribution of Myf5 progenitors to WAT varies tremendously (from 5 to 60%) based on anatomic locale and sex^{##REF##24942009##1##}. The mural cell compartment of the vasculature also appears to be a major contributing source of adipogenic precursors in various WAT compartments.</p>", "<p id=\"P5\">Despite their heterogeneous origins, white and brown adipogenic precursors share some common transcriptional programs. For example, peroxisome proliferator–activated receptor gamma (PPARγ) is a master regulator of adipocyte differentiation and this transcription factor drives expression of a number of genes common to all adipocytes including adiponectin and fatty acid binding protein 4 (FABP4) ^{##REF##23652116##3##, ##REF##8631877##4##}. Moreover, classical brown adipocytes that originate from the dermomyotome and beige adipocytes that are intermingled in WAT depots share PGC1α- and PRDM16- dependent transcriptional programs that drive mitochondrial biogenesis and induce a common subset of BAT-selective genes. These include, among others, Cidea which promotes lipolysis and lipogenesis and uncoupling protein 1 (UCP-1) which is responsible for facilitating thermogenesis through uncoupled respiration ^{##UREF##0##5##}.</p>", "<p id=\"P6\">Importantly, both white and brown adipocytes undergo dynamic and reversible phenotypic conversions in response to environmental cues. For example, thermoneutrality or exposure to high fat diet can induce BAT to undergo reversible WAT-like remodeling. Conversely, long-term cold exposure, agonist-induced activation of the β3 adrenergic pathway, or physical activity can promote WAT to undergo browning. The beneficial physical activity-induced phenotypic changes are mediated, at least in part, by skeletal-muscle derived factors including fibroblast growth factors(FGFs) and bone morphogenic proteins (BMPs) ^{##REF##29052591##6##, ##REF##23805974##7##}.</p>", "<p id=\"P7\">BMP 4, 6, and 7 are particularly strong inducers of brown and beige adipogenesis and, while we do not fully understand the underlying signaling mechanisms, downregulation of RhoA-mediated signaling has been shown to be important ^{##REF##24284793##8##–##REF##24658703##12##}. For example, down-regulation of RhoA in white adipocytes promotes beige cell development and heterozygous germline depletion of the RhoA-dependent kinase, ROCK2, or treatment with Rho kinase inhibitors promoted WAT browning and led to protection from diet-induced obesity and insulin resistance in mice ^{##REF##31914704##13##}. Interestingly, the dynamic regulation of RhoA is also important to control the fate of mesenchymal stem cells and multi-potent progenitors wherein low RhoA activity favors pre-adipocyte specification while high RhoA activity promotes specification towards osteoblast, smooth- or skeletal muscle cell fates ^{##REF##32678016##14##}. Mechanistically, RhoA/ROCK signaling promotes these cell fate conversions by modulating both actin cytoskeleton-dependent shape changes and altering gene transcription (RhoA activity inhibits the pro-adipogenic transcription factor, PPARγ- and activates myogenic SRF/MRTFA- mediated gene transcription) ^{##REF##25579684##15##, ##REF##28125644##16##}.</p>", "<p id=\"P8\">RhoA, like all small GTPases, cycles between an inactive GDP-bound form and an active GTP-bound form and its activity is enhanced by guanine nucleotide exchange factors (GEFs) and inhibited by GTPase-activating proteins (GAPs). To date, two RhoA-specific GAPs have been identified as critical regulators of adipogenesis, p190B and DLC1. P190B limits RhoA activity in myogenic precursors and is necessary for these cells to adopt a pre-adipocyte fate^{##REF##27251423##17##–##REF##23395168##19##}. On the other hand, DLC1 limits RhoA activity in pre-adipocytes and functions to promote white and brown adipogenesis ^{##REF##28358928##20##}. However, a GAP that selectively controls adipocyte browning has not yet been identified. Herein, we show that the multidomain containing RhoA-GAP termed GRAF1 (guanosine triphosphatase (GTPase) regulator associated with FAK-1, also named Arhgap26) ^{##REF##9858476##21##, ##REF##9525907##22##}that was previously reported to be strongly expressed in highly metabolic tissues (including heart, brain, and skeletal muscle) is also strongly expressed in BAT and functions to selectively promote the activation of BAT and the browning of WAT. Our findings highlight the possibility that GRAF1 could be a new therapeutic target to combat obesity and associated morbidity.</p>" ]

[ "<title>Methods</title>", "<title>Animals</title>", "<p id=\"P22\">GRAF1 gene trap mice were generated and obtained from the Texas A&amp;M Institute for Genomic Medicine (College Station, TX) and were described previously ^{##REF##25019370##28##}. ES cells carrying Arhgap26 targeted knockout first conditional-ready alleles were obtained from EuMMCR. Germline transmission of the allele was confirmed from F0 chimeric mice with C57BL/6 genetic background. Next, we crossed Flp recombinase transgenic mice with F1 to generate GRAF1 floxed mice. Finally, we established adipose specific GRAF1 knockout mouse line (GRAF1^AKO) by crossing Adiponectin Cre line (010803, Jackson Lab ^{##REF##21356515##36##}) with GRAF1 floxed mice. All mice were housed in pathogen-free facilities under a 12-hour light/dark cycle with unrestricted access to food and water. Animals were treated in accordance with the approved protocol of the University of North Carolina (Chapel Hill, NC) Institutional Animal Care and Use Committee, which is in compliance with the standards outlined in the guide for the Care and Use of Laboratory Animals. All methods are reported in accordance with ARRIVE (Animal Research: Reporting of In Vivo Experiments) guidelines.</p>", "<p id=\"P23\">To induce GRAF1 knockout <italic toggle=\"yes\">in vivo</italic>, 80mg of Tamoxifen power (Sigma, T5648) was dissolved in 750ul 100% ethanol(molecular biology grade) (ThermoFisher Scientific, T038181000) for 40 mins at 40°C, then was mixed with 3.25 ml filtered corn oil(Sigma, C8267) to make final concentration of 20mg/ml. Tamoxifen was administered at 100mg/kg via oral gavage once every 24 hours for a total of 5 consecutive days. 1 week later, mice were subject to experiment treatment. To induce the browning program of white fat depots, control and Adcre-GRAF1ko (GRAF1^AKO) mice received daily intraperitoneal injections of β3-adrenergic agonist CL316243 at 1mg/kg (Sigma, C5976)for 10 days as described previously ^{##REF##25578880##48##}.</p>", "<title>Cold exposure</title>", "<p id=\"P24\">To assess sensitivity to cold exposure, we monitored rectal temperature at hourly intervals following the placement of mice in a cold room maintained at 6–8 °C. The mice underwent an overnight fast, which continued throughout the duration of the cold exposure. Subsequent to the cold exposure period, mice were humanely euthanized by CO2 inhalation.</p>", "<title>Cell culture and differentiation</title>", "<p id=\"P25\">Human Simpson-Golabi-Behmel syndrome (SGBS) preadipocyte line was provided by Dr. Martin Wabitsch (Ulm University Medical Center, Germany). The cells were maintained in DMEM/F12 (ThermoFisher Scientific, 11330–032) supplemented with 10% fetal calf serum(FCS), biotin 3.3 mM, pantothenate 1.7 mM and antibiotics. When cells were 90–100% confluence, SGBS cells were induced in brown adipocyte differentiation medium for 4 days (DMEM/F12, 3.3 mM biotin, 1.7 mM pantothenate, transferrin 10 μg/ml, 430 nMol insulin, 1nMol T3, 1μMol dexamethasone, 0.5mMol IBMX, 2μMol Rosiglitazone and antibiotics). Then cell medium was changed to brown adipocyte maintenance medium every 4 days for 8 days (DMEM/F12, 3.3 mM biotin, 1.7 mM pantothenate, transferrin 10 μg/ml, 430 nMol insulin, 1nMol T3 and antibiotics).</p>", "<title>Isolation and differentiation of Stromal Vascular Fraction (SVF) Cells</title>", "<p id=\"P26\">Subcutaneous WAT (inguinal WAT) or interscapular BAT tissues were used for Beige cell source, and followed a previously described isolation and differentiation method^{##UREF##3##49##}. In brief, tissues were digested and centrifuged to get SVF cells. SVF cells were cultured to 100% confluence in complete medium (DMEM/F12 containing 10% FPS and P/S). SVF cells were cultured for 2 days in induction medium (complete medium plus 5μg/ml insulin, 1 nM T3, 125 μM Indomethacin, 2 μg/ml Dexamethasone, 0.5 mM IBMX, 0.5 μM Rosiglitazone). Then cell medium was changed to maintenance medium (complete medium plus 5μg/ml insulin, 1 nM T3) with 0.5 μM rosiglitazone for 2 days. At Day 4, cell medium was changed to maintenance medium with 1 μM rosiglitazone for 2 days.</p>", "<p id=\"P27\">WT-1 cells were provided by Dr. Yu-Hua Tseng (Joslin Diabetes Center, Harvard Medical School Affiliate, Boston, MA). Cells were cultured and induced differentiation with the same medium as SVF cells.</p>", "<title>SiRNA treatment</title>", "<p id=\"P28\">Cells were plated at 1.2 × 105/ 2 ml/well in 6 well plate. Next day, cells were transfected GRAF1 targeting siRNA or GFP targeting siRNA (final concentration 10 nM) by Lipofectamine^® RNAiMAX(ThermoFisher Scientific, 13778075) following the manufacturer’s instructions for 72 hours. GRAF1 Stealth RNAi^™ siRNA was synthesized by ThermoFisher with the following sequences: 5’-CGGAAGUUUGCAGAUUCCUUAAAUG-3’ and 5’-CAUUUAAGGAAUCUGCAAACUUCCG-3’. GFP stealth siRNA was used as Control with the following sequence: 5’-GGUGCGCUCCUGGACGUAGCC[dT][dT]-3’ 5’-GGCUACGUCCAGGAGCGCACC[dT][dT]-3’.</p>", "<title>Real time PCR analysis</title>", "<p id=\"P29\">Total RNA was isolated from homogenized whole mouse tissues or cell cultures using RNeasy Mini Kit (Qiagen, 74106) according to manufacturer’s instructions. After homogenizing, samples were placed on ice to remove lipid layer. Complimentary DNA (cDNA) was obtained from 1 μg of RNA isolated using the iScript cDNA Synthesis Kit (Bio-Rad, 1708897), and cDNA was used for qPCR with iTaq Universal SYBR Green Supermix kit (Bio-Rad, 1725124). The relative gene expression levels were calculated using delta-delta Ct method, also known as the 2– ΔΔCt method. Primer sequences were in Supplemental Table 1.</p>", "<title>RhoA activity assay</title>", "<p id=\"P30\">WT-1 cells were serum starved overnight prior to treatment with S1P for indicated times. Rho activity was measured by GST-Rhotekin pulldown assay as previously described ^{##REF##24335996##50##}. Cell lysates were rotated with 40 μg of a GST-rhotekin Rho binding domain fusion protein immobilized to glutathione-Sepharose 4B beads (Cytiva, 17075601) for 15 min at 4°C in binding buffer (50 mM Tris, pH 7.6, 500 mM NaCl, 0.1% SDS, 0.5% deoxycholate, 1% Triton X-100, 0.5 mM MgCl2). Beads were precipitated and washed three times (50 mM Tris, pH 7.6, 150 mM NaCl, 1% Triton X-100, 0.5 mM MgCl2) and resuspended in 2 × Laemmli buffer. Proteins were separated on a 15% SDS–PAGE and transferred to 0.2 μm PVDF membrane (Bio-Rad, 1620177). After blocking in 5% bovine serum albumin/TBST (20 mM Tris–HCl, 500 mM NaCl, 0.05% Tween-20, pH 7.4) for 1 hour at room temperature, blots were probed with 2 μg/ml anti-RhoA (26C4) (Santa Cruz Biotechnology, SC418) overnight at 4°C. Loading controls (typically 10%) were taken from each lysate sample prior to pull downs.</p>", "<title>Western blotting</title>", "<p id=\"P31\">To examine protein levels, lysates from cells or tissues were prepared by lysing in a modified RIPA buffer with 1x HALT phosphatase &amp; protease inhibitor cocktail(ThermoFisher Scientific, 78438 and 78427). Protein concentration was determined by using a colorimetric BCA assay (Pierce, 23227). Lysates were electrophoresed on SDS–polyacrylamide gel, transferred to nitrocellulose and immunoblotted with specific antibodies overnight at 4°C as indicated using a 1:1000 dilution. The following primary antibodies were used in western blot: GAPDH(cell signaling technology, 5174S), b-actin(cell signaling technology, 3700S), α-Tubulin(Sigma, T6199), RhoA(Santa Cruz, SC-418). Rabbit anti-GRAF1 polyclonal antibody is homemade antibody in our lab. Blots were washed in TBST (TBS plus 0.1%Tween20) followed by incubation with horseradish peroxidase conjugated antibody at a 1/1,000 dilution. Blots were visualized after incubation with chemiluminescence reagents (ThermoFisher Scientific, 32106).</p>", "<title>Oil red O staining and quantification</title>", "<p id=\"P32\">Cells were rinsed with PBS and then fixed in 4% PFA for 30 minutes. Then cells were left in the air until completely dry. Oil Red-O working solution (0.3%) was freshly prepared and stained cells on the shaker for 10 minutes. Then cells were rinsed with PBS 3 times and used for imaging. After imaging, liquid was removed from cells completely. To elute the oil red O dye, 100% isopropanol was added to the plates. The plates were incubated for 10 min at room temperature on an orbital shaker. Absorption was measured at 518 nm on a plate reader.</p>", "<title>Statistics</title>", "<p id=\"P33\">Unless stated otherwise, all data represented at least three individual experiments and presented as means ± standard error of the mean (SEM). Means of normally distributed data were compared by two-tailed Student’s t-test, one-way ANOVA (followed by Tukey’s post-hoc correction) or linear regression where indicated and statistical significance was reported as p-values. A p-value &lt; 0.05 was considered significant. Sample sizes were chosen based on an extensive literature search and standard exclusion criterion of two standard deviations from the mean were applied.</p>" ]

[ "<title>Results</title>", "<title>GRAF1 expression correlates with BAT maturation in mouse and human cells.</title>", "<p id=\"P9\">We previously reported that GRAF1 was transiently upregulated during skeletal muscle development (E17-P4) and again following adult muscle injury and that, in these contexts, GRAF1 promoted myoblast differentiation by limiting RhoA activity^{##REF##21622574##23##}. Interestingly, while dissecting various muscles from adult GRAF1 hypomorphic (GRAF1^gt/gt) mice, we noticed that their iBAT depots were relatively pale compared to WT controls. Since myocytes and BAT share common Myf5-expressing precursors^{##REF##24942009##1##, ##REF##18719582##18##, ##REF##22940198##24##, ##REF##18367555##25##} and both cell types require RhoA inactivation for differentiation and maturation^{##REF##12705864##26##},we further investigated a possible role for GRAF1 in BAT development.</p>", "<p id=\"P10\">In mice, iBAT and sBAT depots develop during late embryogenesis, and, upon expression profiling in juvenile mice (3 weeks postnatal), we found that GRAF1 was highly expressed in these major BAT depots. GRAF1 was also expressed (albeit at a lower level) in subcutaneous WAT (scWAT) (##FIG##0##Fig. 1a##). Interestingly, GRAF1 mRNA levels in iBAT increased nearly 20-fold from 1 week to 3 months of age (##FIG##0##Fig. 1b##) and this increase paralleled the expression of BAT maturation genes including PPARγ, the thermogenic protein UCP1, and the mitochondrial marker ND5. In contrast, GRAF1 levels did not significantly change during WAT maturation.</p>", "<p id=\"P11\">We next used cultured cell models to confirm and extend these findings. First, using WT-1 cells (a validated in vitro model of brown adipogenesis)^{##REF##20107496##27##}, we found that GRAF1 expression was dynamically and transiently increased at the onset of differentiation (##FIG##0##Fig. 1c##,##FIG##0##d##) and that its induction occurred prior to, or concomitant with, expression of BAT marker genes UCP1, PPARγ, and ND5 (##FIG##0##Fig. 1d##). This finding is consistent with prior reports indicating RhoA/ROCK signaling is down-regulated upon the induction of adipocyte differentiation in these cells. A similar dramatic increase in GRAF1 expression was observed in human preadipocytes (Simpson-Golabi-Behmel syndrome, SGBS) following exposure to brown adipocyte differentiation medium (Figure S1a). Note that adipogenic differentiation of these cells was accompanied by lipid droplet accumulation and characteristic brown phenotype observed using light microscopy (Fig S1a, bottom panel). Finally, in mouse 3T3L1 adipocytes, which can be induced to form WAT or BAT, treatment with BAT-induction cocktail led to a more robust increase in GRAF1 expression than did treatment with a WAT-induction cocktail (Figure S1b,c). Collectively, these studies indicate that GRAF1 might play an important and conserved role in promoting BAT development.</p>", "<title>GRAF1 is necessary for BAT maturation and function</title>", "<p id=\"P12\">We next compared BAT and WAT marker gene expression in tissue isolated from WT mice or from global GRAF1-deficient mice (GRAF1^gt/gt) which harbors the gene trapping vector VICTR48 within the first intron of Graf1. These mice lines are viable and fertile with no obvious abnormalities under baseline conditions 28. As shown in ##FIG##1##Fig. 2a##, we found no differences in expression of UCP1, PPARγ or ND5 in scWAT or iBAT isolated from 1 week old WT and GRAF1^gt/gt mice, suggesting that GRAF1 does not play an important role in adipose tissue specialization. However, since adipose tissue maturation happens gradually after birth through young adulthood^{##REF##2171932##29##–##REF##31354508##31##} and GRAF1 expression was robustly increased during this timeframe, we reasoned that GRAF1 might impact adipose differentiation/maturation. Indeed, as shown in ##FIG##1##Fig. 2b##, BAT depots isolated from 3 month-old GRAF1-deficient mice exhibited a dramatic reduction in BAT marker gene expression relative to littermate control WT mice. For example, iBAT from GRAF1^gt/gt mice demonstrated a significant reduction in BAT thermogenesis genes (UCP1, Elovl3), mitochondrial genes (ND5, COX7a), and in PPARγ, the master regulator gene of adipocyte differentiation, indicating a differentiation/maturation defect in the iBAT of GRAF1 hypomorphs. Accordingly, H&amp;E staining of iBAT from 2–3 month old GRAF^gt/gt mice revealed a “white like” appearance as demonstrated by a substantial increase in lipid deposition and enlarged lipid droplets compared to iBAT from littermate WT control mice(##FIG##1##Fig. 2c##). scWAT from 2–3 month old GRAF1-deficient mice also exhibited a significant down-regulation of brown marker genes including UCP1 and Elovl3 but upregulation of the general adipose marker adiponectin, suggesting that GRAF1 promotes WAT browning but limits WAT expansion (##FIG##1##Fig. 2d##). Despite these changes, and consistent with our prior report ^{##REF##25019370##28##}, there was no difference in the body weights of GRAF1^gt/gt mice and littermate control mice fed ad libitum (data not shown). This finding is consistent with other mouse models that exhibit defects in brown adipose development, yet do not develop obesity, such as those with the loss of the uncoupling protein UCP-1^{##REF##12569166##32##}, or the fatty acid metabolism gene Adipose acyl-CoA synthetase-1^{##REF##20620995##33##}.</p>", "<p id=\"P13\">To determine the impact of GRAF1-dependent changes in BAT differentiation on BAT function, we next exposed mice housed at sub-thermoneutral temperatures to overnight fasting followed by an acute bout of cold stress (6°C) and measured their body temperatures over time. As shown in ##FIG##1##Fig. 2e##, GRAF1-deficient mice exhibited a remarkable reduction in thermogenic capacity under these conditions. Interestingly, this failure to thermoregulate was associated by a significant reduction in serum lactate levels (##FIG##1##Fig. 2f##). Since circulating lactate is the main fuel that drives the tricarboxylic acid cycle (TCA cycle) in a fasted state ^{##REF##29045397##34##}, the decreased lactate might suggest enhanced TCA cycling due to beta-oxidation defects in GRAF1^gt/gt mice. In support of this possibility, there was a trend towards increased muscle triglyceride levels in the cold-exposed GRAF1^gt/gt mice relative to similarly treated WT mice (##FIG##1##Fig. 2g##). Nonetheless, given our knowledge regarding GRAF1’s role in skeletal muscle maturation, we realized that the drop in temperature coupled with the reduced lactate production observed in GRAF1^gt/gt mice could be due (at least in part) to a reduced capacity for shivering thermogenesis in these germline-deficient mice. To begin to distinguish between these two possibilities, we first sought to determine if GRAF1 can act in a cell autonomous fashion to promote BAT maturation. To this end, we turned to primary pre-brown adipocyte cultures isolated from the stromal vascular fraction (SVF) of iBAT as these cells have been reported to faithfully recapitulate brown adipocyte maturation when exposed to serum-containing media supplemented with insulin, triiodothyronine (T₃), dexamethasone, IBMX and rosiglitazone^{##REF##33439778##35##}. Importantly, upon induction with differentiation media, SVF cells transfected with GRAF1 siRNA exhibited a significant reduction in browning capacity compared to control siRNA treated cells as assessed by UCP1 expression and oil red O staining (##FIG##2##Fig. 3a##–##FIG##2##c##). These findings indicate that GRAF1 levels can directly impact the differentiation of brown adipocytes.</p>", "<p id=\"P14\">Next, to further explore an adipocyte-autonomous role for GRAF1 in BAT formation and function in vivo, we developed a new mouse model using targeting vectors from Eucomm to conditionally target the GRAF1 allele. In this model, Cre-mediated recombination causes a frame shift and early stop codon (Supplemental Fig. 2a) that results in nonsense mediated mRNA decay. Southern blot analysis confirmed successful targeting in embryonic stem cells (ES) and the germline transmission of resulting chimeras(Tm1a). The Tm1a mice were subsequently crossed with Flp recombinase mice to remove the LacZ reporter and a neomycin resistance cassette used for selection of targeted ES cells to generate GRAF1^fl/fl(GRAF1^Tm1c mice; Supplemental Fig. 2b). Finally, we established an adipose-specific GRAF1 knockout mouse line (GRAF1^AKO) by crossing GRAF1^Tm1c mice with the Adiponectin Cre line (010803, Jackson Lab) ^{##REF##21356515##36##} which led to a significant depletion of GRAF1 in scBAT and WAT (##FIG##2##Fig. 3d##). Importantly, while not as dramatic as observed in the GRAF1^gt/gt mice, the thermogenic capacity of GRAF1^AKO mice was significantly blunted when compared to genetic control mice (##FIG##2##Fig. 3e##). Also, activation of a thermogenic gene expression profile in mature BAT was significantly reduced in GRAF1^AKO mice when compared to similarly-treated genetic control mice (##FIG##2##Fig. 3f##). Consistent with the more modest impact on thermogenesis, while BAT from GRAF1^AKO mice exhibited a significant reduction in UCP1 expression, the decrease was not as robust as observed in GRAF1^gt/gt mice (possibly due to incomplete recombination in our model). Nonetheless, GRAF1^AKO BAT also exhibited significantly reduced expression of the mitochondrial genes Cpt1a, Cpt1b and CS. In addition, GRAF1^AKO BAT also exhibited significant reductions in the expression of growth factors known to promote glucose homeostasis including FGF1, which acts in an autocrine fashion to promote glucose uptake in activated BAT and FGF21, an endocrine factor that has been linked to the cardiometabolic benefits of brown fat activation through its ability to promote glucose homeostasis in BAT, skeletal muscle, liver and brain. Collectively, these data confirm that GRAF1 plays an important, adipocyte-autonomous role in BAT formation and function.</p>", "<title>GRAF1 promotes subcutaneous adipose beigeing</title>", "<p id=\"P15\">As noted above, scWAT from adult GRAF1^gt/gt mice also exhibited lower BAT marker gene expression relative to similarly housed WT mice, indicating the possibility that GRAF1 might also be necessary for physiological WAT browning. To further explore a role for GRAF1 in beige fat induction, we treated mice with the browning agent CL316243, a β 3-adrenergic receptor agonist. Treatment with CL316234 for 10 days in WT mice promoted expression of several beige adipose markers in scWAT including the mitochondrial enzyme CPT1A (Carnitine Palmitoyltransferase 1) that is essential for fatty acid beta-oxidation ^{##UREF##1##37##}and the secreted factors FGF1 and FGF21 that improve adipose and systemic glucose and lipid metabolism^{##REF##28605657##38##} (##FIG##3##Fig. 4a##). Interestingly, CL316243 treatment also significantly increased scWAT GRAF1 expression, further suggesting a role for GRAF1 in regulating agonist-induced browning (##FIG##3##Fig. 4a##). Indeed, the induction of each of these marker genes was significantly reduced in scWAT isolated from CL316243-treated GRAF1^AKO mice when compared to scWAT isolated from similarly treated WT mice (##FIG##3##Fig. 4b##). Thus, in addition to promoting classical brown fat maturation and function, GRAF1 also promotes the development of metabolically favorable beige adipocytes in scWAT.</p>", "<title>GRAF1 promotes brown phenotypes by limiting RhoA/ROCK signaling</title>", "<p id=\"P16\">Because previous studies have shown that the downregulation of RhoA-ROCK signaling is necessary and sufficient to promote adipogenesis, we reasoned that GRAF1 might promote BAT differentiation by controlling this pathway. To test this possibility, we quantified RhoA activity in WT-1 brown adipocytes using a standard GST-rhotekin precipitation assay. As shown in ##FIG##4##Fig. 5a## and ##FIG##4##b##, GRAF1-depleted WT-1 cells exhibited significantly higher levels of RhoA activity in comparison with control siRNA-treated cells following treatment with the RhoA agonist, sphingosine-1-phosphate. Importantly, treatment with the ROCK inhibitor, Y27632 at the onset of differentiation completely restored UCP-1 expression in GRAF1-depleted WT-1 cells (##FIG##4##Fig. 5c##) and in GRAF1-depleted SGBS human brown adipocytes (##FIG##4##Fig. 5d##). Collectively, these data indicate that GRAF1 promotes brown adipocyte differentiation by limiting RhoA/ROCK signaling.</p>" ]

[ "<title>Discussion</title>", "<p id=\"P17\">Adipose tissue in mammals is a vital component of the energy regulation system and consists of two primary types: white adipocytes and brown adipocytes. White adipocytes are characterized by their large central lipid droplets, serving as reservoirs for storing excess energy in the form of triglycerides. In contrast, brown adipocytes feature small lipid droplets and a high number of mitochondria, enabling them to dissipate energy as heat through a process known as thermogenesis ^{##UREF##2##39##, ##REF##35120662##40##}. The discovery of metabolically active brown adipose tissue (BAT) in healthy adult humans and beige adipocytes within subcutaneous white adipose tissues(scBAT) has generated significant interest in the biology of brown and beige adipocytes due to their unique ability to improve whole-body glucose and lipid metabolism by consuming substantial amounts of blood glucose and lipids. Consequently, identifying molecules and signals that can regulate brown fat differentiation and function has become an attractive target for potential treatments of obesity and diabetes. Our study demonstrates for the first time that GRAF1 regulates brown adipogenesis. Depletion of GRAF1 reduces brown adipocyte differentiation and thermogenesis function in vivo, highlighting its unique role in mediating BAT cell fate.</p>", "<p id=\"P18\">The differentiation, activation, and maintenance of brown and beige adipocytes are governed by a complex interplay of multiple factors. These factors include endocrine signals such as fibroblast growth factors(FGFs) and bone morphogenic protein factors(BMPs), as well as critical transcription factors like PRDM16, PGC1α, PPARγ, and Foxp1^{##REF##29052591##6##, ##REF##23805974##7##, ##REF##24703692##41##–##REF##17108241##43##}. Of particular interest, the RhoA/ROCK pathway has emerged as a key player in regulating adipogenesis. Previous studies have shown that disrupting this pathway, either through the expression of dominant-negative RhoA or the inhibition of ROCK, promotes differentiation in various adipogenic cell types, highlighting the significance of this pathway. Mechanistically, induction of adipocyte differentiation leads to downregulation of RhoA-ROCK signaling, which promotes disassembly of F-actin stress fibers and results in marked changes in cell shape that are thought to be important for lipid droplet accumulation. Depolymerization of F-actin also leads to accumulation of monomeric G-actin, which binds and sequester MRTFs in the cytosol, preventing their nuclear translocation. As MRTFs and their co-factor, SRF repress PPARγ, this critical step allows for the expression of PPARγ and its target genes, facilitating the development and maintenance of adipocyte characteristics during adipogenic differentiation ^{##REF##25579684##15##, ##REF##28125644##16##, ##REF##21952300##44##}.</p>", "<p id=\"P19\">The Rho GTPase can exist in either an inactive GDP-bound or an active GTP-bound form and is modulated by guanine nucleotide exchange factors (GEFs) and GTPase-activating proteins (GAPs) ^{##REF##12478284##45##}.The comprehensive understanding of the diverse regulators of Rho GTPase/ROCK in adipose tissues and their in vivo functional consequences remains relatively unexplored. Our previous studies have shown that GRAF1 is a bone fide RhoGAP and plays important role in processes like myoblast fusion, which require GAP-dependent actin remodeling ^{##REF##25019370##28##}. Given GRAF1’s established function as a RhoGAP in various tissues, we hypothesized that it might also play a pivotal role in modulating RhoA activity during adipogenesis. Indeed, we observed that GRAF1 is highly and selectively expressed in metabolically active tissues, including brown adipose tissue (BAT), brain, and heart, and its expression profile closely correlated with brown adipocyte differentiation. This association is evidenced by the upregulation of GRAF1 in response to brown adipocyte differentiation medium in various cultured adipocytes and the significant increase in GRAF1 mRNA levels during the maturation stage of BAT in mice, but not in WAT. GRAF1 deficiency, as observed in primary pre-brown adipocyte cultures and in both global and adipose-specific GRAF1-deficient mice, significantly blunted brown and beige adipose differentiation. Furthermore, GRAF1-deficient mice exhibited an inability to efficiently respond to cold challenge-induced thermogenesis. Moreover, development of beige adipocytes in scWAT was compromised in response to browning agent CL316243 stimulation via the β3-adrenergic pathway. Collectively, these data indicate that GRAF1 plays a key cell autonomous role in the development and maturation of beige and brown adipose tissue.</p>", "<p id=\"P20\">Our mechanistic studies showed that GRAF1 depletion in brown preadipocyte cell line increased Rho activity, suggesting that GRAF1 mediates brown adipocyte differentiation by GAP-dependent Rho GTPase inhibition. Interestingly, GRAF1 also possesses a BAR (Bin/amphiphysin/Rvs) and PH (pleckstrin homology) domain that are involved in sensing and inducing membrane curvature and determining membrane binding specificity. Interestingly, these domains in combination with an isoform-specific hydrophobic segment (found in brain-selective GRAF1a) were previously reported to drive GRAF1a association with lipid droplets, an event that promoted lipid droplet clustering and reduced lipolysis ^{##REF##25189622##46##, ##REF##19036340##47##}. While our current study focused on mouse GRAF1.2 (the ortholog of GRAF1b in humans) which does not contain this hydrophobic region, future studies are warranted to determine the extent to which GRAF1.2/GRAF1b may regulate adipocyte lipid droplet homeostasis.</p>", "<p id=\"P21\">Previous studies have demonstrated that p-190 B is a RhoGAP can that shift myogenesis towards adipogenesis by inhibiting Rho GTPase in adipogenic precursor cells^{##REF##27251423##17##–##REF##23395168##19##}. Another RhoGAP, DLC1, has been reported to promote both white and brown adipocyte differentiation and to provide a molecular link between PPARγ and Rho signaling pathways ^{##REF##28358928##20##}. However, it’s important to note that while p-190 B and DLC1 play significant roles in regulating adipocyte differentiation, neither of these RhoGAPs exhibits selective induction of brown adipocyte differentiation. In our current study, we present the unique contribution of GRAF1, a RhoGAP, to the differentiation of brown and beige adipocytes. This distinctive function differentiates GRAF1 from other RhoGAPs. Combining in vitro investigations across various cell types with in vivo experiments in GRAF1 knockout mouse models, we have demonstrated that GRAF1 promotes classical brown fat maturation and the development of beige adipocytes in scWAT. This regulation, at least partly, occurs through GRAF1’s ability to suppress RhoA activity, thereby limiting the RhoA-ROCK signal pathway and promoting brown adipocyte differentiation and maturation. Future research may elucidate the potential for manipulating GRAF1 expression and its GAP activity to improve metabolic profiles, holding promise for the treatment of metabolic diseases, including obesity and insulin resistance.</p>" ]

[]

[ "<p id=\"P1\"><bold>Author contributions:</bold> conceived and performed experiments, data analysis and preliminary manuscript writing XB, final manuscript writing and data analysis, QZ. Data analysis, MEC, development of SBGS cell line, distribution and guidance of usage, MW. Conceptual development of studies and manuscript preparation: CPM. Conceptual development of studies and manuscript preparation: JMT</p>", "<p id=\"P2\">Adipose tissue, which is crucial for the regulation of energy within the body, contains both white and brown adipocytes. White adipose tissue (WAT) primarily stores energy, while brown adipose tissue (BAT) plays a critical role in energy dissipation as heat, offering potential for therapies aimed at enhancing metabolic health. Regulation of the RhoA/ROCK pathway is crucial for appropriate specification, differentiation and maturation of both white and brown adipocytes. However, our knowledge of how this pathway is controlled within specific adipose depots remains unclear, and to date a RhoA regulator that selectively controls adipocyte browning has not been identified. Our study shows that expression of GRAF1, a RhoGAP highly expressed in metabolically active tissues, closely correlates with brown adipocyte differentiation in culture and in vivo. Mice with either global or adipocyte-specific GRAF1 deficiency exhibit impaired BAT maturation, reduced capacity for WAT browning, and compromised cold-induced thermogenesis. Moreover, defects in differentiation of mouse or human GRAF1-deficient brown preadipocytes can be rescued by treatment with a Rho kinase inhibitor. Collectively, these studies indicate that GRAF1 can selectively induce brown and beige adipocyte differentiation and suggest that manipulating GRAF1 activity may hold promise for the future treatment of diseases related to metabolic dysfunction.</p>" ]

[]

[ "<title>Acknowledgments</title>", "<p id=\"P34\">This research is based in part upon work conducted at the Microscopy Services Laboratory, Department of Pathology and Laboratory Medicine, which are supported in part by a National Cancer Institute Cancer Center Core Support Grant to the University of North Carolina Lineberger Comprehensive Cancer Center (P30 CA016086). This work was also supported by NIH including NHLBI/R01HL130367 and 1RO1HL165786 to JMT, 1F31HL145983–01 to M.E.C., and American Heart Association 16PRE30630002 to Q.Z. and a North Carolina Obesity Research Center Pilot Award to XB.</p>", "<title>Data availability</title>", "<p id=\"P35\">All data generated or analysed during this study are included in this published article and its supplementary information files.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><title>GRAF1 expression closely correlates with BAT maturation.</title><p id=\"P37\"><bold>a.</bold> GRAF1 protein levels in 3 week postnatal mouse tissue detected by Western blotting. scWAT, sub-cutaneous white adipose tissue; iBAT and sBAT, inter- and sub-scapular brown adipose tissue, SKM, skeletal muscle. <bold>b</bold>. qRT-PCR of GRAF1 and adipocyte marker genes in indicated adipose depot isolated from 1 week old and 3 month old mice, n=6–7. <bold>c,d.</bold> WT-1 pre-brown adipocytes were exposed to brown fat-inducing insulin differentiation medium for the indicated times followed by Western blotting (<bold>c</bold>) or qRT-PCR (<bold>d</bold>) analysis of GRAF1 or indicated adipocyte markers (n=3). Data(<bold>b,d</bold>) are represented as mean ± SEM, *P &lt; 0.05; **P &lt; 0.01; ****P &lt; 0.0001 by two-tailed student’s t-test.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><title>GRAF1 does not alter adipocyte specification but is required for brown fat formation and function.</title><p id=\"P38\"><bold>a</bold>. iBAT and scWAT isolated from 1 week old WT and GRAF1^gt/gt mice exhibited similar levels in adipose marker gene expression as assessed by qRT-PCR (n=6–9/group), indicating that GRAF1 is not necessary for white or brown adipose tissue specification. <bold>b</bold>, iBAT isolated from 3 month old GRAF1^gt/gt mice exhibited significantly lower levels of brown fat marker genes that mediate thermogenesis compared to littermate control mice as assessed by qRT-PCR (n=7). <bold>c</bold>, Representative H&amp;E stain of iBAT isolated from 3 month old GRAF1^gt/gt and WT mice. Note increased levels of lipid droplets (un-stained white vesicles) indicative of BAT whitening in GRAF1^gt/gt iBAT depot.(n=3/group). <bold>d</bold>, scWAT isolated from 3 month old GRAF1^gt/gt mice exhibited significantly higher levels of white fat marker genes and lower levels of beige-fat associated genes compared to littermate control mice as assessed by qRT-PCR. (n=6–7/group). <bold>e-g</bold>, GRAF1^gt/gt and WT mice were housed at sub-thermoneutral temperatures (22–25°C) and subjected to overnight fasting followed by an acute bout of cold stress (6°C) and their body temperatures were measured over time using a rectal thermometer(<bold>e</bold>) and serum lactate(<bold>f</bold>)and soleus triglyceride(<bold>g</bold>) level were measured. (n=4–5 WT, n=3 GRAF1^gt/gt ). Data(<bold>b,d,e,f,g</bold>) are represented as mean ± SEM, *P &lt; 0.05; **P &lt; 0.01; ***&lt;0.001;****P &lt; 0.0001 by two-tailed student’s t-test.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3</label><caption><title>Adipocyte-autonomous role for GRAF1 in BAT formation and function.</title><p id=\"P39\"><bold>a</bold>. differentiation of primary pre-brown adipocytes isolated from the stromal vascular fraction (SVF) of iBAT was reduced by GRAF1 depletion as assessed by UCP1 expression. n=3–4. <bold>b-c</bold>. Oil red O staining of differentiated SVF cells treated with indicated siRNA(<bold>b</bold>) and quantification of oil red O staining(<bold>c</bold>). (siG1 denotes siGRAF1). n=3/treatment. <bold>d</bold>. Depletion of GRAF1 in iBAT and scWAT in WT and GRAF1^AKO mice as assessed by Western blotting. <bold>e</bold>. GRAF1^AKO mice exhibited cold intolerance when subjected to conditions described in ##FIG##1##Figure 2e##. n=5–6/group. <bold>f</bold>. qRT-PCR data showed significantly decreased markers of brown adipogenesis, mitochondrial components and FGFs in BAT of GRAF1^AKO mice. n=6/group. Data(<bold>a,c,e,f</bold>) are represented as mean ± SEM, *P &lt; 0.05; **P &lt; 0.01; ***&lt;0.001;****P &lt; 0.0001 by two-tailed student’s t-test.</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4</label><caption><title>GRAF1 promotes subcutaneous adipose beigeing.</title><p id=\"P40\"><bold>a.</bold> WT mice were treated for 10 days with CL316243, and, qPCR analysis was performed on scWAT to assess mRNA levels of indicated genes; n=5/group. <bold>b</bold>. qPCR analysis of scWAT from WT and GRAF1^AKO mice revealed decreased Cpt1a, FGF1 and FGF21 in GRAF1^AKO mice. n=5–7/group. Data are represented as mean ± SEM, ns, not significant; *P &lt; 0.05; **P &lt; 0.01 ***&lt;0.001; ****P &lt; 0.0001 by two-tailed student’s t-test.</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5</label><caption><title>GRAF1 promotes BAT differentiation by limiting RhoA/ROCK signaling.</title><p id=\"P41\"><bold>a and b.</bold> RhoA activity in siRNA transfected WT-1 brown adipocytes was assessed by a standard GST-rhotekin precipitation assay(<bold>a</bold>) and quantified(<bold>b</bold>), n=4 independent experiments. <bold>c and d</bold>. ROCK inhibitor Y-27632 restored UCP-1 expression in GRAF1-depleted WT-1 cells (<bold>c</bold>, n=2–3/group) and in GRAF1-depleted SGBS human brown adipocytes(<bold>d</bold>, n=3–6/group).Data(<bold>b,c,d</bold>) are represented as mean ± SEM, ns, not significant; *P &lt; 0.05; **P &lt; 0.01 by two-tailed student’s t-test.</p></caption></fig>" ]

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[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN2\"><p id=\"P36\"><bold>Competing interests:</bold> The authors declare that they have no competing interests.</p></fn></fn-group>" ]

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[ "<title>INTRODUCTION</title>", "<p id=\"P2\">With improved microscopy instrumentation, researchers increasingly produce a wealth of biological data from imaging experiments. Either by mechanical scanning or computational post-processing (e.g., Fourier ptychography, diffraction microscopy), millimeter-scale fields of view can be digitized in sub-cellular detail^{##UREF##0##1##–##REF##31044723##3##}. Moreover, the continued development of fluorescent and colorimetric probes allows numerous sub-cellular structures (e.g., proteins, nucleic acids) to be quantified and spatially visualized. This multi-parametric capacity has been critical to unveiling heterogeneity in biological samples and identifying cell subpopulations with key biological profiles^{##REF##34215862##4##–##UREF##2##10##}. One enabling technology is cyclic multiplexed immunofluorescence microscopy (mIF); through repeated staining, destaining (or quenching), and re-staining of biological samples, this approach significantly expands the number of targets that can be simultaneously detected in the same sample. A considerable number of different technologies of mIF have been developed based on chemical or optical quenching^{##REF##35666645##11##–##REF##36203011##17##}. Resulting molecular data have deepened our understanding of tissue composition, cell-to-cell interactions, and cell signaling^{##REF##35666645##18##, ##REF##34233961##19##}.</p>", "<p id=\"P3\">Analyzing large cyclic-imaging datasets, however, presents computational challenges. For example, imaging a single tissue sample already generates vast amounts of visual data, including i) hundreds of fields of view (FOVs), ii) multiple staining rounds per field of view (often 20–40), and iii) corresponding background intensity images for each marker. As a result, standard steps such as image registration between cycles require intensive computation with standard imaging tools. Similarly, manual cell segmentation on this scale would require several person-hours per sample, severely limiting clinical applications. Off-the-shelf cell segmentation tools (e.g., CellProfiler) reduced some of the workloads, but noise introduced through stain variation and sample debris made consistent performance across samples difficult.</p>", "<p id=\"P4\">Here, we present a computational framework to analyze multi-channel, large FOV cellular images. Termed CycloNET, this framework i) consolidates, aligns, and processes several gigapixels of fluorescent microscope images, ii) segments individual cells and generates their multi-dimensional molecular profiles, and iii) produces visual representations that succinctly summarize cell profiles. As a representative example, we applied CycloNET to profile immune cells in fine-needle aspirate (FNA) samples of head and neck squamous cell carcinoma (HNSCC) patients. CycloNET significantly reduced analysis time when compared to standard computational imaging tools and manual cell segmentation. Further, our pipeline facilitated the identification of rare immune cell clusters, which could be instrumental to future research in cancer management. The developed framework would be readily adapted for applications involving other cell types or tumor structures (i.e., tumor cells, lymphatic or vascular structures). Combined with cyclic imaging, we expect CycloNET will facilitate the development of improved diagnostic and prognostic metrics from complex information-rich imaging data.</p>" ]

[ "<title>METHODS AND MATERIALS</title>", "<title>Human tumor biopsy dataset and image acquisition</title>", "<p id=\"P17\">FNA samples from HNSCC patients were collected, stained, and imaged as previously reported^{##REF##34233961##19##}. The study protocol was reviewed and approved by the Institutional Review Board of Massachusetts General Hospital (IRB number 2014P000559), and the overall procedures followed institutional guidelines. Informed consent was obtained from newly diagnosed and recurrent/metastatic head and neck squamous cell carcinoma (HNSCC) patients, and samples were acquired between Jan 1, 2020 and June 30, 2021. Briefly, samples were collected by aspiration, fixed, and serially stained with three protein markers per cycle, followed by a signal quenching step. This process was repeated until all marker signals (and corresponding quenched background signals) were collected. For each specimen, 20–40 fields of view were imaged by an Olympus BX-63 upright automated epifluorescence microscope. To develop the three modules in this computational pipeline, a total of 17 human samples were collected for training and validation and 5 samples for final evaluation. For analysis, collected images were processed using a graphic user interface (GUI) developed with the Qt GUI toolkit for Python^{##REF##24436451##24##}.</p>", "<title>Image pre-processing</title>", "<p id=\"P18\">Acquired raw images were preprocessed for background subtraction, and imaging cycles were aligned together. To detect immune cells, the CD45 signal, along with a corresponding DAPI image, was processed by a pre-trained neural network to generate single-cell outlines. Once biomarker intensities were recorded for each cell, cell-type labeling for major immune groups was conducted. For each specimen, quenched background signals were subtracted from corresponding fluorescence signals pixel-by-pixel. Pixel intensity for each signal was then normalized using the 25th and the 99th percentiles as lower and upper limits. Next, a Gaussian filter was applied to reduce background impulse noise.</p>", "<title>Image alignment</title>", "<p id=\"P19\">An image registration step was applied between all cycles in a single field of view for a given specimen to account for small translations between imaging steps. First, a reference cycle was chosen for each sample with the provided user interface (##FIG##1##Fig. 2A##). The general registration algorithm structure is outlined in ##FIG##1##Figure 2B##. Signals from the same imaging cycle were collapsed by maximum projection and rescaled by a factor of 0.5. Next, a foreground mask was produced to roughly detect cell-sized objects. Fields of view with no few cell-sized objects were removed from the analysis. Finally, the collapsed signals and corresponding masks were randomly cropped by a factor of 0.75, and the translation shift was calculated by masked cross-correlation using the Python Sci-kit Learn library. The resulting translation was then stored and averaged over three trials to minimize translation errors resulting from signal noise. Finally, a single-cycle translation was applied to all fields of view.</p>", "<title>Cell segmentation</title>", "<p id=\"P20\">A set of 17 human specimen fields of view were used for training and 6 for validation. Each field of view was tiled into 300 × 300 arrays with 20% overlap for each sample. To reduce class imbalance, examples with no labeled cells were removed from the training dataset. The final dataset sizes for training and testing were 7450 and 6302, respectively. The hyper parparameters, <italic toggle=\"yes\">α</italic>, <italic toggle=\"yes\">λ</italic>_{<italic toggle=\"yes\">c</italic>}, <italic toggle=\"yes\">λ</italic>_{<italic toggle=\"yes\">a</italic>}, <italic toggle=\"yes\">β,</italic> were set to 1 × 10³, 1 × 10³, 5 × 10⁻¹, and 1 × 10⁻⁴, respectively. The neural network was defined using the Tensorflow library, trained on two NVIDIA GTX 1080 Ti GPUs for 700 epochs with a batch size of 16, using the Adam optimizer with a learning rate of 10⁻⁵. Additionally, stochastic i) 90° rotation, ii) horizontal and vertical flipping, iii) gaussian noise, and iv) mean shift for data augmentation was employed in the training steps. The best-performing model was saved over five trials, as determined by pixel classification accuracy. To adjust for background-class imbalance, we weighted the cross entropy loss for cell interior (class ii) and cell boundary (class iii) by 3-fold importance compared to the background. Misclassifications of boundary pixels as cell interior were further weighted with two-fold importance to encourage the network to prioritize cell boundaries and avoid merging neighboring cells.</p>", "<title>Clustering</title>", "<p id=\"P21\">CD45-positive immune cell segmentation masks were applied to aligned fluorescent signals to aggregate fluorescent intensity for each marker on a single-cell basis. The aggregate signal was then averaged over each segmented cell area. High-intensity outliers were removed by mean absolute deviation, aggregate intensities were log-transformed, and positivity thresholds for each marker were set to 3 standard deviations below the signal mean. Different immune cell populations were defined based on the expression level of a combination of markers, as in ##TAB##0##Table 1##. Dimensionality reduction was then applied to the sample’s single-cell data by applying uniform manifold approximation and projection (UMAP) using the Sci-kit Learn library for Python and optimized to minimize the distance between predefined cell phenotypes^{##UREF##7##23##, ##UREF##8##25##}. Finally, to define distinct cell clusters, we applied the hierarchical DBSCAN algorithm, a density-based approach that can detect clusters of arbitrary shape^{##UREF##6##22##}.</p>" ]

[ "<title>RESULTS AND DISCUSSIONS</title>", "<title>CycloNET pipeline</title>", "<p id=\"P5\">##FIG##0##Figure 1A## shows CycloNET’s overall workflow. The input data are raw fluorescent images obtained through cyclic immunofluorescent microscopy. The computational pipeline collates acquired multichannel images, performs image registration, detects individual cells, and presents summary statistics. In the current work, we used images generated by the fast analytical screening technique (FAST) as previously reported^{##REF##35666645##18##, ##REF##34233961##19##}; FAST uses custom-designed antibody probes to cyclicly image 20–40 molecular markers within an hour^{##REF##35666645##18##, ##REF##34233961##19##} (##FIG##0##Fig. 1B##). A given specimen underwent rounds of imaging and quenching cycles; at each cycle, several FOVs were captured, resulting in up to 5 gigapixels of visual information per sample. We implemented a graphical user interface to streamline image analysis (##FIG##0##Fig. 1C##). Under the hood, it was equipped with i) a preprocessor that automatically grouped images by cycle and FOV, ii) a registration module that detected translations between imaging cycles, and iii) a pre-trained neural network for task-specific cell segmentation.</p>", "<title>Image registration</title>", "<p id=\"P6\">Between fluorescent staining and destaining cycles, sample slides were manually removed and remounted onto the microscope, introducing spatial offsets. Though negligible for tissue-level assessments, these shifts required careful correction for accurate single-cell metrics. Standard imaging software for spatial alignment faced challenges, including (i) algorithms that were suboptimal for large image sets and (ii) errors in registration due to sample debris, low cellularity, or low signal-to-noise ratio (SNR). We addressed these drawbacks in our custom-designed processing pipeline, optimized for cyclic imaging.</p>", "<title>Pre-processing</title>", "<p id=\"P7\">##FIG##1##Figure 2A## illustrates the CycloNET image registration algorithm. The algorithm processes image datasets derived from several staining cycles (denoted as <italic toggle=\"yes\">N</italic> cycles). Multiple fields of view (FOVs) are collected for each staining cycle to capture as many cells as possible. In this scenario, each FOV tile set can be represented as FOV_{<italic toggle=\"yes\">i</italic>,<italic toggle=\"yes\">j</italic>}, where <italic toggle=\"yes\">i</italic> indicates an individual field of view (<italic toggle=\"yes\">i</italic> = 1, 2, …, <italic toggle=\"yes\">M</italic>) and <italic toggle=\"yes\">j</italic> indicates the particular cycle number (<italic toggle=\"yes\">j</italic> = 1, 2, ..., <italic toggle=\"yes\">N</italic>). We found that images from any two cycles had similar spatial offsets across all FOVs, allowing us to align full cycles by using a single FOV per cycle. Thus, the FOV numbers to be processed for alignment is reduced from (<italic toggle=\"yes\">N</italic> × <italic toggle=\"yes\">M</italic> × <italic toggle=\"yes\">C</italic>) to just (<italic toggle=\"yes\">N</italic> × <italic toggle=\"yes\">C</italic>), where <italic toggle=\"yes\">C</italic> is the number of fluorescent channels used in microscopy. To further minimize dataset size and improve accuracy in low-signal images, the <italic toggle=\"yes\">C</italic>-channel fluorescent signals were normalized and collapsed by maximum projection, resulting in a set of <italic toggle=\"yes\">N</italic> images. By using the maximum projection across individual channels, we enhanced the visibility of cells, as some cells might be visible in one channel and not another, providing more “anchoring points” for accurate alignment. Finally, a reference is selected among these <italic toggle=\"yes\">N</italic> collapsed images, to which all other cycles are aligned. These are labeled in ##FIG##1##Fig 2A## as “Reference” and “Cycle images.”</p>", "<p id=\"P8\">The translations between the N images were calculated by masked cross-correlation. This strategy reduced the impact of cellular debris by focusing only on cell-shaped foreground objects specified by the cell mask (see <bold>Methods</bold> for details). At the time of cross-correlation calculation, images were also downsampled two-fold and randomly cropped to further reduce dataset size. This process was repeated for some <bold><italic toggle=\"yes\">t</italic></bold> trials, with different random crops at each trial. Finally, the mean shift (after removing outliers) is calculated and assigned to all FOVs in a given staining cycle.</p>", "<p id=\"P9\">##FIG##1##Figure 2B## shows snapshots of the image registration process. The maximum projection across individual channels increased the number of cells available as anchoring points for the cross-correlation algorithm. One such anchoring point is indicated by red circles, demonstrating how a cluster of cells shifts to overlap with the reference cycle after alignment. We tested our algorithm on a set of five human specimens (FOVs per sample: 20, 24, 35, 30, 36). On average, a single field of view with seven imaging cycles was aligned in 42 seconds, while a full specimen containing ~20 FOVs typically aligns in under 6 minutes using an Intel^® Core^™ i7–6850K processor. Alignment failed in 5% of 142 FOVs, typically in FOVs with very low contrast in the reference signal or low cellularity.</p>", "<title>Cell segmentation and validation</title>", "<p id=\"P10\">Following alignment, CycloNET generated segmentation masks for individual cells by applying a pre-trained neural network. To detect immune cells in the sample, the algorithm was trained on image stacks consisting of CD45 and nucleus (DAPI) channels, as described in ##FIG##2##Fig. 3A##. Using a modified U-Net structure, we designed a neural network to produce a 2-dimensional array where each pixel was labeled as either: i) background, ii) cell interior, or iii) cell boundary^{##UREF##4##20##} (##FIG##2##Fig. 3A##). The encoded features used to construct segmentation masks were regularized by secondary auto-encoding and cell-counting tasks^{##UREF##5##21##}, resulting in a loss function,\n\nwhere the segmentation loss, , was split into the cross entropy loss () for pixel-by-pixel classification and the per-class Jaccard loss metric and represent the mean squared error loss metrics for the cell counting and autoencoding tasks, respectively. Finally, represents the L2 regularization loss applied to all weights in the network, and assign weights to each loss type. The network was trained using the CD45 and nuclear stains for 17 human specimen FOVs (See Supp 2). At the end of the training, the average F1 score among all pixel classes was 0.94. A single FOV was scanned for CD45+ immune cells in 17.9 seconds (<italic toggle=\"yes\">n</italic> = 134).</p>", "<p id=\"P11\">##FIG##2##Figure 3B## shows the comparison between manual and CycloNET segmentations. Defining the cell boundary class was found to be crucial for separating cells in contact with each other. To evaluate the functional performance of our segmentation network, we interrogated the correlation between biomarker values after automated and manual segmentation on the test set. Among all 134 image FOVs in our test group, the correlation for the number of cells counted was 0.92, with a 0.11 false positive and 0.13 false negative cell detection rate. In single cells, the correlation in cell area was 0.936 (##FIG##2##Fig. 3C##), and the correlation for mean fluorescent intensity was greater than 0.996 for all markers (##FIG##2##Fig. 3D##).</p>", "<title>Single-cell profiling and clustering in patient samples</title>", "<p id=\"P12\">We finally applied the entire CycloNET process to an example patient dataset. The input data were taken from seven cycles of staining (##FIG##3##Fig. 4A##, top row), with each cycle probing three protein markers found in immune cells. The algorithm corrected spatial shifts between cycles (##FIG##3##Fig. 4A##, middle) and segmented immune cells based on CD45 and nucleus stains (##FIG##3##Fig. 4A##, bottom). Single-cell masks were then applied to the aligned image stacks to extract marker expression in individual cells (##FIG##3##Fig. 4B##).</p>", "<p id=\"P13\">To further aid in interpreting the large single-cell dataset, we incorporated dimensionality reduction and an unsupervised clustering protocol. Dimensionality reduction was performed by the uniform manifold approximation and projection (UMAP) algorithm, and potential subpopulation clusters were identified by the Hierarchical density-based spatial clustering of applications with noise (HDBSCAN) algorithm (see Methods for details)^{##UREF##6##22##, ##UREF##7##23##}. ##FIG##4##Figure 5## shows an exemplary profiling result of an HNSCC FNA specimen. CycloNET identified 1528 immune cells in this sample and generated single-cell data for 17 protein markers. After dimensionality reduction, a total of 9 distinct clusters were assigned by the HDBSCAN algorithm (##FIG##4##Fig. 5A##). Each cluster was found to have a unique biomarker profile based on average intensities (##FIG##4##Fig. 5B##).</p>", "<p id=\"P14\">We investigated the detected clusters by identifying common immune cell phenotypes (##TAB##0##Table 1##). Neutrophils and macrophages were among the most abundant, while natural killer and dendritic cells were the least present (##FIG##4##Fig. 5C##). Some of the manually defined phenotype assignments (e.g., CD4+ T cells) closely matched existing clusters, while others spread across different clusters (e.g., macrophages). A close look revealed different phenotypes in a nominal cell type (##FIG##4##Fig. 5D##). For example, neutrophils were found in clusters 1 and 3, separated mainly by p16 expression. Similarly, a sizable percentage of macrophage cells were found to differ in CD11c expression (##FIG##4##Fig. 5E##). Overall, the results demonstrate CycloNET’s capability of streamlining multi-dimensional single-cell phenotyping.</p>" ]

[ "<title>RESULTS AND DISCUSSIONS</title>", "<title>CycloNET pipeline</title>", "<p id=\"P5\">##FIG##0##Figure 1A## shows CycloNET’s overall workflow. The input data are raw fluorescent images obtained through cyclic immunofluorescent microscopy. The computational pipeline collates acquired multichannel images, performs image registration, detects individual cells, and presents summary statistics. In the current work, we used images generated by the fast analytical screening technique (FAST) as previously reported^{##REF##35666645##18##, ##REF##34233961##19##}; FAST uses custom-designed antibody probes to cyclicly image 20–40 molecular markers within an hour^{##REF##35666645##18##, ##REF##34233961##19##} (##FIG##0##Fig. 1B##). A given specimen underwent rounds of imaging and quenching cycles; at each cycle, several FOVs were captured, resulting in up to 5 gigapixels of visual information per sample. We implemented a graphical user interface to streamline image analysis (##FIG##0##Fig. 1C##). Under the hood, it was equipped with i) a preprocessor that automatically grouped images by cycle and FOV, ii) a registration module that detected translations between imaging cycles, and iii) a pre-trained neural network for task-specific cell segmentation.</p>", "<title>Image registration</title>", "<p id=\"P6\">Between fluorescent staining and destaining cycles, sample slides were manually removed and remounted onto the microscope, introducing spatial offsets. Though negligible for tissue-level assessments, these shifts required careful correction for accurate single-cell metrics. Standard imaging software for spatial alignment faced challenges, including (i) algorithms that were suboptimal for large image sets and (ii) errors in registration due to sample debris, low cellularity, or low signal-to-noise ratio (SNR). We addressed these drawbacks in our custom-designed processing pipeline, optimized for cyclic imaging.</p>", "<title>Pre-processing</title>", "<p id=\"P7\">##FIG##1##Figure 2A## illustrates the CycloNET image registration algorithm. The algorithm processes image datasets derived from several staining cycles (denoted as <italic toggle=\"yes\">N</italic> cycles). Multiple fields of view (FOVs) are collected for each staining cycle to capture as many cells as possible. In this scenario, each FOV tile set can be represented as FOV_{<italic toggle=\"yes\">i</italic>,<italic toggle=\"yes\">j</italic>}, where <italic toggle=\"yes\">i</italic> indicates an individual field of view (<italic toggle=\"yes\">i</italic> = 1, 2, …, <italic toggle=\"yes\">M</italic>) and <italic toggle=\"yes\">j</italic> indicates the particular cycle number (<italic toggle=\"yes\">j</italic> = 1, 2, ..., <italic toggle=\"yes\">N</italic>). We found that images from any two cycles had similar spatial offsets across all FOVs, allowing us to align full cycles by using a single FOV per cycle. Thus, the FOV numbers to be processed for alignment is reduced from (<italic toggle=\"yes\">N</italic> × <italic toggle=\"yes\">M</italic> × <italic toggle=\"yes\">C</italic>) to just (<italic toggle=\"yes\">N</italic> × <italic toggle=\"yes\">C</italic>), where <italic toggle=\"yes\">C</italic> is the number of fluorescent channels used in microscopy. To further minimize dataset size and improve accuracy in low-signal images, the <italic toggle=\"yes\">C</italic>-channel fluorescent signals were normalized and collapsed by maximum projection, resulting in a set of <italic toggle=\"yes\">N</italic> images. By using the maximum projection across individual channels, we enhanced the visibility of cells, as some cells might be visible in one channel and not another, providing more “anchoring points” for accurate alignment. Finally, a reference is selected among these <italic toggle=\"yes\">N</italic> collapsed images, to which all other cycles are aligned. These are labeled in ##FIG##1##Fig 2A## as “Reference” and “Cycle images.”</p>", "<p id=\"P8\">The translations between the N images were calculated by masked cross-correlation. This strategy reduced the impact of cellular debris by focusing only on cell-shaped foreground objects specified by the cell mask (see <bold>Methods</bold> for details). At the time of cross-correlation calculation, images were also downsampled two-fold and randomly cropped to further reduce dataset size. This process was repeated for some <bold><italic toggle=\"yes\">t</italic></bold> trials, with different random crops at each trial. Finally, the mean shift (after removing outliers) is calculated and assigned to all FOVs in a given staining cycle.</p>", "<p id=\"P9\">##FIG##1##Figure 2B## shows snapshots of the image registration process. The maximum projection across individual channels increased the number of cells available as anchoring points for the cross-correlation algorithm. One such anchoring point is indicated by red circles, demonstrating how a cluster of cells shifts to overlap with the reference cycle after alignment. We tested our algorithm on a set of five human specimens (FOVs per sample: 20, 24, 35, 30, 36). On average, a single field of view with seven imaging cycles was aligned in 42 seconds, while a full specimen containing ~20 FOVs typically aligns in under 6 minutes using an Intel^® Core^™ i7–6850K processor. Alignment failed in 5% of 142 FOVs, typically in FOVs with very low contrast in the reference signal or low cellularity.</p>", "<title>Cell segmentation and validation</title>", "<p id=\"P10\">Following alignment, CycloNET generated segmentation masks for individual cells by applying a pre-trained neural network. To detect immune cells in the sample, the algorithm was trained on image stacks consisting of CD45 and nucleus (DAPI) channels, as described in ##FIG##2##Fig. 3A##. Using a modified U-Net structure, we designed a neural network to produce a 2-dimensional array where each pixel was labeled as either: i) background, ii) cell interior, or iii) cell boundary^{##UREF##4##20##} (##FIG##2##Fig. 3A##). The encoded features used to construct segmentation masks were regularized by secondary auto-encoding and cell-counting tasks^{##UREF##5##21##}, resulting in a loss function,\n\nwhere the segmentation loss, , was split into the cross entropy loss () for pixel-by-pixel classification and the per-class Jaccard loss metric and represent the mean squared error loss metrics for the cell counting and autoencoding tasks, respectively. Finally, represents the L2 regularization loss applied to all weights in the network, and assign weights to each loss type. The network was trained using the CD45 and nuclear stains for 17 human specimen FOVs (See Supp 2). At the end of the training, the average F1 score among all pixel classes was 0.94. A single FOV was scanned for CD45+ immune cells in 17.9 seconds (<italic toggle=\"yes\">n</italic> = 134).</p>", "<p id=\"P11\">##FIG##2##Figure 3B## shows the comparison between manual and CycloNET segmentations. Defining the cell boundary class was found to be crucial for separating cells in contact with each other. To evaluate the functional performance of our segmentation network, we interrogated the correlation between biomarker values after automated and manual segmentation on the test set. Among all 134 image FOVs in our test group, the correlation for the number of cells counted was 0.92, with a 0.11 false positive and 0.13 false negative cell detection rate. In single cells, the correlation in cell area was 0.936 (##FIG##2##Fig. 3C##), and the correlation for mean fluorescent intensity was greater than 0.996 for all markers (##FIG##2##Fig. 3D##).</p>", "<title>Single-cell profiling and clustering in patient samples</title>", "<p id=\"P12\">We finally applied the entire CycloNET process to an example patient dataset. The input data were taken from seven cycles of staining (##FIG##3##Fig. 4A##, top row), with each cycle probing three protein markers found in immune cells. The algorithm corrected spatial shifts between cycles (##FIG##3##Fig. 4A##, middle) and segmented immune cells based on CD45 and nucleus stains (##FIG##3##Fig. 4A##, bottom). Single-cell masks were then applied to the aligned image stacks to extract marker expression in individual cells (##FIG##3##Fig. 4B##).</p>", "<p id=\"P13\">To further aid in interpreting the large single-cell dataset, we incorporated dimensionality reduction and an unsupervised clustering protocol. Dimensionality reduction was performed by the uniform manifold approximation and projection (UMAP) algorithm, and potential subpopulation clusters were identified by the Hierarchical density-based spatial clustering of applications with noise (HDBSCAN) algorithm (see Methods for details)^{##UREF##6##22##, ##UREF##7##23##}. ##FIG##4##Figure 5## shows an exemplary profiling result of an HNSCC FNA specimen. CycloNET identified 1528 immune cells in this sample and generated single-cell data for 17 protein markers. After dimensionality reduction, a total of 9 distinct clusters were assigned by the HDBSCAN algorithm (##FIG##4##Fig. 5A##). Each cluster was found to have a unique biomarker profile based on average intensities (##FIG##4##Fig. 5B##).</p>", "<p id=\"P14\">We investigated the detected clusters by identifying common immune cell phenotypes (##TAB##0##Table 1##). Neutrophils and macrophages were among the most abundant, while natural killer and dendritic cells were the least present (##FIG##4##Fig. 5C##). Some of the manually defined phenotype assignments (e.g., CD4+ T cells) closely matched existing clusters, while others spread across different clusters (e.g., macrophages). A close look revealed different phenotypes in a nominal cell type (##FIG##4##Fig. 5D##). For example, neutrophils were found in clusters 1 and 3, separated mainly by p16 expression. Similarly, a sizable percentage of macrophage cells were found to differ in CD11c expression (##FIG##4##Fig. 5E##). Overall, the results demonstrate CycloNET’s capability of streamlining multi-dimensional single-cell phenotyping.</p>" ]

[ "<title>CONCLUSIONS</title>", "<p id=\"P15\">High-throughput, multiplexed microscopy generates expansive data (&gt;giga bytes) that are often beyond manual inspection. The developed CycloNET can simplify large-scale dataset analysis and extract biological information. CycloNET integrated three key modules: i) a custom image-alignment algorithm designed to minimize errors and computation time; ii) a trained neural network to provide consistent, accurate segmentation masks for target cells while avoiding cellular debris or non-target cell types; and iii) a dimensionality-reduction to visualize the multi-dimensional single-cell data. Equipped with these technical capacities, CycloNET efficiently processed gigapixels of imaging data (&gt;5,000 cells) and produced a single-cell dataset, all in under 2 minutes. The data produced by CycloNET further revealed sub-clusters within conventional immune cell phenotypes.</p>", "<p id=\"P16\">The neural network that identifies single cells is easily adapted to different tasks. In the presence of novel datasets, the pre-trained network can be re-trained from scratch, fine-tuned, or even combined with other networks. In future studies, we will explore techniques to facilitate transfer learning so that researchers may capture the full mosaic of different sample structures and cell types. We can also explore methods to capture sub-cellular information more efficiently. For example, per single cell, high-resolution details such as the location and concentration of fluorescent staining are lost when utilizing averaging metrics. By contrast, an intelligent state-of-the-art auto-encoding scheme could summarize morphologic information with little supervision. With this expansion in CycloNET’s analytical power, researchers will be better equipped to understand underlying biological mechanisms, improving diagnosis and prognosis metrics and potentially leading to new therapeutic approaches.</p>" ]

[ "<p id=\"P1\">Recent advances in microscopy allow scientists to generate vast amounts of biological data from a single biopsy sample. Cyclic fluorescence microscopy, in particular, enables multiple targets to be detected simultaneously. This, in turn, has deepened our understanding of tissue composition, cell-to-cell interactions, and cell signaling. Unfortunately, analysis of these datasets can be time-prohibitive due to the sheer volume of data. In this paper, we present CycloNET, a computational pipeline tailored for analyzing raw fluorescent images obtained through cyclic immunofluorescence. The automated pipeline pre-processes raw image files, quickly corrects for translation errors between imaging cycles, and leverages a pre-trained neural network to segment individual cells and generate single-cell molecular profiles. We applied CycloNET to a dataset of 22 human samples from head and neck squamous cell carcinoma patients and trained a neural network to segment immune cells. CycloNET efficiently processed a large-scale dataset (17 fields of view per cycle and 13 staining cycles per specimen) in 10 minutes, delivering insights at the single-cell resolution and facilitating the identification of rare immune cell clusters. We expect that this rapid pipeline will serve as a powerful tool to understand complex biological systems at the cellular level, with the potential to facilitate breakthroughs in areas such as developmental biology, disease pathology, and personalized medicine.</p>" ]

[]

[ "<title>Funding:</title>", "<p id=\"P22\">NIH R01CA264363 (H.L.), R01CA239078 (H.L.), R01CA237500 (H.L.), R21CA267222 (H.L.), U01CA284982 (H.L.), R01HL163513 (H.L.), U01CA279858 (H.L.).</p>", "<title>Data availability</title>", "<p id=\"P23\">Imaging data that support the findings of this study are available from the corresponding authors on reasonable request, subject to approval from the Institutional Review Board of the Massachusetts General Hospital. Source codes for the image analysis will be available (<ext-link xlink:href=\"https://csb.mgh.harvard.edu/bme_software\" ext-link-type=\"uri\">https://csb.mgh.harvard.edu/bme_software</ext-link>) upon the publication of the current manuscript.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1.</label><caption><title>Overview of CycloNET analysis pipeline.</title><p id=\"P26\"><bold>(A)</bold> Schematic for computationally assisted biopsy analysis. After immunofluorescent images are collected via cyclic staining, CycloNET automatically aligns images from different cycles, produces a single-cell segmentation mask, and generates 2-dimensional representations for multiplexed biomarker data. <bold>(B)</bold> Cyclic staining protocol. A sample is stained, and several fields of view, <italic toggle=\"yes\">m</italic>, are captured by a microscope. The fluorescent signal is then quenched, and the background “quenched signal” is imaged. This process is repeated <italic toggle=\"yes\">N</italic> times to produce the full dataset. <bold>(C)</bold> Analysis user interface. A simple user interface was developed to group sample images and execute the CycloNET modules described above.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2.</label><caption><title>Correction of inter-cycle image offsets.</title><p id=\"P27\"><bold>(A)</bold> Image registration algorithm. A single specimen undergoes <italic toggle=\"yes\">N</italic> imaging cycles, each cycle producing <italic toggle=\"yes\">m</italic> FOVs. One cycle is chosen as the “reference” to which all other cycles are aligned. For a single FOV, cycles are collapsed by maximum projection, and a threshold mask is computed to roughly determine the presence of cell-sized objects. Each image is then downsampled and randomly cropped. Finally, the masked cross-correlation is computed over <italic toggle=\"yes\">t</italic> trials, and the mean shift is returned. <bold>(B)</bold> Visual examples for cycle registration. The top row shows the original data after maximum projection. The second row demonstrated the threshold mask used to search for cell-sized objects. The heat maps in the third-row show where the correlation between the current cycle and the reference image is highest. The final row shows the aligned images. Red circles highlight a cell cluster of interest. In the final row, this cluster is in the same position for all cycles.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3.</label><caption><title>Single-cell segmentation by a neural network.</title><p id=\"P28\"><bold>(A)</bold> Neural network architecture. The CD45 and DAPI (nuclear) stains were used as the input to a modified U-NET architecture to produce a 3-class segmentation mask. The bottleneck features were also used for a secondary auto-encoding task. <bold>(B)</bold> Segmentation boundaries. Green and red boundaries, respectively, represent cell boundary pixels by manual and CycloNET segmentations. Yellow represents the overlap between the two methods. <bold>(C)</bold> Functional comparisons. (Left) A comparison is made between the areas of manually segmented cells and neural network-predicted areas. (Right) Similarly, the mean fluorescent signal per cell was compared under manual and algorithmic segmentation.</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4.</label><caption><title>Fluorescence Image stack to single-cell metrics.</title><p id=\"P29\"><bold>(A)</bold> We visualize the full image stack of fluorescent signals as single-cell data is extracted. (Top row) For each cycle, we visualize 3 biomarker signals in an RGB image. A single immune cell is highlighted by a red circle, which is at different positions in each imaging cycle. (Middle row) The image stacks are aligned for inter-cycle translations. (Bottom row) Among the many cells seen in the window, immune cells are segmented by the neural network. Each cell is highlighted by a white border and a numeric ID. <bold>(B)</bold> The fluorescent signal within each cell’s interior is mean aggregated, and a z score is calculated for all cells. The heat map shows each cell’s relative signal strength for each of the 18 markers.</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5.</label><caption><title>Patient data visualization.</title><p id=\"P30\"><bold>(A)</bold> Dimensionality reduction. The multi-dimensional single-cell dataset generated from a patient sample was projected onto a two-dimensional manifold by the UMAP algorithm. Cells were clustered by the unsupervised HBDSCAN algorithm. <bold>(B)</bold> Mean marker intensities were calculated for each cluster, and relative marker intensities were visualized in a heat map. <bold>(C)</bold> Common immune cell types were overlaid to individual cells in existing clusters. <bold>(E)</bold> CycloNET subclassified single-cell types according to their distinct phenotypes. For a given immune cell type, its relative presence among unsupervised clusters is summarized. <bold>(D)</bold> Example images show macrophages and neutrophils belonging to different unsupervised clusters.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\"><label>Table 1.</label><caption><p id=\"P31\">Marker combinations for immune cell phenotyping</p></caption><table frame=\"hsides\" rules=\"none\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th rowspan=\"2\" align=\"center\" valign=\"middle\" colspan=\"1\">Cell type</th><th colspan=\"2\" align=\"center\" valign=\"top\" rowspan=\"1\">Markers<hr/></th></tr><tr><th align=\"center\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Positive</th><th align=\"center\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Negative</th></tr></thead><tbody><tr><td colspan=\"3\" align=\"left\" valign=\"top\" rowspan=\"1\">\n<hr/>\n</td></tr><tr><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Immune cells</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD45</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CK56</td></tr><tr><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD4+ T cells</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD45, CD3, CD4</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD11b</td></tr><tr><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD8+ T cells</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD45, CD3. CD8</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD11b</td></tr><tr><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Natural killer cells</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD45, CD56</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD3, CD11b</td></tr><tr><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">B cells</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD45, CD20</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD3, CD11b</td></tr><tr><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Neutrophils</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD45, CD11b, CD66b</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD3</td></tr><tr><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Macrophages</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD45, CD11b, CD68</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD3</td></tr><tr><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Dendritic cells</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD45, CD11b, CD11c</td><td align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">CD3</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group><fn id=\"FN1\"><p id=\"P24\">Competing interests</p><p id=\"P25\">The authors declare no competing interests.</p></fn></fn-group>" ]

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[ "<title>Background</title>", "<p id=\"P7\">Amyotrophic lateral sclerosis (ALS) is a rare neurodegenerative disease characterized by the progressive loss of upper motor neurons in the cortex and lower motor neurons of the brainstem and spinal cord. Even with FDA-approved disease modifying medication and palliation by artificial nutrition and ventilation, the prognosis is poor and death from accumulating paralysis occurs a median of 32 months after symptoms first manifest(##REF##29399045##1##). Over the last 30 years, genetic study of the 5–10% of ALS patients with family history(##REF##23941283##2##, ##REF##24306004##3##) have securely implicated ~20 monogenic causes and showed possible association to a similar number of genes (<ext-link xlink:href=\"https://clinicalgenome.org/affiliation/40096/\" ext-link-type=\"uri\">https://clinicalgenome.org/affiliation/40096/</ext-link>). Causative mutations in the most prevalent ALS genes (<italic toggle=\"yes\">C9ORF72, SOD1</italic>, <italic toggle=\"yes\">TARDBP</italic>, and <italic toggle=\"yes\">FUS)</italic> explain ~70% of familial ALS(##REF##29154141##4##, ##REF##26515623##5##). Due in part to incomplete penetrance, 10% of simplex ALS cases also carry mutations in these same genes(##REF##25700176##6##).</p>", "<p id=\"P8\">A paucity of unsolved ALS pedigrees for family studies has intersected with falling sequencing costs for large-scale sequencing to allow gene discovery studies based on rare variant burden or collapsing methods on cohorts using predominantly simplex patients. Since our group first used this approach to implicate <italic toggle=\"yes\">TBK1</italic> and <italic toggle=\"yes\">NEK1,</italic> others have also identified <italic toggle=\"yes\">DNAJC7, TUBA4A</italic> and several candidates(##REF##25700176##6##–##REF##25374358##9##). These analyses utilized the entire gene or recognizable functional domains as the regions for burden testing (##REF##25700176##6##, ##REF##30940688##7##) and were restricted to cohorts with European ancestry, or with less than 5% non-European ALS cases. Recognizing that power for discovery could be improved by a) increasing case and control numbers, b) diversifying the ancestries of participants, and c) collapsing on domains known to be intolerant of variation, we conducted both standard gene and intolerant domain-based collapsing analyses on 6,970 multi-ethnic ALS cases and ancestry-matched controls. Primary lateral sclerosis (PLS) is also a neurodegenerative disease of motor neurons with clinical features, neuropathology, and some genetics that overlaps with ALS(##REF##17296839##10##–##REF##27066542##12##). PLS is nearly always simplex and 20 times rarer than ALS(##REF##32029539##13##). Because large-scale sequencing studies of ALS often include PLS patients, we were able to conduct a gene-based collapsing analysis in 166 PLS multi-ethnic cases and ancestry matched controls.</p>" ]

[ "<title>Methods</title>", "<title>Study population</title>", "<p id=\"P9\">All samples and data came from participants that provided written, informed consent for genetic studies that had been IRB-approved at each contributing center. The study cohort includes participants from the Genomic Translation for ALS Care (GTAC study), the Columbia University Precision Medicine Initiative for ALS, the ALS COSMOS Study Group, the PLS COSMOS Study Group, the New York Genome Consortium, and the ALS Sequencing Consortium (IRB-approved genetic studies from Columbia University Medical Center, including the Coriell NINDS repository), University of Massachusetts at Worchester, Stanford University (including samples from Emory University School of Medicine, the Johns Hopkins University School of Medicine, and the University of California, San Diego), Massachusetts General Hospital Neurogenetics DNA Diagnostic Lab Repository, Duke University, McGill University (including contributions from Saint-Luc and Notre-Dame Hospital of the Centre Hospitalier de l’Université de Montréal [CHUM], [University of Montreal]), Gui de Chauliac Hospital of the CHU de Montpellier (Montpellier University), Pitié Salpêtrière Hospital, Fleurimont Hospital of the Centre Hospitalier Universitaire de Sherbrooke (CHUS) (University of Sherbrooke), Enfant Jésus Hospital of the Centre hospitalier affilié universitaire de Québec (CHA) (Laval University), Montreal General Hospital, Montreal Neurological Institute and Hospital of the McGill University Health Centre, the University of Edinburgh Scotland, and Washington University in St. Louis (including contributions from Houston Methodist Hospital, Virginia Mason Medical Center, University of Utah, and Cedars Sinai Medical Center). Participants were determined to have ALS or PLS by neuromuscular specialists at tertiary motor neuron disease care centers with expertise in distinguishing between the two. ALS diagnoses were based on the El Escorial Criteria in all cases. For PLS, explicit criteria requiring &gt;3 years of symptoms without conversion to ALS were used for 79 of the 172 PLS participants (##REF##24564738##14##). The criteria used for the remaining PLS diagnoses were not available.</p>", "<p id=\"P10\">Controls were selected from &gt;100,000 whole-exome or -genome sequenced individuals housed in the IGM Data Repository. Individuals with known neurodegenerative disease were excluded. However, none of the controls were screened for neurodegenerative disease. All participants consented to the use of DNA in genetic research.</p>", "<title>Whole exome and genome sequencing</title>", "<p id=\"P11\">DNA sequencing was performed at Columbia University, the New York Genome Center, Duke University, McGill University, Stanford University, HudsonAlpha, and University of Massachusetts, Worcester. Kits used to conduct whole-exome capture are as follows: Agilent All Exon kits (50MB, 65MB, and CRE), Nimblegen SeqCap EZ Exome Enrichment kits (V2.0, V3.0, VCRome, and MedExome), and IDT Exome Enrichment panel. There were 2,185 participants with ALS who were sequenced using Nimblegen SeqCap EZ Exome Enrichment kits and 51 who were sequenced using the IDT Exome Enrichment panel (<underline><bold>Supplemental Table 1</bold></underline>). While 1,272 controls were evaluated using the Aligent All Exon kits, 8,498 with the IDT Exome Enrichment panel, and 11,201 with the Nimblegen SeqCap EZ Exome Enrichment kits. Sequencing was performed using Illumina GAIIx, HiSeq 2000, HiSeq 2500, and NovaSeq 6000 sequencers according to standard protocols. Whole genome sequencing was conducted at the New York Genome Center and in-house at the IGM. Sample-level BAM files were transferred from the New York Genome Center to the IGM (n = 3,418). An additional 1,316 genomes were processed by the IGM. There were 1,553 genomes in our control cohort (<underline><bold>Supplemental Table 1</bold></underline>). Data were aligned to the human reference genome (NCBI Build 37) using DRAGEN (Edico Genome, San Diego, CA, USA). Picard (<ext-link xlink:href=\"http://picard.sourceforge.net/\" ext-link-type=\"uri\">http://picard.sourceforge.net</ext-link>) was used to remove duplicate reads and to process lane-level BAM files to create a sample-level BAM file. GATK was used to recalibrate base quality scores, realign around indels, and call variants utilizing the Best Practices recommendations v3.6 (##REF##20644199##15##). Variants were annotated using ClinEff and the Analysis Tool for Annotated Variants (ATAV), an in-house IGM annotation tool (##REF##33757430##16##). Variants were annotated with the Genome Aggregation Database (gnomAD) v2.1 frequencies, regional-intolerance metrics, and the clinical annotations by the Human Gene Mutation Database (HGMD), ClinVar, and Online Mendelian Inheritance in Man (OMIM). Exonic regions were retained for downstream statistical analyses.</p>", "<title>Sample and variant quality control</title>", "<p id=\"P12\">Samples reporting &gt;2% contamination according to verifyBamID (##REF##23103226##17##) and those with consensus coding sequence (CCDS release 20) &lt;90% were excluded from these analyses. KING (##REF##20926424##18##) was used to test for relatedness. Only unrelated (up to second-degree) individuals were included in these analyses. For related pairs, samples were chosen to prefer cases. Samples where X:Y coverage ratios did not match expected sex were excluded.</p>", "<p id=\"P13\">Only variants within the CCDS or the 2 bp canonical sites were included in these analyses. These variants were also required to have a quality score of at least 50, a quality by depth score of at least 5, genotype quality score of at least 20, read position rank sum of at least −3, mapping quality score of at least 40, mapping quality rank sum greater than −10, and a minimum coverage of at least 10. SNVs had a maximum Fisher’s strand bias of 60, while indels had a maximum of 200. For heterozygous genotypes, the alternative allele ratio was required to be greater than or equal to 30%. Only variants with the GATK Variant Quality Score Recalibration filter “PASS”, “VQSRTrancheSNP90.00to99.00”, or “VQSRTrancheSNP99.00to99.90” were included. Variants were excluded if they were marked by EVS, ExAC, or gnomAD as being failures (<ext-link xlink:href=\"http://evs.gs.washington.edu/EVS/\" ext-link-type=\"uri\">http://evs.gs.washington.edu/EVS/</ext-link>).</p>", "<title>Clustering, ancestry, and coverage harmonization</title>", "<p id=\"P14\">A neural network pre-trained on samples of known ancestry was used to calculate probability estimates for six ancestry groups (African, East Asian, European, Hispanic, Middle Eastern, and South Asian). Methods for characterizing samples into clusters has been previously described (##REF##33326012##19##).</p>", "<p id=\"P15\">To ensure balanced sequencing coverage of evaluated sites between cases and controls, we imposed a statistical test of independence between the case/control status and coverage as previously described (##REF##28099038##20##). Sites were removed where the absolute difference in percentages of cases and controls with at least 10x coverage was greater than 7%. Samples were then pruned using this method on a cluster-by-cluster basis. Through this approach, approximately 7– 11% were removed. Clusters with less than 5 participants were not included in these analyses, thereby removing 6 participants with PLS but none with ALS.</p>", "<title>Variant-level statistical analysis</title>", "<p id=\"P16\">The models that were used to test for associations of nonsynonymous coding or canonical splice variants with outcome included variants with MAF &lt;0.1% for each population represented in gnomad and internal AF of &lt;0.1%. Models tested were a standard gene-unit collapsing analysis, and a domain-unit analysis. The models used for these analyses were previously described (##REF##30940688##7##). A domain-based approach utilizing sub-region Residual Variation Intolerance Score (subRVIS) domain percentage(##REF##30940688##7##) with a threshold of 25 was also used to evaluate case enrichment of rare variants. The full list of 18,653 CCDS genes was analyzed for each model. Genes with at least one qualifying variant were included for analyses. As we are meta analyzing across clusters an exact 2-sided Cochran-Mantel-Haenszel test was used (using the statistical package in R v3.6). Study-wide significance was determined by accounting for 6 nonsynonymous models- multiplicity-adjusted significance threshold α = 4.9 × 10⁻⁷ (<underline><bold>Supplemental Table 2</bold></underline>). Model inflation was calculated using empirical (permutation-based) expected probability distributions as described by Povysil and colleagues (##REF##33326012##19##).</p>", "<title>ALS and PLS rare variant burden testing</title>", "<p id=\"P17\">We conducted both standard gene and intolerant domain-based collapsing analyses on 6,970 multi-ethnic ALS cases (87% European) and 22,534 ancestry-matched controls. Standard gene collapsing analyses identified case enrichment of rare variants (minor allele frequency of 0.001) in an ALS cohort with 12 sub-population groups (<underline><bold>Supplemental Figure 1A</bold></underline>) that correspond to ancestry-based clusters (<underline><bold>Supplemental Figure 1B</bold></underline>; <underline><bold>Supplemental Table 3</bold></underline>). Analyses were conducted on clusters with at least 3 cases. Controls were drawn from individuals sequenced for phenotypes/diseases with no known association with ALS (<underline><bold>Supplemental Table 4</bold></underline>). As expected, a negative control analysis for rare synonymous variants found no case-enrichment (<underline><bold>Supplemental Figure 2</bold></underline>). Because gene-based collapsing considers variation across the entire gene, regions that are tolerant of variation could swamp case-enrichment signals originating from regions that are intolerant of variation (##REF##30940688##7##). To overcome this limitation, we conducted rare variant collapsing on domains that are intolerant to variation as defined as a subRVIS domain score threshold of 25, a cutoff based on threshold testing.</p>", "<p id=\"P18\">As large-scale sequencing studies of ALS often include PLS patients, we were able to conduct a gene-based collapsing analysis in 166 PLS multi-ethnic cases (88% European) and 17,695 ancestry matched controls (<underline><bold>Supplemental Figure 3</bold></underline>; <underline><bold>Supplemental Tables 5</bold></underline>). We expected the study would be underpowered for securely implicating causative genes but used this as an opportunity to generate candidates for future study.</p>", "<title>ALS gene set enrichment analyses</title>", "<p id=\"P19\">An ALS gene set enrichment analysis was conducted using the gene strength association list outlined in ##TAB##0##Table 1##. We utilized the qualifying variants that were associated with ALS in each gene set category and used the exact two-sided CMH test to analyze burden of ALS genes defined by gene set. These lists were curated using data published by Gregory and colleagues (##UREF##0##21##). As outlined, “ALS Confirmed” genes were found to have ample published replication evidence, while ‘ALS Plus’ genes had some replication data and/or functional evidence for an association with ALS. However, ‘ALS Replication Needed’ genes, required additional replication analyses and/or functional data, and ‘ALS Weak Evidence’ genes were genes that overlapped with ALS phenotypically.</p>" ]

[ "<title>Results</title>", "<title>Rare variant burden testing</title>", "<p id=\"P20\">Collapsing analysis of all rare functional variants (missense and protein truncating variants) (<underline><bold>Supplemental Table 2</bold></underline>) found genome-wide and study-wide significant (p &lt; 4.9 × 10⁻⁷) case-enrichment for known ALS genes <italic toggle=\"yes\">SOD1</italic>, <italic toggle=\"yes\">TARDBP</italic>, <italic toggle=\"yes\">TBK1</italic> (OR=19.18, p = 3.67 × 10⁻³⁹; OR=4.73, p = 2 × 10⁻¹⁰; OR=2.3, p = 7.49 × 10⁻⁹, respectively) and control-enrichment for <italic toggle=\"yes\">ALKBH3</italic> (OR=0.26, p = 4.88 × 10⁻⁷) (##FIG##0##Figure 1A##; <underline><bold>Supplemental Data</bold></underline>). Although <italic toggle=\"yes\">SOD1, TBK1</italic> and <italic toggle=\"yes\">TARDBP</italic> are definitive ALS genes, we were intrigued by the identification of controlenriched <italic toggle=\"yes\">ALKBH3.</italic> Control-enrichment was not explained by sequencing methodology, ancestry cluster, or specific phenotype/disease population within the control cohort. Because <italic toggle=\"yes\">ALKBH3</italic> plays a role in DNA repair(##REF##35277482##22##), a mechanism increasingly implicated in ALS pathogenesis(##REF##35572138##23##), we attempted to replicate this novel association by analyzing summary statistics from the Project MinE cohort, which is similar in size to ours (##REF##34873335##24##). None of the available models focused on variation as rare as in our analyses, but at a higher minor allele frequency (MAF) for qualifying variants (0.005), a minor degree of control-enrichment was in fact observed (OR= 0.56, p = 3.96 × 10⁻⁴). This raises the possibility that rare missense and protein truncating variants (PTVs) in <italic toggle=\"yes\">ALKBH3</italic> could protect from ALS, a finding that requires validation in large cohorts.</p>", "<p id=\"P21\">Intolerant domain analyses implicated the same three known ALS genes (<italic toggle=\"yes\">SOD1</italic>, <italic toggle=\"yes\">TARDBP</italic>, and <italic toggle=\"yes\">TBK1</italic> at OR=20.63, p = 1.68 × 10⁻³⁸; OR=10.08, p = 3.62 × 10⁻¹⁶; and OR=3.15, p = 8.38 × 10⁻¹¹, respectively) (##FIG##0##Figure 1B##; <underline><bold>Supplemental Data</bold></underline>). The intolerant domain analysis did not improve over the gene-based analysis for <italic toggle=\"yes\">SOD1</italic> or <italic toggle=\"yes\">TBK1</italic> (##FIG##1##Figure 2##; ##FIG##2##Figure 3##) but doubled the odds ratio and significantly lowered the p-value obtained for <italic toggle=\"yes\">TARDBP.</italic> The improvement of the intolerant domain model (##FIG##0##Figure 1C##, ##FIG##0##1D##) stemmed from a significant drop (one-tailed z-score p=0.031) in the number of qualifying variants found in controls dispersed across tolerant regions, while highlighting qualifying variants in ALS cases predominantly in the intolerant C-terminal region.</p>", "<p id=\"P22\">Although most models showed no significant genes, the dominant PTV model showed significant case enrichment for <italic toggle=\"yes\">ANTXR2</italic> (OR=174.57, p=8.38 × 10⁻⁶) (##FIG##3##Figure 4##; <underline><bold>Supplemental Table 6</bold></underline>; <underline><bold>Supplemental Data</bold></underline>), a gene associated with brain connectivity changes and Alzheimer’s disease(##REF##31996736##25##). Currently, there are no additional large sequencing studies of PLS in which we could attempt replication.</p>", "<title>ALS gene set enrichment analyses</title>", "<p id=\"P23\">A gene set enrichment analysis of genes that were defined as ‘ALS Confirmed’ were significantly associated with the ALS for all dominant models, including PTV only (p = 9.12 × 10⁻²⁴), Missense &amp; PTV (p = 6.63 × 10⁻¹⁹), and Missense only (p = 1.03 × 10⁻¹⁹) (##FIG##4##Figure 5##). The synonymous model, which served as a control, showed no association (p = 0.79) between these genes and ALS. Genes that are weakly associated with ALS, ‘ALS Weak Evidence’, showed no significant enrichment of rare variants for the 4 models that were analyzed. The group of genes that were described as needing additional replication studies, ‘ALS Replication Needed’, showed a significant association with rare variants and ALS for the Missense &amp; PTV model (p = 4.6 × 10⁻³). For all other models, rare variants in these genes were not significantly associated with ALS. An analysis of genes that are characterized as ‘ALS Plus’ showed no significant association of rare variants with ALS for the 4 models that were analyzed.</p>" ]

[ "<title>Discussion</title>", "<title>Burden testing.</title>", "<p id=\"P24\">Conducting genic and intolerant domain based rare variant burden testing in a large multi-ethnic population provides insight into novel and established biological mechanisms in disease manifestations. Additionally, analyzing specific disease subtypes can capture critical disease pathways that could be targets for clinical intervention. Here we show, that performing burden testing in multi-ethnic populations and in disease subtypes found novel genetic associations in individuals diagnosed with ALS and PLS. These analyses implicated ALS genes that have previously been identified (<italic toggle=\"yes\">SOD1</italic>, <italic toggle=\"yes\">TARDBP</italic>, and <italic toggle=\"yes\">TBK1</italic>). We also identified <italic toggle=\"yes\">ALKBH3</italic> as a potentially protective gene that warrants further study in additional cohorts. In addition, we conducted the first rare variant collapsing analysis in PLS, identifying PTVs in <italic toggle=\"yes\">ANTXR2</italic>. This gene will need to be investigated further in larger PLS cohorts or in targeted functional analyses. Lastly, gene set enrichment analyses provide evidence that genes known to be associated with ALS show strong evidence to have a rare variant burden especially for protein truncating variants.</p>", "<title>ALKBH3 associates with ALS.</title>", "<p id=\"P25\">We found that genic burden testing of individuals diagnosed with ALS identified known risk genes (<italic toggle=\"yes\">SOD1</italic>, <italic toggle=\"yes\">TARDBP</italic>, and <italic toggle=\"yes\">TBK1</italic>) and a novel protective gene (<italic toggle=\"yes\">ALKBH3</italic>). <italic toggle=\"yes\">ALKBH3</italic> encodes for AlkB homolog 3, Alpha-Ketoglutarate Dependent Dioxygenase which protects against the cytotoxicity of methylating agents by repair of the specific DNA lesions (##REF##12486230##26##–##REF##37459543##28##). ALKBH3 potentially acts as a putative hyperactive promotor to suppress transcription associated DNA damage of highly expressed genes (##REF##26221185##29##). Genes that play a role in DNA repair and DNA damage response such as <italic toggle=\"yes\">TARDBP</italic>, <italic toggle=\"yes\">FUS</italic>, and <italic toggle=\"yes\">NEK1</italic> (##REF##18309045##30##–##REF##27455347##33##) are known to play a role in ALS potentially through neuronal death pathways.</p>", "<title>ANTXR2 associated with PLS.</title>", "<p id=\"P26\">Genic burden testing of protein truncating variants on individuals with PLS identified a suggestive gene (<italic toggle=\"yes\">ANTXR2</italic>). <italic toggle=\"yes\">ANTXR2</italic> encodes a receptor for anthrax toxin that may be involved in extracellular matrix adhesion. Variants in this gene have been associated with hyaline fibromatosis (##REF##22383261##34##, ##REF##14508707##35##), and has been shown to play a role in angiogenesis (##REF##19901963##36##). This finding adds to the number angiogenic genes that have been implicated in ALS including <italic toggle=\"yes\">VEGF</italic> and <italic toggle=\"yes\">ANG</italic> (##REF##16843725##37##).</p>", "<p id=\"P27\">While we identified a potentially important gene that is associated with PLS, we were limited in our sample size and will therefore need additional cohorts or functional studies to further investigate this finding. Additionally, there are potentially more ALS subtypes that could be investigated to better understand this heterogeneous disease. Lastly, unknown confounders could be contributing to the signal that are found in these association analyses.</p>" ]

[ "<title>Conclusions</title>", "<p id=\"P28\">In summary, we performed the largest rare variant analyses of a multi-ethnic population of patients with ALS to date. Our analysis did not identify new ALS risk genes but demonstrated that collapsing models informed by regions of intolerance can be useful for identifying genes where disease-associated variation is limited to regions with low background variation. This analysis also confirmed the association of the C-terminal domain of <italic toggle=\"yes\">TARDBP.</italic> We also identified <italic toggle=\"yes\">ALKBH3</italic> as a potentially protective gene that warrants further study in additional and larger cohorts. Finally, we conducted the first rare variant collapsing analysis in PLS, identifying PTVs in <italic toggle=\"yes\">ANTXR2</italic> as a candidate disease gene. This association and potential mechanisms for PTVs in this gene will need to be investigated further in larger PLS cohorts.</p>", "<p id=\"P29\">It is important to note that this analysis doubled the number of ALS cases and quadrupled the number of controls from our first study(##REF##25700176##6##) but remained underpowered for the identification of new ALS genes. A recently published rare variant burden analysis with a similar number of ALS cases did not identify new genes(##REF##34873335##24##) either, emphasizing the need for increasingly large genomically characterized ALS cohorts, especially in non-European populations.</p>" ]

[ "<p id=\"P1\">AUTHORS’ CONTRIBUTIONS</p>", "<p id=\"P2\">Conception and design: T.D.P., J.E.M., D.B.G., and M.B.H. Data acquisition: C.A.M.M., H.P., T.J.A., J.S.L, H.M., D.B.G., and M.B.H. Analysis: T.D.P. and M.B.H. Interpretation, draft and review, and final approval: all authors. T.D.P. and M.B.H. had full access to the study data and take responsibility for the integrity of the data and accuracy of analyses. All authors read and approved the final manuscript.</p>", "<title>Background:</title>", "<p id=\"P3\">Amyotrophic lateral sclerosis (ALS) is a neurodegenerative disease affecting over 30,000 people in the United States. It is characterized by the progressive decline of the nervous system that leads to the weakening of muscles which impacts physical function. Approximately, 15% of individuals diagnosed with ALS have a known genetic variant that contributes to their disease. As therapies that slow or prevent symptoms, such as antisense oligonucleotides, continue to develop, it is important to discover novel genes that could be targets for treatment. Additionally, as cohorts continue to grow, performing analyses in ALS subtypes, such as primary lateral sclerosis (PLS), becomes possible due to an increase in power. These analyses could highlight novel pathways in disease manifestation.</p>", "<title>Methods:</title>", "<p id=\"P4\">Building on our previous discoveries using rare variant association analyses, we conducted rare variant burden testing on a substantially larger cohort of 6,970 ALS patients from a large multi-ethnic cohort as well as 166 PLS patients, and 22,524 controls. We used intolerant domain percentiles based on sub-region Residual Variation Intolerance Score (subRVIS) that have been described previously in conjunction with gene based collapsing approaches to conduct burden testing to identify genes that associate with ALS and PLS.</p>", "<title>Results:</title>", "<p id=\"P5\">A gene based collapsing model showed significant associations with <italic toggle=\"yes\">SOD1</italic>, <italic toggle=\"yes\">TARDBP</italic>, and <italic toggle=\"yes\">TBK1</italic> (OR=19.18, p = 3.67 × 10⁻³⁹; OR=4.73, p = 2 × 10⁻¹⁰; OR=2.3, p = 7.49 × 10⁻⁹, respectively). These genes have been previously associated with ALS. Additionally, a significant novel control enriched gene, <italic toggle=\"yes\">ALKBH3</italic> (p = 4.88 × 10⁻⁷), was protective for ALS in this model. An intolerant domain based collapsing model showed a significant improvement in identifying regions in <italic toggle=\"yes\">TARDBP</italic> that associated with ALS (OR=10.08, p = 3.62 × 10⁻¹⁶). Our PLS protein truncating variant collapsing analysis demonstrated significant case enrichment in <italic toggle=\"yes\">ANTXR2</italic> (p=8.38 × 10⁻⁶).</p>", "<title>Conclusions:</title>", "<p id=\"P6\">In a large multi-ethnic cohort of 6,970 ALS patients, rare variant burden testing validated known ALS genes and identified a novel potentially protective gene, <italic toggle=\"yes\">ALKBH3</italic>. A first-ever analysis in 166 patients with PLS found a candidate association with loss-of-function mutations in <italic toggle=\"yes\">ANTXR2</italic>.</p>" ]

[]

[ "<title>ACKNOWLEDGEMENTS:</title>", "<p id=\"P30\">We thank the following groups for contributing ALS/PLS samples, sequencing, or clinical data:</p>", "<p id=\"P31\"><underline><bold><italic toggle=\"yes\">New York Genome Center ALS Consortium:</italic></bold></underline> J. Kwan, D. Sareen, J.R. Broach, Z. Simmons, X. Arcila-Londono, E.B. Lee, V.M. Van Deerlin, E. Fraenkel, L.W. Ostrow, F. Baas, N. Zaitlen, J.D. Berry, A. Malaspina, P. Fratta, G.A. Cox, L.M. Thompson, S. Finkbeiner, E. Dardiotis, T.M. Miller, S. Chandran, S. Pal, E. Hornstein, D.J. MacGowan, T. Heiman-Patterson, M.G. Hammell, N.A. Patsopoulos, J. Dubnau, and A. Nath.</p>", "<p id=\"P32\"><underline><bold><italic toggle=\"yes\">ALS Exome Sequencing Consortium</italic></bold><italic toggle=\"yes\">:</italic></underline> S.H. Appel, R.H. Baloh, R.S. Bedlack, R. Brown, W.K. Chung, S. Gibson, J.D. Glass, A. Gitler, D.B. Goldstein, T.M. Miller, R.M. Myers, S.M. Pulst, J.M. Ravits, G. Rouleau, E. Greene, N. Shneider, and W.W. Xin;</p>", "<p id=\"P33\"><underline><bold><italic toggle=\"yes\">Genomic Translation for ALS Care (GTAC) study</italic></bold></underline><bold>:</bold> S.H. Appel, R.H. Baloh, R.S. Bedlack, S. Chandran, L. Foster, S. Goutman, E. Green, C. Karam, D. Lacomis, G. Manousakis, T.M. Miller, S. Pal, D. Sareen, A. Sherman, Z. Simmons, L. Wang.</p>", "<p id=\"P34\"><underline><bold><italic toggle=\"yes\">ALS COSMOS Study Sites Group:</italic></bold></underline>\n<bold>Columbia University Coordinating Center, NY, NY:</bold> Hiroshi Mitsumoto, MD, DSc, Pam Factor-Litvak, PhD, Regina Santella, PhD, Howard Andrews, PHD; <bold>Texas Neurology, P.A., Dallas, TX:</bold> Daragh Heitzman, MD; <bold>Duke University, Durham, NC:</bold> Richard S. Bedlack, MD, PhD; <bold>California Pacific Medical Center, San Francisco, CA:</bold> Jonathan S. Katz, MD, Robert Miller, MD, Dallas Forshew; <bold>University of Kansas, Kansas City, KS:</bold> Richard J. Barohn, MD, PhD; <bold>Mayo Clinic, Rochester, MN; D</bold>r. Eric J. Sorenson, MD; <bold>University of California - Davis, Sacramento, CA:</bold> Bjorn E. Oskarsson, MD, PhD; <bold>University of Kentucky, Lexington, KY:</bold> Edward J. Kasarskis, MD, PhD; <bold>University of California - San Francisco, San Francisco, CA:</bold> Catherine Lomen-Hoerth, MD, PhD, Jennifer Murphy, PhD; <bold>University of Colorado, Aurora, CO:</bold> Yvonne D. Rollins, MD, PhD; <bold>University of California – Irvine, Orange, CA:</bold> Tahseen Mozaffar, MD; <bold>University of Nebraska, Omaha, NE;</bold> J. Americo M. Fernandes, MD; <bold>University of Iowa, Iowa City, IA:</bold> Andrea J. Swenson, MD; <bold>University of Texas - Southwestern, Dallas:</bold> Sharon P. Nations, MD; <bold>SUNY - Upstate Medical University, Syracuse, NY:</bold> Jeremy M. Shefner, MD, PhD; and <bold>Hospital for Special Care, New Britain, CT:</bold> Jinsy A. Andrews, MD, MS, Dr. Agnes Koczon-Jaremko.</p>", "<p id=\"P35\"><underline><bold><italic toggle=\"yes\">PLS COSMOS Study Group:</italic></bold></underline>\n<bold>Columbia University Irving Medical Center, NY, NY:</bold> Hiroshi Mitsumoto, MD, DSc, Peter L. Nagy, MD, PhD, Pam Factor-Litvak, PhD, PhD, Rejina Santella, PhD, Howard Andrews, PhD, Raymond Goetz, PhD; <bold>Icahn School of Medicine - Mount Sinai, NY, NY:</bold> Chris Gennings, PhD; <bold>University of California - San Francisco, San Fransisco, CA:</bold> Jennifer Murphy, PhD; <bold>National Institute of Neurological Disorders and Stroke, Bethesda, MD:</bold> Mary Kay Floeter, MD, PhD; <bold>University of Kansan Medical Center, Kansas City, KS:</bold> Richard J. Barohn, MD; <bold>University of Texas, Dallus, TX:</bold> Sharon Nations, MD; <bold>Western University, London, Ontario</bold>: Christen Shoesmith, MD; and <bold>University of Kentucky, Louisville, KT</bold>: Edward Kasarskis, MD, PhD.</p>", "<p id=\"P36\">We thank The Washington Heights–Inwood Columbia Aging Project (WHICAP) for the contribution of control samples. We also thank the WHICAP study participants and the WHICAP research and support staff for their contributions to this study: K. Welsh-Bomer, C. Hulette, and J. Burke; D. Valle, J. Hoover-Fong, and N. Sobriera; A. Poduri; S. Palmer; R. Buckley; K. Newby; The Murdock Study Community Registry and Biorepository Pro00011196; National Institute of Allergy and Infectious Diseases Center for HIV/AIDS Vaccine Immunology (CHAVI) (U19-AI067854); National Institute of Allergy and Infectious Diseases Center for HIV/AIDS Vaccine Immunology and Immunogen Discovery (UM1-AI100645); CHAVI Funding; R. Ottman; V. Shashi; S. Berkovic, I. Scheffer, and B. Grinton; The Epi4K Consortium and Epilepsy Phenome/Genome Project; C. Depondt, S. Sisodiya, G. Cavalleri, and N. Delanty; S. Hirose; C. Woods, C. Village, K. Schmader, S. McDonald, M. Yanamadala, and H. White; G. Nestadt, J. Samuels, and Y. Wang; D. Levy; E. Pras, D. Lancet, and Z. Farfel; S. Chen; R. Wapner; C. Moylan, A. Mae Diehl, and M. Abdelmalek; DUHS (Duke University Health System) Nonalcoholic Fatty Liver Disease Research Database and Specimen Repository; M. Winn and R. Gbadegesin; M. Hauser; S. Delaney; A. Need and J. McEvoy; M. Walker; M. Sum; Undiagnosed Diseases Network; National Institute on Aging (R01AG037212, P01AG007232).</p>", "<title>FUNDING SOURCES:</title>", "<p id=\"P37\">The collection of ALS and PLS samples and data was funded in part by: The Scottish Genomes Partnership (Chief Scientist Office of the Scottish Government Health Directorates (SGP/1) and The Medical Research Council Whole Genome Sequencing for Health and Wealth Initiative (MC/PC/15080); The New York Genome Center ALS Consortium (ALS Association 15-LGCA234, 19-SI-459, and the Tow Foundation; The GTAC study (ALS Association 16-LGCA-310 and Biogen Idec); ALS Exome Sequencing Consortium (Biogen Idec). Funding for the ALS 561 COSMOS and PLS COSMOS studies was provided by NIEHS R01ES016348, the Muscular 562 Dystrophy Association, MDA Wings Over Wall Street, Spastic Paraplegia Foundation (SPF), private 563 donations from Mr. and Mrs. Marren, the Adams Foundation, and Ride for Life.</p>", "<p id=\"P38\">The collection of control samples and data was funded in part by: Bryan ADRC NIA P30 AG028377; NIH RO1 HD048805; Gilead Sciences, Inc.; D. Murdock; National Institute of Allergy and Infectious Diseases Center for HIV/AIDS Vaccine Immunology (CHAVI) (U19-AI067854); National Institute of Allergy and Infectious Diseases Center for HIV/AIDS Vaccine Immunology and Immunogen Discovery (UM1-AI100645); Bill and Melinda Gates Foundation; NINDS Award# RC2NS070344; New York-Presbyterian Hospital; The Columbia University College of Physicians and Surgeons; The Columbia University Medical Center; NIH U54 NS078059; NIH P01 HD080642; The J. Willard and Alice S. Marriott Foundation; The Muscular Dystrophy Association; The Nicholas Nunno Foundation; The JDM Fund for Mitochondrial Research; The Arturo Estopinan TK2 Research Fund; UCB; Epilepsy Genetics Initiative, A Signature Program of CURE; Epi4K Gene Discovery in Epilepsy study (NINDS U01-NS077303) and The Epilepsy Genome/Phenome Project (EPGP - NINDS U01-NS053998); The Ellison Medical Foundation New Scholar award AG-NS-0441-08; National Institute Of Mental Health of the National Institutes of Health under Award Number K01MH098126; B57 SAIC-Fredrick Inc. M11-074; OCD Rare 1R01MH097971-01A1. This research was supported in part by funding from Funding from the Duke Chancellor’s Discovery Program Research Fund 2014; an American Academy of Child and Adolescent Psychiatry (AACAP) Pilot Research Award; NIMH Grant RC2MH089915; Endocrine Fellows Foundation Grant; The NIH Clinical and Translational Science Award Program (UL1TR000040); NIH U01HG007672; The Washington Heights Inwood Columbia Aging Project; The Stanley Institute for Cognitive Genomics at Cold Spring Harbor Laboratory and the Utah Foundation for Biomedical Research. Data collection and sharing for the WHICAP project (used as controls in this analysis) was supported by The Washington Heights-Inwood Columbia Aging Project (WHICAP, PO1AG07232, R01AG037212, RF1AG054023) funded by the National Institute on Aging (NIA) and by The National Center for Advancing Translational Sciences, National Institutes of Health, through Grant Number UL1TR001873. This manuscript has been reviewed by WHICAP investigators for scientific content and consistency of data interpretation with previous WHICAP Study publications. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.</p>", "<title>AVAILABLITY OF DATA AND MATERIALS</title>", "<p id=\"P39\">All summary data generated during this study are included in this published article and its supplementary information files.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1.</label><caption><title>Q-Q plots of gene- and domain-level collapsing of ALL functional coding variants in ALS cohort.</title><p id=\"P47\">(A) The results for a standard gene-level collapsing of 6,970 ALS cases and 22,524 controls. P-values were generated using an exact two-sided Cochran-Mantel-Haenszel (CMH) by gene by cluster. The genes with the top associations that achieved study-wide significance of p&lt;4.9×10⁻⁷ (<italic toggle=\"yes\">SOD1</italic> (OR=19.18), <italic toggle=\"yes\">TARDBP</italic> (OR=4.73), <italic toggle=\"yes\">TBK1</italic> (OR=2.3), and <italic toggle=\"yes\">ALKBH3</italic> (OR=0.26)) are labeled. <italic toggle=\"yes\">SOD1, TARDBP, TBK1</italic> have been previously implicated in rare variant association studies of ALS. Yellow and green lines indicate the 2.5^th and 97.5^th percentile of expected p-values, respectively. (B) The results for the domain-based collapsing restricting qualifying variants to those with subRVIS domain percentage score &lt; 25 of 6,970 cases and 22,524 controls. P-values were generated using an exact two-sided Cochran-Mantel-Haenszel (CMH) by gene by cluster. The genes with the top associations (<italic toggle=\"yes\">SOD1</italic> (OR=20.63), <italic toggle=\"yes\">TARDBP</italic> (OR=10.08), and <italic toggle=\"yes\">TBK1</italic> (OR=3.15)) are labeled. (C) Standard gene-level collapsing model showed 44 qualifying variants in cases (red circles) and 31 in controls (blue circles) for <italic toggle=\"yes\">TARDBP</italic> (D) subRVIS domain collapsing improved association by removing control variants (cases = 43; controls = 15). Regions with subRVIS domain percentage below 25 are highlighted in orange while those above this threshold are highlighted in blue. A one tailed z-score showed that there were significantly less controls in the intolerant domain as indicated by subRVIS domain percentage score &lt; 25 (p=0.031).</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2.</label><caption><title>Plot of gene- and domain-level collapsing of ALL <italic toggle=\"yes\">SOD1</italic> functional coding variants.</title><p id=\"P48\">Standard gene-level collapsing model showed 93 qualifying variants in cases (red circles) and 18 in controls (blue circles) for <italic toggle=\"yes\">SOD1</italic>. subRVIS domain collapsing improved association by removing control variants (cases = 90; controls = 16). Regions with subRVIS domain percentage below 25 are highlighted in orange while those above this threshold are highlighted in blue. However, a one tailed z-score showed that the differences in the number of controls in the intolerant domain was not significantly lower than those in the entire gene as indicated by subRVIS domain percentage score &lt; 25 (p=0.4).</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3.</label><caption><title>Plot of gene- and domain-level collapsing of ALL <italic toggle=\"yes\">TBK1</italic> functional coding variants.</title><p id=\"P49\">Standard gene-level collapsing model showed 73 qualifying variants in cases (red circles) and 143 in controls (blue circles) for <italic toggle=\"yes\">TBK1</italic>. subRVIS domain collapsing improved association by removing control variants (cases = 47; controls = 72). Regions with subRVIS domain percentage below 25 are highlighted in orange while those above this threshold are highlighted in blue. However, a one tailed z-score showed that the differences in the number of controls in the intolerant domain was not significantly lower than those in the entire gene as indicated by subRVIS domain percentage score &lt; 25 (p=0.3).</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4.</label><caption><title>Q-Q plot of gene-level collapsing of protein truncating variants (PTV) in PLS cohort.</title><p id=\"P50\">The results for a standard gene-level collapsing of 166 PLS cases and 17,695 controls. P-values were generated using an exact two-sided Cochran-Mantel-Haenszel (CMH) by gene by cluster. The gene with the top associations that achieved genome-wide significance of p&lt;8.38×10⁻⁶ (<italic toggle=\"yes\">ANTXR2</italic> (OR=174.57)) is labeled. <italic toggle=\"yes\">ANTXR2</italic> has not been previously implicated in rare variant association studies of PLS. Yellow and green lines indicate the 2.5^th and 97.5^th percentile of expected p-values, respectively.</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5.</label><caption><title>Forest plot of ALS genes by model.</title><p id=\"P51\">Rare variants in “ALS Confirmed” genes were significantly associated with ALS in all gene-based collapsing models except the control synonymous model. Rare variants in “ALS Plus” genes were associated with ALS in “Missense &amp; PTV” gene-based collapsing model. There was no association with ALS of rare variants in “ALS Replication Needed” and “ALS Weak Evidence” genes. Pooled odds ratio, 95% confidence intervals, and p-values were generated from exact two-sided Cochran-Mantel-Haenszel (CMH) tests.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\"><label>Table 1.</label><caption><p id=\"P52\">ALS gene association strength</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">ALS Confirmed</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">ALS Plus</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">ALS Replication Needed</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">ALS Weak Evidence</th></tr></thead><tbody><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">C9ORF72</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">ALS2</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">ANXA11</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">ANG</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">FUS</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">ATXN2</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">C21ORF2</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">DAO</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">NEK1</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">CHCHD10</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">CCNF</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">EWSR1</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">OPTN</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">CHMP2B</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">DNAJC7</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">NEFH</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">PFN1</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">DCTN1</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">ERBB4</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">PRPH</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">SETX</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">ERLIN2</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">GLT8D1</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">SQSTM1</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">SOD1</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">FIG4</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">hnRNPA1</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">SSI8L1</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">TARDBP</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">GRN</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">hnRNPA2B1</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">TAF15</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">TBK1</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">KIF5A</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">LGALSL</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">UBQLN2</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">MATR3</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">TIA1</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">VAPB</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">SIGMAR1</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">TUBA4A</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">SPG11</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">VCP</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr></tbody></table></table-wrap>" ]

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[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN2\"><p id=\"P40\">COMPETING INTERESTS</p><p id=\"P41\">The authors declare that they have no competing interests</p></fn><fn fn-type=\"COI-statement\" id=\"FN3\"><p id=\"P42\"><bold>Additional Declarations:</bold> No competing interests reported.</p></fn><fn id=\"FN4\"><p id=\"P43\">ETHICS APPROVAL AND CONSENT TO PARTICIPATE</p><p id=\"P44\">All samples and data came from participants that provided written, informed consent for genetic studies that had been IRB-approved at each contributing center.</p></fn><fn id=\"FN5\"><p id=\"P45\">CONSENT FOR PUBLICATION</p><p id=\"P46\">Not applicable</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"P7\">A growing body of literature documents the high prevalence of trauma exposure and PTSD among individuals with substance use disorders (SUD) (##REF##30180003##1##, ##REF##15769558##2##), with rates of co-occurring post-traumatic stress disorder (PTSD) ranging from 33–50% (##UREF##0##3##). However, with the opioid overdose epidemic, more trauma-related research is needed as it relates to opioid use disorder (OUD), and relatively little is known about lifetime trauma exposure among persons engaged in treatment with medication for opioid use disorder (MOUD), critical for improving integrative and comprehensive care for this population (##REF##32522999##4##).</p>", "<p id=\"P8\">An examination of traumatic events across the lifespan allows for differentiation between types of traumatic experiences (e.g., interpersonal, non-interpersonal, childhood), as well the opportunity to examine related sex/gender differences and the relationship to PTSD symptoms, all of which are crucial for understanding the impact of trauma on any health condition or population (##UREF##1##5##). For example, intimate interpersonal trauma is significantly more likely to be associated with symptoms of PTSD, when compared to non-interpersonal trauma and non-intimate interpersonal trauma (e.g., physical assaults perpetrated by non-intimates) (##UREF##2##6##), and there is a significant relationship between the number of traumatic events and the development of PTSD (##REF##17575070##7##). Also, the potential health consequences of childhood trauma are increasingly evident. Systematic reviews consistently show a link between exposure to childhood violence and substance use disorder (##UREF##3##8##), with a 73% increased risk for SUD if there is a history of sexual abuse in childhood and a 74% increased risk if there is a history of physical abuse in childhood (##UREF##4##9##). Sexual trauma is more prevalent among women than men. Women with a history of sexual trauma are at increased risk for SUD compared to men (##UREF##4##9##) and have specific treatment-related needs due to the type of trauma endured and its impact on mental health (##UREF##5##10##).</p>", "<p id=\"P9\">In research specific to trauma for those with opioid use disorder (OUD) (N = 20,522), a recent systematic review examining child maltreatment demonstrated the high prevalence of childhood physical abuse in 43% in the total sample, and significantly more childhood sexual abuse among women (41%) compared to men (16%) (##UREF##6##11##). Studies specific to examining trauma among those treated with MOUD have been relatively limited in scope and/or small in sample size. For example, one study (N = 919) examined interpersonal trauma only (physical, sexual, or emotional abuse) and found that 23% reported sexual abuse, 43% physical abuse, and 58% emotional abuse and that there were no differences by gender on any of these categories (##REF##35403485##12##). Unfortunately, this study by Powers did not distinguish whether the traumatic events occurred in childhood or as adults. Another study (N = 36) examined both interpersonal and non-interpersonal types of trauma (e.g., accidents, natural disasters) and found both to significantly predict OUD (##REF##29569991##13##). A third study (N = 135) exami<italic toggle=\"yes\">n</italic>ed current trauma only (over period of last 12 months) among those engaged in MOUD and found that more than one third reported interpersonal trauma (combining reported interpersonal traumas such as intimate partner violence, sexual assault, physical assault) and found similar overall rates among men (36%) and women (40%) (##UREF##7##14##).</p>", "<p id=\"P10\">Given the high prevalence of chronic pain among those in MOUD (##REF##28476267##15##, ##REF##32088587##16##), it is also highly relevant to examine the relationship between trauma exposure and chronic pain within this population. Prior research demonstrates that trauma exposure is associated with an increased risk of developing chronic pain (##REF##30180003##1##, ##REF##15769558##2##), defined as persistent pain lasting for at least three months that adversely affects the function or well-being of the individual (##REF##17572628##17##). In addition, individuals with a trauma history are approximately three times more likely to develop a chronic pain condition than those without a trauma history (##REF##24336429##18##). Within the population of individuals affected by chronic pain, individuals with a trauma history report more intense pain (##REF##23398939##19##, ##REF##25144169##20##), greater affective distress, and a higher disability (##REF##10798843##21##, ##REF##27641311##22##) than individuals without a trauma history. Previous research has also established a high comorbidity between PTSD and chronic pain in the general adult population (##REF##28959216##23##), in veteran populations (##REF##32427603##24##), and, most recently, among individuals engaged in MOUD (##UREF##8##25##, ##REF##29775955##26##). Importantly, chronic pain has been identified as a significant contributing factor to SUDs, most notably OUD. Patients experiencing comorbid chronic pain and PTSD are reported to have increased odds of OUD compared to individuals with neither a chronic pain condition nor a PTSD diagnosis (##REF##29775955##26##).</p>", "<p id=\"P11\">As noted above, the type of traumatic experience appears to matter; specifically, the type of traumatic experience appears to be differentially associated with the development of chronic pain. The relationship between exposure to non-interpersonal trauma (e.g., traumatic accidents) and the development of chronic pain is well-established in individuals with and without SUD, with research demonstrating that accident-related pain is associated with greater pain severity and related disability in those with vs. without SUD (##REF##23548492##27##). The relationship between exposure to interpersonal trauma, childhood trauma in particular, and the development of chronic pain has also been established in the general population (##REF##19654389##28##–##UREF##9##30##), replicated in SUD populations (##REF##33131372##31##) and documented in OUD populations (##UREF##10##32##–##REF##16336480##36##). However, in most studies examining chronic pain or OUD, childhood trauma exposure has been defined and limited to single types of childhood abuse or neglect (##REF##21745041##33##, ##REF##24360527##37##). Different types of trauma (e.g., interpersonal, non-interpersonal, adult and/or child, etc.) have yet to be investigated among persons with OUD. Doing so may illuminate important risk factors for those with co-occurring chronic pain and OUD (##REF##33405314##38##).</p>", "<p id=\"P12\">The purpose of this study is to comprehensively examine lifetime trauma exposure among individuals engaged in treatment with MOUD. The four aims of this study are to: 1) examine prevalence of different types of trauma exposure among individuals in MOUD; 2) identify gender differences in lifetime trauma exposure; 3) examine whether trauma exposure and number of traumatic events predict PTSD diagnostic status and PTSD symptoms, and 4) compare types of trauma exposure among those with and without chronic pain.</p>" ]

[ "<title>Methods</title>", "<title>Study Design and Enrollment</title>", "<p id=\"P13\">A NCCIH-funded randomized controlled trial to examine mindful body awareness training in individuals engaged in MOUD treatment provided the opportunity to examine the prevalence of self-reported lifetime trauma exposure and differences in trauma exposure by gender and among those with and without chronic pain. This study received Human Subjects Institutional Review Board approval from the University of Washington. Data for this project was collected at baseline, prior to randomization to study treatment groups. Study participants were recruited from five community clinics offering MOUD in Washington state.</p>", "<p id=\"P14\">Recruitment was based on referral of interested and potentially eligible patients by clinic staff (i.e., nurses, physicians, and counselors). The Research Coordinator at each clinical site screened for eligibility and enrolled patients interested in study participation. Screening criteria aimed to select patients with adequate treatment engagement and clinical stability to participate in the mindful body awareness intervention sessions. Evidence of medication dose stability: for buprenorphine/naloxone, this was defined as at least four weeks of medication treatment and an appointment frequency of less than once weekly. For methadone, this was defined as at least 90 days in treatment with a minimum dose of 60mg and no more than three missed doses or any missed dose evaluation appointments in the past 30 days. Patients also needed to speak English and be willing to attend intervention sessions when offered. They were excluded if they were unwilling or unable to remain in MOUD treatment for the one-year trial or if they showed evidence of overt psychosis or cognitive impairment.</p>", "<title>Measures</title>", "<p id=\"P15\">Demographic Characteristics, Health History and Substance Use History</p>", "<p id=\"P16\">Demographic characteristics, including self-identified gender, along with other information specific to health history, was collected by patient self-report. Substance use was assessed using the Timeline Follow-Back Interview (TLFB) (##REF##23276315##39##); a calendar method used to identify substance use over the 90 days prior to study enrollment.</p>", "<title>Trauma History</title>", "<p id=\"P17\">The Trauma Life Events Questionnaire (TLEQ) was used to assess the prevalence and number of traumatic events across the lifespan (##REF##10887767##40##). The TLEQ is a 23-item self-report measure to assess lifetime exposure to a broad range of potentially traumatic events (see appendix for items). Two sex-specific items were removed and not administered to participants: one specific to miscarriage and one specific to abortion. Participants are asked to report the number of times they experienced each event (event frequency) on a 7-point scale ranging from <italic toggle=\"yes\">never</italic> to <italic toggle=\"yes\">more than 5 times</italic>.</p>", "<p id=\"P18\">Based on the responses to the 21 item TLEQ, we chose to categorize the items as adult interpersonal trauma, adult non-interpersonal trauma, or childhood trauma, and then examined the type of event to determine if any could be combined conceptually to minimize the number of total categories for analysis (for example we combined natural disaster with other types of accidents to create a non-interpersonal category titled “accident”). We excluded 6 items from the original measure for which the response rate was relatively low; these were items 4 (military trauma), 6 (the survival of someone you loved after a life-threatening accident or illness), 7 (having had a life-threatening illness), 11 (witnessing a stranger beat, attack or kill someone), 19 (subjected to uninvited or unwanted sexual attention other than sexual contact covered by items 15, 16, 17, or 18), and 21 (experienced other events that were highly distressing such as lost in the wilderness; a serious animal bite; violent death of a pet; being kidnapped or held hostage; seeing a mutilated body or body parts). Our final set of 15 items and 11 categorizations are listed in ##TAB##0##Table 1##.</p>", "<title>PTSD Symptom Severity</title>", "<p id=\"P19\">The Posttraumatic Stress Disorder Checklist for DSM 5 (PCL-5) assesses PTSD symptom severity (##REF##26606250##41##). Participants were asked to indicate how much they have been bothered by each PTSD symptom in the past month. It includes 20 items with a 5-point scale ranging from 0 (not at all) to 4 (extremely). We used a screening cut-off of &gt; 31, indicative of probable PTSD (##UREF##11##42##). The reliability of the PCL-5 in this sample was .93.</p>", "<title>Chronic Pain</title>", "<p id=\"P20\">The Brief Pain Inventory (BPI) (##REF##15322437##43##) is a well-validated questionnaire comprising 11 items designed to evaluate the severity of pain and its impact on daily activities (i.e., pain interference) (##REF##8080219##44##). The scale’s reliability in this sample was .88 for pain severity and .93 for pain interference.</p>", "<title>Analyses</title>", "<p id=\"P21\">Descriptive statistics (counts, percentages, mean values, and SDs) were used to summarize sample demographics, self-report indices, and survey scales. Independent sample t-tests were used to examine differences in trauma exposure between men and women and between those with and without chronic pain. Linear regression was used to examine whether the number of trauma events predicted PTSD symptoms. Logistic regression was used to examine whether the number of trauma events predicted PTSD status (scoring above the screening cut-point for PTSD). All analyses were conducted using Stata version 18.0 (College Station, TX, USA)</p>" ]

[ "<title>Results</title>", "<title>Participants</title>", "<p id=\"P22\">This sample (N = 303) had a median age of 40, with ages ranging from 21–73. Self-report gender in the sample was 144 male, 157 female, and two non-binary. The majority (79%) of the sample identified as White, 9% as mixed-race, 5% as Black, 4% as Native American, 1% as Asian, and 1% as Native Hawaiian or Pacific Islander. Nine percent identified as Hispanic. The highest level of education was high school for 66% of the sample. Socioeconomic status was low, reflected in the overall low employment rate (34% employed (at either full or half-time) and high public insurance rate (72%) on Medicaid. Chronic pain was reported in 57% of the sample. Before study enrollment, most participants (67%) were engaged in MOUD treatment for over 12 months, reported high levels of abstinence from opioids and other substances, and had engaged in relatively few lifetime mental health services (see ##TAB##1##Table 2##).</p>", "<title>Lifetime Trauma Exposure</title>", "<p id=\"P23\">All participants in the sample, with one exception, reported at least one lifetime traumatic event. Over 70% of the sample reported exposure to five types of traumatic events. Within the category of adult interpersonal trauma: 71% reported physical assault (e.g., robbed or witnessing a robbery when a weapon was used, or physically assaulted by a stranger), 79% reported intimate partner violence (IPV), and 89% reported the experience of a sudden and unexpected death of a close friend or loved one. Within the category of adult non-interpersonal trauma: 86% reported an accident (e.g., a natural disaster or injurious accident. Within the category of childhood trauma: 89% reported at least one type of traumatic event (see ##TAB##2##Table 3##).</p>", "<title>Trauma Exposure and Gender</title>", "<p id=\"P24\">Women reported significantly more trauma than men in many categories (IPV, sexual assault, being stalked, total childhood violence, childhood witness of IPV, childhood sexual abuse, and sudden death of a loved one). Men reported significantly more trauma than women in witnessing a traumatic event, physical assault, and childhood physical abuse. Notably, despite gender differences the prevalence of exposure to some of these events was very high for both men and women; for example, IPV (men 73%; women 85%), physical assault (women 61%; men 81%), total childhood violence (men 72%; women 89%), and sudden death of a loved one (men 83%; women 94%). There was equivalent exposure to accidents across genders (see ##TAB##2##Table 3##).</p>", "<title>Trauma Exposure and PTSD Status</title>", "<p id=\"P25\">In this study sample, 41% (n = 124) met the screening criteria for PTSD. Exposure to trauma was significantly higher across all categories of trauma for those positive for PTSD compared to those without, with the exception of childhood witnessing of IPV, accidents, or sudden death of a loved one (see ##TAB##3##Table 4##). Notably, those with subthreshold symptoms of PTSD still reported exposure to a great deal of trauma; for example, 72% experienced IPV, 77% experienced childhood violence, and 64% reported physical assault.</p>", "<title>Number of Trauma Exposure Events and PTSD Symptoms and Status</title>", "<p id=\"P26\">The number of reported traumatic events (i.e., the total number of events reported within each trauma category) predicted PTSD symptoms. Results from the univariate linear regression model showed that for every 1-point increase in number of trauma events, there was an increase of 6.5 on the PTSD symptom scale (β = 6.5, 95% CI, 4.8–8.2; ##FIG##0##Fig. 1##). Likewise, the number of traumatic events predicted PTSD status (scoring above the PTSD screening cut-off on the PCL-5; OR = 2.1; 95% CI, 1.6–27.7; ##FIG##1##Fig. 2##).</p>", "<title>Trauma Exposure and Chronic Pain</title>", "<p id=\"P27\">Individuals with chronic pain, compared to those without chronic pain, reported significantly more trauma in the following categories: accidents (n = 155, 91%), childhood violence (total; n = 147, 85%), childhood physical abuse (n = 82, 48%), witnessing IPV in childhood (n = 144, 78%), childhood sexual abuse (n = 106, 62%: ##TAB##4##Table 5##).</p>" ]

[ "<title>Discussion</title>", "<p id=\"P28\">This is the first study to examine lifetime trauma experiences among a large sample of individuals in MOUD. The results highlight the high prevalence of trauma in both childhood and in adulthood, as well as both interpersonal and non-interpersonal traumatic events in both men and women. While differences across gender and chronic pain status are notable, the remarkable prevalence of exposure to all trauma categories across all groups points to the critical need for both trauma assessment and mental health services that are accessible and integrated into MOUD treatment. Individuals in this sample were stabilized on MOUD for a substantial amount of time and reported high levels of abstinence from substance use yet were not accessing a level of mental health care commensurate with their need. Also notable is the particularly high report of sudden and unexpected death of a close friend or loved one – reflecting the tragic experience of loss among this sample likely due to drug overdose in their communities.</p>", "<p id=\"P29\">There were distinct gender differences in trauma exposure, the most striking being the higher number of women who reported sexual abuse in childhood and sexual assault in adulthood compared to men. This finding aligns with prior research and the identified need for women-specific programs in SUD treatment to address the high prevalence of sexual trauma (##UREF##4##9##, ##UREF##5##10##). Perhaps unexpected, although similar to study findings examining interpersonal trauma in the past 12 months among those in MOUD (##UREF##7##14##), was the high number of men who reported being victims of intimate partner violence (IPV); while not as high as the report of IPV among women, this finding warrants further research and clinical attention as it points to the need for more assessment and clinical support for IPV, for everyone regardless of gender/sex. Overall, these results point to the need to ensure that support services and trauma treatment are available and integrated into treatment to optimize outcomes for those receiving MOUD.</p>", "<p id=\"P30\">In this study, 41% of participants screened positive for PTSD, congruent with previously published literature (##UREF##12##45##, ##REF##25792193##46##). Given the high prevalence of many types of traumatic experiences across the participants in this sample, we could not link PTSD diagnostic status to particular types of traumatic event (i.e., whether they occurred during childhood or as an adult; whether interpersonal or non-interpersonal). However, the results demonstrate the link between the number of traumatic events experienced and PTSD symptomatology and diagnosis. These findings align with previous studies (##REF##19252066##47##), and the understanding that traumatic events in both childhood or adulthood can impact symptom severity, expression, and complexity (##REF##19795402##48##).</p>", "<p id=\"P31\">The high prevalence of chronic pain in MOUD populations allowed us to examine the relationship between trauma exposure and chronic pain. Congruent with previous studies among individuals with and without SUD, our study found that individuals with OUD and chronic pain were more likely to report traumatic accidents (e.g., car accidents, falls, natural disasters) (##REF##23548492##27##–##UREF##9##30##, ##UREF##10##32##, ##REF##21745041##33##, ##REF##31103834##35##–##REF##24360527##37##, ##REF##12670615##49##). Impaired cortisol secretion and psychological stress in response to a traumatic injury/ accident has been associated with development of chronic pain over time (##REF##23548492##27##). Prior life circumstances that result in sustained, long term cortisol surges or activations, are known to contribute to cortisol dysfunction, and may then increase risk the risk of development of chronic pain (##UREF##13##50##). The relationship between abnormal physiological stress reactivity (i.e., heart rate, blood pressure, respiration rate, cortisol secretion) on negative health outcomes is well-established (##REF##34323531##51##), and linked to pain somatization disorders (##REF##11454437##52##, ##REF##25035267##53##).</p>", "<p id=\"P32\">We also found that individuals who endorsed chronic pain were more likely to report childhood violence, including physical abuse, sexual abuse, and witnessing IPV in childhood. Most prior studies that have examined chronic pain, OUD, and childhood trauma exposure have been limited to single types of childhood abuse or neglect (##REF##21745041##33##, ##REF##16336480##36##). Our findings align with prior research showing a link between childhood trauma and chronic pain in community and SUD samples, highlighting the importance of assessing PTSD among those with chronic pain in MOUD and the potential need for psychological treatment in the context of recovery.</p>", "<p id=\"P33\">Providing trauma-focused therapy alongside treatment for opioid use disorder (##REF##25792193##46##, ##REF##23815424##54##), may prove to be beneficial. There is evidence that patients with chronic pain and a co-occurring history of physical trauma demonstrate a diminished response to treatment, when compared with a cohort of patients without a history of trauma. Moreover, recent clinical reports have described the indirect and successful treatment of intractable and chronic pain in patients with comorbid PTSD, only after instituting behavioral therapy targeting the PTSD symptoms. Cognitive-behavioral therapies with proven efficacy for the treatment of PTSD are now available to pain practitioners, and it is noteworthy that these interventions are now being tailored within comprehensive pain rehabilitation programs. Incorporating novel mindfulness and body therapy approaches to increase sensory and emotional awareness may also benefit individuals with PTSD and co-occurring OUD, and further research is needed in this area.</p>", "<p id=\"P34\">There are important related clinical implications of these findings for medical providers. Given the high prevalence of trauma exposure and PTSD among individuals with OUD, evidence-based PTSD screenings, assessments, and treatments should be provided alongside MOUD (##UREF##14##55##). Although calls to lower barriers and increase access to MOUD treatment have resulted in more primary care providers treating people with OUD (##UREF##15##56##–##REF##29972748##59##) and national guidelines recommend that primary care clinics screen for depression (##REF##37338872##60##) and anxiety (##REF##37338866##61##), there is not a similar recommendation for universal PTSD screening (##REF##23712724##62##) and, thus, detection rates are low (##REF##26868222##63##, ##REF##36707910##64##).</p>", "<p id=\"P35\">Study limitations include the characteristics of the sample: the majority were white, low SES, and from one region of the United States. The findings may not generalize to a more racially, ethnically or economically diverse population. Also, only two individuals in this study identified as non-binary, limiting our ability to learn more about this population and highlighting an important line of future research. The TLEQ, the questionnaire we used to collect trauma exposure data, is comprehensive and has been used in prior research; however, until there is a more standard measure used consistently across studies, it will continue to be challenging to compare findings from one study to another in order to gather a more subtle understanding of the sequelae of trauma exposure across the lifespan (##UREF##1##5##). This study has multiple strengths. First, it is a multi-site study including participants from urban and rural areas and multiple practice settings (opioid treatment program, mental health clinic, addiction clinic, and primary care clinic.) Patients reported a high proportion of days abstinent, and the majority had been in prolonged MOUD treatment, reducing the possibility that mental health symptoms were primarily substance-induced.</p>" ]

[ "<title>Conclusions</title>", "<p id=\"P36\">In conclusion, the findings highlight the complex connection between trauma exposure, OUD, gender, PTSD symptoms, and chronic pain. This study provides valuable insights into the prevalence of trauma across genders and points to the potential impact on individuals engaged in MOUD. These findings may inform the development of enhanced gender-specific interventions and approaches for patients engaged in MOUD treatment, potentially addressing the interconnectedness of trauma, prolonged pain, and psychological issues in this population.</p>" ]

[ "<p id=\"P1\"><bold>Authors’ contributions:</bold> CJP and JOM obtained funding and oversaw data collection; CJP and MNR conceptualized the manuscript; KP was responsible for data analysis; MNR, DDC, SL, and CJP wrote the article; JOM manuscript review. All authors read and approved the final manuscript.</p>", "<title>Background</title>", "<p id=\"P2\">There is little study of lifetime trauma exposure among individuals engaged in medication treatment for opioid use disorder (MOUD). A multisite study provided the opportunity to examine the prevalence of lifetime trauma and differences by gender, PTSD status, and chronic pain.</p>", "<title>Methods</title>", "<p id=\"P3\">A cross-sectional study examined baseline data from participants (N = 303) enrolled in a randomized controlled trial of a mind-body intervention as an adjunct to MOUD. All participants were stabilized on MOUD. Measures included the Trauma Life Events Questionnaire (TLEQ), the Brief Pain Inventory (BPI), and the Posttraumatic Stress Disorder Checklist (PCL-5). Analyses involved descriptive statistics, independent sample t-tests, and linear and logistic regression.</p>", "<title>Results</title>", "<p id=\"P4\">Participants were self-identified as women (<italic toggle=\"yes\">n</italic> = 157), men (<bold>n</bold> = 144), and non-binary (<italic toggle=\"yes\">n</italic> = 2). Fifty-seven percent (<italic toggle=\"yes\">n</italic> = 172) self-reported chronic pain, and 41% (n = 124) scored above the screening cut-off for PTSD. Women reported significantly more intimate partner violence (85%) vs 73%) and adult sexual assault (57% vs 13%), while men reported more physical assault (81% vs 61%) and witnessing trauma (66% vs 48%). Men and women experienced substantial childhood physical abuse, witnessed intimate partner violence as children, and reported an equivalent exposure to accidents as adults. The number of traumatic events predicted PTSD symptom severity and PTSD diagnostic status. Participants with chronic pain, compared to those without chronic pain, had significantly more traumatic events in childhood (85% vs 75%).</p>", "<title>Conclusions</title>", "<p id=\"P5\">The study found a high prevalence of lifetime trauma among people in MOUD. Results highlight the need for comprehensive assessment and mental health services to address trauma among those in MOUD treatment.</p>", "<title>Trial Registration:</title>", "<p id=\"P6\">\n<ext-link xlink:href=\"https://clinicaltrials.gov/ct2/show/NCT04082637\" ext-link-type=\"uri\">NCT04082637</ext-link>\n</p>" ]

[]

[ "<title>Funding:</title>", "<p id=\"P37\">This research was made possible by Grant Numbers R33AT009932 from the National Center for Complementary and Integrative Health (NCCIH) and R01 AT010742 from the National Center for Complementary and Integrative Health (NCCIH) and the National Institute of Neurological Disorders and Stroke (NINDS). Its contents are solely the authors’ responsibility and do not necessarily represent the official views of the NCCIH, NINDS, or the National Institutes of Health.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><p id=\"P50\">Legend not included with this version.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><p id=\"P51\">Legend not included with this version.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\"><label>Table 1</label><caption><p id=\"P52\">Trauma Categories and Corresponding TLEQ Items</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"top\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Adult Interpersonal Trauma</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Witness Trauma</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">10. Have you seen a stranger attack or beat up someone and seriously injure or kill them?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Intimate Partner Violence (IPV)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">14. Have you ever been slapped, punched, kicked, or beaten up or physically hurt by your spouse or other intimate partner?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Physical Assault</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">8. Have you been robbed or present during a robbery where the robber used a weapon?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9. Have you ever been hit or beat up by a stranger or someone you didn’t know very well?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Adult Sexual Assault</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">18. Over 18 years old: did anyone touch sexual parts of your body or make you touch theirs without your consent?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Adult Stalking</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">20. Has anyone stalked you-in other words: followed you or kept track of your activities causing you to feel intimidated or concerned for your safety?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Sudden Death</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5. Have you experienced the sudden unexpected death of a loved one?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Adult Non-Interpersonal Trauma</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Accidents</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1. Have you ever experienced a natural disaster?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2. Were you involved in a motor vehicle accident for which you received medical attention or that badly injured or killed someone?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3. Have you been involved in any other kind of accident where you or someone else was badly hurt?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Childhood Trauma</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Childhood Physical Abuse</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12. While growing up: Were you physically punished in a way that resulted in bruises, burns, cuts or broken bones?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Childhood Witnessing IPV</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">13. While growing up: Did you see or hear family violence?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Childhood Sexual Abuse</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">15. Before your 13th birthday: did anyone who was 5 years older than you, touch or fondle your body in a sexual way?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">16. Before your 13th birthday: did anyone close to your age touch sexual parts of your body without your consent?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">17. Between 13–18 yrs. old: did anyone touch sexual parts of your body or make you touch theirs without your consent?</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Total Childhood Trauma</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Items 12, 13, 15, 16, 17</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T2\"><label>Table 2.</label><caption><p id=\"P53\">Sample Demographics</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mean</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Percent</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Age, median (range)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">40</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(21–73)</td></tr><tr><td colspan=\"3\" align=\"left\" valign=\"top\" rowspan=\"1\">Gender Identity</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Male</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">144</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">48%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Female</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">157</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">52%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Non-binary</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Hispanic</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">27</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9%</td></tr><tr><td colspan=\"3\" align=\"left\" valign=\"top\" rowspan=\"1\">Race</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Native American</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">13</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">4%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Asian</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Black or African American</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">16</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Hawaiian or Pacific Islander</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">4</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> White</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">238</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">79%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> More than one race</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">28</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9%</td></tr><tr><td colspan=\"3\" align=\"left\" valign=\"top\" rowspan=\"1\">Marital Status</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Married</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">33</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">11%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Single</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">215</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">71%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Domestic Partnership</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">18</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Unknown (Endorsed “Other”)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">36</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12%</td></tr><tr><td colspan=\"3\" align=\"left\" valign=\"top\" rowspan=\"1\">Highest Education Level</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> High school or less</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">168</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">44%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Two-year college/technical school</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">103</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">34%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> College or advanced degree</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">32</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">11%</td></tr><tr><td colspan=\"3\" align=\"left\" valign=\"top\" rowspan=\"1\">Monthly Income</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Less than $1000</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">179</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">59%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> $1000 or more</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">124</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">41%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Employed</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">104</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">34%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Full time</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">69</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">66%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Part time</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">35</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">34%</td></tr><tr><td colspan=\"3\" align=\"left\" valign=\"top\" rowspan=\"1\">Insurance^{<xref rid=\"TFN1\" ref-type=\"table-fn\">a</xref>}</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Medicaid</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">219</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">72%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Medicare</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">69</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">23%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Private</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">36</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> None</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Chronic Pain 3 Months or More</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">172</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">57%</td></tr><tr><td colspan=\"3\" align=\"left\" valign=\"top\" rowspan=\"1\">Mental Health Services in Lifetime</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> 0–10 therapy sessions</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">99</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">33%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> 11–30 therapy sessions</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">64</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">21%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> &gt;31 therapy sessions</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">140</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">46%</td></tr><tr><td colspan=\"3\" align=\"left\" valign=\"top\" rowspan=\"1\">Time in MOUD Treatment Prior to Study Enrollment</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> &lt; 3 months</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">28</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> 3–6 months</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">26</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> 6–12 months</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">46</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">15%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> &gt; 12 months</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">203</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">67%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Percent days abstinent from any opioid</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">96%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Percent days abstinent from any substance<xref rid=\"TFN2\" ref-type=\"table-fn\">**</xref></td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">88%</td></tr><tr><td colspan=\"3\" align=\"left\" valign=\"top\" rowspan=\"1\">Medication for Opioid Use Disorder</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Methadone</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">35</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12%</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Buprenorphine</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">268</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">88%</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T3\"><label>Table 3.</label><caption><p id=\"P56\">Endorsed Trauma Categories</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"top\" span=\"1\"/><col align=\"left\" valign=\"top\" span=\"1\"/><col align=\"left\" valign=\"top\" span=\"1\"/><col align=\"left\" valign=\"top\" span=\"1\"/><col align=\"left\" valign=\"top\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">N (%)</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Female</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Male</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"><italic toggle=\"yes\">P</italic> value</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">N</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">301</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">157</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">144</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Witness Trauma</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">172 (57%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">75 (48%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">95 (66%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.001</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Intimate Partner Violence (IPV)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">238 (79%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">132 (85%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">104 (73%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.012</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Physical Assault</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">214 (71%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">96 (61%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">116 (81%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Adult Sexual Assault</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">110 (37%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">90 (57%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">19 (13%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Stalking</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">161 (53%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">106 (68%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">53 (37%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Total Childhood Trauma</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">245 (81%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">140 (89%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">103 (72%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">  Childhood Physical Abuse</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">127 (42%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">61 (39%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">64 (44%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.373</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">  Witnessing IPV</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">212 (70%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">117 (75%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">94 (66%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.079</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">  Childhood Sexual Abuse</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">165 (54%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">115 (73%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">48 (33%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Accidents</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">260 (86%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">135 (86%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">123 (85%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.888</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Sudden Death</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">269 (89%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">148 (94%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">120 (83%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.002</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T4\"><label>Table 4.</label><caption><p id=\"P59\">Endorsed Trauma Categories and PTSD Status</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Subthreshold Symptoms</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">PTSD</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">P value</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">N</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">179</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">124</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Witness Trauma</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">91 (51%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">81 (66%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.011</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Intimate Partner Violence (IPV)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">128 (72%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">110 (90%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Physical Assault</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">115 (64%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">99 (80%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.003</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Adult Sexual Assault</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">55 (31%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">55 (45%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.011</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Stalking</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">85 (48%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">76 (61%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.023</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Total Childhood Trauma</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">137 (77%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">108 (87%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.022</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Childhood Physical Abuse</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">64 (36%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">63 (51%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.011</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Witnessing IPV</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">121 (68%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">91 (75%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.192</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Childhood Sexual Abuse</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">83 (46%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">82 (66%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Accidents</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">151 (84%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">109 (88%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.384</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Sudden Death</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">156 (87%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">113 (91%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.281</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T5\"><label>Table 5.</label><caption><p id=\"P62\">Trauma Categories Endorsed by Those With and Without Chronic Pain</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">No Chronic Pain</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Chronic Pain</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">P value</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">N</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">131</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">172</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Witness Trauma</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">70 (54%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">102 (60%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.314</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Intimate Partner Violence (IPV)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">101 (78%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">137 (80%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.609</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Physical Assault</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">90 (69%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">124 (72%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.521</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Adult Sexual Assault</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">43 (33%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">67 (39%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.276</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Stalking</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">65 (50%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">96 (56%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.290</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Total Childhood Trauma</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">98 (75%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">147 (85%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.020</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Childhood Physical Abuse</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">45 (35%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">82 (48%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.020</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Witnessing IPV</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">79 (61%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">133 (78%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.001</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"> Childhood Sexual Abuse</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">59 (45%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">106 (62%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.004</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Accidents</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">105 (80%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">155 (90%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.014</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Sudden Death</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">113 (86%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">156 (91%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.225</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN3\"><p id=\"P38\"><bold>Competing interests:</bold> The authors declare that they have no competing interests.</p></fn><fn id=\"FN4\"><p id=\"P39\"><bold>Ethics approval and consent to participate.</bold> This study received Human Subjects Institutional Review Board approval from the University of Washington. All participants involved in this study provided consent prior to their participation. Participants were provided detailed information about the study’s purpose, procedures, potential risks, benefits, and confidentiality measures. They were also informed about their right to withdraw from the study at any point without facing any consequences.</p></fn><fn id=\"FN6\"><p id=\"P40\"><bold>Availability of data and materials:</bold> The datasets generated and/or analysed during the current study are available in the Dryad Data Repository, [link to be available before copy editing].</p></fn></fn-group>", "<table-wrap-foot><fn id=\"TFN1\"><label>a</label><p id=\"P54\">Respondents could select multiple responses.</p></fn><fn id=\"TFN2\"><label>**</label><p id=\"P55\">Percent Days Abstinent excludes cannabis, and prescribed buprenorphine or methadone</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"TFN3\"><p id=\"P57\">Binary variables, endorsed implies ≥ 1 on original 0–6 metric</p></fn><fn id=\"TFN4\"><p id=\"P58\">Frequency (Percent%): p-value from chi-square test</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"TFN5\"><p id=\"P60\">Binary variables, endorsed implies ≥ 1 on original 0–6 metric</p></fn><fn id=\"TFN6\"><p id=\"P61\">Frequency (Percent%): p-value from chi-square test</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"TFN7\"><p id=\"P63\">Binary variables, endorsed implies ≥ 1 on original 0–6 metric</p></fn><fn id=\"TFN8\"><p id=\"P64\">Frequency (Percent%): p-value from chi-square test</p></fn></table-wrap-foot>" ]

[ "<graphic xlink:href=\"nihpp-rs3750143v1-f0001\" position=\"float\"/>", "<graphic xlink:href=\"nihpp-rs3750143v1-f0002\" position=\"float\"/>" ]

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{ "acronym": [ "SUD", "PTSD", "OUD", "MOUD", "TLFB", "TLEQ", "PCL 5", "BPI", "IPV" ], "definition": [ "Substance use disorder", "Post–traumatic stress disorder", "Opioid use disorder", "Medication for opioid use disorder", "Timeline Follow–Back Interview", "Trauma Life Events Questionnaire", "Posttraumatic Stress Disorder Checklist for DSM 5", "Brief Pain Inventory", "Intimate partner violence" ] }

[ "<title>INTRODUCTION</title>", "<p id=\"P11\">Prostate cancer (<bold>PC</bold>) ranks as the second leading cause of cancer-related deaths among American men, claiming over 34,700 lives annually ^{##REF##36633525##1##}. While androgen deprivation therapy (<bold>ADT</bold>) is initially effective in most men with advanced PC, the emergence of castration-resistant prostate cancer (<bold>CRPC</bold>) and resistance to androgen receptor (<bold>AR)</bold> signaling inhibitors (<bold>ARSIs</bold>) develops in almost all patients ^{##UREF##0##2##,##REF##34370973##3##}.</p>", "<p id=\"P12\">It is increasingly recognized that resistance to contemporary AR targeting therapies is associated with a diverse spectrum of disease phenotypes characterized by morphologic and molecular changes, which are often associated with a loss of prostate lineage features (such as the expression of AR) and the gain of more stem-like and neuronal features ^{##REF##31363002##4##–##REF##33328650##7##}. Therefore, disease subclassifications were proposed that are based on the assessment of AR and neuroendocrine marker expression ^{##REF##31361600##5##,##REF##30772358##8##,##UREF##1##9##}. Among these molecular subtypes, neuroendocrine prostate cancer (<bold>NEPC</bold>), characterized by the absence of AR signaling and gain of neuroendocrine features, represents the most aggressive disease subtype, with chemotherapy as the only available treatment option ^{##REF##31363002##4##,##REF##34111024##10##,##REF##29460922##11##}. There is, therefore, a critical clinical need for novel therapeutics in this difficult-to-treat and prognostically poor subset of patients.</p>", "<p id=\"P13\">Targeting cell-surface antigens through the delivery of cytotoxic agents directly to cancer sites or by generating anti-tumor immune responses are promising therapeutic approaches for advanced cancers ^{##REF##32243795##12##–##REF##35318309##17##}. Prostate-specific membrane antigen (<bold>PSMA</bold>) is currently the most extensively validated theranostic cell surface target in PC ^{##REF##34167925##18##,##REF##32865839##19##}. Although PSMA shows a favorable and relatively prostate lineage-restricted expression, up to 40% of CRPC patients show loss or heterogeneous PSMA expression ^{##REF##34167925##18##,##UREF##2##20##–##REF##31345636##22##}. In particular, the absence of PSMA expression is nearly universal in NEPC ^{##UREF##2##20##,##REF##36721073##21##}. To maximize the therapeutic benefit, there is a great need to understand the expression patterns of other cell surface proteins.</p>", "<p id=\"P14\">Of the constantly expanding spectrum of cell-surface targets in oncology, delta-like ligand 3 (<bold>DLL3</bold>), carcinoembryonic antigen-related cell adhesion molecule 5 (<bold>CEACAM5</bold>), and trophoblast cell-surface antigen 2 (<bold>TROP2</bold>) have been a focus for pre-clinical and clinical drug development efforts for advanced PC ^{##REF##33203642##23##–##UREF##6##28##}.</p>", "<p id=\"P15\">DLL3 is a ligand that inhibits the Notch signaling pathway and is expressed in the spinal cord and nervous system during embryonic development ^{##UREF##3##24##}. Importantly, DLL3 is expressed at high levels in the majority of tumors that exhibit high-grade neuroendocrine/small cell carcinoma features, making it a potentially valuable target for NEPC ^{##REF##33203642##23##–##UREF##4##25##,##REF##34354225##29##}. Similarly, CEACAM5, a member of the carcinoembryonic antigen family, is overexpressed in a larger fraction of solid tumors, with high expression observed in NEPC ^{##REF##33199493##26##,##UREF##7##30##}. Notably, several antibody-drug conjugates (<bold>ADCs</bold>) targeting CEACAM5 have been developed and explored in the context of different solid tumors ^{##REF##33199493##26##,##REF##33046521##31##,##REF##35026412##32##}. TROP2 is a transmembrane protein that is expressed in multiple malignancies^{##UREF##5##27##,##UREF##8##33##–##REF##33929895##36##}. Clinical trials using TROP2-targeting agents have shown efficacy and a TROP2 ADC sacituzumab govitecan has been approved for triple-negative breast cancer and urothelial carcinoma, and phase 2 studies in CRPC are currently ongoing ^{##REF##33882206##35##,##REF##33929895##36##}.</p>", "<p id=\"P16\">Since the efficacy of DLL3-, CEACAM5-, and TROP2-targeting strategies will in part depend on the expression of these antigens, it is informative to examine their expression in clinically relevant and well-annotated metastatic CRPC (<bold>mCRPC</bold>) cohorts. From a clinical perspective, it is particularly relevant to understand antigen expression across different molecular subtypes of PC and to establish the inter- and intra-patient expression variability. While prior studies have established the expression of these proteins in smaller PC cohorts, their patterns of expression and intra- and inter-tumoral heterogeneity have not been rigorously studied in metastatic CRPC. This is largely due to the difficulties of accessing biospecimen cohorts across diverse metastatic sites that provide a comprehensive representation of the metastatic tumor burden within and across different patients ^{##REF##33328650##7##}.</p>", "<p id=\"P17\">In this study, we determined the expression of DLL3, CEACAM5 and TROP2 in 753 tissue samples from 52 mCRPC patients. Leveraging the unique biospecimens from the University of Washington rapid autopsy cohort, we show that DLL3 and CEACAM5 expression is mostly restricted to tumors lacking AR signaling activity and expressing neuroendocrine markers. Conversely, TROP2 is expressed at high levels in most tumors except for NEPC. Despite these molecular subtype-specific expression differences, we demonstrate that TROP2 and DLL3 show relatively limited inter-tumoral heterogeneity. In addition, we show a relative enrichment of cell surface antigen expression in certain somatic genomic backgrounds, and we highlight a novel epigenetic mechanism involved in the regulation of DLL3, CEACAM5, and TROP2. These data provide valuable information on therapeutic target expression in CRPC and present a rationale for informed co-targeting strategies.</p>" ]

[ "<title>MATERIALS AND METHODS</title>", "<title>Human tissue samples</title>", "<p id=\"P36\">This study was approved by the Institutional Review Board of the University of Washington (protocol no. 2341). Formalin-fixed, paraffin-embedded tissues from 52 patients were used to construct tissue microarrays as described previously ^{##UREF##2##20##}.</p>", "<title>Immunohistochemical staining</title>", "<p id=\"P37\">Slides were deparaffinized and steamed for 45 min in Target Retrieval Solution (Dako Cat. S169984–2). Primary antibodies and dilutions used were as follows: TROP2 (Abcam, ab214488, 1:200), CEACAM5 (Agilent, M7072, 1:20), and PSMA (Agilent, M3620, 1:20). PV Poly-HRP Anti-Mouse IgG (Leica Microsystems Cat. PV6114) or Anti-Rabbit IgG (Leica Microsystems Cat. PV6119) was used as secondary antibody. Further signal amplification was done for CEACAM5 immunostains by using the Biotin XX Tyramide SuperBoost kit (Life Tech Cat. B40931). DLL3 staining was carried out on a Roche Benchmark Ultra instrument (Roche) using DLL3 (Ventana, SP347, 790–7016, 1μg/ml) and the CC1 module. DAB was used as the chromogen and counterstaining was done with hematoxylin and slides were digitized on a Ventana DP 200 Slide Scanner (Roche). Immunoreactivity was scored in a blinded manner by two pathologists (M. P. R., E. S.), whereby the staining level (“0” for no brown color, “1” for faint and fine brown chromogen deposition, and “2” for prominent chromogen deposition) was multiplied by the percentage of cells at each staining level, resulting in a total H-score with a range of 0–200. Note that PSMA and CEACAM5 expression in this cohort were detailed previously ^{##UREF##2##20##}.</p>", "<title>Genomic and epigenomic studies</title>", "<p id=\"P38\">Somatic alterations of the University of Washington rapid autopsy samples ^{##REF##31361600##5##,##REF##29017058##6##,##REF##26928463##38##,##REF##32460015##50##,##REF##34877933##51##} and genomics calls from the SU2C-WCDT were derived from published sources ^{##REF##35943799##52##,##REF##30033370##53##}. ChIP-seq and whole genome bisulfite sequencing data were published previously and analyzed as described previously ^{##UREF##2##20##,##REF##33785741##54##,##UREF##13##55##}.</p>", "<title>Statistics</title>", "<p id=\"P39\">Mean H-scores for each cell surface antigen were estimated using linear mixed models with fixed effects for anatomical site and random effects for patients to account for repeated sampling. Associations between expression (dichotomized FPKM) and genomic mutations were evaluated using logistic regressions with random effects for patients to account for repeated sampling. Intra-tumoral and inter-tumoral heterogeneity were estimated by bootstrap random sampling of 1000 pairs of tissue samples from the same tumor block or from the same patient and evaluating whether H-scores were both above or both below a pre-specified threshold of ≥20. Bias-corrected and accelerated 95% confidence limits used the R package Bootstrap ^{##REF##32460015##50##}. In all analyses, a p-value &lt;0.05 was considered statistically significant.</p>" ]

[ "<title>RESULTS</title>", "<title>Patterns of DLL3, CEACAM5, PSMA and TROP2 protein expression across molecular subtypes of mCRPC</title>", "<p id=\"P18\">To contextualize the expression patterns of cell surface antigens in mCRPC, we utilized a recently published molecular subgrouping framework based on AR signaling and neuroendocrine (<bold>NE</bold>) marker expression ^{##REF##31361600##5##,##REF##29017058##6##,##UREF##2##20##,##REF##36337169##37##}. This approach allows for the classification of tumors into four clinically relevant subtypes: prostatic adenocarcinoma (AR+/NE−), NEPC (AR−/NE+), amphicrine carcinoma (AR+/NE+) and double negative CRPC (AR−/NE−) ^{##UREF##2##20##,##REF##36337169##37##}. To investigate the expression of DLL3, CEACAM5, and TROP2, we employed previously validated antibodies and immunohistochemistry (<bold>IHC</bold>) assays on a dataset consisting of 753 samples from 372 distinct metastatic sites of 52 patients who underwent a rapid autopsy as part of the University of Washington Tissue Acquisition Necropsy (<bold>UW-TAN</bold>) cohort ^{##REF##31361600##5##,##REF##29017058##6##}.</p>", "<p id=\"P19\">DLL3, CEACAM5, and TROP2 exhibited membranous and cytoplasmic reactivity, with substantial differences in semiquantitative expression levels (H-score) across different molecular subtypes (##FIG##0##Figure 1A##). Consistent with prior reports, we observed the highest levels of DLL3 expression in AR−/NE+ tumors (median H-score: 90; range, 0–180) (##FIG##0##Figure 1A##,##FIG##0##B##) ^{##UREF##3##24##,##UREF##4##25##}. Similarly, CEACAM5 expression was high in AR−/NE+ tumors (median, 60; range, 0–200) (##FIG##0##Figure 1A##,##FIG##0##C##) ^{##REF##33199493##26##}. Of note, we also observed CEACAM5 reactivity in AR−/NE− tumors (##FIG##0##Figure 1C##). TROP2 expression was consistently present in AR+/NE− (median H-score: 200; range, 0–200), AR+/NE+ (median H-score: 180; range, 0–200), and AR−/NE− (median H-score: 200; range, 0–200) tumors (##FIG##0##Figure 1A##,##FIG##0##D##), whereas AR−/NE+ tumors were mostly negative (median, 0; range, 0–200). Notably, TROP2 showed more uniform expression compared to PSMA in the same cohort of AR+/NE− (median H-score: 120; range, 0–200) and AR−/NE− (median H-score: 12; range, 0–160) tumors ^{##UREF##2##20##}.</p>", "<p id=\"P20\">Next, we determined the co-expression patterns of cell surface antigens across patients. Applying a cut-off for positive expression of an H-score of ≥20 (<underline><bold>Supplementary Figure 1</bold></underline>), we found that in AR+/NE− tumors 233/304 (77%) of lesions showed expression of both TROP2 and PSMA, 7/304 (2%) were positive only for PSMA, 61/304 (20%) were positive only for TROP2 and 3/304 (1%) showed neither PSMA nor TROP2 (##FIG##0##Figure 1E##, <underline><bold>Supplementary Table 2</bold></underline>, <underline><bold>Supplementary Figure 2</bold></underline>). Similarly, in AR−/NE+ tumors, we found DLL3 and CEACAM5 co-expression in 38/71 (54%) tumors, DLL3 expression alone in 21/71 (30%), CEACAM5 expression alone in 3/71 (4.2%) and expression of neither target in 9/71 (13%) (##FIG##0##Figure 1F##, <underline><bold>Supplementary Table 2</bold></underline>, <underline><bold>Supplementary Figure 2</bold></underline>).</p>", "<title>Anatomic site distribution and inter- and intra-tumoral heterogeneity of TROP2, DLL3, and CEACAM5 expression</title>", "<p id=\"P21\">Prior studies suggested differences in cell surface protein expression based on the tumor microenvironment in different anatomic locations ^{##REF##36721073##21##}. Indeed, lower levels of PSMA expression were observed in liver metastases ^{##UREF##2##20##,##REF##36721073##21##}. To examine the association between anatomic location and the level of cell surface antigen expression, we assessed DLL3, CEACAM5, and TROP2 expression across 11 major anatomic sites of CRPC metastases (##FIG##1##Figure 2A##). While bone was the most common metastatic site in this cohort, we observed a high frequency of liver and soft tissue metastases, irrespective of the molecular tumor phenotype (##FIG##1##Figure 2A##). We observed significantly lower TROP2 expression in liver (mean H-score difference: −17; 95% CI −31 to −3.0; p=0.02) and lung (mean H-score difference: −40; 95% CI −65 to −15; p=0.001) than in vertebral bone metastases (mean H-score: 131; 95% CI 110 to 151). CEACAM5 expression in the prostate was significantly higher (mean H-score difference: 19; 95% CI 9.2 to 28; p&lt;0.001) than in vertebral bone (mean H-score:19; 95% CI 5.6 to 33), whereas DLL3 expression was higher in liver (mean H-score difference: 11; 95% CI 5.5 to 17; p&lt;0.001) and lung (mean H-score difference: 14; 95% CI 4.3 to 23; p=0.005) compared to vertebral bone metastases (mean H-score: 12; 95% CI 1.4 to 22). Note, while these differences were statistically significant, estimated differences in mean H-scores were very modest in magnitude and unlikely to be biologically relevant (##FIG##1##Figure 2A##).</p>", "<p id=\"P22\">CRPC is known to be a heterogeneous disease, often showing phenotypic differences between different metastatic sites in a given patient ^{##REF##33328650##7##,##UREF##2##20##,##REF##26928463##38##}. To characterize the heterogeneity of TROP2, DLL3, and CEACAM5 expression, we quantified the hypergeometric probability of concordant binarized H-scores (both ≥20 or both &lt;20) for random pairs of samples from a given patient (intra-patient, inter-tumoral) or from the same tumor (intra-tumoral) (##FIG##1##Figure 2B##). Estimated heterogeneity was highest for PSMA (intra-patient 17% and intra-tumoral 5%, previously reported^{##UREF##2##20##}), then CEACAM5 (14% and 6%), then TROP2 (8% and 2%), and finally DLL3 (7% and 2%).</p>", "<p id=\"P23\">We analyzed TROP2, DLL3, and CEACAM5 expression levels across different metastatic sites and classified patients into three groups: non-expressors, heterogeneous expressors, and high-expressors. In 39/52 (75%) of cases, DLL3 showed no expression, while in 9/52 (17%) of cases, it showed heterogenous expression and in 4/52 (8%) of cases, it showed homogeneous high expression. Of note, most cases with heterogeneous expression displayed different molecular subtypes across metastatic sites. Furthermore, except for two cases (##FIG##1##Figure 2C##), DLL3 labeling was present in AR−/NE+ metastases, even in the context of metastases of other molecular subtypes, confirming the tight association between DLL3 expression and neuroendocrine differentiation even in admixed molecular phenotype backgrounds. TROP2 showed the most consistent expression among the three analytes tested in this study. Only 6/52 (12%) cases showed no expression, and negative cases were enriched for NE+ tumors, with only one AR+/NE− dominant case lacking TROP2 reactivity (##FIG##1##Figure 2D##). Heterogenous TROP2 expression was present in at least one tumor in 12/52 (23%) cases, while tumors in 34/52 cases (65%) were uniformly positive. This high rate of TROP2 expression compares favorably to the expression of PSMA in the same cohort (25% no expression, 44% heterogeneous, and 31% uniformly positive) ^{##UREF##2##20##}. CEACAM5, on the other hand, was not expressed in 26/52 (51%) of cases, heterogeneously expressed in 22/51 (43%) of cases, and uniformly positive in only 3/51 (6%) of cases (##FIG##1##Figure 2E##). Notably, even in some cases which showed AR−/NE+ disease in the majority of metastases, CEACAM5 expression was low; conversely, a subset of tumors that lacked neuroendocrine features showed reactivity, suggesting that molecular subtype alone might not be sufficient to determine CEACAM5 expression.</p>", "<title>Genomic and epigenetic determinants of TROP2, PSMA, DLL3, and CEACAM5 expression</title>", "<p id=\"P24\">To explore associations between TROP2, PSMA, DLL3, and CEACAM5 expression and somatic genomic alterations, we evaluated logistic regressions and found statistically significantly higher odds of PSMA expression (OR 25; 95% CI 2.4 to 260; p=0.007) in tumors with <italic toggle=\"yes\">AR</italic> amplification but lower odds of PSMA expression in tumors with <italic toggle=\"yes\">RB1</italic> homozygous loss (OR 0.02; 95% CI 0.0 to 0.3; p=0.006) (##FIG##2##Figure 3A##, <underline><bold>Supplementary Table 3</bold></underline>). We also found lower odds of TROP2 expression in tumors with <italic toggle=\"yes\">RB1</italic> homozygous loss (OR 0; 95% CI 0.0 to 0.02; p&lt;0.001) and in tumors with <italic toggle=\"yes\">PTEN</italic> alterations (OR 0.77; 95% CI 0.77 to 0.77; p&lt;0.001). In independent publicly available transcriptomics and genomics data from 99 mCRPC cases from the StandUp2Cancer West Coast Dream Team (<bold>SU2C-WCDT</bold>), we observed higher odds of CEACAM5 expression in tumors with <italic toggle=\"yes\">PTEN</italic> deletions (OR 4.9; 95% CI 1.3 to 21; p=0.02), lower odds of CEACAM5 expression in tumors with <italic toggle=\"yes\">AR</italic> amplifications (OR 0.2; 95% CI 0.05 to 0.74; p=0.02), and lower odds of DLL3 in tumors with <italic toggle=\"yes\">AR</italic> amplifications (OR 0.06; 95% CI 0.0 to 0.4; p=0.01), and higher odds of TACSTD2 expression with <italic toggle=\"yes\">AR</italic> amplification (OR 13; 95% CI 1.8 to 260; p=0.03) (##FIG##2##Figure 3B##, <underline><bold>Supplementary Table 4</bold></underline>).</p>", "<p id=\"P25\">Prior studies have shown that <italic toggle=\"yes\">FOLH1</italic> (PSMA) expression is regulated by an orchestrated interaction between DNA methylation and histone acetylation changes ^{##UREF##2##20##}. To study the epigenetic configuration of TROP2 (encoded by <italic toggle=\"yes\">TACSTD2</italic>), <italic toggle=\"yes\">DLL3</italic>, and <italic toggle=\"yes\">CEACAM5</italic> in tumors with variable levels of target expression, we evaluated previously published whole-genome bisulfite sequencing (<bold>WGBS</bold>) and histone H3 lysine 27 acetyl (<bold>H3K27ac</bold>) and histone H3 lysine 27 tri-methyl (<bold>H3K27me3</bold>) chromatin immunoprecipitation sequencing (<bold>ChIP-seq</bold>) from CRPC patient-derived xenograft (<bold>PDX</bold>) models. We observed that in AR+/NE− PDX lines the <italic toggle=\"yes\">TACSTD2</italic> locus was enriched for H3K27ac marks, consistent with an actively transcribed gene locus (##FIG##2##Figure 3C##). AR−/NE+ tumors, however, showed gain of the repressive polycomb mark H3K27me3. No consistent DNA methylation changes associated with TACSTD2 were observed (<underline><bold>Supplementary Figure 3</bold></underline>). We further investigated the chromatin patterns at the <italic toggle=\"yes\">DLL3</italic> and <italic toggle=\"yes\">CEACAM5</italic> locus in AR+/NE− and AR−/NE+ tumors and observed H3K27ac enrichment in AR−/NE+ lines. DLL3- and CEACAM5-negative tumors were characterized by enrichment for H3K27me3. Collectively, these data demonstrate that distinct chromatin states are associated with <italic toggle=\"yes\">TROP2</italic>, <italic toggle=\"yes\">DLL3</italic>, and <italic toggle=\"yes\">CEACAM5</italic> expression.</p>" ]

[ "<title>DISCUSSION</title>", "<p id=\"P26\">Targeting cell-surface proteins has opened novel avenues for cancer therapy ^{##REF##32243795##12##–##REF##35318309##17##}. In advanced metastatic PC, PSMA-directed agents have demonstrated encouraging clinical activity, which culminated in the recent approval of PSMA-directed radioligand therapy 177-Lu-PSMA-617 ^{##REF##34161051##39##,##REF##33581798##40##}. However, a notable fraction of mCRPC tumors exhibit insufficient levels of PSMA expression for effective targeting ^{##REF##36261050##41##}. Furthermore, heterogeneity in expression that may not be detected on molecular imaging can drive treatment resistance. While experimental approaches to augment PSMA expression are being explored ^{##UREF##2##20##,##REF##35552383##42##}, it is crucial to investigate alternative cell-surface antigens to overcome primary or secondary resistance to PSMA-directed therapies and optimize therapeutic outcomes.</p>", "<p id=\"P27\">An additional challenge in the treatment of CRPC is the presence of molecular subtypes, which show distinct phenotypic and expression differences ^{##REF##31361600##5##}. Importantly, the expression patterns of cell surface proteins vary across these subtypes; for example, PSMA is rarely found in NEPC ^{##UREF##2##20##}, and even within a subtype, there can be substantial heterogeneity in cell surface protein expression ^{##UREF##2##20##,##REF##31345636##22##}.</p>", "<p id=\"P28\">Pan-cancer analyses have determined that cell surface proteins are being expressed in a lineage-independent manner across multiple tumor types. TROP2 is one such protein that has been shown to be present in multiple epithelial-derived tumors ^{##REF##21372224##34##–##REF##33929895##36##}. Sacituzumab govitecan and other TROP2-directed ADCs, including datopotamab deruxtecan, are currently in clinical development ^{##REF##36302269##43##}. Sacituzumab govitecan, which has already been approved for triple-negative breast cancer and urothelial carcinoma, has also displayed activity across multiple tumor types and is presently under evaluation in mCRPC (<ext-link xlink:href=\"https://clinicaltrials.gov/ct2/show/NCT03725761\" ext-link-type=\"uri\">NCT03725761</ext-link>) ^{##UREF##5##27##}.</p>", "<p id=\"P29\">Given the lack of detailed TROP2 protein expression data in CRPC, we determined TROP2 levels in 52 patients of the UW rapid autopsy cohort. Our analyses revealed that TROP2 protein expression is present in 88% of cases, with 34/52 patients (65%) showing TROP2 expression in all metastatic sites. Prior studies have suggested that TROP2 expression induces a neuroendocrine phenotype and that TROP2 is enriched in NEPC ^{##UREF##9##44##}. However, subsequent <italic toggle=\"yes\">in silico</italic> analyses have demonstrated low TACSTD2 (which encodes for TROP2) transcript levels in NEPC ^{##UREF##2##20##,##UREF##5##27##}. Similarly, our data show that TROP2 expression is absent in most NEPC (AR−/NE+) tumors. Collectively, TROP2 expression does not appear to be associated with the prognostically poor neuroendocrine subtype, and therefore, TROP2 targeting approaches are likely not effective in NEPC. Notably, compared to PSMA, which we previously analyzed in the same set of tissues ^{##UREF##2##20##}, TROP2 demonstrated more robust and uniform reactivity in most other CRPC tumors. Of particular interest is the high expression of TROP2 in AR−/NE− tumors, a molecular tumor subtype for which there are presently only limited specific therapies ^{##REF##29017058##6##}.</p>", "<p id=\"P30\">Clinically, NEPC represents a major challenge and novel therapies for this aggressive variant of CRPC are needed ^{##REF##34370973##3##,##REF##34111024##10##,##REF##29460922##11##}. DLL3 is an inhibitory ligand of the Notch signaling pathway, which has been found to be expressed on the surface of a variety of different neuroendocrine neoplasms, including NEPC ^{##UREF##3##24##,##UREF##4##25##,##UREF##10##45##}. Although some early clinical trials with DLL3 ADCs (Rova-T and SC-002) were impeded by systemic toxicities due to payload conjugation concerns, recent studies using bispecific T-cell engagers (such as tarlatamab, BI 764532, and HPN328) have demonstrated encouraging early results ^{##REF##33203642##23##,##UREF##3##24##,##REF##32554516##46##–##UREF##12##48##}. This expanding spectrum of targeting agents make DLL3 a very interesting and potentially relevant target in NEPC. Our protein expression data corroborated that DLL3 is primarily expressed in AR−/NE+ (NEPC) tumors. Notably, when considering all tumors, DLL3 positivity was limited, but 83% (69/83) of NEPC tumors showed protein expression, while no AR+/NE− tumors exhibiting positivity, indicating that DLL3 is a sensitive and specific marker for NEPC. This information is relevant for the development of DLL3 targeting agents for NEPC imaging.</p>", "<p id=\"P31\">In addition to DLL3, CEACAM5 has been shown to be expressed at high levels in NEPC ^{##REF##33199493##26##}. Although CEACAM5 expression can also be found in gastrointestinal, genitourinary, breast and lung cancers, a recent unbiased surface profiling effort showed a strong enrichment of CEACAM5 expression in NEPC and subsequent <italic toggle=\"yes\">in vivo</italic> models demonstrated activity of a CEACAM5 ADC in NEPC PDX models ^{##REF##33199493##26##,##UREF##7##30##}. While our study confirmed the expression of CEACAM5 in NEPC, we also noted expression in AR−/NE− tumors. Of note, 4/52 (8%) patients showed no expression of TROP2, CEACAM5, DLL3 and PSMA.</p>", "<p id=\"P32\">Our somatic genomic association studies showed that lower levels of TROP2 and PSMA were present in tumors with homozygous <italic toggle=\"yes\">RB1</italic> loss, whereas higher levels were seen in tumors with <italic toggle=\"yes\">AR</italic> amplification. Conversely, high DLL3 expression was seen in <italic toggle=\"yes\">RB1</italic> deleted cases. While these data present intriguing novel insights between the expression of TROP2, DLL3 and PSMA with common genomic alterations in CRPC, it is important to note that these associations are also tightly associated with tumor phenotype (i.e., <italic toggle=\"yes\">AR</italic> amplification is seen in AR+/NE− tumors, whereas <italic toggle=\"yes\">RB1</italic> loss is enriched in NEPC). Therefore, it is challenging to untangle the genomic alteration from broader cellular state shifts that contribute to differential expression patterns ^{##REF##30772358##8##,##REF##29460922##11##,##REF##28059768##49##}.</p>", "<p id=\"P33\">DLL3 and CEACAM5 have been shown to be regulated by the neuronal transcription factor ASCL1 ^{##UREF##4##25##,##REF##33199493##26##}. Here, we further determined the epigenetic context of these gene loci in tumors with high and low DLL3 and CEACAM5 expression. We observe that the repressive polycomb mark H3K27me3 shows strong enrichment at transcriptional start sites and gene bodies of both genes in PDX tumors with low DLL3 and CEACAM5 expression. Similarly, we show that PDX lines that lack TROP2 expression also showed enrichment for H3K27me3. This contrasts with our prior findings demonstrating that DNA methylation alterations, rather than polycomb marks, are associated with PSMA repression. Thus, polycomb repressive marks, which are established by Enhancer of zeste homolog 2 (<bold>EZH2</bold>), are likely an important epigenetic determinant of TROP2, DLL3 and CEACAM5 expression. It will therefore be important to test in future studies if EZH2 inhibitors, which are currently in clinical development for prostate cancer, can be used to pharmacologically enhance the expression of these cell surface antigens and, therefore, increase tumor targeting.</p>", "<p id=\"P34\">It is essential to consider several limitations of our study. First, this autopsy-based, single-institution study included only patients with extensive pretreatment. Thus, it remains to be established how our findings would apply to patients in earlier stages of the disease, including castration-sensitive disease. Second, the use of tissue microarray sampling may not entirely capture the intra-tumoral heterogeneity of individual lesions. Additionally, pre-analytical variables must be taken into account, particularly when evaluating bone lesions, as with all studies using formalin-fixed, paraffin-embedded tissues. Despite this potential limitation, it’s worth noting that we did not observe a trend towards lower expression in bone metastasis.</p>", "<p id=\"P35\">In summary, we have investigated the expression of clinically relevant cell surface targets in mCRPC, providing the most comprehensive tissue-based assessment of TROP2, DLL3, and CEACAM5 in CRPC to date. Our findings highlight the molecular subtype-specific expression of these proteins and provide crucial insights for the future clinical development of these drug targets.</p>" ]

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[ "<p id=\"P1\">These authors contributed equally</p>", "<p id=\"P2\"><bold><italic toggle=\"yes\">Corresponding Author:</italic></bold> Michael C. Haffner, Fred Hutchinson Cancer Center, 1100 Fairview Avenue, Seattle, WA 98109; (206) 667-6769 (phone), <email>[email protected]</email>.</p>", "<p id=\"P3\">AUTHORS’ CONTRIBUTIONS</p>", "<p id=\"P4\"><bold>Designing research studies</bold>: A. Ajkunic, E. Sayar, C. Morrissey, R. Gulati, M. T. Schweizer, P. S. Nelson, M. C. Haffner</p>", "<p id=\"P5\"><bold>Development of methodology</bold>: A. Ajkunic, E. Sayar, R. A. Patel, I. M. Coleman, B. Hanratty, M. P. Roudier, P. S. Nelson, R. Gulati, M. C. Haffner.</p>", "<p id=\"P6\"><bold>Conducting experiments</bold>: A. Ajkunic, E. Sayar, M. P. Roudier, R. A. Patel, E. Corey, I. Coleman, J. Zhao, S. Zaidi, B. Hanratty, M. Adil, L. D. True, J. K. Lee, C. Morrissey, P. S. Nelson, M. C. Haffner.</p>", "<p id=\"P7\"><bold>Acquiring and analyzing data</bold>: A. Ajkunic, E. Sayar, N. De Sarkar, R. Gulati, J. Zhao, M. P. Roudier, E. Corey, R. A. Patel, M. T. Schweizer, I. Coleman, B. Hanratty, C. Morrissey, P. S. Nelson, M. C. Haffner.</p>", "<p id=\"P8\"><bold>Providing reagents/data</bold>: J. M. Sperger, H. H. Cheng, E. Y. Yu, R. B. Montgomery, J. E. Hawley, G. Ha, J. K. Lee, S. A. Harmon, E. Corey, J. M. Lang, C. L. Sawyers, L. D. True, C. Morrissey, P. S. Nelson, M. C. Haffner.</p>", "<p id=\"P9\"><bold>Writing the manuscript</bold>: All authors.</p>", "<p id=\"P10\">Therapeutic approaches targeting proteins on the surface of cancer cells have emerged as an important strategy for precision oncology. To fully capitalize on the potential impact of drugs targeting surface proteins, detailed knowledge about the expression patterns of the target proteins in tumor tissues is required. In castration-resistant prostate cancer (CRPC), agents targeting prostate-specific membrane antigen (PSMA) have demonstrated clinical activity. However, PSMA expression is lost in a significant number of CRPC tumors, and the identification of additional cell surface targets is necessary in order to develop new therapeutic approaches. Here, we performed a comprehensive analysis of the expression and co-expression patterns of trophoblast cell-surface antigen 2 (TROP2), delta-like ligand 3 (DLL3), and carcinoembryonic antigen-related cell adhesion molecule 5 (CEACAM5) in CRPC samples from a rapid autopsy cohort. We show that DLL3 and CEACAM5 exhibit the highest expression in neuroendocrine prostate cancer (NEPC), while TROP2 is expressed across different CRPC molecular subtypes, except for NEPC. We observed variable intra-tumoral and inter-tumoral heterogeneity and no dominant metastatic site predilections for TROP2, DLL3, and CEACAM5. We further show that <italic toggle=\"yes\">AR</italic> amplifications were associated with higher expression of PSMA and TROP2 but lower DLL3 and CEACAM5 levels. Conversely, PSMA and TROP2 expression was lower in <italic toggle=\"yes\">RB1</italic>-altered tumors. In addition to genomic alterations, we demonstrate a tight correlation between epigenetic states, particularly histone H3 lysine 27 methylation (H3K27me3) at the transcriptional start site and gene body of <italic toggle=\"yes\">TACSTD2 (encoding TROP2)</italic>, <italic toggle=\"yes\">DLL3</italic>, and <italic toggle=\"yes\">CEACAM5</italic>, and their respective protein expression in CRPC patient-derived xenografts. Collectively, these findings provide novel insights into the patterns and determinants of expression of TROP2, DLL3, and CEACAM5 with important implications for the clinical development of cell surface targeting agents in CRPC.</p>" ]

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[ "<title>ACKNOWLEDGEMENTS</title>", "<p id=\"P40\">We are grateful to the patients and their families, and the rapid autopsy teams and Dr’s. Andrew Hsieh, Jonathan Wright, Funda Vakar-Lopez, Daniel Lin, and Celestia Higano for their contributions to the University of Washington Medical Center Prostate Cancer Donor Rapid Autopsy Program. We also thank the members of the Haffner, Lee, and Nelson laboratories for their constructive suggestions. This work was supported by the NIH/NCI (P30CA15704, P50CA097186, R01CA234715-03, R01CA266452, R50CA221836, PO1CA163227) NIH Office of Research Infrastructure Programs (ORIP) (S10OD028685), the U.S. Department of Defense Prostate Cancer Research Program (W81XWH-20-1-0111, W81XWH-21-1-0229, W81XWH-22-1-0278, W81XWH-18-1-0347, W81XWH-18-1-0689, W81XWH-21-1-0264), Grant 2021184 from the Doris Duke Charitable Foundation, the V Foundation, the Prostate Cancer Foundation, the Safeway Foundation, the Richard M. Lucas Foundation the Fred Hutch/UW Cancer Consortium, the Brotman Baty Institute for Precision Medicine, and the UW/FHCC Institute for Prostate Cancer Research.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1.</label><caption><title>Distribution and co-expression patterns of DLL3, CEACAM5, PSMA, and TROP2 expressions across different molecular subtypes of mCRPC.</title><p id=\"P43\"><bold>A.</bold> Representative images of cell surface antigen expressions (determined by IHC) across different molecular subtypes (AR+/NE− [green], AR−/NE+ [yellow], AR+/NE+ [red], and AR−/NE− [blue]). Molecular subtypes were defined by expression of AR signaling markers (AR, NKX3.1) and NE markers (SYP, INSM1) as described previously ^{##UREF##2##20##}. Box plots show the distribution of <bold>B.</bold> DLL3, <bold>C.</bold> CEACAM5, and <bold>D.</bold> TROP2 expressions based on H-score in the UW-TAN cohort (N=753). Box and dot colors indicate molecular phenotypes as above. <bold>E.</bold> Top, micrographs of PSMA and TROP2 in AR+/NE− tumors. Bottom, donut chart shows the distribution of PSMA and TROP2 reactivity. <bold>F.</bold> Top, micrographs of DLL3 and CEACAM5 in AR−/NE+ tumors. Bottom, donut chart shows the distribution of DLL3 and CEACAM5 reactivity. (See <underline><bold>Supplementary Table 2</bold></underline> for all co-expression profiles). Scale bars denote 50 μm.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2.</label><caption><title>Anatomic site distribution and inter- and intra-tumoral heterogeneity of TROP2, DLL3 and CEACAM5 expression in mCRPC.</title><p id=\"P44\"><bold>A.</bold> Distribution of DLL3, TROP2, and CEACAM5 protein expression across different organ sites based on IHC H-scores. Dot colors indicate molecular phenotypes. Each dot represents a tumor sample; the color codes indicate the molecular subtype (AR+/NE− [green], AR−/NE+ [yellow], AR+/NE+ [red], and AR−/NE− [blue]). <bold>B.</bold> Inter- and intra-tumoral heterogeneity of TROP2, PSMA, CEACAM5 and DLL3 expression. Mean (95% confidence interval) hypergeometric expression heterogeneity indices across different metastatic sites in a given patient (inter-tumoral heterogeneity, red) and within a metastatic site (intra-tumoral heterogeneity, gray). Dot and box plots showing the distribution of <bold>C.</bold> DLL3, <bold>D.</bold> TROP2, and <bold>E.</bold> CEACAM5 protein expression IHC H-scores in 52 cases from the UW-TAN cohort. Each dot represents a tumor sample; the color codes indicate the molecular subtype (AR+/NE− [green], AR−/NE+ [yellow], AR+/NE+ [red], and AR−/NE− [blue]). Gray shadings show interquartile ranges. Percentages show the frequencies of cell surface antigens in cases with uniformly low/negative expression (all sites H-score &lt;20), heterogeneous expression (both H-scores &lt;20 and H-score ≥20) and uniformly high expression (all sites H-scores ≥20).</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3.</label><caption><title>Genetic and epigenetic determinants of TROP2, PSMA, DLL3 and CEACAM5 expression in CRPC.</title><p id=\"P45\"><bold>A.</bold> Mosaic plots show the frequencies of TROP2, PSMA, DLL3, and CEACAM5 protein expression determined by IHC (1, expressed; 0, not expressed) as a function of the genomic status of <italic toggle=\"yes\">AR</italic>, <italic toggle=\"yes\">CHD1</italic>, <italic toggle=\"yes\">PTEN</italic>, <italic toggle=\"yes\">RB1</italic>, and <italic toggle=\"yes\">TP53</italic> (1, altered; 0, not altered) in 44 cases of the UW-TAN cohort. <bold>B.</bold> Mosaic plots show the frequencies of <italic toggle=\"yes\">TACSTD2</italic>, <italic toggle=\"yes\">FOLH1</italic>, <italic toggle=\"yes\">DLL3</italic> and CEACAM5 mRNA expression determined by RNA-seq (1, expressed; 0, not expressed) as a function of the genomic status of <italic toggle=\"yes\">AR</italic>, <italic toggle=\"yes\">BRCA2</italic>, <italic toggle=\"yes\">CHD1</italic>, <italic toggle=\"yes\">PTEN</italic>, <italic toggle=\"yes\">RB1, SPOP</italic>, and <italic toggle=\"yes\">TP53</italic> (1, altered; 0, not altered) in 99 cases of the SU2C-WCDT. <bold>C.</bold> Representative H3K27ac (gray) and H3K27me3 ChIP-seq tracks from AR+/NE− (LuCaP 77 and LuCaP 78) and AR−/NE+ (LuCaP 93 and LuCaP 145.1) PDX lines. Note the inverse differential enrichment pattern of H3K27ac and H3K27me3 (yellow box) in the upstream regulatory regions of <italic toggle=\"yes\">TACSTD2</italic>, <italic toggle=\"yes\">DLL3</italic>, and <italic toggle=\"yes\">CEACAM5</italic>.</p></caption></fig>" ]

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[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN4\"><p id=\"P41\">Conflict of interest statement:</p><p id=\"P42\">J. Zhao is currently an employee at Astra Zeneca. L.D. True is a co-founder and has equity in Alpenglow Biosciences. H.H. Cheng was a paid consultant to AstraZeneca in the past and has received research funding from Astellas, Clovis Oncology, Color Foundation, Janssen, Medivation, Promontory Therapeutics, Sanofi, and royalties from UpToDate. E.Y.Yu served as paid consultant/received honoraria Amgen, AstraZeneca, Bayer, Churchill, Dendreon, EMD Serono, Incyte, Janssen, Merck, Clovis, Pharmacyclics, QED, Seattle Genetics, and Tolmar and received research funding from Bayer, Daiichi-Sankyo, Dendreon, Merck, Taiho, and Seattle Genetics. R.B. Montgomery received research funding from AstraZeneca, ESSA, Ferring and Janssen Oncology. J.E. Hawley received consulting fees from ImmunityBio and research funding to her institution from Bristol Myers Squibb, Astra Zeneca, Vaccitech, Crescendo and Macrogenics. J. K. Lee has received research funding from Immunomedics and serves as a scientific advisor for and has equity in PromiCell Therapeutics. E. Corey received sponsored research funding from AbbVie, Astra Zeneca, Foghorn, Kronos, MacroGenics, Bayer Pharmaceuticals, Forma Pharmaceuticals, Janssen Research, Gilead, Arvina, and Zenith Epigenetics. J.M. Lang served as paid consultant/received honoraria from Sanofi, AstraZeneca, Gilead, Pfizer, Astellas, Seattle Genetics, Janssen, and Immunomedics. C.L. Sawyers serves on the board of directors of Novartis, is a cofounder of ORIC Pharmaceuticals, and is a co-inventor of enzalutamide and apalutamide. He is a science adviser to Arsenal, Beigene, Blueprint, Column Group, Foghorn, Housey Pharma, Nextech, KSQ, and PMV. M.T. Schweizer is a paid consultant/received honoraria from Sanofi, AstraZeneca, PharmaIn, and Resverlogix and has received research funding from Novartis, Zenith Epigenetics, Bristol Myers Squibb, Merck, Immunomedics, Janssen, AstraZeneca, Pfizer, Hoffman-La Roche, Tmunity, SignalOne Bio, Epigenetix, Xencor, Incyte and Ambrx, Inc. P.S. Nelson served as a paid advisor for Bristol Myers Squibb, Pfizer, and Janssen. All other authors declare no potential conflicts of interest. M.C. Haffner served as paid consultant/received honoraria from Pfizer and has received research funding from Merck, Genentech, Promicell and Bristol Myers Squibb.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"P7\">Hepatitis B virus (HBV) and HIV are the leading cause of chronic viral infection globally and it’s among the infections with prominence in the clinical evaluation of pregnancy [##UREF##0##1##]. HBV infection in pregnancy poses serious implications such as the risk of development of chronic HBV, perinatal transmission of HBV and accelerated HIV-related liver damage [##REF##33359947##2##].</p>", "<p id=\"P8\">Globally, there were about 2 billion people infected with HBV one-third of whom reside in China [##REF##29599078##3##]. China is also one of the countries with a higher prevalence of Hepatitis C virus (HCV) infection with an estimated burden of 9.8 million infections [##REF##35239716##5##]. According to the Joint United Nations programs on HIV/AIDS (UNAIDS) and World Health organization (WHO), about 38 million people were diagnosed with HIV, 257 million had HBV and 71 million had HCV by the end of 2020 [##REF##32920236##6##]. In addition, the WHO and UNAIDS implemented strategies to promote the global elimination of these viruses by 2030 [##REF##32920236##7##]. The success of the strategies relies on testing and diagnosis of at least 90% of all persons living with HBV, HCV and HIV infection, and steps towards engagement in care and treatment [##REF##34996450##8##]. In Nepal the burden of HBV/HIV, HIV/HCV, and HIV/HBV/HCV remain very high at a rate of 2.995%,18.14%, and 2.53% respectively [##REF##35239716##5##].</p>", "<p id=\"P9\">The African continent is currently facing a double burden of HBV and HIV which has affected approximately 5 to 8% of the continent’s population [##REF##30430114##9##]. In addition to HIV, HBV remains a significant public health issue characterized by high prevalence, morbidity, and mortality [##REF##27031352##10##]. Moreso, Africa accounts for 26% of the global burden of hepatitis B and C and 125,000 associated deaths [##UREF##1##11##]. Similarly in Africa, the coverage for routine childhood vaccination against hepatitis was 72% well below the target of 90% required to ensure that the virus is no longer a public health menace. In addition, HBV birth dose is administered in only 14 countries in Africa with an overall coverage of 10%. Relatedly, about 2% of persons infected with hepatitis B were diagnosed while only 0.1% were treated and 5% of persons infected with HCV were diagnosed while close to 0% were treated in Africa[##UREF##1##11##].</p>", "<p id=\"P10\">In Kenya, the prevalence of HBV co-infection with HIV was 5.8% (95%CI:2.65 to 8.9%) while HCV co-infection was 4.2% (95%CI:1.65 to 7.4%)[##REF##34308161##12##].</p>", "<p id=\"P11\">In Uganda, there is a regional variation in the prevalence of both HIV and HBV infections. In Northern Uganda, the prevalence of HIV was 3.1–7.2% and the one of HBV was 6–12%. in contrast, there was lower HBV prevalence in Southern Uganda ranging from 4.7–8%[##REF##23297754##13##–##UREF##2##14##]. Despite the high prevalence of HBV and HIV in the country, a study conducted on the uptake of HBV vaccines at birth established (39.68%) in north Uganda compared to 0.23% in the central and eastern parts of Uganda %)[##UREF##3##15##].</p>", "<p id=\"P12\">Uganda like any other sub-Saharan African country has a functional infrastructure and service delivery system for HIV care except for HBV [##UREF##4##16##]. As of now, both HIV and HBV patients are managed in separate clinics with separate staff teams even though they all receive antiretroviral treatment. Patients with HBV do not receive routine counselling and education, there are limited resources for laboratory investigation, and a high loss to follow up. HBV-infected persons are not being tested for HIV and they do not undergo routine hepatitis testing or education, yet HIV services have been integrated into routine services at the primary care level in most African countries [##UREF##4##16##].</p>", "<p id=\"P13\">The key component to reducing morbidity and mortality among people living with HBV is to receive early diagnosis and clinical care. In addition, integrated care models include follow-up of patients, knowledge on triple elimination of HBV, HCV and HIV, diagnosis of HBV, HCV and HIV, the effectiveness of diagnostic methods, vaccination, the effectiveness of treatment and prevention of HBV, HCV, and HIV [##REF##36593483##17##].</p>", "<p id=\"P14\">Against this background, the study sought to assess barriers and facilitators of integrated viral hepatitis B C and HIV care model to optimize screening uptake among mothers and newborns at health facilities in Koboko district, Uganda.</p>" ]

[ "<title>Methods</title>", "<title>Study setting</title>", "<p id=\"P15\">A cross-sectional grounded theory qualitative approach was employed in an institutional setting to obtain data on facilitators and barriers of integrated viral hepatitis B, C and HIV screening among pregnant mothers and their newborns in selected health facilities in Koboko, district, West Nile sub-region. Data was audio recorded using a recording device during the key informant interviews and was transcribed after all interviews were conducted. This type of approach was used because it provides explicit, chronological guidelines for conducting qualitative research, offers specific strategies for handling the analytical phases of inquiry, streamlines and integrates data collection and analysis and advances conceptual analysis of qualitative data [##REF##30474023##18##].</p>", "<title>Study design and sample selection</title>", "<p id=\"P16\">This study was conducted at the level of HCIIIs and the general district hospital. In Uganda, a health center III is managed by a Clinical Officer, it serves a population of 10,000 people and acts as a referral for Health Centre IIs as well as offering in-patient care, and simple diagnostic and maternal health services [##UREF##5##19##]. The study population comprised of participants drawn from; health facilities in-charges, research assistants (midwives/nurses), DHO, district EPI focal person, ART coordinator, district medical laboratory coordinator, and district cold chain technician.</p>", "<title>Diagnostics</title>", "<p id=\"P17\">The HBV and HCV testing kits were supplied for testing pregnant women in their first trimester. The HIV testing kits (DUO), coupled with the vaccines were already in the study sites supplied by MoH.</p>", "<p id=\"P18\">Respondents for this study were selected purposively based on their involvement in the delivery of HBV, HCV, and HIV care services to both neonates and mothers. Qualitative samples are purposive by virtue of the respondents having the capacity to provide rich-textured information that is relevant to integrated health services delivery [##UREF##6##20##]. The sample size determination was guided by the criterion of informational redundancy, when no information is generated by sampling more units, the sampling of respondents can be terminated. This follows the logic of informational comprehensiveness, which that suggests the more information power the sample provides, the smaller the sample size is required during the study [##REF##22058580##21##]. Therefore, 20 key informants participated in this study. A researcher-administered technique through face-to-face interaction was used for collecting data during the key informant interviews (KIIs).</p>", "<title>Study tools and data collection</title>", "<p id=\"P19\">The KII was used for collecting data from the study participants. The triple elimination for HBV, HCV, and HIV was measured using the key indicators.</p>", "<title>Data analysis</title>", "<p id=\"P20\">The data was audio recorded using phones during the KIIs, then it was transcribed after all the interviews were conducted. The data was then analyzed using framework analysis as it allows systematic data review and allows researchers to ensure that they handle the data per predetermined procedures [##UREF##7##22##]. The coding process was done using an inductive method that generates emerging themes while a deductive approach was used for pre-selected themes.</p>", "<p id=\"P21\">The study protocol was reviewed, and approved by Clarke International University Research Ethics Committee, as part of a bigger study on integrating HBV, HCV and HIV, and then registered with Uganda National Council for Science and Technology under registration numbers: CLARKE-2022–388, and HS2706ES respectively.</p>", "<p id=\"P22\">Permission to access the study participants and the health facilities was granted by the Chief Administrative Officer of Koboko district. Written voluntary informed consent was obtained from each respondent and the purpose of the study was. Participation in the study was voluntary and one could withdraw from the interview at any stage. Confidentiality of data collected was ensured to protect the privacy of the study participants. Utmost respect was accorded to the study participants.</p>" ]

[ "<title>Results</title>", "<p id=\"P23\">A total of (20 KIs) participated in this study; (12) were male, while (8) were female. Similarly, (10) respondents were aged 25 to 30 years while the remaining (10) were aged 31 to 45 years and above. In terms of the cadre (2) of the study participants were enrolled midwives, 8 were clinical officers, (1) was a district laboratory focal person, and (1) was the ART Focal Person at Koboko District General Hospital 8 were nurses (enrolled and registered). In terms of their positions at the time of the interview, (5) were maternity in-charges, (5) facility in-charges, (1) was a District Laboratory Focal person, another (1) ART/Hepatitis Focal Person at Koboko district general hospital, and (1) was the District Cold chain focal person.</p>", "<p id=\"P24\">Facilitators and Barriers of integrated viral hepatitis B, C and HIV screening among mothers and newborns</p>", "<p id=\"P25\">The facilitators and Barriers of integrated HBV, HCV, and HIV at one point of care were assessed using the WHO Health Systems Framework based on the six building blocks of a healthcare system that include; service delivery, health workforce, health management information system, medical products vaccines and technology, financing, and leadership &amp; governance. Several sub-themes emerged as facilitators and barriers to integrated HBV, HCV, and HIV as discussed under each study objective and building block below.</p>", "<p id=\"P26\">Facilitators of integrated viral hepatitis B, C, and HIV screening among mothers and newborns</p>", "<title>Service delivery</title>", "<title>Awareness and general knowledge about integrated health services delivery</title>", "<p id=\"P27\">Overall, the study participants were aware of the integrated health service delivery, and the common themes that emerged were offering integrated and a variety of health services in the health facilities. Integration also involves bringing different services together to achieve one goal, and a means of offering various packages of health services to a particular patient, and or merging health services together.</p>", "<p id=\"P28\">A respondent was quoted saying:</p>", "<p id=\"P29\"><italic toggle=\"yes\">The initial plan can be HBV but we bring HIV, syphilis, HB estimation (check blood level), check syphilis, and blood grouping. Knowing the HIV status of the mother is important so as to protect the child and have an early intervention.</italic> Another respondent also stated <italic toggle=\"yes\">“It means offering various packages of health services to a particular patient for example” physical examination of mothers, screening for HIV, HBV, malaria, syphilis, HCV, offer appropriate treatment to those found diseased.</italic> In addition, <italic toggle=\"yes\">“we also give them health education, talk to them about the importance of conducting screening at the maternity and out-patient department (KI).</italic></p>", "<p id=\"P30\">This study established that the majority of the respondents had ever heard about integrated services for viral diseases, and they acknowledged offering services to pregnant mothers, especially for HIV and HBV services. Services provided include; screening of mothers at ANC and maternity for; HIV, and HBV, linking HIV positive clients to care, and referral of HBV patients for further investigation to Koboko General Hospital. One respondent noted that</p>", "<p id=\"P31\">For the three viral infections, we screen for clients who come for health care services depending on the case history/assessment for, HBV and HIV. We do confirmatory test in the OPD laboratory for clients who test HIV positive and link them to care in the ART clinic. However, under public service delivery system, we do not have test kits for HCV” we have only accessed them through this project” (KI).</p>", "<p id=\"P32\">On the contrary, one participant disagreed that the integration of the three viral diseases (HBV, HCV, and HIV) is less talked about to the patients. The respondent was quoted saying <italic toggle=\"yes\">“At my facility, HBV is one of the conditions that are less talked about and we normally suspect HBV when there is yellowing of the eyes, abdominal distention”. “We do not have a clinic for HBV, Our ART clinic only incorporates TB and leprosy in the ART clinic (KI).</italic></p>", "<title>Diagnostics (screening), health education, and linkage to care</title>", "<p id=\"P33\">The severity of hepatitis B infection in the region and serving humanity to protect the mother and the newborn are some of the reasons why health workers promote integrated HBV, HCV, and HIV. It was also reported that HBV services are integrated into routine immunization services mainly for newborns.</p>", "<p id=\"P34\">“Involvement of implementing partners such as Infectious Diseases Institute (IDI) which offered onsite mentorship and support supervision of the health care workers has increased the number of patients seeking HBV services (KI).</p>", "<p id=\"P35\">The study participants further elaborated that for clients who test HBV (+) positive, they always subject them to other preliminary tests like antigen tests, and abdominal scans to establish whether there is an impact on the liver. One respondent said; <italic toggle=\"yes\">Depending on the outcome, we automatically start them on either medication or lifestyle counselling”. However, “under the current study, we only screen pregnant mothers during their first ANC and first trimester, if we find them positive, we refer them to the general district hospital for further investigation to reaffirm the positivity (KI).</italic></p>", "<p id=\"P36\">Another respondent stated that <italic toggle=\"yes\">“In a nutshell, our West Nile region is where viral hepatitis is still a problem as a region and we feel we have a burden that we should manage. In addition, “Integration has become policy of Ministry of Health and it has to become part of us, as we offer services to the clients.</italic></p>", "<p id=\"P37\">Integration has built our capacity as health care workers, we used to think HBV testing was meant for laboratory personnel but through integration, the midwives can now do HBV testing (improved level of knowledge for midwives)” and it has given us opportunity for early detection of viral diseases.</p>", "<title>Linkage to care</title>", "<p id=\"P38\">The following were established on linkage to care for those who test positive for HBV and HIV;</p>", "<p id=\"P39\"><italic toggle=\"yes\">We do confirmatory test and link them to care and they are initiated on ART</italic> For HBV-positive clients, <italic toggle=\"yes\">we refer them for confirmatory test to the district general hospital for example, the ART clinic at the hospital conducts antigen. We do viral load twice a year to determine replication of the virus (KI).</italic></p>", "<p id=\"P40\">While for exposed infants to HIV and HBV, nevirapine syrup is given to them to prevent mother-to-child transmission. <italic toggle=\"yes\">Children exposed to high-risk mothers are initiated on nevirapine syrup and PCR test are done at 6 and 8 months to establish whether the child acquired the virus and if PCR results turn negative, we discontinue the syrup”</italic> Meanwhile, <italic toggle=\"yes\">for children exposed to HBV positive mothers, we do not vaccinate the baby at birth but we establish whether the baby has acquired the virus, we get samples and send them to the district general hospital for further investigation (KI).</italic></p>", "<title>Health workforce</title>", "<p id=\"P41\">This study established that team spirit from health care workers facilitated integrated services for viral diseases care. This is because fellow health care workers “cover up” to fill in the gap for their colleagues in case, they are taken by other duties outside the health facility. Additionally, personal commitment from health care workers’ perspective and involvement of implementing partners through training of health care workers, supervision and supply of testing kits enhanced integrated health services delivery. To affirm that statement, one participant elaborated this further</p>", "<p id=\"P42\"><italic toggle=\"yes\">Different stakeholders and implementing partners supported health care workers for the good work done in various ways; for example, some patients gave a vote of thanks to the staff, while implementing partners provided capacity building, on-site mentorship during integrated out reaches, provision of ‘safari day’ allowance, transport allowance and salary increments by the ministry of health Uganda, that positively influenced health care workers attitude towards integrated viral hepatitis B, C and HIV (KI).</italic> On team spirit one respondent stated that</p>", "<p id=\"P43\">When the ANC department is overloaded with work, the general laboratory personnel always come in to support especially in testing”. Also, “External quality assurance checks on the competence of health care workers to establish whether the reagents are working very well, we then give feedback which makes health care workers very happy” (KI)</p>", "<title>Medical products, supplies and vaccines.</title>", "<p id=\"P44\">The majority of the KIs acknowledged that medical products such as testing kits for HBV and HIV were available in most of the health facilities and were accessible to every patient. However, no HCV RDTs are currently supplied by MoH. A respondent stated</p>", "<p id=\"P45\">We get supplies from national medical stores every two months; the consignment is first taken to the district but each facility has its package labeled (KI).</p>", "<title>Health management information system (HMIS)</title>", "<p id=\"P46\">This study established that at source document and point of care level data is managed by the respective departments such as; Outpatient (OPD), laboratory, ANC and maternity. The responsible persons have been assigned to handle data in the respective departments. It was noted that; <italic toggle=\"yes\">There are no challenges at all, the test results delivery depends on turnaround time for liver function and renal function tests, we normally collect enough samples (blood) for testing from the patients and run tests at the same time like in the following day (KI).</italic></p>", "<title>Leadership and governance</title>", "<p id=\"P47\">Regarding the leadership and governance building block; the health facility In-charges have the mandate to make requisitions for test kits, and vaccines on behalf of their respective lower health facilities such as the Health Centre IIIs in case of shortages. While the District Health Office makes requisitions on behalf of all the health facilities including Koboko General Hospital. <italic toggle=\"yes\">We make requisition, sometimes we borrow from other nearby facilities when there is a shortage, and also the implementing partner Infectious Disease Institute (IDI) usually help us with some of the supplies (KI).</italic></p>", "<p id=\"P48\">Barriers to integrated viral hepatitis B C and HIV care among mothers and newborns.</p>", "<title>Service delivery (health education, screening, linkage to care)</title>", "<title>Health education</title>", "<p id=\"P49\">There are several sub-themes that emerged as barriers to integrated service delivery. These included; a shortage of testing kits, lack of Information Education materials (IEC), language barriers, limited knowledge on HBV/C, and inadequate staffing. While barriers to linkage to care included; a lack of funds for follow-up of patients, transport challenges to support referral of patients to the next level, and inadequate knowledge among the village health teams at the grassroots level on integrated HBC, HCV, and HIV. It was noted that</p>", "<p id=\"P50\">When it comes to the issue of testing, the midwife feels it should be done by laboratory personnel and the laboratory personnel says, since the client is pregnant, she should be tested by the midwife and eventually leading to long waiting time from both the maternity and main laboratory (KI)</p>", "<p id=\"P51\">In addition, it was revealed that <italic toggle=\"yes\">“For HCV; we have knowledge gap because we are used to doing integrated testing for HIV and HBV and when HCV was introduced, in this project we had a gap and we took some time without doing the test” While another respondent noted that “Health care providers lack information on HBV/ C, they do not offer sufficient information to the clients.</italic> Consequently, they (midwives and nurses) are not willing to test pregnant mothers regularly since health talks are organised on different topics and offering targeted health talk is not sufficient at integrated care points. Thus, a respondent stated that <italic toggle=\"yes\">“There is inadequate knowledge among the village health teams on HBV, HCV and HIV during health visit to the pregnant mothers and the community (KI).</italic></p>", "<title>Shortage of test kits and vaccines</title>", "<title>Shortage of test kits;</title>", "<p id=\"P52\">Shortage of test kits emerged as a significant barrier to the integration of viral diseases. This is because the government stopped supplying them in 2018. It was during this study project that we got some HBV/HCV test kits but are only meant for pregnant mothers in their first trimester. One respondent stated <italic toggle=\"yes\">“government does not supply testing kits for HCV but HBV is supplied occasionally and the available testing kits only target pregnant mothers yet there are other people who need the services (KI).</italic></p>", "<title>Language barriers;</title>", "<p id=\"P53\">Relatedly, the study established that language barrier is a real challenge during health education. This is because patients in Koboko district (study area) speak various languages such as; Lugbara, Kakwa, Lingala, and Arabic which the healthcare workers cannot speak fluently during health education sessions. As a result, when there is no interpreter, health education sessions are not conducted, and health care workers start offering services without appropriate or adequate information.</p>", "<p id=\"P54\">We have VHTs who help us interpret but the days VHTs are not there, we start providing services without health education because there is no one to interpret. We also lack IEC materials in local language such as in South Sudanese and Congolese commonly spoken languages (KI).</p>", "<title>Diagnosis/screening and linkage to care</title>", "<p id=\"P55\">Linkage to care mostly at the referral level was found to be a barrier, and this is evidenced in the following narrative <italic toggle=\"yes\">Patients take long to come to the general hospital upon referral from the lower-level facility due to high transport cost. Initially we had challenges linking clients from maternity to OPD. For some patients we had to escort them to the point of care (KI).</italic></p>", "<title>Human resources for health</title>", "<title>Inadequate staffing</title>", "<p id=\"P56\">Human resources for health emerged as a significant barrier to the integration of viral hepatitis B, C, and HIV services in all the health Centre’s III in Koboko district. The respondents experienced inadequate staffing at different points of care mostly during community outreaches, at maternity, OPD and during health education. For instance, one respondent had this to say; <italic toggle=\"yes\">We have limited number of healthcare workers, for maternity, we have two volunteers who do not come every day and when they come, they spend 20–30 minutes, and they leave the health facility because they are not paid salary. (KI)</italic></p>", "<title>Health management information system</title>", "<p id=\"P57\">Generally, all respondents had issues with the health management information system because the HMIS 2 does not have a provision/column for entering information on HCV and as a result, HCV records are captured in counter books and sometimes papers improvised by the health care workers. Moreover, some details of the patients are taken at the point of care for example, place of residence, and the ANC preliminary care makes follow up very difficult. Also, HBV and HCV are not integrated into the Electronic Medical Records which makes updating records in the system very difficult.</p>", "<title>Leadership and governance</title>", "<p id=\"P58\">At the leadership and governance level, respondents experienced several barriers including delays in re-stocking of HBV adult vaccines. Rampant transfer of health workers providers already trained in the integration of the three viral diseases as well as political influence and objection of referrals, loss to follow up, and negative attitude towards external quality assurance by health care providers. One respondent stated that <italic toggle=\"yes\">“Limited human resource causes complaints due to long waiting time, when I am away, any services offered by me are put on hold. Similarly, the hospital management committee is chosen based on technical know-how and some have limited knowledge and understanding making it difficult to work with them (KI).</italic> Administratively, health facilities operate on political boundaries, some areas are outside their catchment area, thus communities seek services from nearby facilities. Integration of skills through internal and external approaches is difficult to manage due to limited resources.</p>", "<title>Recommendations by the Health workers</title>", "<title>Promote Integration of HBV, HCV, and HIV;</title>", "<p id=\"P59\">All HBV, HCV HIV, must be integrated into routine services since hepatitis has non-specific symptoms. It was also suggested that implementing partners must promote integration and provide funding for all three viral diseases (HBV, HCV, and HIV). Similarly, there is a need for the office of the DHO to provide transport to facilitate timely referrals. It was also suggested that pregnant mothers must be prioritized at the point of care to avoid delays in accessing services by the duo (mother and baby).</p>", "<title>Constant supply of HBV, and HCV test kits;</title>", "<p id=\"P60\">The study participants recommended sufficient and constant supply of HBV and HCV test kits. Similarly, it was suggested that; pregnant mothers and those on first ANC must be tested for HCV; organize outreach programs to offer sensitization for HBV and HCV to create demand for the services, and an additional token or incentives must be given to health workers who handle testing of mothers for HIV, HBV and HCV. It was also suggested that the national medical store should adhere to order cycles and deliver according to schedule.</p>", "<title>Engaging VHTs/Community health volunteers:</title>", "<p id=\"P61\">The study participants suggested that VHTs be attached to villages or parishes to support follow-up of patients and conduct health care workers performance reviews. This is meant to establish performance of individual health workers to enhance improvement.</p>", "<title>Human resources:</title>", "<p id=\"P62\">It was generally agreed that there is a need to address the issue of human resource. <italic toggle=\"yes\">If we have sufficient number of midwives on ground, we shall offer all required services to pregnant mothers.</italic></p>", "<p id=\"P63\">Additionally, training of health workers through on-the-job mentorship and assigning them responsibilities such as providing reports on integration of viral diseases is key. Finally, the involvement of sub-county staff who are key in mobilization during training and mentorship will increase uptake of screening and other HBV, and HCV services.</p>", "<title>Dealing with misconceptions at community level;</title>", "<p id=\"P64\">Community dialogues to remove the misconceptions and myths, such as “the baby might not grow well if you go to the health facility at early stages of the pregnancy”. More engagement of key stakeholders such the DHO, CAO, RDC political and religious leaders to support sensitization will increase uptake.</p>" ]

[ "<title>Discussion</title>", "<title>Facilitators of integrated HBC, HCV and HIV in health facilities</title>", "<p id=\"P65\">The West Nile region in Uganda has a high burden of HBV and HIV [##REF##23297754##13##]. This study revealed that health workers were aware of integrated health service delivery but reported low knowledge levels on the one-point-of-care approach. They were able to explain aspects of integrated health services by citing several examples including the HBV, HIV and Syphilis combo testing of mothers at ANC. Similarly, participants in this study noted that the severity of hepatitis B infection in the region; the protection of the mother and the newborn from HBV, HCV and HIV infection as some of the reasons that motivated them to integrate screening for the three diseases. Other reasons are the team spirit exhibited among health care workers, involvement of implementing partners in training of health care workers, and support supervision, facilitated integrated services for viral diseases. In a study conducted in an Indian hospital it was found out that health workers were aware of the hepatitis B, and C infection through blood and blood products as a mode of transmission, but awareness in relation to other modes of transmission, and integrated service delivery was dissatisfactory [##REF##23297754##23##]. The HBV and HCV disease burden, variation in the level of healthcare cadre, levels of healthcare facility, differences in testing and treatment protocols may account for the similarities and differences. On the contrary, a study conducted in US identified a knowledge gap on HBV among healthcare providers. The healthcare workers were not aware of the outcome of HBV infection, who should be screened and vaccinated against HBV, the appropriate screening methods as well as interpretation of serologic tests for HBV and proper treatment for the HBV infected [##UREF##8##24##]. This difference is possible because HBV prevalence in the US is only 0.4% [##UREF##9##25##]. this is below the WHO endemicity threshold and therefore integration may not be a priority [##UREF##10##26##].</p>", "<p id=\"P66\">The availability of HBV and HCV testing kits supplied by this project targeting pregnant women in the first trimester and HIV testing kits (DUO), coupled with the vaccines supplied by the NMS, facilitated the integration of viral diseases service delivery to the pregnant women and newborns. This is consistent with the MoH guidelines were HBV Prevention (vaccination of mothers not infected with HBV given at 0, 1 and 6 months is recommended as well as the introduction of the HBV birth dose [##UREF##11##27##]. Khan and Rose reported in a study conducted in South Africa that the most cost-effective way to prevent and control hepatitis B is through exposure to vaccine and the perception that the vaccine is safe and effective and can protect for a lifetime [##UREF##12##28##]. Similarly, a study conducted in the informal settlement of Kampala indicated that having the belief that the hepatitis B vaccine is effective in the prevention of HBV infection influenced uptake of HBV vaccines [##UREF##13##29##]. Therefore, availability of the medical supplies, the willingness of the healthcare workers to screen, vaccinate and refer patients for treatment, coupled with the positive perception of pregnant mothers promoted integration of HBC, HCV and HIV in the health facilities.</p>", "<p id=\"P67\">The HBV, HCV, and HIV cascade of care involves several steps such as; screening, diagnosis, linkage to care, assessment of liver disease stage and treatment eligibility. Laboratory blood tests are required at every step of the care cascade, to monitor the health outcomes of the patient [##REF##35588113##30##]. This study established that clients who test HBV (+) positive, are always referred to the general district hospital so other preliminary tests like antigen tests, and abdominal scans can be carried out to establish whether there is an impact on the liver. [##UREF##14##31##] A study conducted in Australia established that PoC testing theoretically removes several access barriers to services including referrals for continuum of care. Whereas there is available data and evidence for PMTCT for HIV, there is no sufficient data about HBV and HCV especially in the West Nile region and Uganda in general.</p>", "<p id=\"P68\">In this study, participants reported that availability of the HMIS2 registers and forms, electronic data management information system (EDMIS), dedicated personnel responsible for routine patient data entry into the system by both using the physical HMIS2 form or electronically, contribute to enhanced integrated health services delivery. [##UREF##15##32##] conducted a study on PMTCT in six provinces in Indonesia, and established that the majority of the districts had insufficient resources to implement PMTCT in the health facilities, including data collection infrastructure. For instance, Health facilities which provided PMTCT were limited and uncoordinated; screening mothers for HIV, HBV and Syphilis was done in Puskesmas (public health centre), while test results that required confirmation as well as study participants who required enrollment on treatment were referred to hospitals or HIV clinics elsewhere. Pregnant women had to come back to Puskesmas for post counselling. More importantly it was difficult to capture, store and transmit data for decision making on integrated services.</p>", "<p id=\"P69\">In a country with a high prevalence of HBV and lacking nationwide services to prevent mother-to-child transmission, such as Uganda, it becomes imperative to explore innovative approaches, including the integration of HBV and HCV screening and prevention into antenatal care and maternity services. Moreover, gaps exist in availability of literature and evidence on the barriers and facilitators of integrated HBV and HCV in health facilities [##UREF##16##33##]. In terms of leadership roles by the health facility and maternity In-charges, this study established that requisitions for drugs, test kits, and other medical supplies, and products are made by the health facility In-charges from the district medical store. This eases the bureaucracy and any delays that may be associated with stock-outs, thus enhancing integration</p>", "<title>Barriers to integrated HBC, HCV and HIV in health facilities</title>", "<p id=\"P70\">This study revealed that a lack of information and education materials (IEC) regarding integrated HBV, HCV, and HIV, along with language barriers stemming from the diverse languages spoken by patients in Koboko district, such as Kakwa, Arabic, Swahili, Lingala, and Lugbara, poses a significant challenge for most healthcare workers. Consequently, conducting effective health education sessions becomes problematic in the absence of an interpreter. Consequently, healthcare workers often provide health services without adequate health education. This, in turn, results in communication gaps during both pre- and post-test counseling, affecting clients who receive positive test results until they enter the continuum of care. A scoping review conducted by Mohanty et al., [##UREF##17##34##] also established that poor communication between health care providers and caregivers, limited availability and access to health facility-based immunization and cost of vaccines were barriers to integration of viral HBV, HCV and HIV.</p>", "<p id=\"P71\">Our results demonstrated that healthcare workers exhibited an insufficient level of knowledge regarding the management and treatment of viral diseases, particularly HBV and HCV. This knowledge gap was also evident in their ability to conduct HBV and HCV tests, as they lacked access to comprehensive information and had not received adequate training in viral diseases. Consequently, this deficiency in knowledge and skills hindered the incorporation of HCV screening into pregnancy-related care. [##REF##33369439##35##]. Whereas pregnant women are screened for HBV and HCV, they are not provided with health education particularly at the ANC and maternity. But these challenges seem not to be only an issue in developing countries such as Uganda; for instance, a study conducted in the US identified knowledge gap on HBV among health workers, and reported that they were not aware of HBV infection outcome, who should be screened and vaccinated against HBV, the screening algorithm as well as interpretation of serologic tests for HBV and appropriate treatment for the HBV infected [##UREF##8##24##]. Additionally, lack of public awareness significantly contributed to poor outcomes from infection among risk persons due to continued transmission to susceptible individuals [##REF##36399489##36##]. Therefore, lack of knowledge was a strong barrier to testing, prevention, and care.</p>", "<p id=\"P72\">Shortage of test kits for HBV in the health facilities significantly affected integration of viral disease care. Lower health facilities normally have a challenge of capacity to order and store medical products, vaccines and other supplies. A qualitative study conducted in Democratic Republic of Congo established that untimely vaccinations, regular logistics challenges mostly stock-outs, and inability to store vaccines were key barriers to integrated HBV, HCV and HIV screening [##REF##17570807##37##]. Moreso, there were complex and unsynchronized vaccines fees across health facilities, inadequate communication across delivery and vaccination wards, limited and incorrect understanding of vaccines among mothers and community members. A related study conducted in Lira district northern Uganda found that lack of access to HBV screening services at the government health facilities contributed to HBV infection [##UREF##18##38##]. Medical products, vaccines and technology are a key building block of a healthcare system according to WHO especially in creating access to critical services such as HBV and HCV for mothers.</p>", "<p id=\"P73\">Shortage of human resources for health to provide HBV and HVC services emerged as a barrier to integration of viral disease care in all the study sites. All study participants highlighted this as a persistent problem both in the maternity and out-patient department. The shortage of human resources for health is a global issue that affects delivery of quality healthcare services. Findings from this study are consistent with those from a study conducted in Indonesia, where researchers established that the majority of the health facilities had insufficient human and other resources to implement PMTCT [##UREF##18##38##]. In addition, health facilities that offered PMTCT services were limited and uncoordinated. For example, screening mothers for HIV, HBV and syphilis was done in public health facilities while the test results confirmation and treatment initiation were done in other HIV clinics [##UREF##19##41##] This implies that pregnant mothers had lost opportunities in accessing screening services, treatment and care due to long distance to health facilities, and parallel testing systems due to the insufficient human resources.</p>", "<p id=\"P74\">This study also established that respondents faced leadership and governance challenges during integration of viral disease cares. Delays were experienced in re-stocking of HBV vaccines for adults, frequent transfer of already trained health care workers in the integration of viral diseases and political influence on decisions to screen every patient for viral disease even when testing kits are meant for pregnant mothers and newborns. Resource mobilization and allocation to facilitate delivery of healthcare services is a function of leadership and governance, the first building block of a healthcare system. A study conducted to assess recommendations for prevention and control of hepatitis B and C revealed that inadequate resources allocated for the prevention, control and surveillance program increased continued transmission of HBV and HCV [##UREF##8##24##]. Similarly, Ajuwon et al (X) reported in a study conducted in Nigeria that nine in every ten of its population that live with chronic HBV are not aware of their infection status and are not captured in the global health statistics due to lack of resources and lack of political will to address the HBV burden [##REF##31349838##39##].</p>", "<p id=\"P75\">Finally, loss to follow-up was identified as another challenge in the continuum of care for HBV, HCV and HIV integration. The study participants experienced loss to follow-up of patients due to lack of funds, but also partly due to negative attitude towards clients by the health workers, which negatively affected integration of viral disease care. In a study carried out by Mitchell <italic toggle=\"yes\">et al</italic>., it was established that inadequate resources allocated affected referral mechanism and follow-up of clients [##UREF##8##24##]. Generally, the majority of health programmes funded by partners tend to focus mainly on procurement of test kits, and renumerations for health workers. However, there is no due consideration to support referral pathways as an important component of promoting the continuum of care.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"P76\">The facilitators to integrated viral diseases (HBV, HCV, and HIV) were; high prevalence of hepatitis B infection in West Nile region, team spirit from health workers perspective, reduced long waiting time, availability of medical products such as HBV and HCV test kits, integration of HBV and HIV into HMIS2 form and availability of support from implementing partners such as IDI which offer mentorship and training on integration and support supervision. Barriers to integration included; knowledge gap from health care workers perspective, and lack of transport, language barriers during health education, inadequate human resource for health, stock-out of testing kits for HBV and HCV, lack of HMIS 2 that integrated HCV, lack of fund to facilitate follow up of patients after referral further investigation upon suspected cases of HBV and HCV.</p>" ]

[ "<p id=\"P1\">Authors’ Contributions</p>", "<p id=\"P2\">John Bosco Alege conceptualized the study, collected data, and wrote the first draft research report, including the draft manuscript with substantial guidance and mentorship from; Orago, Oyore, and Musoke (supervisors). Nanyonga reviewed the draft manuscript and provided additional technical guidance that led to the writing of this final manuscript. Alege remains the primary and corresponding author.</p>", "<title>Background</title>", "<p id=\"P3\">HIV and HBV remain significant public health challenges characterized by high prevalence, morbidity, and mortality, especially among women of reproductive age in Uganda. However, both HIV and HBV patients are managed in separate clinics with separate staff even though they all receive ART. Patients with HBV do not receive routine counselling and education, and there are limited resources for laboratory investigation coupled with a high loss to follow-up. This study set out to “assess barriers and facilitators of integrated viral hepatitis B C and HIV care model to optimize screening uptake among mothers and newborns at health facilities in Koboko District, west Nile sub-region, Uganda”.</p>", "<title>Methods</title>", "<p id=\"P4\">A cross-sectional grounded theory qualitative approach was employed in an institutional setting (HC IIIs). Data was audio recorded using a recording device during the key informant interviews and was transcribed after all interviews were conducted. Data was then analyzed using framework analysis.</p>", "<title>Results</title>", "<p id=\"P5\">The following facilitated integration: High prevalence, and therefore burden of hepatitis B infection in West Nile region, team spirit by the health workers, reduced long waiting time, availability of medical products such as HBV and HCV test kits, integration of HBV and HIV into HMIS2 form and availability of support from implementing partners such as Infectious Dease Institute which offered mentorship and training on integration and support supervision.</p>", "<title>Conclusion</title>", "<p id=\"P6\">Barriers to integration included; knowledge gap among health care workers, lack of transport for patients, language barriers during health education, inadequate human resources for health, stock-out of testing kits for HBV and HCV, lack of HMIS 2 column to capture HCV data, lack of funds to facilitate follow up of patients after referral for further investigation upon suspected cases of HBV and HCV. The study participants recommended; Promoting the integration of HBV, HCV, and HIV into routine health services; ensuring a constant supply of HBV, and HCV test kits to avoid stock-out; Engaging VHTs/Community health volunteers to support follow-up of patients and conducting health care workers performance reviews; addressing the issue of inadequate human resource; and finally dealing with misconceptions at community level about HBV and HCV diseases which hinder access to services.</p>" ]

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[ "<title>Acknowledgments</title>", "<p id=\"P77\">The authors would like to acknowledge Lugei Foundation, Clarke International University, the School of Public Health and Applied Human Sciences, Kenyatta University, Nairobi, Kenya, to which the authors are affiliated, and their support in this implementation research project.</p>", "<title>Funding</title>", "<p id=\"P78\">The Research reported in this publication was supported by the Fogarty International Center of the National Institutes of Health, U.S. Department of State’s Office of the U.S. Global AIDS Coordinator and Health Diplomacy (S/GAC), and President’s Emergency Plan for AIDS Relief (PEPFAR) under Award Number 1R25TW011213. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.”</p>", "<title>Data Availability</title>", "<p id=\"P79\">The raw data for this study are available and shall be shared upon request.</p>" ]

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[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN2\"><p id=\"P80\">Declarations</p><p id=\"P81\">Competing Interests</p><p id=\"P82\">All authors declared that they had no competing interests</p></fn><fn fn-type=\"COI-statement\" id=\"FN3\"><p id=\"P83\"><bold>Additional Declarations:</bold> No competing interests reported.</p></fn><fn id=\"FN4\"><p id=\"P84\">Ethical Approval</p><p id=\"P85\">This study was approved by Clarke International University Research Ethics Committee and UNCST.</p></fn><fn id=\"FN5\"><p id=\"P86\">Consent for publication</p><p id=\"P87\">Participants provided consent to publish the findings of this study</p></fn></fn-group>" ]

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{ "acronym": [ "ANC", "CHB", "GHSS", "HBsAg", "HBV", "HCC", "MoH", "REC", "UNCST" ], "definition": [ "Antenatal Clinic", "Chronic hepatitis B", "Global health sector strategy", "Hepatitis B surface antigen", "Hepatitis B virus", "Hepatocellular carcinoma", "Ministry of Health", "Research and Ethics Committee", "Uganda Council for Science and Technology" ] }

[ "<title>INTRODUCTION</title>", "<p id=\"P2\">Ebola virus (EBOV; family <italic toggle=\"yes\">Filoviridae</italic>: species <italic toggle=\"yes\">Orthoebolavirus zairense</italic>) causes severe and frequently fatal acute human disease in outbreaks that can cause thousands of deaths. Complicating containment efforts, EBOV may persist subclinically in survivors for years and reignite outbreaks. Ebola virus disease (EVD) can be prevented with two licensed vaccines and treated with approved monoclonal antibody (mAb)-based therapeutics^{##UREF##0##1##}. However, even with approved mAbs, acute outcomes remain poor in EVD patients with high viral loads and/or advanced disease, and the impact of on viral persistence is unknown. As such, identification and optimization of novel mAbs is needed to address these gaps.</p>", "<p id=\"P3\">All advanced anti-EBOV mAb therapeutics and vaccines target protein spikes protruding from virion envelopes ^{##REF##35335698##2##}. Each spike comprises a single EBOV-encoded glycoprotein (GP_1,2), synthesized by translation of a preproprotein that is cleaved in the Golgi apparatus into GP₁ and GP₂ subunits. A disulfide bond links these two subunits to form heterodimers (##FIG##0##Fig. 1A## top) that assemble into GP_1,2 trimers ^{##REF##9658086##3##,##REF##9576958##4##}. GP₁ contains a heavily glycosylated mucin-like domain (MLD) that obscures the upper and outer portions of GP_1,2 and a glycan cap domain that shields the virion receptor-binding site in the GP₁ core from the host immune response (##FIG##0##Fig. 1A## top). Upon virion entry, host-cell cathepsins proteolytically process GP_1,2 in the endosome to remove the MLD and glycan cap domain to expose the GP₁ core and binding site for the virion receptor, NPC intracellular cholesterol transporter 1 (NPC1) ^{##REF##26771495##5##–##REF##18615077##7##}. GP₂, a typical class I fusogen, mediates fusion of virion and endosomal membranes to release viral ribonucleocomplexes into the target cell ^{##REF##28874543##8##}. GP₂ contains an internal fusion loop (IFL), two consecutive heptad repeat regions (HR1 and HR2), a membrane proximal external region (MPER), and a C-terminal transmembrane (TM) domain (##FIG##0##Fig. 1A## top). HR2, also called the stalk, is largely alpha-helical and connects the GP₁ core to the MPER and TM domain ^{##REF##18615077##7##}.</p>", "<p id=\"P4\">Major recognition sites for anti-EBOV mAbs are the GP₁ MLD and glycan cap domain; the GP_1,2 trimer base; and the GP₂ IFL, stalk, and MPER ^{##REF##26912366##9##–##REF##36708708##17##}. The stalk–MPER is of special interest for therapeutic and vaccine design. The mAbs that bind the stalk–MPER have potent neutralization activity and target a region with high amino acid sequence conservation across orthoebolaviruses. Indeed, the region has 70% sequence conservation among all six orthoebolaviruses and for the three viruses that can cause fatal disease, Bundibugyo virus (BDBV), EBOV, and Sudan virus (SUDV), the sequence conservation increases to ~ 90% ^{##REF##26912366##9##,##REF##31104840##11##,##REF##26806128##13##}. Despite these important features, mAbs against the stalk–MPER are the least well-characterized of anti-orthoebolavirus mAbs in part because of a comparative lack of structural information for this site: to increase solubility and stability, MPER was deleted from GP_1,2 constructs used for all high-resolution GP_1,2 structures determined to date. Only one structure, of a stalk–MPER targeted mAb, the BDBV223, against BDBV, is available ^{##REF##29736037##18##,##REF##30996276##19##}. Curiously, BDBV223 binds to a GP₂ epitope that in current models of the EBOV GP_1,2 structure is predicted to be occluded by the virion membrane ^{##REF##30996276##19##}. As such, the mechanism by which mAbs that target the stalk–MPER access their epitope, and whether accessibility is associated with therapeutic efficacy for patients infected with EBOV or related viruses, remains unclear.</p>" ]

[ "<title>Methods</title>", "<title>Cell lines</title>", "<p id=\"P20\">Human embryonic kidney (HEK) epithelial Expi293F cells (Thermo Fisher Scientific, Waltham, MA, USA) were cultured on orbital shakers in Expi293 expression medium (Thermo Fisher Scientific) at 37°C in a humidified atmosphere containing 8% CO₂. HEK 293T cells (American Type Culture Collection [ATCC] Manassas, VA, USA; #CRL-3216) were cultured in high-glucose Dulbecco’s modified Eagle’s medium (DMEM) containing L-glutamine (Invitrogen, Carlsbad, CA, USA), supplemented with 10% heat-inactivated fetal bovine serum (FBS; Omega Scientific, Tarzana, CA) and 1% penicillin–streptomycin solution (Thermo Fisher Scientific). Cells were maintained at 37°C in a humidified atmosphere with 5% CO₂. <italic toggle=\"yes\">Drosophila</italic> Schneider 2 (S2) cells (Thermo Fisher Scientific) were cultured with Schneider’s <italic toggle=\"yes\">Drosophila</italic> medium (Thermo Fisher Scientific) in stationary flasks at 27°C. Stable cell lines were adapted to serum-free conditions and maintained on orbital shakers at 27°C.</p>", "<title>Antibody and antibody fragment expression, purification, crystallization, and visualization</title>", "<p id=\"P21\">Protein fragment generation, protein and protein fragment purification, crystallization, X-ray structure determination, and negative-stain electron microscopy were performed following standard protocols.</p>", "<title>Protein expression and purification</title>", "<p id=\"P22\">The Expi293 Expression System (Thermo Fisher Scientific) was used for expression of immunoglobulins. Light and heavy chain-encoding plasmids were prepared using an endotoxin free kit (Takara Bio, NucleoBond Xtra Midi Plus EF) and used to transfect Expi293 cells at a 2:1 ratio of light chain to heavy chain using Expifectamine 293 transfection reagent (Thermo Fisher Scientific) according to the manufacturer’s instructions. Monoclonal antibodies (mAbs) containing supernatants from transfected cells were clarified by centrifugation and then incubated with protein A agarose resin (GenScript, Piscataway, NJ, USA) in batch format overnight, followed by washing, elution, and buffer exchange into Dulbecco’s phosphate-buffered saline (DPBS; Thermo Fisher Scientific) as previously described ^{##REF##19247287##29##}. Antibodies used <italic toggle=\"yes\">in vivo</italic> were verified to be endotoxin-free using a commercial kit (Thermo Fisher Scientific). All antibodies produced in this study were expressed as human IgG1.</p>", "<p id=\"P23\">Fragment antigen binding (Fab) fragments were generated from purified IgG1s through digestion with 3% immobilized papain (Thermo Fisher Scientific) for 2 h, followed by purification with a Mono Q anion exchange chromatography column (GE Healthcare, Chicago, Illinois, USA) and size-exclusion chromatography with a Healthcare Superdex 75 Increase 10/300 GL column (GE Healthcare) in 1X tris-buffered saline (TBS, Thermo Fisher Scientific). Fractions with pure Fab were concentrated using Ultra Centrifugal Filter Units (Amicon, Miami, Florida, USA). Epitope peptide representing glycoprotein (GP_1,2) residues 626–640 was chemically custom-synthesized by Thermo Fisher Scientific and purified via high-performance liquid chromatography (HPLC).</p>", "<p id=\"P24\">Recombinant wild-type (WT) and Ebola virus/H.sapiens-tc/COD/1976/Yambuku-Mayinga GP_1,2 ectodomain variants, the latter lacking residues 312–462 of the mucin-like domain and residues 644–676 of the transmembrane domain (GP_{1,2ΔTM/ΔMLD}) and, were expressed in S2 cells. All constructs carried a C-terminal double-strep tag for affinity purification. Stably transfected cells were selected with 6 μg/mL puromycin (InvivoGen, San Diego, California, USA). The resulting strep-tagged proteins were purified using a 5-mL StrepTrap column (Cytiva, Marlborough, MA, USA) following the manufacturer’s protocol and then further purified with size-exclusion chromatography (SEC) using a Superdex 200 column (Cytiva) in 1X TBS.</p>", "<title>Protein crystallization</title>", "<p id=\"P25\">3A6 Fab was crystallized in 28% PEG400, 0.1 M 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid (HEPES)–sodium hydroxide (NaOH) pH 7.5 buffer, and 0.2 M calcium chloride (CaCl₂) (Hampton Research, Viejo, California, USA) at 20°C. To form the 3A6 Fab–peptide complex, purified 3A6 Fab was concentrated to 5 mg/mL, combined with a five-fold excess of peptide, and incubated at 4°C for 18 h. The Fab–peptide complex was crystallized in 30% polyethylene glycol (PEG) 3000, 0.1 M tris(hydroxymethyl)aminomethane (Tris), pH 7.0, and 0.2 M sodium chloride (NaCl) (Hampton Research) at 20°C. Crystals were flash-cooled in liquid nitrogen, with 15% ethylene glycol (Hampton Research) as a cryoprotectant.</p>", "<title>X-ray data collection and protein structure determination</title>", "<p id=\"P26\">X-ray diffraction data of Fab–peptide complexes were collected on beamline 12 − 2 at the Stanford Synchrotron Radiation Lightsource, and Fab diffraction data were collected on beamline 23ID-B at the Advanced Photon Source ^{##REF##7943563##30##,##REF##20439981##31##}. One dataset for the Fab crystal was used, and two datasets from separate Fab–peptide complex crystals were merged for processing using AutoPROC with XDS ^{##REF##20124692##32##,##REF##21460447##33##} for indexing and integration, followed by POINTLESS ^{##REF##21460446##34##} and AIMLESS ^{##REF##23793146##35##}, programs for data reduction, scaling, merging, and calculation of structure factor amplitudes and intensity statistics. One Fab–peptide complex per asymmetric unit was present in space group P1 21 1 (a = 52.3 Å, b = 66.4 Å, c = 68 Å, α = γ = 90°, β = 104.2°), and four Fabs were present in the asymmetric unit of the Fab structure in space group P1 (a = 53.7 Å, b = 65.7 Å, c = 125.6 Å, α = 98.7°, β = 91.4°, γ = 96.0°). Crystal structures were determined by molecular replacement using Phaser ^{##REF##19461840##36##} within the CCP4 package ^{##REF##21460441##37##}, with a homology model predicted with SWISS-MODEL ^{##REF##29788355##38##} as a starting model. Iterative manual model rebuilding was performed using Coot^{##REF##15572765##39##} and refined with <italic toggle=\"yes\">Phenix</italic>\n^{##REF##31588918##40##}. The peptide was built into different Fourier maps and calculated prior to inclusion of the respective structural elements. Final atomic coordinates and structure factors of the Fab–peptide complex and Fab were deposited in the Protein Data Bank (PDB) under identification numbers (IDs) 7RPU and 7RPT, respectively. Figures were created in PyMOL (<ext-link xlink:href=\"http://www.pymol.org/\" ext-link-type=\"uri\">http://www.pymol.org/</ext-link>).</p>", "<title>Size-exclusion chromatography coupled to multi-angle light scattering</title>", "<p id=\"P27\">Size-exclusion chromatography coupled to multi-angle light scattering (SEC-MALS) experiments were performed using a Superdex 200 Increase 10/300 column (Cytiva), and an ÄKTA FPLC purifier in line with a Wyatt miniDAWN MALS detector and a Wyatt Optilab digital refractive index (dRI) detector (Amersham Biosciences). All experiments were performed in 1X TBS. ASTRA VI software was used to combine these measurements and enable the absolute molar mass and extinction coefficient of the eluting glycoprotein, Fab, or glycoprotein–Fab complex to be determined ^{##UREF##2##41##,##REF##8811899##42##}.</p>", "<title>Composition gradient multi-angle light scattering</title>", "<p id=\"P28\">Composition gradient multi-angle light scattering (CG-MALS) experiments were performed with a Calypso II composition gradient system (Wyatt) to prepare different compositions of buffer, glycoprotein, and antibody and deliver to the miniDAWN detector and an online ultraviolet (UV) detector (Cytiva). The extinction coefficient obtained from the SEC-MALS experiment was used to measure the concentration of the glycoprotein during CG-MALS experiments. Polycarbonate filter membranes with 0.1-μM pore size (Millipore Sigma, Burlington, MA, USA) were installed in the Calypso system for sample and buffer filtration. Glycoprotein was diluted to a stock concentration of 40–60 μg/mL in TBS. Fab was diluted to a stock concentration of 50–60 μg/mL in TBS. The automated Calypso method consisted of a dual-component “crossover” gradient to assess hetero-association between the glycoprotein and Fab. For each composition, 0.7 mL of protein solution were injected into the UV and MALS detectors until an equilibrium was reached within the MALS flow cell and the flow stopped for 300–800 s. Data were collected, and analyses were performed with CALYPSO software. GP₂–3A6 Fab association was measured in triplicate with 2 different preparations of glycoprotein and Fab.</p>", "<title>Protein assays</title>", "<p id=\"P29\">Enzyme-linked immunosorbent and cell-based antibody binding assay were performed with wild-type virus glycoproteins or variants created via alanine scanning following standard protocols.</p>", "<title>Cell-based antibody binding assay</title>", "<p id=\"P30\">To evaluate binding of mAbs to glycoprotein variants, HEK 293T cells expressing full-length GP₁,2 or variants thereof were incubated with unlabeled mAbs at 10 μg/mL, followed by staining with DyLight 488 anti-human IgG and detection of fluorescence by microscopy. The binding of a control conformational mAb (ADI-15878 Ig) ^{##REF##26912366##9##,##REF##30184505##43##,##REF##30206174##44##} was used as a control for glycoprotein expression levels. Secondary antibody binding only was used as a negative control to assess background binding. In detail, HEK 293T cells were plated at ≈ 1×10⁵ cells per well in 24-well plates treated with Poly-L-lysine (Millipore Sigma) 1 d prior to transfection. Cells were transiently transfected with 0.5 μg DNA per well using <italic toggle=\"yes\">Trans</italic>IT-LT1 transfection reagent (Mirus Bio, Madison, WI, USA). At 48 h post-transfection, cells were fixed with 4% paraformaldehyde (PFA, Electron Microscopy Sciences) in DPBS for 20 min. Cells were then incubated for 1 h at room temperature with 10 μg/mL primary mAbs in DPBS supplemented with 1% BSA (Millipore Sigma). Cells were subsequently incubated at room temperature for 1 h with DyLight 488 anti-human IgG secondary antibody (SA5–10126; Thermo Fisher Scientific) and Hoechst 33342 (Invitrogen) in DPBS supplemented with 1% BSA. Finally, cells were imaged on a widefield fluorescence Axiovert 200M Marianas microscope with a 10x/0.3 dry objective (ZEISS, Feasterville, PA, USA) or a confocal LSM780 microscope with a 10x/0.3 dry objective (ZEISS). Images were analyzed in QuPath ^{##REF##29203879##45##}. Nuclei were segmented using the Hoechst image, and the objects were expanded by 5 μm to locate approximate cell boundaries. DyLight 488-positive and DyLight 488-negative cells were counted using a trained object classifier. The classifier was optimized for the widefield and confocal images separately. Then, all data from 3 biological replicates were combined, and the 3A6-positive cell percentage was normalized against that obtained with ADI-15878.</p>", "<title>Enzyme-linked immunosorbent assay</title>", "<p id=\"P31\">Microtiter plate wells were coated with purified recombinant WT or mutant GP_{1,2ΔTM/ΔMLD} and incubated at room temperature for 1 h before blocking with 3% bovine serum albumin (BSA; Millipore Sigma) in DPBS containing 0.05% TWEEN-20 (Fisher Scientific) for 1 h. Serial dilutions of mAb were applied to the wells and incubated for 1 h at room temperature. The bound antibodies were detected using Jackson Immuno Research Labs peroxidase-conjugated goat anti-human IgG (Thermo Fisher Scientific; #109036006) with horseradish peroxidase (diluted 1:4,000) and 3,3’,5,5”-tetramethylbenzidine (TMB) substrate (Thermo Fisher Scientific) before 50 μl of 1 N sulfuric acid (Fisher Scientific) was added to stop the reaction. Absorbance at 450 nm was then measured using a Spark microplate reader (Tecan, Männedorf, Switzerland). Half-maximal response (EC₅₀) values for mAb binding were determined using Prism 7 (GraphPad Software, Boston, Massachusetts, USA) after log-transformation of antibody concentrations using EC₅₀ shift nonlinear regression analysis.</p>", "<title>Plaque reduction assay using biologically contained EBOV</title>", "<p id=\"P32\">A biologically contained EBOV, EbolaΔVP30 virus (Halfmann et al., 2008), was used to assess the impact of a P636S mutation on 3A6-mediated neutralization as previously described (Davis et al., 2019). Briefly, Ebola-GP-P636SΔVP30-eGFP virus was incubated with 10 μg/mL of monoclonal antibody (mAb) at 37°C for 60 min. The virus/mAb mixture was then inoculated onto Vero VP30 cells, seeded the previous day in 12-well plates. After a 60 min incubation, cells were washed to remove any unbound virus, and overlaid with 1.25% methylcellulose media to allow for plaque formation. Seven days after infection, the number of plaques was quantified after immunochemistry staining with an antibody against the VP40 protein.</p>", "<title>Negative-stain electron microscopy</title>", "<p id=\"P33\">GP2–3A6 complexes were obtained by incubating GP_{1,2ΔTM/ΔMLD} with a three-fold molar excess of 3A6 Fab overnight followed by purification using a Superdex 6 Increase 10/300 GL SEC column. The complexes were diluted to 0.01 mg/mL, and 4 μL of the complex solutions were each applied to freshly plasma-cleaned carbon-coated 400-mesh copper grids (Electron Microscopy Sciences, Hatfield, PA, USA) for 1 min. The solutions were blotted from the grids, followed by staining with 1% uranyl formate (Electron Microscopy Sciences, Hatfield, PA, USA) for 1 min. The stain was then blotted from the grids, which were then air-dried before imaging. Images were collected on an FEI Titan Halo 300 kV electron microscope (Thermo Fisher, Waltham, MA, USA) at a magnification of ×57,000 with a Falcon II camera. Contrast transfer function (CTF) correction, particle picking, 2-dimensional class averaging, and 3-dimensional reconstruction and refinement were all performed using cryoSPARC v3.1.0 ^{##REF##28165473##46##}.</p>", "<title>Virion-like particle preparation, purification, and visualization</title>", "<p id=\"P34\">Ebola virus virion-like particles (VLPs) were prepared from HEK 293T cells by co-expression of full-length GP_1,2 and matrix protein (VP40) essentially as described previously ^{##REF##33016878##47##} except that phosphate-buffer saline (Gibco PBS; Thermo Fischer Scientific) instead of TNE buffer was used. Clarification of supernatants from 4 × 150 mm dishes was performed at 3,000 ×g for 15 min at 4°C. Final pellets after density gradient purification were resuspended in 200 μL Gibco PBS.</p>", "<title>Cryogenic electron tomography</title>", "<title>Sample preparation, data collection, and tomogram reconstruction</title>", "<p id=\"P35\">Fabs (1 mg/mL) were mixed with purified VLPs and 10-nm colloidal gold and incubated for 30–60 min at 4°C. Different combinations of Fabs were prepared, imaged, and processed in parallel. The different mixtures (4 μL) were added to C-Flat 2/2 EM grids (Protochips) and vitrified by back-side blotting (4-s blotting time) using a LeicaGP cryo plunger (Leica; Deerfield, IL, USA) and stored in liquid nitrogen until imaging.</p>", "<p id=\"P36\">Cryogenic electron tomography data collection was performed essentially as described previously ^{##REF##32805734##48##} on a Titan Krios electron microscope equipped with Gatan Bioquantum energy filter and K3 direct electron camera (Thermo Fisher). The nominal magnification was 64,000×, giving a pixel size of 1.39 Å on the specimen. The defocus range was − 2.0 to −4.5 μm, with a 0.25-μm step size (Supplementary Table 2).</p>", "<title>Subtomogram averaging</title>", "<p id=\"P37\">Tomograms were reconstructed using IMOD ^{##REF##8742726##49##}, and the initial steps of subtomogram alignment and averaging were implemented using MATLAB (MathWorks) scripts and subTOM package (<ext-link xlink:href=\"https://github.com/DustinMorado/subTOM/releases/tag/v1.1.0\" ext-link-type=\"uri\">https://github.com/DustinMorado/subTOM/releases/tag/v1.1.0</ext-link>), which were derived from the TOM ^{##REF##15721576##50##} and AV3 ^{##REF##15774580##51##} packages as described previously ^{##REF##32805734##48##}.</p>", "<p id=\"P38\">To generate an initial starting model for each structure, 50–100 copies of glycoprotein were manually identified from VLP filaments that were down-scaled by 6× binning of the voxels and subjected to reference-free subtomogram alignment. To identify the positions of all the particles on the viral surface of viral filaments, markers were placed manually along the filament of the tube using the Volume Tracer function in UCSF Chimera (v.1.13.1) ^{##REF##15264254##52##} and the radius of the filament was determined centered at the membrane using the Pick Particle Chimera Plugin ^{##REF##30478053##53##}. An oversampled cylindrical grid of points was generated on the filament surface with ≈ 8 nm spacing, and subtomograms were extracted for all grid points with a box size of 64 pixels (approximately 50 nm) centered at a radius 10 nm above these grid positions. Initial Euler angles were assigned to each subtomogram based on the orientation of the normal vectors relative to the cylinder surface.</p>", "<p id=\"P39\">Subsequent processing was performed in RELION ^{##REF##26256537##54##}, as described previously ^{##REF##32805734##48##}. The VLP-GP_1,2–3A6 Fab structure was refined to 17.7 Å with 9,602 particles from 26 tomograms; the VLP-GP_1,2–KZ52 Fab structure was refined to 8.7 Å with 13,520 particles from 18 tomograms; and the VLP-GP_1,2–KZ52 Fab–3A6 Fab structure was refined to 7.5 Å with 40,072 particles from 42 tomograms.</p>", "<title>Alanine scanning and antibody binding test</title>", "<p id=\"P40\">Alanine scanning was performed by introducing alanyls (alanyls were changed to seryls) into GP_1,2ΔMLD region 627–639 via site-directed mutagenesis of GP_1,2ΔMLD-encoding plasmid. The plasmid clones were individually arrayed into 384-well plates and transfected into HEK 293T cells. Protein variants were cell surface-expressed for 22 h ^{##REF##26311869##21##}. The indicated mAbs were incubated with the cells for 1 h before an Alexa Fluor 488-conjugated secondary antibody (Thermo Fischer Scientific) was added. Antibody binding was assessed by detection of cellular fluorescence with a high throughput flow cytometer (Intellicyt, Albuquerque, NM, USA). Background fluorescence was measured in vector-transfected control cells and mAb reactivity against the variants was calculated with respect to reactivity with GP_1,2ΔMLD by subtracting the signal from mock-transfected controls and normalized to signals from wild-type GP_1,2ΔMLD-transfected controls. Residues predicted to be involved in the epitope were identified when mAb and variant did not react, but when reactivity of other control mAbs was observed, which excludes glycoproteins variants that were misfolded or were expressed at low levels.</p>", "<title>Animal studies</title>", "<p id=\"P41\">Animal exposure and treatment experiments using infectious EBOV were performed in the biosafety level 4 (BSL-4) laboratory at the Integrated Research Facility at Fort Detrick (IRF-Frederick), Division of Clinical Research (DCR), National Institutes for Allergy and Infectious Diseases (NIAID), National Institutes of Health (NIH) under accreditation (000777) by the Association for Assessment and Accreditation of Laboratory Animal Care (AAALAC), Laboratory Animal Welfare approval (D16–00602) by the Public Health Service (PHS), and United States Department of Agriculture (USDA) registration (51-F-0016). Animal experiments were approved by the NIAID DCR Animal Care and Use Committee (ACUC) and followed the recommendations provided in the Guide for the Care and Use of Laboratory Animals.</p>", "<title>Domesticated guinea pigs</title>", "<p id=\"P42\">Hartley strain domesticated guinea pigs (<italic toggle=\"yes\">Cavia porcellus</italic> (Linnaeus, 1758)) of both sexes, aged 6–8 weeks, were acquired from Charles River Laboratories and six animals (three males and three females each) assigned to five groups. All animals were exposed intraperitoneally (IP) to 1,000 plaque forming units (PFU) of domesticated guinea pig-adapted Ebola virus/UTMB/C.porcellus-lab/COD/1976/Yambuku-Mayinga-GPA on Day 0. Animals in each group were injected IP on Day 3 with 5 mg of i) 3A6, ii) 1A2, iii) 7G7, or iv) 42–2D2 anti-influenza A virus (FLUAV) IgG in 3 mL of DPBS or received no treatment. Animals were observed daily for clinical signs of disease and were assigned a clinical score of 0–3 (0 = none; 1 = mild; 2 = moderate; 3 = severe). Animals reaching endpoint criteria (score of 3) were euthanized. Weight was recorded daily starting 1 d before exposure until all animals recovered from disease (Day 15), then twice weekly until the study endpoint on Day 28. Blood was collected twice, on Day 6 after exposure and at the time of euthanasia.</p>", "<title>Rhesus monkeys</title>", "<p id=\"P43\">Four rhesus monkeys (<italic toggle=\"yes\">Macaca mulatta</italic> (Zimmermann, 1780)) of both sexes (WorldWide Primates, Miami, FL, USA) were single-housed and acclimated to BSL-4 conditions prior to virus exposure. On Day 0, monkeys were sedated using intramuscular (IM) injection of 15 mg/kg of Ketamine HCl (KetaThesia, Henry Schein, USA), and injected IM with a target dose of 1,000 PFU of Ebola virus/H.sapiens-tc/COD/1995/Kikwit-9510621 (NR-50306, Lot 9510621, ≥ 95% 7U abundance at the <italic toggle=\"yes\">GP</italic> editing site, BEI Resources, USA; the same dose and lot of this virus previously resulted in death at days 5–8 post-exposure in 12 out of 12 untreated rhesus monkeys (“historical controls” ^{##REF##32674252##55##}). On Day 4 and Day 7 after exposure, monkeys 1–3 received 25 mg/kg of 3A6 IgG in DPBS (kindly provided by Chakravarthy Reddyvia) by intravenous infusion, and the control monkey received an equivalent volume of DPBS. All monkeys were observed for the development of clinicals signs of EBOV infection and scored daily according to a four-point scoring scale. Physical examination and blood collections were conducted on the monkeys once prior to exposure (Day – 1) and at 4, 7, 9, 12, 21, and 28 d after exposure. Complete blood counts with reticulocytes and differential were analyzed on a Sysmex XT-2000iV hematology instrument (Sysmex America, New York, NY, USA). Sera were obtained after separation at room temperature and centrifugation for 15 min at 1,500 ×<italic toggle=\"yes\">g</italic> followed by analysis using the Piccolo general chemistry 13 panel on a Piccolo Xpress analyzer (Abaxis, NJ, USA). Prothrombin and activated partial thromboplastin times were measured on a CS-2500 system automated coagulation analyzer (Sysmex America). Infectious titers were determined in sera using an Avicel-based crystal violet stain plaque assay on Vero E6 cell culture monolayers (BEI Resources) as previously described ^{##REF##23223188##56##}. Sera were inactivated in TRIzol LS according to the manufacturer’s instructions (Thermo Fisher Scientific), and nucleic acid extracted using the QIAamp Viral RNA Mini Kit (Qiagen, Germantown, MD, USA). The BEI Resources Critical Reagents Program (CRP) EZ1 RT-PCR kit assay was used in accordance with manufacturer’s instructions ^{##REF##20439981##31##} on an Applied Biosystems 7500 FastDx Real-Time PCR instrument (Thermo Fisher Scientific) to quantify EBOV nucleic acids in sera and to transform results into log₁₀ genome equivalents (GE) per mL of sample. The control monkey died on Day 8 whereas the treated monkeys underwent elective euthanasia approximately 3 mo after virus exposure.</p>", "<title>Statistical analysis</title>", "<p id=\"P44\">Statistical details of experiments, including numbers of replicates and measures of precision (standard deviation, SD), can be found in the figure legends, figures, results, and methods. Dose-response ELISA curves were fit to a EC₅₀ shift by nonlinear regression analysis. All analyses were performed with Prism 7.</p>" ]

[ "<title>RESULTS</title>", "<title>Crystal structures of unbound human mAb 3A6 Fab bound to the Ebola virus glycoprotein stalk–MPER</title>", "<p id=\"P5\">mAb 9.6.3A6 (henceforth abbreviated as “3A6 IgG”) was isolated from a human survivor of the 2013–2016 Western African EVD outbreak 6 months after hospital discharge ^{##REF##31104840##11##}. The predicted linear epitope of 3A6 IgG encompasses GP_1,2 residues 626–640 and extends from the C-terminal end of the stalk to the start of the MPER ^{##REF##31104840##11##} (##FIG##0##Fig. 1A## bottom/inset, orange box, outlined in purple in the sequence alignment). To determine the mode of molecular recognition and neutralization of EBOV by 3A6 IgG, we crystallized the 3A6 Fab fragment alone (Supplementary Fig. 1) and in complex with a 14-amino acid peptide having a sequence corresponding to the EBOV GP_1,2 stalk–MPER epitope (aa 626–640; ##FIG##0##Fig. 1B##–##FIG##0##C##). Crystals of 3A6 Fab diffracted to 2.5 Å and had an asymmetric unit containing four Fab fragments. Meanwhile, crystals of the 3A6 Fab-stalk–MPER peptide complex diffracted to 1.27 Å and had one Fab fragment in the asymmetric unit (Table S1). The 3A6 Fab structure was essentially identical in the unbound and peptide-bound states as evidenced by the 0.46Å root-mean-square deviation (RMSD) (##FIG##0##Fig. 1C##, Supplementary Fig. 1). Residues I627-G639 of the 3A6 IgG-peptide epitope (##FIG##0##Fig. 1A## bottom/inset) are visible (Supplementary Fig. 1). The EBOV stalk–MPER peptide is -helical from its N terminus (I627) to residue T634 and then slightly unravels through the visible terminus at residue G639 (Supplementary Fig. 1).</p>", "<p id=\"P6\">The heavy (H) and light (L) chains of 3A6 Fab both participate in epitope binding (##FIG##0##Fig. 1C##). At the N-terminal helical end of the stalk–MPER peptide, antibody residues R98, S100, T101 (complementarity determining region [CDR] H3), and E31 (CDR H1) form hydrogen bonds to GP₂ stalk residues H628 and D629 and MPER residue D632 (##FIG##0##Fig. 1D##; Supplementary Fig. 1). CDRs L2, H1, and H3 form additional hydrophobic interactions with the peptide (##FIG##0##Fig. 1E##). At the C-terminal end of the peptide, antibody residues H31 and S32 (CDR L1), T97 (CDR L3), and N59 (CDR H2) engage MPER residues L635, P636, D637, and Q638 (##FIG##0##Fig. 1F##; Supplementary Fig. 1). The CDRs L1, L3, and H2 contribute additional hydrophobic interactions with MPER residues T634-Q638 (##FIG##0##Fig. 1G##). MPER residues D632 and P636 are particularly key and form five polar and ten nonpolar interactions with the Fab CDRs (##FIG##0##Fig. 1D##–##FIG##0##G##).</p>", "<title>Ebola virus glycoprotein MPER residues D632 and P636 are critical to mAb 3A6 binding</title>", "<p id=\"P7\">3A6 IgG binds to and neutralizes EBOV but not SUDV <italic toggle=\"yes\">in vitro</italic>\n^{##REF##31104840##11##}. In the 3A6 IgG–peptide epitope that includes stalk–MPER residues I627-G639, four residues differ substantially between EBOV and SUDV GP₂: K633 vs. N, T634 vs. P, D637 vs. N, and G639 vs. D (##FIG##0##Fig. 1A## bottom/inset). Using a cell-based antibody-binding assay we next compared binding of 3A6 IgG to full-length (MLD-containing) EBOV GP_1,2 with each of these four residues changed individually or in combination with the corresponding SUDV residues (##FIG##1##Fig. 2A##). We also measured binding with an ELISA using purified EBOV GP_{1,2ΔTM/ΔMLD} containing the same amino acid changes (##FIG##1##Fig. 2B##). None of the individual mutations affected 3A6 IgG binding to cell-surface GP_1,2 or GP_{1,2ΔTM/ΔMLD}, but binding was completely inhibited when all four residues were changed to the SUDV counterparts (##FIG##1##Fig. 2A##, ##FIG##1##2B##; Supplementary Fig. 2).</p>", "<p id=\"P8\">Next, we used alanine scanning mutagenesis of GP_1,2 to identify individual residues throughout the epitope that are critical for 3A6 binding. We made alanine point mutations (wild type alanines were mutated to serines) at each amino acid residue between positions 627 and 639 of EBOV GP_1,2ΔMLD (GP_1,2 lacking the mucin-like domain; ##FIG##0##Fig. 1A##) and analyzed each resulting EBOV GP_1,2ΔMLD for 3A6 reactivity by flow cytometry (Supplementary Fig. 2 and Supplementary Table 2). Notably, D632A and P636A mutations produced a ≤ 20% reduction in 3A6 Fab binding relative to wild-type (WT) GP_1,2ΔMLD (##FIG##1##Fig. 2C##). Both residues are identical in EBOV and SUDV GP₂ (##FIG##0##Fig. 1A## bottom/inset), and changes at these sites do not substantially affect binding of control mAbs KZ52, 1H3, or 4G7 that target conformational epitopes on the base (KZ52, 4G7) and glycan cap (1H3) of EBOV GP₁\n^{##REF##18615077##7##,##REF##25404321##20##,##REF##26311869##21##} (##FIG##1##Fig. 2C##). These results are consistent with those of previous studies in which morphologically authentic “biologically-contained” EbolaΔVP30 virions ^{##REF##18212124##22##} passaged in the presence of 3A6 IgG led to the emergence of glycoproteins bearing P636S and P636Q mutations ^{##REF##31104840##11##}. Therefore, we next evaluated neutralization of P636S-bearing EbolaΔVP30-eGFP virions by multiple mAbs using a plaque-reduction assay. The P636S mutation abolished neutralization activity of both 3A6 and 1E6, another stalk-binding mAb, but did not affect neutralization of mAbs targeting the glycoprotein core (Supplementary Table 3).</p>", "<title>Binding of mAb 3A6 lifts Ebola virus glycoprotein relative to the membrane surface</title>", "<p id=\"P9\">We superimposed the 3A6 Fab-stalk–MPER structure onto trimeric GP_{1,2ΔTM/ΔMLD} using the overlapping portion of the epitope as a guide (Protein Data Bank [PDB] identification number [ID]: 5JQ7; ##FIG##2##Fig. 3A##). This superimposition revealed steric clashes of the bound 3A6 Fab with the other two GP₂ monomers of stalk–MPER such that three Fabs could not simultaneously bind to the tightly bundled conformation of GP₂ observed in previous crystal structures (##FIG##2##Fig. 3B##). The EBOV GP in the previously published high-resolution crystal structure (resides 32–312 fused to 464–632; compare to ##FIG##0##Fig. 1A##, top) was stabilized by fusion of a fibritin trimerization motif to the C terminus ^{##REF##27362232##23##}. A structure of GP₂ of the related Marburg virus that was not fused to any trimerization motif also demonstrated the same close bundling of the three GP₂ monomers in the trimer ^{##REF##29324225##24##}. Hence, these structures may represent the native bundled conformation and binding of three copies of 3A6 IgG to stalk–MPER would likely promote or stabilize a more open conformation of the GP₂ trimeric interface.</p>", "<p id=\"P10\">Consequently, we assessed whether the EBOV GP_1,2 trimer is sufficiently flexible to accommodate simultaneous binding of three copies of 3A6 IgG by evaluating Fab binding to both full-length, transmembrane EBOV GP_1,2 and purified GP_{1,2ΔTM/ΔMLD}. We first used size-exclusion chromatography coupled to multi-angle light scattering (SEC-MALS) to determine the number of 3A6-Fabs that one trimeric EBOV GP_{1,2ΔTM/ΔMLD} can accommodate by measuring the molecular weights of 3A6 alone, EBOV GP_{1,2ΔTM/ΔMLD} trimer alone, and the GP₂-3A6 Fab complex. The 3A6 Fab alone registered a molecular weight of 48.6 (± 3.7) kDa and EBOV GP_{1,2ΔTM/ΔMLD} alone registered the expected molecular weight for a trimer (186.3 (± 1.5) kDa). The GP_{1,2ΔTM/ΔMLD}-3A6 Fab complex exhibited a single peak corresponding to a molecular weight of 326 (± 2.6) kDa. No other peaks, corresponding to binding of one or two 3A6 Fabs to GP_{1,2ΔTM/ΔMLD}, were observed (Supplementary Fig. 3). These results indicate that the GP_{1,2ΔTM/ΔMLD}–3A6 complex indeed contains three copies of 3A6 Fab bound to stalk–MPER. Composition gradient multi-angle light scattering (CG-MALS) supported this result, revealing that three 3A6 Fabs consistently bind to one GP_{1,2ΔTM/ΔMLD} trimer with equal affinities (K_D=52.15 (± 1.3) nM; Supplementary Fig. 3). Furthermore, negative stain EM (nsEM) analysis of the GP–3A6 complexes also showed binding of three 3A6 Fabs to the stalk–MPER (##FIG##2##Fig. 3C##).</p>", "<p id=\"P11\">The GP₂ subunits in the natural membrane-anchored form have less freedom to open and separate from each other than would the free GP₂ C-termini of the ectodomain. To image 3A6 Fab bound to GP₂ in its natural transmembrane form, we produced filamentous EBOV-like particles consisting of EBOV matrix protein (VP40) and full-length GP_1,2. These virion-like particles (VLPs) were incubated with 3A6 Fab for cryogenic electron tomography (cryo-ET) and subtomogram averaging analysis (Supplementary Table 4). Tomogram reconstructions showed extra densities anchored to stalk–MPER that indicated the presence of 3A6 Fabs (##FIG##3##Fig. 4A##). We also complexed VLPs with the Fab of a well-characterized anti-EBOV control mAb, KZ52 ^{##REF##18615077##7##}, to represent the state of GP_1,2 in the absence of 3A6 Fab (##FIG##2##Fig. 3##). We used this core-binding mAb instead of unbound GP_1,2 because antibody binding generates a larger structure that can be more accurately aligned during subtomogram averaging. The KZ52 epitope at the base of GP₁ is sufficiently separated from the stalk–MPER 3A6 binding site and its binding does not disrupt the native structure of GP_{1,2 7}.</p>", "<p id=\"P12\">Comparison of the structures of VLP-GP_1,2–3A6 (18 Å) and VLP-GP_1,2–KZ52 (8.7 Å) revealed that in the presence of the 3A6 Fab the GP_1,2 body was displaced vertically away from the VLP membrane by approximately 3 nm (##FIG##3##Fig. 4B##). We conclude that 3A6 binding induces this vertical displacement since VLP- GP_1,2 complexed with both 3A6 and KZ52 Fabs together resulted in a similarly lifted GP_1,2 (##FIG##3##Fig. 4C##) with densities that overlapped with VLP-GP_1,2–3A6 (##FIG##3##Fig. 4D##). Steric hindrance of 3A6 Fab with the lipid membrane likely necessitates the vertical lift of GP_1,2 from its natural position upon 3A6 Fab binding. This lift could inhibit conformational changes that stalk–MPER must undergo for GP_1,2 to mediate Ebola virion entry into target cells.</p>", "<p id=\"P13\">Together, these data unveil a novel mechanism-of-action for antibodies and identify 3A6 IgG as a first-in-class antibody that appears to perform physical work.</p>", "<title>Low-dose mAb 3A6 monotherapy is efficacious in domesticated guinea pigs and rhesus monkeys exposed to Ebola virus</title>", "<p id=\"P14\"><italic toggle=\"yes\">In vitro</italic>, 3A6 IgG neutralized EBOV at a concentration of 0.33 nM (50% plaque reduction neutralization test [PRNT₅₀]) ^{##REF##31104840##11##}. <italic toggle=\"yes\">In vivo</italic>, prophylactic administration of 3A6 IgG protected laboratory mice from fatal outcome after exposure to a typically lethal dose of mouse-adapted EBOV (100% protection after a 100 μg dose [~ 5 mg/kg] and 50% protection at a 25 μg dose [~ 1.25 mg/kg]) ^{##REF##31104840##11##}. To increase stringency, in this study we evaluated the efficacy of 3A6 IgG in a post-exposure domesticated guinea pig model. Groups of six (three male and three female) guinea pigs were exposed intraperitoneally (IP) to 1,000 plaque-forming units (PFU) of domesticated guinea-pig-adapted EBOV (Day 0). On Day 3, the guinea pigs were either left untreated or treated IP with a single 5 mg dose of the EBOV antibodies 3A6 IgG, 1A2 IgG (targets the EBOV GP₂ fusion loop), 7G7 IgG (targeting an unknown epitope on EBOV GP_{1,2 11}), or the anti-influenza A virus (FLUAV) antibody 42–2D2. All animals in the untreated group and those treated with 1A2 or 7G7 succumbed to EBOV infection. All but two of the anti-FLUAV 42–2D2 treated control animals succumbed to EBOV infection. In contrast, all guinea pigs treated with 3A6 survived and exhibited few or no clinical signs of disease (##FIG##4##Fig. 5A##, Supplementary Fig. 4).</p>", "<p id=\"P15\">The rhesus monkey model recapitulates key features of EVD and is generally preferred over rodent models for development of EVD medical countermeasures ^{##UREF##0##1##}. We randomized four rhesus monkeys into treatment (n = 3, rhesus monkeys 1–3) and no treatment (n = 1) groups. All monkeys were given an intramuscular (IM) injection of a typically lethal 1,000-PFU dose of EBOV (day 0). On Day 4 and 7 after infection, the treatment group monkeys received 25 mg/kg of 3A6 in PBS intravenously, whereas the control monkey received intravenous PBS only. EBOV replication was confirmed in all monkeys on Day 4 by plaque assay titration and quantitative real-time reverse transcription polymerase chain reaction (RT-qPCR), with 10⁴–10⁶ EBOV PFU per mL and 10⁸–10¹⁰ EBOV glycoprotein gene equivalents per mL of serum (Supplementary Fig. 4). Notably, these high levels of viremia could still be reversed by 3A6 administration, as evidenced by a decrease in viral load after the first dose on Day 4 and continued reduction to below the limit of detection by Day 21 (Supplementary Fig. 4). Clinical signs consistent with EVD were observed in monkeys 1 and 3 as early as Day 4 but resolved by Day 13 (Supplementary Fig. 4). Notably, monkey 3 had significantly elevated AST activity and rapidly rising serum creatinine levels suggesting a marked reduction in the glomerular filtration rate (the initial rise was similar to the control animal) with pathologic evidence of EBOV-induced liver injury. Nonetheless, this animal still recovered after 3A6 IgG treatment (Supplementary Fig. 5). All three monkeys in the treatment group survived, whereas the control monkey was found dead on Day 8 (##FIG##4##Fig. 5B##). These results demonstrate that post-exposure dosing of 3A6 IgG alone reverses the course of EBOV infection and protects animals of different species from fatal outcome.</p>" ]

[ "<title>Discussion</title>", "<p id=\"P16\">Vaccines are currently only approved for prevention of EBOV infection ^{##UREF##1##25##}, but not for infections caused by other filoviruses. Antibody therapeutics that can be used at a low dose to reverse advanced disease are urgently needed to treat people with filovirus infections who live in countries with limited resources. The EBOV GP_1,2 stalk–MPER is of interest for therapeutic/vaccine design due to its relatively high amino acid sequence conservation among all orthoebolaviruses, indicating that a single mAb targeting this region could have therapeutic activity against infections by any of these viruses. Moreover, known anti-<italic toggle=\"yes\">orthoebolavirus</italic> stalk and/or MPER mAbs are highly potent neutralizers in vitro ^{##REF##31104840##11##,##REF##29736037##18##,##REF##30996276##19##}, suggesting that they may be applied in much lower doses compared to mAbs that are currently used in the clinic.</p>", "<p id=\"P17\">Here we built on previous <italic toggle=\"yes\">in vitro</italic> and prophylactic laboratory mouse efficacy studies of the EBOV GP_1,2 stalk–MPER-binding 3A6 IgG by demonstrating complete post-exposure protection in stringent models of EVD in domesticated guinea pigs and rhesus monkeys. 3A6 IgG showed unprecedented potency in the rhesus monkey model of disease. A single 25-mg/kg dose of 3A6 IgG rescued monkeys that had extremely high viral loads in pilot studies, reducing viral loads on Day 4 of 10⁹–10¹⁰ PFU per mL to undetectable levels by Day 21. Concomitantly, 3A6 IgG reversed clinical signs of advanced disease and decreased elevated liver enzyme activities and serum creatinine concentrations to baseline. The administered dose is half that used for the human standard-of-care ansuvimab (mAb114) monotherapy and one-sixth of that used for the human standard-of-care mAb combination atoltivimab/maftivimab/odesivimab (REGN-EB3) in the same animal model (albeit administered on different days) ^{##REF##26917593##10##,##REF##29860496##26##}. Our data therefore pave the way for development of novel therapeutics that potentially expand the treatment window for effective intervention in highly viremic patients and later in the EVD course. Such therapeutics could increase the likelihood of survival for this group of patients seen relative to currently approved mAb therapeutics.</p>", "<p id=\"P18\">We previously hypothesized that binding of BDBV223, an anti-stalk antibody targeting a similarly occluded epitope in the EBOV-related Bundibugyo virion ^{##REF##30996276##19##}, requires either bending or lifting of GP_1,2. In this study, we experimentally addressed this hypothesis using 3A6, which binds an epitope that is closer to the C terminus (i.e., even more occluded) than that bound by BDBV223. The linear epitope of 3A6 spans residues I627-G639 in the lower region of the EBOV GP_1,2 stalk and our structural studies suggest that a portion of the 3A6 epitope is embedded within the membrane prior to antibody binding. Our structural data further suggest that 3A6 Fab first binds to the exposed stalk polypeptide above the membrane and then displaces and separates the GP_1,2 monomer stalk bundles. The first and second 3A6 Fab likely promote gradual stabilization of intermediate conformational states to tilt the GP_1,2 relative to the membrane surface and increase stalk–MPER exposure as it partially elevates above the membrane surface. Due to the steric hindrance between the 3A6 Fab and the membrane, binding of the Fab at the third stalk–MPER site on the GP_1,2 trimer vertically lifts GP_1,2 relative to the membrane surface. We hypothesize that following binding 3A6 IgG achieves potent neutralization activity by blocking conformational changes needed to drive fusion of virion and cell membranes. Human immunodeficiency virus 1 glycoprotein and FLUAV hemagglutinin can also be tilted by binding of anti-MPER antibodies ^{##REF##32348769##27##,##REF##30224494##28##}, indicating that positional flexibility is a common property of class I fusogens.</p>", "<p id=\"P19\">In conclusion, our studies establish 3A6 IgG as the founding member of a new group of immunotherapeutics against Ebola virus that achieves complete protection against advanced disease at the lowest dose yet observed for a monotherapy via a novel mechanism of action. The next desired feature of this new group is breadth: 3A6-like antibodies against stalk–MPER epitopes that have pan-orthoebolavirus activity likely exist. Such antibodies could be used at even lower concentrations and at more advanced stages across the filovirus disease spectrum.</p>" ]

[]

[ "<p id=\"P1\">Monoclonal antibodies (mAbs) against Ebola virus (EBOV) glycoprotein (GP_1,2) are the standard of care for Ebola virus disease (EVD). Anti-GP_1,2 mAbs targeting the stalk and membrane proximal external region (MPER) potently neutralize EBOV <italic toggle=\"yes\">in vitro</italic>. However, their neutralization mechanism is poorly understood because they target a GP_1,2 epitope that has evaded structural characterization. Moreover, their <italic toggle=\"yes\">in vivo</italic> efficacy has only been evaluated in the mouse model of EVD. Using x-ray crystallography and cryo-electron tomography of 3A6 complexed with its stalk– GP_1,2 MPER epitope we reveal a novel mechanism in which 3A6 elevates the stalk or stabilizes a conformation of GP_1,2 that is lifted from the virion membrane. In domestic guinea pig and rhesus monkey EVD models, 3A6 provides therapeutic benefit at high viremia levels, advanced disease stages, and at the lowest dose yet demonstrated for any anti-EBOV mAb-based monotherapy. These findings can guide design of next-generation, highly potent anti-EBOV mAbs.</p>" ]

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[ "<title>Acknowledgements</title>", "<p id=\"P45\">We are grateful to Dustin Morado, Kun Qu, and Joaquin Oton (Medical Research Council Laboratory of Molecular Biology) for support in collecting and/or processing cryogenic electron tomography data and to Stanford Synchrotron Light Source (SSRL) and Advanced Photon Source (APS) for assistance in collection of X-ray data. We thank Anya Crane (National Institutes of Health [NIH], National Institute of Allergy and Infectious Diseases [NIAID], Integrated Research Facility at Fort Detrick, Frederick, MD, USA [IRF-Frederick]) for critically editing the manuscript and Jiro Wada (NIH NIAID IRF-Frederick) for creating figures. We gratefully acknowledge Peter B. Jahrling and Lisa E. Hensley (NIH NIAID IRF-Frederick) for supporting the studies involving animals.</p>", "<p id=\"P46\">The work for this study was supported by NIH NIAID U19 AI142790 (E.O.S. and C.W.D.), DARPA contract W31P4Q-14-1-0010 (C.W.D. and R.A.), NIH NIAID contract HHSN272201400058C to B.J.D., and UK Medical Research Council (MC_UP_1201/16), the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (ERC-CoG-648432 MEMBRANEFUSION), and the Max Planck Society (J.B.). This research was further supported through the NIH/NIAID prime contract with Battelle Memorial Institute (Contract No. HHSN272200700016I) and subsequently with Laulima Government Solutions (Contract No. HHSN272201800013C). G.W. and M.R.H. performed this work as employee under these contracts. J.H.K. performed this work partly as an employee under the Battelle Memorial Institute contract and partly as an employee of Tunnell Government Services, formerly a subcontractor of Battelle Memorial Institute and now of Laulima Government Solutions. This work has been further funded in whole or in part with federal funds from the National Cancer Institute, National Institutes of Health, under Contract No. HHSN261201500003I, Task Order No. HHSN26100043 and Contract No. 75N91019D00024, Task Order No. 75N91019F00130 (I.C.). S.M. was funded by an Imaging Scientist grant (2019-198153) from the Chan Zuckerberg Initiative.</p>", "<p id=\"P47\">Use of the SSRL SLAC National Accelerator Laboratory is supported by the US Department of Energy (DOE) Office of Science Basic Energy Sciences (BES) program under Contract No. DE-AC02-76SF00515. The SSRL Structural Molecular Biology Program is supported by the DOE Biological and Environmental Research (BER) program and by the NIH National Institute of General Medical Sciences (NIGMS; P41GM103393). This research also used resources of the Advanced Photon Source (APS), a DOE Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory (ANL) under Contract No. DE-AC02-06CH11357. The General Medical Sciences and Cancer Institute Structural Biology Facility at the Advanced Photon Source (GM/CA) project has been funded in whole or in part with Federal funds from the National Cancer Institute (ACB-12002) and the NIGMS (AGM-12006). The Eiger 16M detector at the ANL X-ray Science Division (XSD) was funded by NIH grant S10 OD012289.</p>", "<p id=\"P48\">Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the U.S. Army. The funding sources had no involvement in study design; in the collection, analysis, and interpretation of data; in the writing of the report; and in the decision to submit the paper for publication.</p>", "<p id=\"P49\">The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the U.S. Department of Health and Human Services or of the institutions and companies affiliated with the authors, nor does mention of trade names, commercial products, or organizations imply endorsement by the U.S. Government.</p>", "<title>Data availability</title>", "<p id=\"P50\">Structure factors and associated model coordinates have been deposited in the Protein Data Bank (PDB; <ext-link xlink:href=\"http://www.rcsb.org/\" ext-link-type=\"uri\">http://www.rcsb.org</ext-link>) under PDB accession numbers 7RPU (3A6 Fab-stalk–MPER) and 7RPT (3A6 Fab structure).</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><p id=\"P62\">Crystal structures of unbound human mAb 3A6 Fab and mAb 3A6 Fab bound to the Ebola virus glycoprotein stalk–MPER (A) Top: Schematic of proteolytically processed mature EBOV GP_1,2 using the amino acid residue numbering of its uncleaved precursor minus signal peptide residues. Middle and bottom: EBOV GP_1,2 constructs used in this study. Inset: Sequence alignment of GP₂ stalk–MPER region amino-acid sequences. Aligned are the stalk–MPER transition areas (with the two regions separated by a vertical black line) of all six known orthoebolaviruses. The predicted linear epitope of 3A6 ^{##REF##31104840##11##} is indicated by a purple box. The EBOV residues observed to interact with 3A6 in the crystal structure highlighted in dark orange and the corresponding regions in glycoproteins of the other orthoebolaviruses highlighted in light orange. Orthoebolaviruses associated with fatal human disease are indicated in bold. (B) Top and front view of the 3A6 Fab fragment (grey) bound to the EBOV stalk–MPER peptide (orange). The heavy and light chains of 3A6 are highlighted in dark and light grey, respectively. (C) View of 3A6 highlighting the molecular surface contributed by the heavy (H1, H2, H3) and light chain (L1, L2, L3) CDRs in the paratope. Polar interactions (D, E) and hydrophobic interactions (F, G) (atoms within 4 Å) are shown as black dotted lines and red surfaces. P636, associated with escape from 3A6 after change P636S, is boxed in red in (A, F, G). BDBV, Bundibugyo virus; BOMV, Bombali virus; CDR, complementarity determining region; EBOV, Ebola virus; GP_1,2, glycoprotein; GP₂, glycoprotein subunit 2; HR, heptad repeat regions; IFL, internal fusion loop; MLD, mucin-like domain; MPER, membrane proximal external region; RESTV, Reston virus; SUDV, Sudan virus; TAFV, Taï Forest virus; TM, transmembrane domain.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><p id=\"P63\">Residues D632 and P636 of the Ebola virus glycoprotein MPER are key for mAb 3A6 binding (A) Cells expressing WT EBOV GP_1,2 or recombinant EBOV GP_1,2 with the indicated changes to cognate amino-acid residues found in SUDV GP₂ were incubated with 3A6 IgG and then stained with DyLight 488 anti-human IgG for detection by fluorescence microscopy, followed by quantification of antibody-positive cells. (B) ELISA binding curves for 3A6 IgG to purified EBOV GP_{1,2ΔTM/ΔMLD} or variants thereof containing the indicated amino-acid residue changes. (C) Flow cytometry analysis of mAb binding to cell-surface expressed EBOV GP bearing a D632A or P636A mutation. Error bars represent the mean ± standard deviation of triplicates (A and B) and duplicates (C); black dots in (A) indicate the values for the individual experiments. EBOV, Ebola virus; ELISA, enzyme-linked immunosorbent assay; Fab, fragment antigen binding; GP_1,2, glycoprotein; GP₂, glycoprotein subunit 2; Ig, immunoglobulin; SUDV, Sudan virus; WT, wild-type.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3</label><caption><p id=\"P64\">Binding of mAb 3A6 lifts Ebola virus glycoprotein relative to the membrane surface (A) Alignment of the EBOV stalk–MPER peptide (orange) to the full-length EBOV GP_1,2 structure (PDB ID: 5JQ7; GP₁ in blue, GP₂ in yellow) illustrates anchoring of 3A6 Fab to the C-terminus of the ectodomain of the GP₂ stalk. (B) The close trimeric bundle arrangement of GP₂ is incompatible with the GP₂–3A6 complex structure, as the bound antibody sterically clashes with neighboring monomers. (C) Negative-stain EM reference-free two-dimensional class averages of 3A6 Fab in complex with trimeric EBOV GP_{1,2ΔTM/ΔMLD}, showing representative side and tilted views. Scale bar, 20 nm. EBOV, Ebola virus; EM, electron microscopy; GP_1,2, glycoprotein; GP₁, glycoprotein subunit 1; GP₂, glycoprotein subunit 2; ID, identification number; MPER, membrane proximal external region; PDB, Protein Data Bank; VLP, virion-like particle; VP40, EBOV matrix protein.</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4</label><caption><p id=\"P65\">Binding of mAb 3A6 lifts Ebola virus glycoprotein relative to the membrane surface (A) Representative tomographic slices of filamentous EBOV VLPs bound to mAb Fabs. VP40 of VLPs is visible as a dotted layer underneath the lipid bilayer. The bottom row corresponds to magnified view of areas enclosed by red boxes in the top row. (B–D) VLP GP_1,2-Fab complexes were imaged by cryogenic electron microscopy, followed by subtomogram averaging. (B) Isosurface representations of reconstructions of GP_1,2 on the surface of VLPs bound to 3A6 Fab (gold) or KZ52 Fab (blue) superimposed to align with GP_1,2 (top) or the outer layer of the VLP membrane (bottom). Density corresponding to the VP40 layer has been removed for clarity. (C) Reconstructions of GP_1,2 on the surface of VLPs bound to 3A6 and KZ52 Fabs (magenta) and KZ52 Fab alone (blue) were superimposed to align with GP_1,2 (top) or the outer layer of the VLP membrane (bottom). (D) GP_1,2 on the surface of VLPs bound to 3A6 Fab and bound to KZ52 and 3A6 Fabs overlap well both in GP_1,2 and membrane positions. EBOV, Ebola virus; VLP, virion-like particles; mAb, monoclonal antibody; VP40, EBOV matrix protein.</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5</label><caption><p id=\"P66\">Low-dose mAb 3A6 monotherapy is efficacious in domesticated guinea pigs and rhesus monkeys exposed to Ebola virus (A) Domesticated guinea pigs (n=6 per group) were exposed to a typically lethal 1,000-PFU dose of domesticated guinea-pig-adapted EBOV on Day 0. On Day 3, the indicated mAbs were administered at 5 mg each in DPBS. Control guinea pigs were either given a FLUAV-specific human immunoglobulin G1 or were untreated. (B) Rhesus monkeys (n=3) were exposed to a typically lethal 1,000-PFU dose of EBOV on Day 0. On Day 4 and Day 7, 25 mg/kg of 3A6 was administered in PBS. One control monkey was given PBS. Treatment days are indicated by dotted lines. mAb, monoclonal antibody; FLUAV, influenza A virus; PBS, phosphate-buffered saline.</p></caption></fig>" ]

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[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN1\"><p id=\"P51\">Competing interests</p><p id=\"P52\">A.G., E.D., and B.J.D. are employees of Integral Molecular, and B.J.D.is a shareholder in that company.</p></fn><fn fn-type=\"COI-statement\" id=\"FN2\"><p id=\"P53\"><bold>Additional Declarations:</bold> There is <bold>NO</bold> Competing Interest.</p></fn><fn id=\"FN3\"><p id=\"P54\">Declarations</p><p id=\"P55\">Resource availability</p><p id=\"P56\">Further information and requests for resources and reagents should be directed to and will be fulfilled by the lead contact, Erica Ollmann Saphire (<email>[email protected]</email>).</p></fn><fn id=\"FN4\"><p id=\"P57\">Code availability</p><p id=\"P58\">Not applicable.</p></fn><fn id=\"FN5\"><p id=\"P59\">Ethics declarations</p><p id=\"P60\">Study compliance</p></fn><fn id=\"FN6\"><p id=\"P61\">Animal exposure and treatment experiments using infectious EBOV were performed in registered and accredited biosafety level 4 (BSL-4) laboratories with appropriate approvals.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"P4\">Currently, over 100,000 individuals anxiously await a lifesaving organ transplant, yet thousands of these will die every year while on the waitlist due to a severe organ shortage. Current clinical standards for organ handling for transplantation include a period of cold ischemia during transport, which is aimed at suppressing metabolism to extend <italic toggle=\"yes\">ex vivo</italic> graft survival by several hours. Human livers, for instance, can be stored for up to 8 hours in static cold storage (SCS) immersed in a preservation solution without additional dissolved oxygen^{##UREF##0##1##,##REF##35313073##2##}. Following cold ischemia, grafts are then exposed to a sudden burst of oxygen supply during implantation, resulting in ischemia-reperfusion injury (IRI). IRI is characterized by a reduction in the ATP levels, accumulation of metabolic waste products^{##REF##15919422##3##,##REF##30843314##4##}, and production of reactive oxidative species (ROS), which damages the vascular endothelium and activates a cascade of events that may lead to cell death and organ dysfunction^{##UREF##1##5##,##UREF##2##6##}. IRI remains a significant challenge in transplantation that is responsible for posttransplant graft dysfunction^{##REF##32713098##7##,##UREF##3##8##} and is poised to limit the utilization of grafts that are exposed to warm ischemic injury, including donation after cardiac death (DCD)^{##REF##37438690##9##,##UREF##4##10##}.</p>", "<p id=\"P5\">Several strategies currently exist to mitigate IRI with the goal of reducing (or eliminating) ischemia while minimizing reperfusion injury. <italic toggle=\"yes\">Ex situ</italic> machine perfusion (MP) is currently the leading strategy that maintains organs in an active state and delivers oxygen and nutrients via a perfusate prior to transplantation. The metabolic activity and oxygen demand during MP can vary considerably depending on the temperature of perfusion. Simultaneously, different perfusate mediums can be altered to supply the desired amounts of oxygen and nutrients. For example, Normothermic Machine Perfusion (NMP), an FDA approved protocol, involves maintaining the organ at a physiological temperature of 37°C and supplementing the perfusate with an oxygen carrier, such as packed red blood cells (pRBC)^{##REF##29670285##11##} or synthetic hemoglobin from swine (HBOC201^{##UREF##7##19##}) and murine sources (Hemarina M101^{##REF##37127632##20##}). While NMP minimizes ischemia time, grafts are continuously maintained in a hyper-oxygenated state^{##UREF##5##12##}. Sub-normothermic machine perfusion (SNMP) is another popular method that maintains organs at 21°C with an acellular perfusate carrying only dissolved oxygen that should be sufficient to meet lower metabolic demands. Like NMP, SNMP reduces ischemia; however, uses a more gradual increase in metabolism that may protect injured organs^{##REF##24758155##13##} from a sudden burst in oxygen consumption at 37°C^{##REF##30588774##14##}. Finally, hypothermic oxygenated machine perfusion (HOPE) uses dissolved oxygen in a perfusate pumped at 4–10°C as a means to prevent early mitochondrial injury upon reperfusion^{##REF##23063573##15##,##REF##24295869##16##}, although it represents a departure from physiology.</p>", "<p id=\"P6\">Despite the protective benefits of these techniques, mechanistic insights into IRI and the optimal oxygenation strategy to improve metabolic recovery while minimizing ROS injury during machine perfusion is lacking in the field. Mitochondria are attractive targets to observe reoxygenation due to their extensive role in metabolism and IRI^{##REF##15253700##27##–##REF##16093501##29##}. Measurement of mitochondrial activity can be used to study energetic recovery via ATP production after ischemia and reperfusion. Further, injury to mitochondria may also predict cellular damage. Despite the importance of mitochondria, current technologies to measure mitochondrial function during machine perfusion lack sensitivity/specificity, do not capture the dynamic nature of oxygenation as a function of IRI, or rely on destructive tissue sampling. Instead, we developed a metric of mitochondrial function using a real-time, non-destructive measurement of mitochondrial redox state using resonance Raman spectroscopy (RRS)^{##UREF##11##30##}.</p>", "<p id=\"P7\">The reduced and oxidized states of mitochondrial cytochromes have unique resonance Raman spectral signatures when excited with a 441 nm laser. This spectrum is recorded from the surface of organs during MP, and simultaneously cross referenced with fully oxidized and reduced mitochondrial cytochrome spectral libraries in real-time, to derive the resonance Raman reduced mitochondrial ratio (3RMR). 3RMR is defined as a ratio of reduced to total mitochondrial complex redox states. In a healthy condition there is a steady transfer of electrons from the mitochondrial complexes to oxygen, which keeps them in an oxidized state, i.e., with low 3RMR. However, such transfer can be disrupted during ischemia and reperfusion and lead to a higher reduced state, i.e., higher 3RMR. In contrast to other methods, the 3RMR level can be used to directly measure metabolic utilization of oxygen for ATP production with high sensitivity and specificity. Moreover, it works independent of the presence of hemoglobin yet, provides complementary hemoglobin saturation measurement; and thus, can be used with both RBC-based and acellular perfusates. Finally, it is a non-invasive and real-time assay free of tissue biopsies and cumbersome mitochondrial isolation procedures^{##REF##19816421##31##}, thus making it an ideal platform for assessment during machine perfusion.</p>", "<p id=\"P8\">To leverage this powerful technology to understand IRI and develop and assay of mitochondrial health, we measured 3RMR during normothermic machine perfusion of rodent livers that were exposed to minimal versus extended cold ischemia durations (0-hour cold ischemia vs 24-hour cold ischemia). We also varied the NMP protocol to include either an acellular perfusate or packed RBCs to understand the impact of oxygen delivery methods on reperfusion injury. As expected, the 3RMR values reflected the oxygen consumption and metabolism patterns consistent with the expected trends for each storage condition and oxygenation strategy. The red blood cells provided sufficient oxygen supply to support the starving organ after 24 hours of ischemia, however dealt higher endothelial damage. On the other hand, the acellular perfusate provided insufficient oxygen after ischemia but treated the endothelial layer gently. We also support these observations with other markers of perfusion and metabolic recovery as well as injury, highlighting the robustness of this index in predicting mitochondrial health and organ recovery.</p>" ]

[ "<title>MATERIALS AND METHODS</title>", "<title>Liver procurement and storage</title>", "<p id=\"P21\">All animals for the experiments were maintained in accordance with National Research Council guidelines and were approved by the Institutional Animal Care and Use Committee (IACUC) at Massachusetts General Hospital (Boston, MA, USA). Female Lewis rats (200g, Charles River Laboratories) were sedated in an induction chamber using isoflurane set to 3–5%. The excess vapor was filtered through with activated charcoal (Vet Equip Vapor Guard Activated Charcoal, 931401). The animal was then removed from the chamber and placed on a surgical table in a supine position. Depth of anesthesia was deemed adequate when muscular contraction was absent following toe pinch. The abdominal region was shaved, and abdomen is opened with a transverse abdominal incision. The hepatic artery and branches of portal vein were ligated and 200U of sodium heparin (MGH Pharmacy) was injected into the supra-hepatic vena cava. The bile duct was cannulated with a 22G polyethylene tube and a 16G catheter (BD Insyte Authoguard, 381454) was inserted into the portal vein, and the IVC transected. The liver was immediately flushed with 40mL of heparinized saline solution and an additional 20mL to remove any residual blood. Perfusion was started within 5 minutes of harvesting, except for livers exposed to durations of static cold storage events which were further flushed with 25mL of chilled University of Wisconsin and stored in 50mL of UW solution at 4°C.</p>", "<title>Normothermic machine perfusion setup</title>", "<p id=\"P22\">The perfusion set-up provides oxygenated continuous flow through the portal vein at temperature of 37°C. The specifics of the machine perfusion protocol used were previously covered elsewhere^{##REF##34307768##43##}. Perfusion was pressure and flow controlled; the flow gradually increased from 6 ml/min to a maximum perfusion rate of 30 ml/min.</p>", "<p id=\"P23\">Acellular perfusate was composed of base William’s E medium containing 1.022g of BSA (Sigma-Aldrich, A7906), 2.4mg of dexamethasone (Sigma-Aldrich, D2915), 1.02mL of penicillin streptomycin (Sigma-Aldrich, P4458), 1.02mL of Glutamax (Thermo Fisher Scientific, 35050061), 0.5U of insulin and 200U of heparin. Perfusate was sterile filtered, and pH stabilized at 7.4 by adding sodium bicarbonate (MGH Pharmacy). For the cellular perfusate group, 80mL of acellular perfusate was added to 20mL packed red blood cells (hematocrit 15–20%) to bring total volume up to 100mL. Prior to perfusion of the liver, the perfusate was cycled for about 15 minutes in the perfusion system to oxygenate and warm it to 37°C.</p>", "<title>Red blood cell collection and preparation of cellular perfusate</title>", "<p id=\"P24\">The donor rats were anesthetized by inhalation of isoflurane (3–5%) in an induction chamber and depth of anesthesia was confirmed by lack of response following toe pinch. The animal was then placed on heating pad in a supine position, and the area around the lower rib was shaved. A small midline incision was made, and the underlying tissue was separated. The heart was located and a 23G needle attached to a 10mL syringe containing xml of sodium heparin was inserted into the left ventricle, and the slowly retracted until no blood was available. A total of 6–8mL of blood was collected and the rat was euthanized by exsanguination.</p>", "<p id=\"P25\">Whole blood collected in sodium heparin was spun down at 2.200 gm for 10 mins at 20°C. The plasma and buffy coat layers are aspirated, and red blood cells are resuspended in perfusate and spun down two more times for the wash process. The washed RBCs are transferred into sterile storage tubes and stored at 4°C for up to 5 days.</p>", "<title>Resonance Raman Spectroscopy Setup</title>", "<p id=\"P26\">The portable RRS System (Pendar Technologies) was developed as a compact 441 nm laser device housing a power laser source of 8.9mw and compact probe with a flexible fiber optic bundle. The probe head is an 8mm × 12mm × 60mm which delivers single mode laser light about 1.5 mm diameter on the tissue. The light is passed through the collection optics, and the filtered RR photons are collected at the temperature-controlled two-dimensional charge coupled device (CCD) detector array. Measurements are recorded for total time of 180s with isolated signals from mitochondrial in the 700–1700 cm⁻¹ spectral range. The readout from the CCD is recorded and analyzed using software developed in LabView.</p>", "<title>Experimental Design</title>", "<p id=\"P27\">Our experimental groups that consist of livers with low and high demand of oxygen (0hCI and 24hCI livers) were perfused with perfusates carrying a low and high supply (acellular and pRBC perfusate) of oxygen as summarized in ##FIG##1##Fig. 2a##. For each of these groups, we obtained the 3RMR value every 30 minutes starting roughly 5 minutes after the beginning of machine perfusion. We also performed an analysis of the blood gases and chemistry of the perfusate at every hour of perfusion to validate the perfusion technique. At the end of the perfusion, wedge biopsies were flash frozen for energetic analysis including ATP, ADP, AMP ratios, and NAD:NADH ratio that show mitochondrial electron transport chain activity both downstream (i.e., at complex V) and upstream (i.e., at complex I) of 3RMR. Finally, we also compared injury to the graft using markers including flow resistance during perfusion, hemolysis, potassium level, ALT/AST levels every 60 minutes during perfusion, and histological markers of injury at the end of perfusion.</p>", "<title>Perfusion data acquisition and processing</title>", "<p id=\"P28\">Blood gas analysis by perfusate sampling every thirty minutes using the RAPIDPOINT 500 Blood gas analyzer (Siemens Healthineers, Munich, Germany) for measurements of blood gases and chemistries including pH, pO2, electrolytes (Na⁺, K⁺, Ca⁺⁺, CI⁻), and metabolites (lactate). AST and ALT concentrations were measured hourly from the venous outflow using the Piccolo Xpress Chemistry Analyzer (Abbott, Illinois, USA). For hematological analysis, perfusate samples collected were centrifuged at 4000g for 10 mins. Supernatant were collected and stored at − 20°C which were later recovered for free hemoglobin assessment using the NanoDrop One Microvolume UV-Vis Spectrophotometer (ThermoFisher Scientific, Waltham, MA, USA) at 414nm. Hematocrit levels were checked prior to perfusion using the Sysmex XP-300^™ Automated Hematology Analyzer (Sysmex, Kobe, Japan).</p>", "<p id=\"P29\">The liver was weighed immediately after harvesting and after machine perfusion. The pressures and flow rates were recorded every 30 mins during the 3 hours perfusion run. Liver tissues were flash frozen in liquid nitrogen immediately after perfusion and stored at −80°C. The concentrations of NAD, NADH, NADP and NADPH in liver tissue were analyzed by mass spectrometry core of Shriners’ Children’s Boston (Boston, MA, USA). Liver biopsies were also fixed in 10% formaldehyde for a maximum of 96hrs before transferring into 70% ethanol. Fixed tissues were processed for TUNEL and H&amp;E staining, which was performed by the Massachusetts General Hospital Histology Molecular Pathology Core (Charlestown, MA, USA).</p>", "<title>Statistical Analysis</title>", "<p id=\"P30\">All statistical analyses were performed in Prism 9 (GraphPad Software Inc., La Jolla, CA). Two-way ANOVA based multiple comparison tests in GraphPad were used to derive statistical conclusions.</p>" ]

[ "<title>Results</title>", "<title>RRS allows quantification of mitochondrial redox state with different perfusate compositions.</title>", "<p id=\"P9\">RRS allows accurate quantification of the redox state of mitochondria as well as hemoglobin in the perfusate since both mitochondrial complexes and hemoglobin molecules possess a porphyrin ring structure with a strong Soret absorption band at 441 nm wavelength. Excitation near the Soret band results in a resonant enhancement of the vibrational modes that result in Raman spectra. When excited by this laser during machine perfusion (##FIG##0##Fig. 1a##), the vibrational scattering spectrum from the liver surface is carried via optical fibers to a Charge Coupled Device (CCD) array, and eventually analyzed using custom LabVIEW program (##FIG##0##Fig. 1b##). This program deconvolves the spectrum into its components by using pre-recorded libraries of fully reduced and oxidized mitochondrial complexes and hemoglobin (##FIG##0##Fig. 1c##, ##FIG##0##d##) by minimizing error in a statistical regression fit. Depending on the perfusate composition, either only the mitochondrial libraries, or both mitochondrial and hemoglobin libraries are used for calculating 3RMR. Upon deconvolution, weights are assigned to each complex which are then used to calculate the Resonance Raman Reduced Mitochondrial Ratio (3RMR) which is the ratio of the weights of reduced to total mitochondrial 3RMR (##FIG##0##Fig. 1e##)^{##UREF##11##30##}. ##FIG##0##Figure 1e## also shows representative images of rodent liver perfusions with and without the packed RBCs, and representative spectrum from each type of perfusion. A more detailed description of the perfusion system and the Raman device is available in the methods section.</p>", "<title>3RMR reflects oxygenation dynamics and energetic recovery of 0-hour cold ischemic &amp; 24-hour cold ischemic livers in real time during perfusion.</title>", "<p id=\"P10\">Our study to observe ischemia and reoxygenation during machine perfusion using 3RMR consisted of groups that reflect different degrees of ischemic stress in the form of short or long durations of storage at 4°C – one group of livers had minimal ischemia (&lt; 15 minutes), referred to as the 0 hour cold ischemic (0h-CI) livers; and a second group of livers that were stored for 24 hours in cold ischemic condition (24h-CI) livers. These groups were chosen because the former mimics fresh transplantations with minimal ischemia, while latter mimics the longest cold ischemic duration that maintains 100% survival after transplantation based on previous studies by our group^{##UREF##12##32##}. We also further tested two perfusate compositions – one that contained packed RBCs as oxygen carriers (pRBC group) and a second group where no oxygen carriers were used (acellular group). These perfusate compositions were chosen to understand the effects of different levels of oxygen stress and test the usefulness of the RRS device in each condition. A summary of the storage and perfusion conditions is shown in ##FIG##1##Fig. 2a##. It was expected that the 24h-CI livers would be starved for oxygen and benefit from high oxygen supply, however, also experience higher IRI. 0h-CI livers on the other hand, are expected to experience a lower basal metabolic demand for oxygen and experience minimal IRI.</p>", "<p id=\"P11\">As shown in ##FIG##1##Fig. 2b##, we observed a low 3RMR value for 0h-CI livers irrespective of the mode of oxygenation. At the start of perfusion, the acellular perfusate group showed a 3RMR of 21.75 ± 2.25, and the pRBC group showed a 3RMR of 16.38 ± 5.121 which were statistically similar (p = 0.1031). The values remained low throughout the 3 hours of perfusion, with acellular group 3RMR 13.13 ± 3.09, and pRBC 3RMR 22.63 ± 8.97 (p = 0.09) at the end of perfusion. This shows sufficient oxygenation with both perfusates, indicating that the dissolved oxygen at a higher partial pressure (500–600 mm Hg) is sufficient for meeting the oxygen demand of these livers. However, this was different from the trend in 24h-CI group of livers, as shown in ##FIG##1##Fig. 2c##. The 3RMR value was high immediately upon reperfusion in the 24h-CI acellular group with a mean of 62.33 ± 4.25, which was in contrast with the low 3RMR in the 24hCI-pRBC group at 23 ± 5.48 (p-value = 0.0002). The high 3RMR likely indicates a high demand of a starving liver which is insufficiently satisfied by the acellular perfusate carrying lower oxygen than the pRBC perfusate, however, the exact mechanism for observation of reduced cytochromes remains to be studied. Interestingly, the 3RMR for 24hCI-acellular livers decreases during perfusion to 34.22 ± 22.48 at 90 minutes, which becomes statistically indistinguishable (p = 0.9426) from the 24hCI-pRBC group by the end of perfusion (33.25 ± 11.62). This appears consistent with the observation that 24 hour cold stored livers can be fully recovered with machine perfusion^{##REF##24338652##33##}. The 3RMR values at the beginning and the end of perfusions for each group are also summarized in ##FIG##1##Fig. 2d## highlighting significant improvement in the 24h-CI acellular group.</p>", "<p id=\"P12\">The initial period of hypoxia and subsequent recovery of 24h-CI groups was also reflected in other perfusion-based biochemical markers of metabolism. ##FIG##1##Figure 2e## shows the high venous lactate concentration in the 24hCI-pRBC group (4,828 ± 2.64 mM) compared to 24h-CI acellular (2 ± 1 mM), 0h-CI pRBC (2.175 ± 1.026 mM) and 0h-CI acellular (1.44 ± 0.39 mM) groups. This may occur due to lactate production in the red blood cells or from liver cells as a result of high oxygen starvation due to long duration of cold ischemia. The lactate buildup in 24h-CI pRBC group gets cleared (1.44 ± 0.5 mM) by the first 30 minutes of perfusion and becomes statistically similar to 24hCI-acellular (0.95 ± 0.9 mM) and 0hCI-pRBC (1.07 ± 0.41 mM) groups (p = 0.404 &amp; p = 0.302 respectively). This trend of low lactate values continues until the end of 3 hours of perfusion when the lactate values are as follows: 0h-CI acellular-1.47 ± 0.425 mM; 0h-CI pRBC-1.14 ± 0.118 mM; 24h-CI acellular – 1.203 ± 0.546 mM; 24h-CI pRBC- 1.45 ± 0.764 mM. ##FIG##1##Figure 2f## shows oxygen uptake rate (OUR) by livers in all groups. Here, we observe higher oxygen uptake by the 24hCI-pRBC group compared to other groups in the initial phase of perfusion- for instance, the oxygen uptake rate at the beginning of perfusion were as follows for the following groups-0hCI acellular, 0hCI pRBC, 24hCI acellular, and 24hCI pRBC: 0.014 ± 0.002, 0.027±, 0.018 ± 0.005, and 0.13 ± 0.164 ml/min respectively (p = 0.97 0hCI pRBC vs 0hCI acellular; p = 0.0082 24hCI pRBC vs 24hCI acellular); and 90 minutes were 0.052 ± 0.004 ml/min, 0.1 ± 0.064 ml/min, 0.052 ± 0.014 ml/min, and 0.177 ± 0.075 and ml/min respectively (p = 0.9, for 0hCI pRBC vs acellular; p = 0.0116 for 24hCI pRBC vs acellular). However, levels became lower to statistically similar level as all other groups by the end of perfusion with OUR as follows at 3 hours of perfusion: 0.052 ± 0.005 ml/min, 0.057 ± 0.011 ml/min, 0.054 ± 0.01 ml/min, and 0.126 ± 0.03 ml/min respectively ((p = 0.99, for 0hCI pRBC vs acellular; p = 0.16 for 24hCI pRBC vs acellular). The lower OUR during perfusion in the 24hCI acellular group compared to 24hCI pRBC group likely reflects limited oxygen availability and causes some damage to the liver. Finally, ##FIG##1##Fig. 2g## shows energy charge (EC), defined as ratio (ATP + 0.5*ADP) / (ATP + ADP + AMP) ^{##REF##18416733##34##} and NAD:NADH ratio for all groups. Energy charge is an indicator of energetic recovery at the end of perfusion, while NAD:NADH ratio is an indicator of utilization of substrates at complex I of the electron transport chain during machine perfusion. EC shows statistically similar levels in all the groups of livers at end of the 3 hours of perfusion- 0hCI acellular-0.455 ± 0.06, 0hCI pRBC- 0.442 ± 0.024, 24hCI acellular 0.369 ± 0.075, and 24hCI pRBC- 0.462 ± 0.03 (p = 0.99 for all comparisons). This may indicate a restoration of ATP across all experimental groups. On the other hand, NAD:NADH ratios for each group were 7.98 ± 2.74, 8.25 ± 1.41, 6.03 ± 1.19, and 8.92 ± 0.96, respectively with p &gt; 0.05 for all pairwise comparisons, except 24hCI acellular vs 24hCI pRBC where the difference between groups was statistically significant with p = 0.029. Based on these metabolic markers, it may be hypothesized that 3RMR, ATP, and NAD:NADH ratios confirm energetic recovery for all groups of livers, albeit sub-optimally in the 24hCI acellular perfusion group where the livers may be experiencing minimal levels of hypoxia due to an incomplete recovery from ischemic stress.</p>", "<title>Increased levels of injury markers indicate suboptimal recovery of livers perfused with pRBCs</title>", "<p id=\"P13\">To quantify injury due to different ischemic durations and perfusate composition, as well as correlation with 3RMR, we looked at clinical markers of injury such as portal resistance, ALT &amp; AST hemolysis for pRBC perfusate, alanine aminotransferase (ALT) &amp; aspartate aminotransferase (AST) levels, potassium, and histological markers of injury during and at the end of 3 hours of perfusion for all groups. We observed a higher portal resistance after 3 hours in the packed RBC groups (&gt; 0.01 mmHg*min/ml/g) compared to acellular groups (&lt; 0.01 mmHg*min/ml/g), however, only reached significance for the 0hCI acellular vs 24hCI pRBC groups with p = 0.0086 (##FIG##2##Fig. 3a##). ALT and AST levels are considered important markers of liver injury, especially during machine perfusion^{##REF##33963657##35##}. Their levels were also higher for the 0hCI pRBC and 24hCI pRBC groups, compared to the 0hCI acellular and 24hCI acellular groups at the beginning (ALT: 10, 11 ± 6.557, 0, and 0 U/L; AST: 32 ± 4, 34 ± 9.17, 3 ± 3.83, and 9.33 ± 8.37 U/L) as well as the end (ALT: 27.3 ± 12.7, 51.33 ± 18.037, 4 ± 3.65, and 4 ± 3.46 U/L; AST: 114, 169.3 ± 65.187, 23 ± 12.91, and 40.67 ± 11.015 U/L) of 3 hours of perfusion. The absolute values of ALT and AST for all the groups of livers are shown in ##FIG##2##Figs. 3b## and ##FIG##2##3c##, respectively. Finally, hyperkalemia (high potassium) is also a marker of injury to the organ that measures release of potassium into the perfusate due to cell death. The potassium concentration stayed between 4–8 mmol/L for all groups during perfusion. Neither drastic shifts over time, nor significant differences between groups at each time were observed as shown in ##FIG##2##Fig. 3d##.</p>", "<p id=\"P14\">Hemolysis is a marker of injury to RBCs that affects oxygenation capacity as well as microcirculatory resistance of the organs, as the RBC in the perfusate are continuously exposed to mechanical stresses. It may cause endothelial dysfunction and increased resistance to flow in the capillaries^{##REF##16291595##36##}. Thus, we quantified hemolysis by analyzing the absorbance properties of the perfusate at 414 nm wavelength (detailed protocol in methods section). We observed increasing hemolysis with the duration of perfusion in both 0hCI pRBC and 24hCI pRBC groups as seen in ##FIG##2##Fig. 3e##, where the absorbance of 414 nm wavelength of light increases at a steady rate over time. The increasing hemolysis can also be observed from the increasing intensity of red coloration of the perfusate, where RBCs have been centrifugally separated (as shown in ##FIG##2##Fig. 3f##). Furthermore, an analysis of histological staining using hematoxylin &amp; eosin (H&amp;E) stain, as well as terminal deoxynucleotide transferase (dUTP) nick end labeling (TUNEL) followed by a blinded analysis by a pathologist was performed. It indicated higher endothelial damage in the 24hCI-pRBC group of livers (as seen in the predominant TUNEL staining of endothelial cells) compared to 24hCI-acellular group where higher damage to hepatocytes (as seen in the predominant TUNEL staining of hepatocytes) was observed. Representative histological micrographs are shown in ##FIG##2##Fig. 3g##. Comparatively, only marginal cell death was observed in the 0hCI-acellular and 0hCI-pRBC groups, indicating sufficient recovery of organs in these groups of livers. Other functional metrics, blood gases, and chemistries were also measured including pH, bile production, weight gain, Ca⁺⁺, Na⁺, and HCO₃, and are summarized in Supplementary Fig. 1. We observed significantly lower pH in the 24hCI pRBC group of livers (p &lt; 0.0001). We also observed significantly higher Na⁺, but significantly lower Ca⁺⁺ in the pRBC groups. Interestingly, there was no statistical difference in bile production, indicating full recovery of function in all livers. However, the 24hCI livers perfused with both compositions gained weight, indicating the presence of endothelial injury leading to edema, especially in the 24hCI pRBC group of livers.</p>" ]

[ "<title>Discussion</title>", "<p id=\"P15\">Solid organ transplantation is a lifesaving procedure, yet severely limited in its capacity to save lives due to ischemia-reperfusion injury (IRI) related complications. Machine perfusion (MP) allows minimization of and recovery from IRI when compared to the current clinical standard of static cold storage. Different machine perfusion methods use different temperatures and perfusate compositions. However, the direct impact of various machine perfusion protocols on mitochondrial injury are incompletely understood. To further enhance the understanding of organ quality during machine perfusion and to increase the overall number of transplants by better assessment of marginal organs, there is a significant push to develop highly specific, sensitive, and quantifiable markers to understand organ health during perfusion. Such technologies help further optimize conditions of perfusion, may provide a better understanding of the underlying mechanisms of IRI, and identify new therapeutic interventions to overcome mitochondrial injury.</p>", "<p id=\"P16\">We present a novel platform for non-invasive, real-time assessment of oxygenation of mitochondria during <italic toggle=\"yes\">ex vivo</italic> liver perfusion using resonance Raman spectroscopy. Through experiments in rat livers, we aimed to provide mechanistic insights into oxygen demand and supply at the electron transport chain during normothermic machine perfusion-based recovery from cold ischemia. For instance, when a minimally ischemic (&lt; 15 minutes) rat liver is perfused with either an acellular perfusate or packed RBC based perfusate, 3RMR is low indicating sufficient oxygenation (##FIG##1##Fig. 2b##). However, when a 24-hour cold ischemic (24hCI) liver is perfused, the pRBC perfusate group shows a low 3RMR (&lt; 25, sufficiently oxygenated) while the acellular perfusate group shows a high 3RMR (&gt; 40, insufficiently oxygenated) (##FIG##1##Fig. 2c##), which is also corroborated by significant hepatocyte death as observed via TUNEL staining. It is likely that there is an accumulation of electrons at the mitochondrial cytochromes during cold ischemia, which can be rapidly transferred to the abundant oxygen molecules in case of pRBC based perfusate. However, this process is comparatively slower in the case of acellular perfusates due to a lower concentration of oxygen. This trend is also supported by the high lactate levels and high oxygen uptake rate in the 24hCI groups immediately upon reperfusion (##FIG##1##Figs. 2e## and ##FIG##1##2f##).</p>", "<p id=\"P17\">While sufficient reoxygenation after ischemia is necessary for recovery, a sudden temperature change and oxygen burst may also exacerbate IRI. Indeed, this suspicion is confirmed by the injury markers ALT and AST, which show higher injury in organs that are perfused with pRBC compared to the organs perfused with an acellular perfusate (##FIG##2##Figs. 3b## and ##FIG##2##3c##). Quite interestingly, a marginal increase in 3RMR at the end of the perfusion for the 24hCI pRBC group is also observed without achieving statistical significance (3RMR of 30–40), indicating the possibility of rising injury over time. This damage is also confirmed by histology with TUNEL, where higher endothelial cell death in livers perfused with packed RBCs is observed, as compared to acellular perfusate. It is possible that such a response is triggered by the innate coagulability of pRBCs^{##UREF##13##39##}, RBC damage due to hemolysis (as a result of mechanical stresses in the perfusion circuit) leading to endothelial dysfunction, vasculopathy and reduced bioavailability of nitric oxide^{##REF##16291595##36##}, or hyperoxygenation induced breakdown of NO leading to vasoconstriction^{##REF##33262362##40##}. Alternatively, it may be hypothesized that the higher supply of oxygen that is available upon reperfusion with pRBCs may lead to ROS generation and reperfusion injury^{##REF##25383517##38##}.</p>", "<p id=\"P18\">Despite these initial high values, the 3RMR for 24hCI-acellular group decreases gradually until about 90 minutes of reperfusion when it becomes comparable to 3RMR in 24hCI-pRBC group as shown in ##FIG##1##Fig. 2c##. It is likely that the accumulated electrons are lowered to physiologic levels by 90 minutes due to the continuous supply of oxygen in the perfusate, albeit at a slower rate compared to the 24hCI pRBC group and are available for ATP production promoting energetic recovery which is an essential criterion to accept organs for transplant. This is confirmed by the energy charge values and NAD:NADH ratios (##FIG##1##Fig. 2g##) which were statistically similar in all groups at the end of 3 hours of NMP. Furthermore, lactate levels were also similar for most of the duration of perfusion, indicating that all livers likely remain transplantable including those in the acellular perfusate group.</p>", "<p id=\"P19\">3RMR may indeed provide an opportunity for dynamic optimization of recovery of livers, by real-time optimization of temperature and perfusate conditions during machine perfusion. For instance, further optimization of the recovery of cold stored organs using MP before transplant may involve a controlled rate of increasing the temperature of the organ after cold storage. Such controlled increase in temperature may slow down the recovery of electron transport chain function, and thus avoid the higher stress associated with sudden change in temperature and ROS generation upon reperfusion. Indeed, in our previous study, when 24hCI livers were recovered with acellular perfusate using sub-normothermic machine perfusion (SNMP), we observed that the 3RMR levels falls to low level within 30 minutes of perfusion^{##REF##34705828##37##}. This may be explained by a slower rate of accumulation of electrons at the complexes III and IV at sub-normothermic temperatures compared to the normothermic temperatures due to the relatively slower rate of electron transport chain activity at this temperature. A slower push and pull of electrons at this temperature in the initial phase of perfusion is likely to be favorable for the overall health of the mitochondria and lead to overall reduced stress on the electron transport chain while allowing ATP production and better functioning of repair mechanisms.</p>", "<p id=\"P20\">In conclusion, there is an urgent need for mechanistic, non-invasive, and real-time assessment of IRI during machine perfusion due to the low specificity and sensitivity of currently used markers^{##REF##31592808##41##}. We successfully developed a novel approach that provides a reporter free highly specific readout of functional oxidation state of mitochondrial cytochromes during machine perfusion in real-time. Better understanding of mitochondrial injury may not only help in better optimization of storage and perfusion conditions, but also assess the extent of injury and function of organs before transplant^{##UREF##14##42##}. Thus, optimal assessment may allow the use of organs from donors deceased due to cardiac death (DCD) and other extended criteria donor (ECD) organs. This will help increase the number of organs available for transplant and improve the quality of organs that are transplanted leading to an overall improved standard of care for patients requiring this life saving treatment. Based on our observations of 3RMR at different temperatures (SNMP and NMP), perfusates (acellular and pRBC perfusate), and durations of storage (fresh, 24-hour cold storage, 72-hour cold storage) in rat livers, we observe that ischemic livers have a higher demand for oxygen at the beginning of perfusion, yet packed RBCs may exacerbate liver damage by the end of machine perfusion.</p>" ]

[]

[ "<p id=\"P1\">Author Contribution</p>", "<p id=\"P2\">R.J. - Planning and execution of the experiments, data analysis, making figures and writing the manuscript.E.O.A. - Planning and execution of the experiments, manuscript editing.P.R. - Design and optimization of the Raman spectroscopy device, critical analysis of the results, manuscript editing.E.O.A.H. - Histological analysis and manuscript editing.S.N.T. - Conception of the project and planning of experiments, manuscript editing, supervision of all aspects of the work.</p>", "<p id=\"P3\">Organ transplantation is a life-saving procedure affecting over 100,000 people on the transplant waitlist. Ischemia reperfusion injury is a major challenge in the field as it can cause post-transplantation complications and limits the use of organs from extended criteria donors. Machine perfusion technology is used to repair organs before transplant, however, currently fails to achieve its full potential due to a lack of highly sensitive and specific assays to predict organ quality during perfusion. We developed a real-time and non-invasive method of assessing organ function and injury based on mitochondrial oxygenation using resonance Raman spectroscopy. It uses a 441 nm laser and a high-resolution spectrometer to predict the oxidation state of mitochondrial cytochromes during perfusion, which vary due to differences in storage compositions and perfusate compositions. This index of mitochondrial oxidation, or 3RMR, was found to predict organ health based on clinically utilized markers of perfusion quality, tissue metabolism, and organ injury. It also revealed differences in oxygenation with perfusates that may or may not be supplemented with packed red blood cells as oxygen carriers. This study emphasizes the need for further refinement of a reoxygenation strategy during machine perfusion that is based on a gradual recovery from storage. Thus, we present a novel platform that provides a real-time and quantitative assessment of mitochondrial health during machine perfusion of livers, which is easy to translate to the clinic.</p>" ]

[]

[ "<title>Acknowledgements</title>", "<p id=\"P31\">This work was supported by generous funding to S.N.T. from the US National Institutes of Health (R01DK134590). We also gratefully acknowledge funding from the US National Institute of Health (K99/R00 HL1431149; R01HL157803; R24OD034189), National Science Foundation (EEC 1941543), American Heart Association (18CDA34110049), Harvard Medical School Eleanor and Miles Shore Fellowship, Polsky Family Foundation, the Claflin Distinguished Scholar Award on behalf of the MGH Executive Committee on Research, and Shriners Children’s Boston (Grant #BOS-85115). Further, we acknowledge the salary support to R.J. provided by the Massachusetts General Hospital’s Executive Committee on Research - Fund for Medical Discovery (FMD) award, and the Afdhal/McHutchison LIFER award by the American Association for the Study of Liver Diseases (AASLD). The authors would also like to thank the Center for Comparative Medicine at Massachusetts General Hospital for animal management. Finally, we thank the Mass Spectroscopy, Genomics and Proteomics, and Morphology facilities at Shriners Children’s Boston.</p>", "<title>Data availability statement</title>", "<p id=\"P32\">The authors affirm that the primary data supporting the conclusions of this research are represented within the article. Any other information not contained within the article will be made available by the corresponding author upon reasonable request.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><title>Resonance Raman Spectroscopy predicts functional oxygenation of liver tissue during machine perfusion.</title><p id=\"P35\"><bold>a</bold>, schematic of machine perfusion of livers where a perfusate(I) with or without red blood cells as oxygen carriers is circulated by a pump (II) to an oxygenator (III). The oxygenator consists of a gas-permeable tubing that carries the perfusate surrounded by oxygen at a higher ambient partial pressure than atmosphere. This allows the perfusate to be oxygenated before it is supplied to a cannulated liver (IV). <bold>b,</bold> the principle of our custom approach to measuring mitochondrial redox state in the tissue non-invasively using a resonance Raman spectroscopy device. A 441nm excitation laser is used to excite molecules with a porphyrin ring (such as mitochondrial complexes, cytochromes, and hemoglobin) that produce a resonance Raman spectrum. <bold>c,</bold> shows the spectrum of oxidized and reduced isolated mitochondria. <bold>d,</bold> shows the spectrum of reduced and oxidized hemoglobin. <bold>e,</bold> shows two rat livers that are sitting in the perfusion bowl that are supplied with oxygen either with or without RBCs. Directly below the livers are the RRS spectrum from each liver. This spectrum is deconvoluted into its constituent molecular signatures using pre-recorded libraries as shown in c and d. The deconvolution coefficients for the reduced mitochondria is averaged with the sum of reduced and oxidized mitochondria to quantify the redox state of tissue and called the 3RMR value. Similarly, the oxygen saturation of hemoglobin in the tissue is also obtained by taking the ratio of oxidized hemoglobin coefficient with the total coefficients of oxidized and reduced hemoglobin.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><title>3RMR with lactate, energy charge, and NAD:NADH ratio predict metabolic recovery dynamics during machine perfusion.</title><p id=\"P36\"><bold>a,</bold> Schematic of the experiments and test groups. Each liver is either stored at 4°C for 24 hours or perfused immediately after recovery from the rat. It may then be machine perfused with either a packed RBC based perfusate or an acellular perfusate for 3 hours at 37°C. During this time the perfusate is sampled every 60 minutes for blood gas, blood chemistry, and ALT/AST analysis. Raman measurements are also taken from the surface of the livers every 30 minutes. At the end of the perfusion, wedge biopsies from the liver are either flash frozen for energetic tests or stored in formalin for histology. <bold>b,</bold> shows low 3RMR values throughout the 3 hours of perfusion that are statistically indistinguishable between 0hCI acellular and 0hCI pRBC groups. <bold>c,</bold> shows significantly higher 3RMR value immediately after reperfusion of 24hCI acellular group compared to 24hCI pRBC group. The 3RMR value decreases continuously during perfusion and becomes statistically indistinguishable around 90 minutes of perfusion. This trend reverses after 90 minutes indicating possible dysfunction of mitochondrial electron transport chain, however remaining statistically the same. <bold>d,</bold> shows that the comparison between 3RMR values at the start of perfusion and at the end of perfusion for each of the tested conditions. The p-values that are marked for each pair show statistically significant lower 3RMR values for acellular perfusate while those with packed RBCs are statistically the same. <bold>e,</bold> shows lactate levels during the 3 hours of perfusion for all four conditions. that remain low and indistinguishable for all the tested conditions during perfusion. <bold>f,</bold> shows oxygen uptake rate for all four groups during perfusion. The oxygen uptake for 24hCI pRBC group is the highest which reflects the higher oxygen demand that is satisfied by the higher supply of oxygen bound to RBCs. OUR between the other three groups is comparable with only slightly higher values in the 0hCI pRBC group. <bold>g,</bold> shows energy charge and NAD:NADH values for all the four conditions tested. These were obtained from flash frozen wedge biopsies from the livers at the end of 3 hours of perfusion. While not statistically significant, the ratios for machine perfusion of 24-hour cold ischemic livers perfused with acellular perfusate were slightly lower than the other conditions, indicating potential slower recovery of oxidative phosphorylation. Statistical significance levels - * p&lt;0.033, ** p&lt;0.0021, *** p&lt;0.0002, **** p&lt;0.0001.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3</label><caption><title>Portal vein pressure, ALT, and AST levels indicate higher injury to livers perfused with packed RBCs compared to acellular perfusate for the 3 hours of perfusion.</title><p id=\"P37\"><bold>a,</bold> Shows the trends in resistance for the tested conditions. It is significantly higher in the pRBC groups than the acellular perfusate groups after 2 hours of perfusion. There is no significant difference between the fresh and cold storage groups for each type of perfusate. <bold>b,c,</bold> Show trends in ALT and AST levels between the four groups, where pRBC perfusate groups show higher values compared to the acellular perfusate groups. The statistically significant difference becomes more prominent over time for these groups. <bold>d,</bold>Shows no significant difference in IVC potassium for all four groups during perfusion. <bold>e,</bold> Shows hemolysis over the duration of perfusion. There is no significant difference at any time during the perfusion between 0hCI and 24hCI groups. The hemolysis is also evident from the increasing red coloration of the perfusate samples with time as shown in <bold>f.</bold> Finally, <bold>g,</bold> shows representative histology to show the patters of injury in 24hCI livers from acellular and pRBC groups stained with H&amp;E and TUNEL stains (Magnification 20x). The blue arrows indicate hepatocyte death while the black arrows show endothelial death as observed via TUNEL staining of apoptotic cells. Statistical significance levels - * p&lt;0.033, ** p&lt;0.0021, *** p&lt;0.0002, **** p&lt;0.0001.</p></caption></fig>" ]

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[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN2\"><p id=\"P33\">Competing Interests</p><p id=\"P34\">The authors declare competing interests. Dr. Tessier and Mr. Romfh have provisional patent applications relevant to this study. P.R. is an employee and shareholder of Pendar Technologies. S.N.T.’s competing interests are managed by the MGH and Partners HealthCare in accordance with their conflict-of-interest policies, and P. R.’s competing interests are subject to the Research Integrity Policy of Pendar Technologies. The following patented technologies have been used in this study: US2020/0281474A1 In-vivo monitoring of cellular energetics with Raman spectroscopy (application). Additional patent applications for use in ophthalmology, tissue viability, and burn injury assessment using Resonance Raman Spectroscopy have been submitted where R.J. is also an inventor. This does not alter our adherence to Scientific Reports’ policies on sharing data and materials. All other authors do not have competing interests.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Background</title>", "<p id=\"P15\">Myocarditis is an inflammation of the myocardium, or muscle tissue of the heart, and a leading cause of sudden cardiac death in persons under 50 years of age [##REF##30638108##1##, ##REF##33309175##2##]. The Global Burden of Disease (GBD) study from 2019 reported 1.8 million cases of myocarditis world-wide [##REF##33309175##2##]. A 2014 Swedish study reported myocarditis at an incidence of 8.6 people per 100,000 [##REF##35110692##3##]. Several epidemiological studies estimated at least a 15-fold increased incidence of myocarditis/perimyocarditis from severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) infection during the coronavirus disease 2019 (COVID-19) pandemic [##REF##34473684##4##, ##REF##37167363##5##]. Prior to the pandemic, coxsackievirus B3 (CVB3) was the leading suspected cause of myocarditis in the United States. Myocarditis can progress to dilated cardiomyopathy (DCM) in susceptible individuals and in mouse models of viral and autoimmune myocarditis [##UREF##0##6##–##REF##36937911##8##]. In this model, upregulated profibrotic remodeling genes during acute myocarditis (day 10 post infection/pi) lead to the development of fibrosis and ventricular dilation during DCM (day 35 pi and onwards) [##REF##15579433##9##, ##REF##22328081##10##]. Chronic heart failure during DCM leads to heart transplants in a significant proportion of patients [##REF##21884947##11##, ##REF##31073128##12##]. A lack of disease-specific therapies aside from heart failure medications provides impetus for identification of novel biomarkers and therapeutic targets with the goal of earlier detection and more targeted treatment.</p>", "<p id=\"P16\">The incidence and severity of myocarditis is greater in cis-males (referred to hereafter as males) than cis-females (referred to hereafter as females) in humans and mouse models [##REF##36937911##8##]. The GBD study reported a mortality rate in patients with myocarditis aged 35–39 of 4.4 per 100,000 in women and 6.1 per 100,000 in men, indicating that more men die of myocarditis than women worldwide [##REF##33309175##2##]. We previously reported a sex ratio of 3.5 males to 1 female among patients with biopsy confirmed myocarditis [##REF##30638108##1##]. Men are more likely to develop cardiac fibrosis and progress to DCM after myocarditis compared to women [##REF##22328081##10##, ##REF##21884947##11##, ##REF##19710029##13##]. We, and others, previously reported that testosterone promotes a proinflammatory and profibrotic response in an autoimmune model of CVB3 myocarditis while estrogen is cardioprotective [##REF##30638108##1##, ##REF##22328081##10##, ##REF##18586295##14##, ##UREF##1##15##]. The goal of this study was to better understand sex differences in CVB3 myocarditis using bulk-tissue RNA sequencing (RNAseq). We found major sex differences in transcriptional programming related to cardiac mitochondrial biogenesis.</p>" ]

[ "<title>Methods</title>", "<title>Myocarditis Model</title>", "<p id=\"P46\">Male and female 6–8 week-old BALB/cJ mice (stock# 651) were obtained from Jackson Laboratory (Bar Harbor, ME). Mice were inoculated with sterile phosphate buffered saline (PBS) (vehicle control) or 10³ plaque forming units (PFU) of heart-passaged CVB3 intraperitoneally (ip) on day 0 and hearts collected on day 10 pi, as previously described [##UREF##11##76##]. This is an autoimmune model of myocarditis using live virus as the adjuvant that closely resembles experimental autoimmune myocarditis and human disease (reviewed in [##UREF##6##54##, ##REF##22488075##55##]). The Nancy strain of CVB3 was originally obtained from the American Type Culture Collection (ATCC; Manassas, VA) and grown in Vero cells (ATCC), to create a tissue culture-derived virus stock as previously described [##UREF##11##76##]. Briefly, 100mL of tissue culture virus (10³ PFU) was injected ip into 4-week-old female BALB/c mice and virus obtained from hearts at day 3 pi by homogenization in Gibco Minimum Essential Media (Thermo-Scientific, Waltham, MA, 11095–080) supplemented with 2% heat inactivated FBS. Homogenized hearts were centrifuged at 4C for 20 min at 795g. Homogenized supernatant that contains infectious virus and damaged heart proteins (heart-passaged virus) was stored at −80 until used to induce myocarditis, as described in [##UREF##11##76##].</p>", "<title>Histology</title>", "<p id=\"P47\">Mouse hearts were cut longitudinally and fixed in 10% phosphate-buffered formalin and embedded in paraffin for histological analysis. 5 μm sections were stained with hematoxylin and eosin (H&amp;E) to detect inflammation. Myocarditis was assessed as the percentage of the heart with inflammation compared to the overall size of the heart section using a microscope eyepiece grid, as previously [##REF##15579433##9##, ##REF##37612820##77##]. Sections were scored by two individuals blinded to the treatment group.</p>", "<title>Immunohistochemistry</title>", "<p id=\"P48\">Heart sections (5 mm) were stained with ERRa (ThermoFisher/Invitrogen, 1:1,000, cat<bold># PA5–28749</bold>). An Envision+ anti-rabbit labeled polymer (K4003) and rat-on-rodent kit (RT517) (Biocare, Pacheco, CA) were used as secondary antibodies for rat antibodies. Stained slides were scanned using an Aperio AT2 slide scanner (Leica, Wetzlar, Germany). Representative fields of view from the apex, mid and base of ventricles were manually selected.</p>", "<title>Quantitative Real Time PCR</title>", "<p id=\"P49\">RNA was isolated from mouse hearts using Qiagen’s Fibrous Tissue Mini Kit (Qiagen 74704) before concentration (Abs. 260) and quality (Abs. 260/280) of preps was assessed using a Nanodrop. cDNA was generated using the iScript cDNA synthesis kit (Biorad, #1708891). Quantitative real time PCR (qRT-PCR) was assessed with Taqman probes (CD45/<italic toggle=\"yes\">Ptprc</italic> Mm01293577_m1; CD11b/<italic toggle=\"yes\">Itgam</italic> Mm00434455_m1; F4/80/<italic toggle=\"yes\">Adgre1</italic> Mm00802529_m1; peroxisome proliferator-activated receptor gamma coactivator 1 (PGC1α)/<italic toggle=\"yes\">Ppargc1a</italic> Mm01208835_m1; nuclear respiratory factor 1 (NRF1)/<italic toggle=\"yes\">Nrf1</italic> Mm01135606_m1; estrogen-related receptor-a (ERRa)/<italic toggle=\"yes\">Esrra</italic> Mm00433143_m1) and normalized against hypoxanthine phosphoribosyltransferase 1 (HPRT) (<italic toggle=\"yes\">Hprt</italic>, Mm03024075_m1) to determine relative gene expression (RGE) using ΔΔCt as previously [##REF##37612820##77##, ##UREF##12##78##].</p>", "<title>RNA Sequencing</title>", "<p id=\"P50\">At the time of harvest, half of the heart was collected for histological evaluation using H&amp;E to determine the severity of myocardial inflammation and the other half was snap frozen in liquid nitrogen. Histology shown in ##FIG##0##Figure 1## is combined data from 3 separate experiments. The investigator selected 3 histologically representative samples of the overall dataset from a single experiment (see Figure S1; <italic toggle=\"yes\">n</italic> = 3/group) and sent to the Mayo Clinic Genome Analysis Core for library preparation and bulk-tissue RNA sequencing. Libraries for this study were prepared using the core’s standard mRNAseq prep which uses poly A selection. RNA libraries were prepared using 200 ng of total RNA according to the manufacturer’s instructions for the TruSeq Stranded mRNA Sample Prep Kit (Illumina, San Diego, CA). The concentration and size distribution of the completed libraries was determined using an Agilent Bioanalyzer DNA 1000 chip (Santa Clara, CA) and Qubit fluorometry (Invitrogen, Carlsbad, CA). Libraries were sequenced at 50 million fragment reads per sample following Illumina’s standard protocol using the Illumina cBot and HiSeq 3000/4000 PE Cluster Kit. The flow cells were sequenced as 100 × 2 paired end reads on an Illumina HiSeq 4000 using HiSeq 3000/4000 sequencing kit and HiSeq Control Software HD 3.4.0.38 collection software. Base-calling was performed using Illumina’s RTA version 2.7.7</p>", "<title>RNA Sequencing Analysis</title>", "<p id=\"P51\">After next-generation RNA sequencing, the Mayo Clinic Genome Analysis Core provided differential expression data. We compared PBS control females vs. females with myocarditis, PBS control males vs. males with myocarditis, and females with myocarditis vs. males with myocarditis. Because of small group size, we assessed intra-group variability using <italic toggle=\"yes\">ClustVis</italic> [##REF##25969447##79##] by performing unsupervised hierarchical clustering using Euclidean row and column distances and principal component analysis (PCA).</p>", "<p id=\"P52\">Differential expression analysis was performed by the Mayo Clinic Genome Analysis Core and gene names were converted to murine ensemble IDs (ENSMUSG) for analysis. We performed enrichment analysis as in Reimand et al. [##REF##30664679##80##]. For gProfiler, transcripts with nominal p-value &lt; 0.05 were ordered from most to least significant and an ordered query was run (gene sets with 5–350 entities were included). At the time of analysis, we excluded duplicate transcripts or those not recognized by gProfiler. The same set of transcripts used for gProfiler were ordered by logFC to perform gene set enrichment analysis (GSEA) pre-ranked utilizing a combined gene matrix transposed (GMT) from gProfiler. GSEA pre-ranked was performed with default gene set size restriction (15–500) and permutation parameters (1,000).</p>", "<p id=\"P53\">GSEA results were plotted in Cytoscape (Version 3.7.23) using <italic toggle=\"yes\">Enrichment Map</italic> with a node (i.e., gene set/pathway) cutoff of FDR(Q) value &lt; 0.1 and edge cutoff of 0.375. Nodes were clustered based on shared genes and <italic toggle=\"yes\">AutoAnnotate</italic> was used to identify clusters of nodes (sometimes referred to as “super-clusters” in this text). We selected all nodes to create a combined heat map of the top 273 genes and additionally selected mitochondrial-related nodes to create a combined heat map of the top 132 mitochondrial genes by group (row-normalized by Cytoscape). Combined heat maps for top NES pathways in F-CON vs F-MYO and M-CON vs M-MYO comparisons and mitochondrial pathways in F-MYO vs M-MYO comparison (including the combined auto-annotated cluster of gene sets “respiratory complex mitochondrial) were generated from Cytoscape (row-normalized). For Metascape enrichment analysis, we excluded all genes with <italic toggle=\"yes\">p</italic> &gt; 0.05, and ran corresponding gene sets for each phenotype with the <italic toggle=\"yes\">Express Analysis</italic> option (<ext-link xlink:href=\"https://metascape.org/\" ext-link-type=\"uri\">Metascape.org</ext-link>)[##REF##30944313##17##] for the following comparisons: F-CON vs F-MYO, M-CON vs M-MYO, and F-MYO vs M-MYO. We averaged MCODE clusters’ Log10p values (rounded to nearest whole number) to obtain values listed in ##FIG##4##Figures 5c## and ##FIG##5##6c##.</p>", "<p id=\"P54\">A list of the murine nuclear encoded mitochondrial respiratory chain transcripts was generated from the Mouse Genome Informatics (MGI) database from Jackson Laboratories for respiratory chain (<ext-link xlink:href=\"https://www.informatics.jax.org/go/term/GO:0005746\" ext-link-type=\"uri\">https://www.informatics.jax.org/go/term/GO:0005746</ext-link>) and the ATP synthase (<ext-link xlink:href=\"https://www.informatics.jax.org/go/term/GO:0005753\" ext-link-type=\"uri\">https://www.informatics.jax.org/go/term/GO:0005753</ext-link>). Duplicates were removed and only transcripts of the mitochondrial respiratory chain were included. Transcripts and related data were used to create a combined gene expression matrix for each complex with row normalization (using the STANDARDIZE function in Excel). Transcripts not expressed across all four groups (PBS control females, PBS control males, myocarditis females, and myocarditis males) were excluded. The final list of transcripts was used to determine percent of transcripts predicted to be regulated via estrogen related receptors using TRANSFAC in gProfiler.</p>", "<title>ELISA</title>", "<p id=\"P55\">Frozen hearts were rapidly thawed and weighed to obtain tissue wet weight before homogenizing using a polytron homogenizer in minimum essential media (MEM) with 2% fetal bovine serum (FBS). Homogenized tissue was centrifuged at 3,000 rpm at 6°C for 20 minutes and the supernatant was collected for analysis. Whole heart ERRa protein expression was quantified using the Mouse Estrogen-Related Receptor Alpha ELISA Kit from MyBioSource (cat# MBS080310, San Diego, CA). Absorbance was used to calculate concentration relative to a standard curve and normalized to tissue wet weight, as previously [##REF##12707353##46##, ##REF##37612820##77##, ##UREF##12##78##, ##REF##32006209##81##]. The lowest detection limit for the ERRa kit was 0.1ng/mL with a detection range of 0.25–8 ng/mL.</p>", "<title>Echocardiography</title>", "<p id=\"P56\">Cardiac function was determined by transthoracic echocardiography performed using the Vevo 3100 Ultrasound machine equipped with a MX550D 40MHz transducer mounted to a “3D Motor” (VisualSonics Inc., Toronto, Canada) Mice were sedated with 3% isoflurane, hair across the abdominal cavity was removed using Nair while isoflurane sedation was continued at 1–3% depending on animal heart rate, and ultrasonic transmission gel (Parker Laboratories, Fairfield, NJ) was applied to the thorax.^{##REF##30944313##17##,##UREF##3##29##,##REF##35440636##44##}. A heart rate of 350–450 beats per minute (bpm) was maintained during the procedure. Two-dimensional (2D) parasternal long-axis (LAX) of the left-ventricle (LV) were acquired in B-mode. For echocardiography-derived global longitudinal strain (GLS), LAX images were analyzed using Vevo Strain analysis software (within Vevo LAB) with three cardiac cycles. Strain measures were derived from the formula for cardiac strain which is defined by the difference in movement of a wall from its starting position at end-diastolic diameter to its end position at end-systolic diameter divided by the original position of the wall. This effectively represents a percent-change in wall position composed of individual component vectors. For 2D LAX this includes the longitudinal and radial movement vectors.</p>", "<title>Statistical Analysis</title>", "<p id=\"P57\">Normally distributed data comparing two groups, determined with Prism, were analyzed using a 2-tailed Student’s t test. Multiple comparison analysis was performed by ANOVA with each group compared to the corresponding control group; 2-way ANOVA with repeated measures was used to determine the effect of sex vs. disease (myocarditis) using a main effects model. Multiple comparisons were performed with Holm-Sidak. Outlier analysis/exclusion was performed with ROUT (Q = 2%). Violin plots display mean and quartiles, other data are displayed as mean ± SEM. A value of <italic toggle=\"yes\">p</italic> &lt; 0.05 was considered significant. Adjusted p-values (from Prism) were used for multiple comparisons.</p>" ]

[ "<title>Results</title>", "<title>Myocardial inflammation is increased in males compared to females</title>", "<p id=\"P17\">We first examined inflammation in males versus females and uninfected PBS vehicle controls to confirm sex differences and to select samples for RNA sequencing. As we have shown previously [##REF##17513715##16##], males in our autoimmune CVB3 model develop significantly more inflammation than females (<italic toggle=\"yes\">p</italic> = 0.0001) according to histological assessment whereas vehicle controls did not develop myocarditis (##FIG##0##Figure 1a##). Representative examples of histology for each group are shown in ##FIG##0##Figure 1b##. We confirmed major immune cell types in the heart of males vs. females with myocarditis compared to controls using qRT-PCR. We found total immune cells (CD45, <italic toggle=\"yes\">p</italic> &lt; 0.0001), complement 3 activated myeloid cells (CD11b, <italic toggle=\"yes\">p</italic> &lt; 0.0001), and macrophages (F4/80, <italic toggle=\"yes\">p</italic> &lt; 0.0001) were increased in males with myocarditis compared to females with myocarditis (##FIG##0##Figure 1c##–##FIG##0##h##), as expected [##REF##17513715##16##]. Thus, males have greater cardiac inflammation during autoimmune CVB3 myocarditis than females.</p>", "<title>Females upregulate while males downregulate gene pathways related to mitochondrial homeostasis during myocarditis</title>", "<p id=\"P18\">We then examined sex differences in myocarditis using bulk-tissue RNAseq (an overview of the experimental design is illustrated in ##FIG##1##Figure 2a##). PCA analysis revealed good separation between groups and high similarity within groups (##FIG##1##Figure 2b##). To better understand the mechanisms underlying sex differences in myocarditis, we performed gene set enrichment analysis (GSEA) of RNAseq data comparing males and females with myocarditis versus controls. When we compared females and males with myocarditis, we found males with myocarditis (blue) had significantly enriched clusters (i.e., auto-annotated grouped gene sets) for the following gene pathways compared to females: regulation mediated (immune) response, viral life cycle, presentation MHC antigen, Fc-receptor complement cascade, nucleoside activity anhydrides, abnormal thrombocyte morphology, and activity of serine peptidases (##FIG##1##Figure 2c##). Females with myocarditis (pink) had significantly enriched clusters for the following gene pathways compared to males: respiratory complex mitochondrial, generation precursor energy, and serum lactate levels (##FIG##1##Figure 2c##). Non-super-clustered gene sets (i.e., nodes) and their identities are displayed in <bold>Additional File 1: Figure S2.</bold> These data indicate that females with myocarditis have higher expression of mitochondrial respiratory transcripts than males. In contrast, males have higher expression of immune system genes compared to females, which is consistent with histology findings (##FIG##0##Figure 1##).</p>", "<p id=\"P19\">We used Cytoscape to generate heat maps from RNA sequencing data for the top 273 differentially expressed genes between males and females with myocarditis or controls (##FIG##1##Figure 2d##). These data revealed distinct gene profiles between each group with 200 of the top 273 genes being increased in females during myocarditis compared to males with myocarditis (##FIG##1##Figure 2d##). Females with myocarditis upregulated 216 genes compared to female controls in contrast to males with myocarditis that downregulated 210 genes compared to male controls (##FIG##1##Figure 2d##).</p>", "<p id=\"P20\">To better understand sex differences in mitochondrial genes, we selected mitochondrial gene sets in Cytoscape to generate a heat map containing 132 differentially expressed and mitochondrial-specific genes comparing females to males with myocarditis or controls (##FIG##1##Figure 2e##). The mitochondrial gene expression differences between controls and by sex were very similar to the findings of the top 273 genes. Females with myocarditis had higher expression of 118 of 132 mitochondrial genes compared to males with myocarditis (##FIG##1##Figure 2e##). Females upregulated 119 of 132 mitochondrial genes during myocarditis compared to female controls while males downregulated 120 mitochondrial genes compared to male controls (##FIG##1##Figure 2e##). These data demonstrate that males with myocarditis have decreased mitochondrial-related transcriptional support whereas females with myocarditis have increased mitochondrial-related transcript support during myocarditis.</p>", "<p id=\"P21\">To better understand gene pathways that differed by sex during myocarditis we plotted the top ten significant gene pathways from GSEA ranked by normalized enrichment score (NES) for control versus myocarditis (##FIG##2##Figure 3a##,##FIG##2##b##) and by sex (##FIG##2##Figure 3c##). We found that uninfected female hearts were enriched for gene sets related to mitochondrial and cellular homeostasis (##FIG##2##Figure 3a##). Increased expression of transcripts related to immune activation such as “antigen processing and interaction” were found in females during myocarditis (##FIG##2##Figure 3a##). Males with myocarditis transitioned from mitochondrial homeostasis in uninfected hearts to a proinflammatory immune response during myocarditis (##FIG##2##Figure 3b##). We generated heatmaps corresponding to highlighted top significantly enriched pathways in the female control versus female myocarditis comparison (extracellular matrix structural constituent and antigen binding, respectively) and for the male control versus male myocarditis comparison (inner mitochondrial membrane protein complex and immune response, respectively) which can be found with NES and false discovery rate (FDRq) values in ##FIG##2##Figure 3c##.</p>", "<p id=\"P22\">A direct comparison of females to males with myocarditis revealed that females were enriched for pathways related to mitochondrial homeostasis and anti-oxidant responses while males were enriched for pathways related to the innate and adaptive immune responses (##FIG##3##Figure 4a##). We generated a heatmap showing all four groups of genes and gene sets in the super-cluster auto annotated as “respiratory complex mitochondrial”, which contained enriched gene sets related to the mitochondrial respiratory chain. Similar to the findings of the heatmap of all mitochondrial genes in ##FIG##1##Figure 2e##, females had higher expression of 114 of 127 genes in this super-cluster (##FIG##3##Figure 4a##). Females with myocarditis upregulated 115 genes compared to female controls and males with myocarditis downregulated 116 genes compared to male controls (##FIG##3##Figure 4a##). We generated heatmaps and highlighted mitochondrial enriched gene sets, which were significantly more enriched in females with myocarditis compared to males with myocarditis including mitochondrial protein complex (NES = −2.6, <italic toggle=\"yes\">FDRq &lt; 0.0001</italic>) and mitochondrial inner membrane (NES = −2.4, <italic toggle=\"yes\">FDRq = 0.0007</italic>) (##FIG##3##Figure 4b##). Inner mitochondrial membrane protein complex (NES = −2.4, <italic toggle=\"yes\">FDRq = 0.0007</italic>) and respirasome (NES = −2.3, <italic toggle=\"yes\">FDRq = 0.004</italic>) were also significantly enriched in females with myocarditis and mostly contained common genes with the pathways shown in ##FIG##3##Figure 4b##; these heatmaps can be found in <bold>Additional File 1: Figure S3.</bold> Thus, aside from sex differences in immune changes during myocarditis, which have been well characterized in the past, the main difference in cardiac transcript enrichment between males and females with myocarditis occurred in pathways related to mitochondrial function.</p>", "<p id=\"P23\">To ensure that the observed sex differences in mitochondrial transcriptional enrichment were not a result of the enrichment method performed (i.e., GSEA Pre-Ranked), we additionally performed enrichment analysis using Metascape [##REF##30944313##17##]. Metascape enrichment for females with myocarditis compared to controls versus males with myocarditis compared to controls were broadly similar to GSEA enrichment findings in ##FIG##1##Figure 2## and can be found in <bold>Additional File 1: Figures S4–7</bold>. When we directly compared females with myocarditis to males with myocarditis, we confirmed that females with myocarditis were significantly enriched for pathways supporting mitochondria homeostasis and cell energetics (##FIG##4##Figure 5##) and males with myocarditis were enriched for pathways related to upregulation of the immune response (##FIG##5##Figure 6##). Importantly, MCODE_1 clustering of protein-protein interaction analysis by pathway, comprised of <ext-link xlink:href=\"https://reactome.org/\" ext-link-type=\"uri\">Reactome.org</ext-link> enrichment terms for respiratory electron transport (R-MMU-611105, and R-MMU-163200) and Gene Ontology oxidative phosphorylation (GO:0006119) pathways, were all enriched at Log10p values of −100 further highlighting that mitochondrial pathways were the primary enrichment signature in females with myocarditis compared to males with myocarditis (##FIG##4##Figure 5c##).</p>", "<p id=\"P24\">Enrichment quality control metrics from Metascape revealed cell-specific signatures for males with myocarditis compared to females with myocarditis which mirrored the known prevalence of immune cells in the heart with highest to lowest being macrophages, T cells, natural killer (NK cells), B cells and mast cells (##FIG##6##Figure 7a##) [##REF##17513715##16##]. The most enriched transcription factor in males with myocarditis was signal transducer and activator of transcription (STAT)1, which is known to mount interferon (IFN) and T helper (Th)1/M1 immune responses that clear CVB3 infection during myocarditis [##REF##15611248##18##, ##REF##37660839##19##] (##FIG##6##Figure 7a##). In contrast, females with myocarditis were more enriched for myoblasts (c2c12), myocytes and other cardiac cell components with an absence of immune cells compared to males with myocarditis (##FIG##6##Figure 7b##). In contrast, the top enriched transcriptional regulator in females with myocarditis was peroxisome proliferator-activated receptor gamma (PParg), a transcriptional regulator associated with metabolic shift to integrate glycolysis and lipid anabolism in the failing heart [##REF##19490906##20##].</p>", "<p id=\"P25\">In <bold>Table S1</bold>, we display the top 5 most enriched gene sets from gProfiler comparing females and males with myocarditis, which were primarily comprised of gene sets related to mitochondrial homeostasis. The top enriched pathways from gProfiler for each source were: electron transfer activity (Gene Ontology (GO): Molecular Functions/MF), electron transport chain (GO: Biological Process/BP), mitochondrial inner membrane (GO: Cellular Component/CC), oxidative phosphorylation (Kyoto Encyclopedia of Genes and Genomes/ KEGG), and the citric acid cycle and respiratory electron transport (Reactome/<ext-link xlink:href=\"https://reactome.org/\" ext-link-type=\"uri\">Reactome.org</ext-link>) (<bold>Additional File 1: Table S1</bold>). Overall, these results indicate that females with myocarditis have a stronger mitochondrial gene signature compared to males.</p>", "<title>Males with myocarditis have lower expression of electron transport chain genes compared to females</title>", "<p id=\"P26\">Based on our finding of sex differences in pathways related to mitochondrial respiration during myocarditis, we next focused our analysis on mitochondrial electron transport chain (ETC) genes (##FIG##7##Figure 8##). Using reads per kilobase per million (RPKM) from RNAseq results, we compared the expression of murine nuclear encoded ETC transcripts for each complex in the ETC. We found that 36 of 45 genes (80%) that form Complex I were significantly lower in males with myocarditis compared to females with myocarditis (##FIG##7##Figure 8a##,##FIG##7##b##), suggesting Complex I dysfunction in males with myocarditis. Significant differences in genes comparing males to females with myocarditis are indicated by asterisks (##FIG##7##Figure 8##). Males with myocarditis also had significant decreases in 3 genes out of 6 (50%) in Complex II (##FIG##7##Figure 8c##), 8 genes out of 11 (73%) in Complex III (##FIG##7##Figure 8d##), 14 genes out of 23 (61%) in Complex IV (##FIG##7##Figure 8e##), and 12 genes out of 18 (67%) in the ATP synthase compared to females with myocarditis (##FIG##7##Figure 8f##). These findings show that expression of nuclear encoded mitochondrial respiratory chain transcripts increase in females during myocarditis whereas they decrease in males.</p>", "<title>ERRa identified using TRANSFAC as a candidate transcription factor that may regulate mitochondrial genes</title>", "<p id=\"P27\">To identify transcriptional regulators that might globally affect the major changes in cellular mitochondrial energetic pathways according to sex that we observed, we used the gProfiler “<italic toggle=\"yes\">TRANScription FACtor database</italic>” (TRANSFAC) to identify candidate transcription factors. TRANSFAC analysis identified interferon regulatory factors (IRFs) and estrogen-related receptors (ERRs) as the top potential regulators of gene differences between males and females with myocarditis (##FIG##8##Figure 9a##). We compared expression of all nine IRFs (##FIG##8##Figure 9b##) and all three ERRs (##FIG##8##Figure 9c##) from RNA sequencing data and found that none of the nine IRFs were significantly different by sex but ERRα was significantly higher in females with myocarditis compared to males (FDR = 0.03).</p>", "<p id=\"P28\">We assessed the predicted binding capacity of ERRs among the nuclear encoded mitochondrial respiratory chain complexes examined in ##FIG##7##Figure 8## using TRANSFAC. ERRα and ERRγ shared a core predicted binding motif of TCAAGGTCA with ERRα present in the proximal promoter of around 30% of the nuclear encoded mitochondrial respiratory chain complex transcripts (##FIG##8##Figure 9d##). This is in line with a previous study that showed that both ERRa and ERRg target a common set of promoters involved in mitochondrial respiration and ATP production in the hearts of male mice [##REF##17488637##21##]. Indeed, respiratory chain complex genes that were predicted to be bound by ERRa (or ERRg) are indicated by green boxes and those that were significantly different by sex in ##FIG##7##Figure 8## are indicated with bold blue lettering in ##FIG##8##Figure 9e##. These findings suggest that ERRs may influence the sex differences in mitochondrial gene expression that were observed during myocarditis.</p>", "<title>Females upregulate master regulators of mitochondrial homeostasis during myocarditis</title>", "<p id=\"P29\">Because we found sex differences in the expression of mitochondrial ETC genes (##FIG##7##Figure 8##), we examined whether sex differences existed in global regulators of mitochondrial metabolism including PGC1α and nuclear respiratory factor 1 (NRF1). Using qRT-PCR, we found that PGC1a levels were significantly increased in the heart during myocarditis when males and females with myocarditis were combined compared to controls (<italic toggle=\"yes\">p</italic> &lt; 0.0001) (##FIG##9##Figure 10a##) or examined individually compared to controls (females <italic toggle=\"yes\">p</italic> &lt; 0.0001, males <italic toggle=\"yes\">p</italic> = 0.0003) (##FIG##9##Figure 10b##). Comparing males to females with myocarditis, females with myocarditis had significantly higher levels of PGC1a in the heart compared to males (<italic toggle=\"yes\">p</italic> = 0.0458) (##FIG##9##Figure 10b##).</p>", "<p id=\"P30\">PGC1α interacts with NRF1 leading to transcription of mitochondrial genes including ATP synthase, cytochrome-c, cytochrome-c-oxidase subunit IV, and mitochondrial transcription factor A which activates mitochondrial DNA replication and transcription [##REF##8816484##22##–##REF##12588810##24##]. We found that NRF1 RNA levels were significantly decreased when males and females with myocarditis were combined compared to controls (<italic toggle=\"yes\">p</italic> &lt; 0.0001) (##FIG##9##Figure 10c##). This was also observed when NRF1 levels were examined in males with myocarditis versus controls (<italic toggle=\"yes\">p</italic> &lt; 0.0001) or females with myocarditis versus controls (<italic toggle=\"yes\">p</italic> &lt; 0.0001) (##FIG##9##Figure 10d##). However, NRF1 levels in the heart during myocarditis were significantly higher in females compared to males (<italic toggle=\"yes\">p</italic> = 0.0315) (##FIG##9##Figure 10d##).</p>", "<p id=\"P31\">We also assessed ERRa (ESRRA mRNA) by sex, which had been identified using TRANSFAC (##FIG##8##Figure 9##). ERRa is designated as an orphan nuclear receptor [##UREF##2##25##–##REF##18778951##27##] but recent evidence suggests its endogenous ligand may be cholesterol [##REF##26777690##28##–##UREF##4##30##]. ERRα displays some basal activity during nominal cell states but transcriptional activity is enhanced by co-activator interaction with PGC1α [##REF##12588810##24##]. PGC1a is a transcriptional co-activator protein that binds ERRa (not as a ligand but as a co-factor [##REF##14530391##31##]) and promotes its transcriptional activity [##UREF##2##25##]. ERRa-PGC1a have been found to regulate hundreds of genes involved in mitochondrial oxidative phosphorylation, the tricarboxylic acid (TCA) cycle, fatty acid beta-oxidation, and glucose and lipid metabolism [##UREF##2##25##, ##REF##17229846##26##, ##REF##31024446##32##–##REF##26037200##35##]. When we examined ESRRA mRNA levels by qRT-PCR during myocarditis, we found that ESRRA was significantly decreased in mice with myocarditis compared to controls (<italic toggle=\"yes\">p</italic> &lt; 0.0001) (##FIG##10##Figure 11a##). This was also true when we examined females with myocarditis compared to controls (<italic toggle=\"yes\">p</italic> &lt; 0.0001) or males with myocarditis were compared to controls (<italic toggle=\"yes\">p</italic> &lt; 0.0001) (##FIG##10##Figure 11b##). Similar to PGC1a and NRF1, we found that females had significantly higher levels of ESRRA during myocarditis compared to males (<italic toggle=\"yes\">p</italic> = 0.0128) (##FIG##10##Figure 11b##).</p>", "<p id=\"P32\">To further investigate ERRa levels during myocarditis we examined heart protein levels of ERRα by ELISA. At the protein level, we found that ERRα was significantly increased when males and females with myocarditis were combined compared to controls (<italic toggle=\"yes\">p</italic> = 0.0486) (##FIG##10##Figure 11c##) and in all females compared to all males regardless of disease status (<italic toggle=\"yes\">p</italic> = 0.0094) (##FIG##10##Figure 11d##). To determine the effect of sex versus myocarditis in ERRα protein expression, we performed two-way ANOVA and found a significant effect of sex (<italic toggle=\"yes\">p</italic> = 0.006) and myocarditis (<italic toggle=\"yes\">p</italic> = 0.017), indicating sex differences drive the main effect (##FIG##10##Figure 11e##). We also found that ERRa protein levels were significantly increased in females with myocarditis compared to controls (<italic toggle=\"yes\">p</italic> = 0.0340) and in males with myocarditis compared to controls (<italic toggle=\"yes\">p</italic> = 0.0340) (##FIG##10##Figure 11e##). Importantly, ERRa protein levels were significantly increased in females with myocarditis compared to males with myocarditis (<italic toggle=\"yes\">p</italic> = 0.0234) (##FIG##10##Figure 11e##). Interestingly, ERRa protein was also significantly increased in females without myocarditis compared to males without myocarditis (<italic toggle=\"yes\">p</italic> = 0.0234) (##FIG##10##Figure 11e##). Thus overall, ERRα protein levels were elevated in the heart of females compared to males during myocarditis.</p>", "<p id=\"P33\">To examine cardiac expression of ERRα in the murine heart, we selected representative slides based on average inflammation scores and performed immunohistochemistry (IHC). We selected representative images at the base, mid, and apical myocardium for each heart section. We observed higher ERRa staining intensity in the female control hearts compared to male controls where staining appeared to be most concentrated in cardiomyocyte nuclei (##FIG##11##Figure 12##). Increased staining intensity was also found for ERRα in females compared to males with myocarditis (##FIG##11##Figure 12##). ERRα expression was observed for individual immune cells and inflammatory foci in males and females as well as cardiomyocyte nuclei during myocarditis (##FIG##11##Figure 12##).</p>", "<p id=\"P34\">To better understand function and disease relevance of ESRRA expression in the context of myocarditis, we performed two-tailed correlation analysis comparing ESRRA transcript levels and global longitudinal strain (GLS) obtained using echocardiography (##FIG##10##Figure 11f##) and ESRRA transcript levels and cardiac inflammation (scored histologically) (##FIG##10##Figure 11g##). We did not find a significant correlation when assessing ESRRA transcript levels to GLS when we included males and females with myocarditis together (<italic toggle=\"yes\">p</italic> = 0.077) but found a significant correlation in females (<italic toggle=\"yes\">p</italic> = 0.011) but not males (<italic toggle=\"yes\">p</italic> = 0.487) (##FIG##10##Figure 11f##). In contrast, ESRRA transcript levels negatively correlated with cardiac inflammation when including males and females in the comparison (<italic toggle=\"yes\">p</italic> = 0.008), but this effect was only significant in males (<italic toggle=\"yes\">p</italic> = 0.035) and not females (<italic toggle=\"yes\">p</italic> = 0.848) with myocarditis (##FIG##10##Figure 11g##). These findings indicate that ESRRA levels directly impact inflammation and cardiac function during myocarditis in a sex-specific manner.</p>" ]

[ "<title>Discussion</title>", "<p id=\"P35\">In this study we show for the first time that male mice with CVB3 myocarditis have reduced mitochondrial transcription compared to females using an autoimmune model of CVB3 myocarditis that is highly translational to human disease [##REF##34042389##36##]. We show that females with myocarditis have higher expression of several master regulators of mitochondrial homeostasis including PGC1a, NRF1 and ERRa compared to males. Females with CVB3 myocarditis had transcriptional evidence of better mitochondrial function and significantly less myocardial inflammation than males. A sex-specific effect of ERRa on inflammation and cardiac function suggests a potential regulatory mechanism for our observed sex differences in mitochondrial gene transcription.</p>", "<p id=\"P36\">PGC1a was originally identified as a regulator of mitochondrial function in brown adipose tissue, but was later also found to be expressed at high levels in cardiac tissue where it influences cardiovascular health and disease [##UREF##2##25##, ##REF##11018072##37##]. PGC1a globally regulates mitochondrial pathways in response to stresses such as cold, fasting and infection [##REF##26037200##35##, ##REF##25500872##38##–##REF##18664618##41##]. Thus, the metabolic stress of CVB3 infection is a likely explanation for the elevated levels of PGC1a that we observed in males and females with myocarditis compared to controls (##FIG##9##Fig. 10a##,##FIG##9##b##). Additionally, a study by Dufour et al. using ERRα deficient mice, found that when ERRa was low in the heart PGC1a was elevated as a compensation mechanism [##REF##17488637##21##].</p>", "<p id=\"P37\">ERRa has been found to be critical in regulating mitochondrial homeostasis in the heart demonstrated by Dufour et al. using male ERRa deficient (KO) C57BL/6 mice [##REF##17488637##21##]. They found that ERRa targeted mitochondrial NRF1, cyclic AMP-response element binding protein (CREB), and STAT3 [##REF##37620559##42##]. Surprisingly, we observed an inverse relationship between mRNA and protein levels of ERRα in the heart with increased levels by ELISA and IHC in males and females with myocarditis. Regardless of mRNA levels, ERRα protein levels can be highly regulated by post-translational modifications and metabolic stress [##REF##32079653##43##]. Previously, it was notably shown that under specific cellular metabolic stress conditions, such as reactive oxygen species (ROS) exposure, ERRα protein levels can be dramatically altered in a proteasome-dependent manner [##REF##32079653##43##]. Similarly, insulin or glucose stimulation increased ERRα protein levels without altering mRNA expression in hepatocytes [##REF##35440636##44##], further strengthening the hypothesis that ERRα protein levels can be altered under metabolic pressure independently of gene expression. Indeed, transcription factor expression often does not necessarily provide detailed information as to the direct actions of that transcription factor; and in this case, how ERRα activity may differ by sex during myocarditis. Future studies utilizing methods to characterize the genomic interactions of ERRα would be useful in elucidating sex-specific transcriptional activity.</p>", "<p id=\"P38\">Interleukin (IL)-1a, IL-1b, and tumor necrosis factor (TNF)a are known to activate the transcriptional activity of PGC1a through direct phosphorylation of p38 mitogen-activated protein (MAP) kinase [##REF##12588810##24##, ##REF##11741533##45##]. We found previously that IL-1b levels are increased in the heart of males with myocarditis, while cardiac levels of TNFa are increased in females in our CVB3 model of myocarditis [##REF##22328081##10##, ##REF##12707353##46##]. Most cardiac inflammatory cells during acute myocarditis at day 10 pi are CD11b+ (macrophages and mast cells) that express TLR4 and release IL-1b [##REF##17513715##16##]. We showed previously that elevated IL-1b levels in the heart directly correlate to elevated cardiac inflammation in males with myocarditis and poor cardiac function by echocardiography [##REF##22328081##10##]. Importantly, Remels et al. showed that elevated TNFa levels in cardiomyocytes in culture following CVB3 infection were directly associated with decreased PGC1a mRNA levels [##REF##29730342##47##].</p>", "<p id=\"P39\">Additional evidence of the negative effect that IL-1b can have on mitochondrial gene expression in males was found in studies by Ge et al. [##REF##37660839##19##, ##REF##34365571##48##, ##REF##35997820##49##]. Calpain is a calcium-dependent protease that facilitates apoptotic signaling and localizes to the mitochondria during CVB3 infection to proteolyze mitochondrial substrates, leading to increased mitochondrial fission (mitochondrial fragmentation due to pathological or physiological stress). Inhibition of calpain reduced mitochondrial fission and cardiomyocyte apoptosis during myocarditis [##REF##34365571##48##]. Liu et al. showed that mitochondrial calpain-1 induces mitochondrial dysfunction and ROS production which activated the NLRP3 inflammasome, which leads to IL-1b production [##REF##35997820##49##]. Macrophages, which are the predominant infiltrating immune cells during myocarditis, were found to respond to CVB3 infection by upregulation of calpain-4; RNA sequencing of CVB3 infected macrophages in vitro revealed predominant enrichment for pathways related to macrophage maturation and interleukin signaling, and loss of calpain-4 reduced IL-1b expression [##REF##37660839##19##]. Although we did not specifically examine IL-1b in this study, our previous findings may be relevant to the current results that suggest that elevated inflammatory cells and cytokines, especially IL-1b, in the heart of males during acute CVB3 myocarditis [##REF##22328081##10##] may directly contribute to lower PGC1a levels in males than females leading to decreased mitochondrial gene expression in the heart at that timepoint.</p>", "<p id=\"P40\">In general, sex differences are known to exist in mitochondrial bioenergetics [##UREF##5##50##, ##REF##29513564##51##], but we provide sex-specific information in the context of viral myocarditis. Similar to previous studies that examined gene changes in the heart during CVB3 myocarditis in male mice [##REF##22328081##10##, ##REF##29730342##47##, ##REF##21127995##52##], we found that the predominant gene expression changes aside from immune pathways were mitochondrial genes. Previously, Remels et al. reported that PGC1a mRNA and NRF1 protein levels were significantly decreased in the heart of male mice with CVB3 myocarditis compared to controls from day 4 to 7 pi [##REF##29730342##47##]. They also found decreased gene expression profiles for ETC genes during myocarditis in males [##REF##29730342##47##], similar to our results, but they did not examine females with myocarditis.</p>", "<p id=\"P41\">Ebermann et al. also examined gene expression in males with CVB3 myocarditis comparing C57BL/6 (B6) to A.SW/SnJ mice [##REF##21968812##53##]. They used a tissue culture CVB3-induced model that produces similar inflammation in these two strains of mice but different cytokine profiles [##REF##21968812##53##]. This tissue-culture CVB3 model produces a completely different myocardial immune profile than our model of autoimmune CVB3-myocarditis comparing BALB/c to B6 mice [##REF##15579433##9##, ##UREF##6##54##, ##REF##22488075##55##]. However, Ebermann et al. found that A.Sw/SnJ male mice with myocarditis have significantly lower ETC gene expression compared to controls that was directly related to the level of viral replication in the heart [##REF##21968812##53##]. In our model of CVB3 myocarditis there are no sex differences in VP1 RNA levels or viral replication based on plaque assay during acute myocarditis [##REF##17513715##16##]. The findings of Ebermann et al. may reflect, however, the finding of Sin et al. who showed in cultured cardiomyocytes that CVB3 localizes to mitochondria, induces mitophagy, and disseminates from the cell in an extracellular autophagosome-bound virus-laden mitochondrial complex [##UREF##7##56##]. Sin et al. showed that upstream suppression of the mitophagy pathway in HL-1 cardiomyocytes using small interfering RNA (siRNA) targeted to dynamin-related protien-1 (DRP1) or mitochondrial division inhibitor (Mdivi-1) significantly reduced virus production from cardiomyocytes [##UREF##7##56##] (as mitochondrial fission is an early stage of mitophagy). Other viruses that cause myocarditis such as human immunodeficiency virus (HIV), hepatitis B and C, influenza, Epstein-Barr virus and SARS-CoV-2 have been found to localize to mitochondria and hijack aspects of the mitochondrial machinery for replicationn [##REF##37167363##5##, ##REF##34599743##57##–##UREF##8##59##]. This might explain why so many diverse viruses without specific tropism for cardiac tissue (i.e., murine cytomegalovirus/ MCMV, SARS-CoV-2, CVB3) are able to cause myocarditis, since they can target a mitochondria-rich environment for replicatory advantage.</p>", "<p id=\"P42\">TRANSFAC analysis identified IRFs and ERRs as key transcription factors that could mediate sex differences in gene expression in our model. CVB3 infection strongly activates type I interferons (IFNas and IFNb) and type II (IFNg) IFN production during myocarditis to reduce viral replication via Toll-like receptor (TLR) activation including TLR3, TLR4, TLR7 and TLR9 and the transcription factor TIR domain-containing adaptor inducing interferon-β (TRIF) which is downstream of TLR3 and TLR4 [##REF##22328081##10##, ##REF##12707353##46##, ##REF##21338512##60##–##REF##23255589##62##]. Although IFNg is increased in our model of CVB3 myocarditis in males [##REF##17513715##16##, ##REF##15611248##18##], we showed that elevated IFN levels in male BALB/c mice with myocarditis are mediated by IL-18, which is downstream from TLR4, rather than traditional STAT4/IL-12 transcriptional activity [##REF##15611248##18##, ##REF##16949558##63##]. We do not observe a sex difference in viral levels in the heart during myocarditis in our CVB3 mouse model and sex differences in IFNg are not mediated by classic IFN signaling. Therefore, is not surprising that we did not observe a significant difference by sex of the nine IFN transcription factors (##FIG##4##Fig. 5b##).</p>", "<p id=\"P43\">Sex hormones are known to strongly drive the innate and adaptive immune response to infections in general and during myocarditis [##REF##36937911##8##, ##UREF##9##64##, ##REF##23158412##65##], and to confer sex differences in mitochondrial morphology and function via estrogen receptor (ER) nuclear and mitochondrial transcription factor activity [##REF##32197947##39##, ##REF##28424375##66##, ##REF##31474941##67##]. The heart of females is known to have greater mitochondrial efficiency, fatty acid utilization during exercise, and calcium retention whereas males have more mitochondrial content, reactive oxygen species production, and higher calcium uptake rate for example [##REF##32197947##39##, ##REF##28424375##66##]. A summary of these known sex differences in mitochondrial-related genes and pathways can be found in <bold>Additional File 1: Table S2.</bold> ERRα was originally named based on its sequence homology to ERα [##REF##3267207##68##]. Although sex differences in some mitochondrial gene expression pathways during CVB3 myocarditis may be explained by sex hormones, specifically estrogen via ERs, 17b-estradiol and other natural estrogens are not endogenous ligands for ERRα [##REF##3267207##68##, ##REF##8621448##69##]. Two groups have provided evidence that support the hypothesis that cholesterol is the endogenous ligand for ERRα with <italic toggle=\"yes\">in vitro</italic> and <italic toggle=\"yes\">in vivo</italic> data [##REF##26777690##28##–##UREF##4##30##]. During nominal cellular states and unbound by its ligand, ERRα displays some transcriptional activity [##UREF##2##25##, ##REF##31024446##32##, ##REF##17053040##70##]. Based on structural homology, ERRs are speculated to share target genes, coregulatory proteins, and sites of action with ERs and therefore actively influence the estrogenic response [##REF##12185669##71##]. The genotype-tissue expression (GTEx) project identified ERRα as a “sex-biased” transcriptional regulator in humans [##UREF##10##72##]. Overall, this could explain sex differences in ERRa expression.</p>", "<p id=\"P44\">Lee et al. found sex differences in ERRa levels in the brains of 4-week-old immature female but not male mice that had been treated with a chemical known to reduce mitochondrial function [##REF##35879031##73##], suggesting sex differences in ERRa function prior to the production of circulating hormone production. De Jesus-Cortez et al. found that ERRa deficient adult female mice had defects in neural function in a mouse model of eating disorders, which mainly affect women, which was not observed in ERRa deficient male mice, and they concluded that ERRa was required for optimal mitochondrial function in females [##REF##27155145##74##]. Watson et al. found sex-specific effects of ERRa expression in the hearts of female but not male mice in a model of heart failure [##REF##20935148##75##]. We are the first to report sex differences in ERRα expression in the hearts of healthy mice and mice with viral myocarditis. Subsequent studies are needed to further characterize sex-specific effects of ERRα on mitochondrial function during CVB3 myocarditis. However, to fully characterize the sex differences in ERRa effects on gene regulation in healthy and mice with myocarditis, an analysis of gene-specific transcription factor (TF)-DNA interaction of ERRa is needed using chromatin immunoprecipitation (ChIP) or similar methods.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"P45\">In this study we show for the first time that males with CVB3 myocarditis have reduced mitochondrial gene expression of nuclear-encoded electron transport chain genes compared to females. Females had higher levels of global regulators of mitochondrial function compared to males which may promote mitochondrial homeostasis that protects females from cardiac damage following infection and inflammation. Future studies should characterize the direct transcriptional activity of the sex-differentially expressed orphan nuclear receptor ERRα.</p>" ]

[ "<p id=\"P1\">Contributions</p>", "<p id=\"P2\">Conceptualization: DND, KAB, DF</p>", "<p id=\"P3\">Data Acquisition: DND, KAB, DF</p>", "<p id=\"P4\">Sample Acquisition: DND, KAB, DF</p>", "<p id=\"P5\">Methodology: DND, DJG, EJM, IC, KAS, AJ, KAB, GRS, DJB, PGG, SK, NEB-H, ERW, VB, GJW, JAF, SCK, CJM, EA-W, MJC, JS, DF</p>", "<p id=\"P6\">Data Analysis: DND, DJG, EJM, IC, KAS, AJ, KAB, GRS, DJB, PGG, SK, NEB-H, ERW, VB, GJW, SCK, JAF, CJM, EA-W, MJC, JS, DF</p>", "<p id=\"P7\">Data Curation: DND</p>", "<p id=\"P8\">Project Administration: DF</p>", "<p id=\"P9\">Writing original draft: DND, DF</p>", "<title>Background</title>", "<p id=\"P10\">Myocarditis is an inflammation of the heart muscle most often caused by an immune response to viral infections. Sex differences in the immune response during myocarditis have been well described but upstream mechanisms in the heart that might influence sex differences in disease are not completely understood.</p>", "<title>Methods</title>", "<p id=\"P11\">Male and female BALB/c wild type mice received an intraperitoneal injection of heart-passaged coxsackievirus B3 (CVB3) or vehicle control. Bulk-tissue RNA-sequencing was conducted to better understand sex differences in CVB3 myocarditis. We performed enrichment analysis to understand sex differences in the transcriptional landscape of myocarditis and identify candidate transcription factors that might drive sex differences in myocarditis.</p>", "<title>Results</title>", "<p id=\"P12\">The hearts of male and female mice with myocarditis were significantly enriched for pathways related to an innate and adaptive immune response compared to uninfected controls. When comparing females to males with myocarditis, males were enriched for inflammatory pathways and gene changes that suggested worse mitochondrial transcriptional support (e.g., mitochondrial electron transport genes). In contrast, females were enriched for pathways related to mitochondrial respiration and bioenergetics, which were confirmed by higher transcript levels of master regulators of mitochondrial function including peroxisome proliferator-activated receptor gamma coactivator 1 (PGC1α), nuclear respiratory factor 1 (NRF1) and estrogen-related receptor alpha (ERRα). TRANSFAC analysis identified ERRa as a transcription factor that may mediate sex differences in mitochondrial function during myocarditis.</p>", "<title>Conclusions</title>", "<p id=\"P13\">Master regulators of mitochondrial function were elevated in females with myocarditis compared to males and may promote sex differences in mitochondrial respiratory transcript expression during viral myocarditis resulting in less severe myocarditis in females following viral infection.</p>", "<title>Plain English Summary</title>", "<p id=\"P14\">Many viruses can infect the heart. Immune cells go to the heart to get rid of the virus but sometimes this response can damage the heart. When immune cells go to the heart it is called “myocarditis,” which means ‘myo’ for muscle and ‘carditis’ for heart inflammation. More men get myocarditis after viral infections than women. When we study viral infection in mice as a model of human disease, we see the same sex differences as in humans. To know why this happens, we gave mice a viral infection and looked at how their hearts changed. We found that a special gene acts like a like a <italic toggle=\"yes\">light switch</italic> to turn on the genes that help cell energy factories, called “mitochondria.” Females used this light switch to turn on the genes that help mitochondria, but males did not and turned most of these genes off. We learned that the light switch works better in females compared to males, so it may play an important role in protecting female’s hearts during myocarditis. More researched is needed to better understand how this light switch works. Understanding how to turn on mitochondria genes in the heart could help doctors also do this in men after a viral infection to prevent myocarditis and save patients’ lives.</p>" ]

[]

[ "<title>Acknowledgements</title>", "<p id=\"P58\">We thank the Dickson Histology Group for their work embedding and staining slides for this project. This group includes Dr. Dennis W. Dickson, Linda Rousseau, Virginia Phillips, Ariston Libraro, and Monica Castanedes. We thank Dr. Laura Lewis-Tuffin and the Mayo Microscopy and Cell Analysis Core for experimental and technical support. We thank the Mayo Clinic Genome Analysis core for help with experimental design, library preparation, sequencing, and initial differential expression analysis. We thank Aishe Kurti for consultation with design for Cytoscape enrichment and heatmap displays of GSEA data.</p>", "<title>Funding</title>", "<p id=\"P59\">This work is partially supported by the, National Institutes of Health (NIH) TL1 TR002380 (DND, AJ, DJB, ERW, DF), National Institute of Allergy and Infectious Diseases under award numbers R21 AI152318 (DF), R21 AI145356 (DF), R21 AI154927 (DF), R21 AI163302 (KAB), National Heart Lung and Blood Institute under award number R01 HL164520 (DF), National Institute of Diabetes and Digestive and Kidney Diseases under award number R01 DK125692 (JS), the American Heart Association under award number 20TPA35490415 (DF), the American Heart Association under award number 23SCEFIA1153413 (KAB), the American Heart Association under award number 19CDA34770083 (JS), and the Mayo Clinic Center for Regenerative Medicine in Florida (DF). The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding agencies.</p>", "<title>Availability of Data and Materials</title>", "<p id=\"P60\">The data that support the findings of this study are available from the corresponding author upon reasonable request.</p>", "<p id=\"P61\">Data will be uploaded to an approved repository upon acceptance to journal.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><title>Myocardial inflammation is increased in males compared to females</title><p id=\"P72\"><bold>a,</bold> Myocarditis severity (% inflammation) between female controls (F-CON, <italic toggle=\"yes\">n</italic>= 27), females with myocarditis (F-MYO, <italic toggle=\"yes\">n</italic> = 41), male controls (M-CON, <italic toggle=\"yes\">n</italic>= 30), and males with myocarditis (M-MYO, <italic toggle=\"yes\">n</italic> = 40); <bold>b,</bold> representative heart sections (scale bars = 80μm); <bold>c-h,</bold> relative gene expression (RGE) for controls (CON, <italic toggle=\"yes\">n</italic> = 31–34) versus mice with myocarditis (MYO, <italic toggle=\"yes\">n</italic> = 32–41) and for F-CON (<italic toggle=\"yes\">n</italic> = 15–16), F-MYO (<italic toggle=\"yes\">n</italic> = 15–21), M-CON (<italic toggle=\"yes\">n</italic> = 18), and M-MYO (<italic toggle=\"yes\">n</italic> = 15–20).</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><title>RNA sequencing reveals sex differences in immune and mitochondrial genes</title><p id=\"P73\"><bold>a,</bold> RNA-sequencing experimental pipeline; <bold>b,</bold> principal component analysis plot showing female controls (F-CON), females with myocarditis (F-MYO), male controls (M-CON), and males with myocarditis (M-MYO); <bold>c,</bold> results from GSEA pre-ranked plotted on Cytoscape with Enrichment Map and AutoAnnotate, pink = F-MYO and blue = M-MYO, nodes circled in black are mitochondrial-related pathways; heat map for <bold>d,</bold> the top 273 most differentially expressed genes and <bold>e,</bold> mitochondrial genes between all four groups.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3</label><caption><title>Sex-specific gene expression pathways during myocarditis</title><p id=\"P74\">The top ten most enriched pathways from GSEA ranked by normalized enrichment score comparing <bold>a,</bold> female controls (F-CON) to females with myocarditis (F-MYO), <bold>b,</bold> male controls (M-CON) to males with myocarditis (M-MYO) (pathway text marked with astricts indicate abbreviated pathway names, see supplement for abbreviations and full pathway names) <bold>c,</bold> Row normalized heatmaps for the most enriched pathways from F-CON vs F-MYO and M-CON vs M-MYO, respective to colors highlighting pathways in (a) and (b). NES = normalized enrichment score, Extracellular Matrix Structural (Struct.) Constituent (Const.). *<italic toggle=\"yes\">FDRq</italic>&lt;0.05,**<italic toggle=\"yes\">FDRq</italic>&lt;0.01, *** <italic toggle=\"yes\">FDRq</italic>&lt;0.001, **** <italic toggle=\"yes\">FDRq</italic>&lt;0.00001</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4</label><caption><title>Sex differences in differential gene expression pathways during myocarditis</title><p id=\"P75\">The top ten most enriched pathways from GSEA ranked by normalized enrichment score comparing <bold>a,</bold> females with myocarditis (F-MYO) and males with myocarditis (M-MYO) and heatmap of auto-annotated cluster of pathways/nodes describing the mitochondrial respiratory complex, <bold>b,</bold> Row normalized heatmaps for pathways highlighted in pink for F-MYO (from F-MYO vs M-MYO comparison; other 2 pathways share common genes with those in (b) and are available in Supplemental Figure 7). *<italic toggle=\"yes\">FDRq</italic>&lt;0.05, ** <italic toggle=\"yes\">FDRq</italic>&lt;0.01, *** <italic toggle=\"yes\">FDRq</italic>&lt;0.001, **** <italic toggle=\"yes\">FDRq</italic>&lt;0.00001</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5</label><caption><title>Females with myocarditis are enriched for pathways related to mitochondrial respiration compared to males with myocarditis</title><p id=\"P76\">Metascape enrichment results for females with myocarditis (comparing females and males with myocarditis) <bold>a,</bold> the top enriched pathways colored by cluster/pathway and <bold>b,</bold> by p-value. Top enriched pathways <bold>c,</bold> Protein-protein interaction analysis clustered by interaction outside of pathways (i) and interaction inside pathways (ii). Log10(P) vals are derived by averaging the Log10(P) vals for the 3 MCODE annotations, rounded to whole number with colors indicating respective pathways. Images in (i) are cropped to show the bulk of pathways and interactors and the top 3 pathways only are shown in (ii).</p></caption></fig>", "<fig position=\"float\" id=\"F6\"><label>Figure 6</label><caption><title>Males with myocarditis are enriched for pathways related immune activation compared to females with myocarditis</title><p id=\"P77\">Metascape enrichment results for males with myocarditis (comparing females and males with myocarditis) <bold>a,</bold> the top enriched pathways colored by cluster/pathway and <bold>b,</bold> by p-value. Top enriched pathways <bold>c,</bold> Protein-protein interaction analysis clustered by interaction outside of pathways (i) and interaction inside pathways (ii). Log10(P) vals are derived by averaging the Log10(P) vals for the 3 MCODE annotations, rounded to whole number with colors indicating respective pathways. Images in (i) are cropped to show the bulk of pathways and interactors and the top 3 pathways only are shown in (ii).</p></caption></fig>", "<fig position=\"float\" id=\"F7\"><label>Figure 7</label><caption><title>Metascape Enrichment QC shows cell specific signals and identifies potential transcriptional regulators of sex differences in myocarditis</title><p id=\"P78\"><bold>a,</bold> Enrichment quality control (QC) for males with myocarditis (M-MYO) shows cell specific enrichment signals from Pattern Gene Database (PaGenBase) and suggested transcription factors from Transcriptional Regulatory Relationships Unraveled by Sentence-based Text mining (TRRUST) and for <bold>b,</bold> females with myocarditis (F-MYO)</p></caption></fig>", "<fig position=\"float\" id=\"F8\"><label>Figure 8</label><caption><title>Sex differences in mitochondrial electron transport genes during myocarditis</title><p id=\"P79\">Row normalized RPKM comparing female controls (FC), females with myocarditis (FM), male controls (MC) and males with myocarditis (MM) for nuclear encoded genes for <bold>a,</bold> complex I, <bold>b,</bold> color-coded illustration of the mitochondrial electron transport chain; <bold>c,</bold> complex II, <bold>d,</bold> complex III, <bold>e,</bold> complex IV, and <bold>f,</bold> ATP synthase. *<italic toggle=\"yes\">p</italic>&lt;0.05, ** <italic toggle=\"yes\">p</italic>&lt;0.01, *** <italic toggle=\"yes\">p</italic>&lt;0.001,****<italic toggle=\"yes\">p</italic>&lt;0.0001</p></caption></fig>", "<fig position=\"float\" id=\"F9\"><label>Figure 9</label><caption><title>TRANSFAC analysis identifies interferon regulatory factors and estrogen-related receptors as potential mediators of sex difference during myocarditis</title><p id=\"P80\"><bold>a,</bold> TRANSFAC results comparing females with myocarditis (F-MYO) and males with myocarditis (M-MYO); RPKM (reads per kilobase per million) using false discovery rate to compare F-MYO and M-MYO for <bold>b,</bold> interferon regulatory factors (IRFs) and <bold>c,</bold> estrogen-related receptors (ERRs); <bold>d,</bold> predicted binding capacity of ERRs for electron transport chain transcripts; <bold>e,</bold> significantly different transcripts by sex in electron transport genes are indicated by bold blue lettering, green boxes indicate genes that ERRa predicted to bind to.</p></caption></fig>", "<fig position=\"float\" id=\"F10\"><label>Figure 10</label><caption><title>Females with myocarditis express higher levels of mitochondrial master regulators PGC1a and NRF1</title><p id=\"P81\">Relative gene expression (RGE) for controls (CON, <italic toggle=\"yes\">n</italic> = 33–35) versus mice with myocarditis (MYO, <italic toggle=\"yes\">n</italic> = 35–39) and for F-CON (<italic toggle=\"yes\">n</italic> = 15), F-MYO (<italic toggle=\"yes\">n</italic> = 20–21), M-CON (<italic toggle=\"yes\">n</italic> = 17–18), and M-MYO <italic toggle=\"yes\">n</italic> = 19–20) for <bold>a-b,</bold> PGC1a; and <bold>c,d,</bold> NRF1.</p></caption></fig>", "<fig position=\"float\" id=\"F11\"><label>Figure 11</label><caption><title>Females with myocarditis express higher levels of ERRαcompared to males</title><p id=\"P82\">Relative gene expression (RGE) for controls (CON, <italic toggle=\"yes\">n</italic> = 33–35) versus mice with myocarditis (MYO, <italic toggle=\"yes\">n</italic> = 35–39) and for F-CON (<italic toggle=\"yes\">n</italic> = 15), F-MYO (<italic toggle=\"yes\">n</italic> = 20–21), M-CON (<italic toggle=\"yes\">n</italic> = 17–18), and M-MYO <italic toggle=\"yes\">n</italic> = 19–20) for <bold>a,b,</bold> ERRa ELISA of ERRa protein from whole heart homogenate supernatant comparing <bold>c,</bold> CON (<italic toggle=\"yes\">n</italic> = 14) to MYO (n= 42); <bold>d,</bold> all females (F, <italic toggle=\"yes\">n</italic> = 27) to all males (M, <italic toggle=\"yes\">n</italic> = 29) regardless of disease state; <bold>e,</bold> two-way ANOVA F-CON (<italic toggle=\"yes\">n</italic> = 7), F-MYO (<italic toggle=\"yes\">n</italic> = 20), M-CON (<italic toggle=\"yes\">n</italic> = 7), and M-MYO (<italic toggle=\"yes\">n</italic> = 22). Two-tailed assessment of correlations for, <bold>f</bold> ERRα gene expression and global longitudinal strain (GLS (%)), and <bold>g</bold> for ERRα gene expression and myocarditis severity (Inflammation %) scored by H+E stain.</p></caption></fig>", "<fig position=\"float\" id=\"F12\"><label>Figure 12</label><caption><title>In situ expression of ERRα demonstrates sex differences</title><p id=\"P83\">Representative heart sections (based on H&amp;E scores) for F-CON, M-CON, F-MYO, and M-MYO stained for ERRα. Images of the myocardium were taken at the base, middle (mid) and apex of the heart for each sample. Sale bars = 70 μm.</p></caption></fig>" ]

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[ "<boxed-text id=\"BX1\" position=\"float\"><caption><title>Highlights</title></caption><list list-type=\"bullet\" id=\"L1\"><list-item><p id=\"P84\">Viral myocarditis is more severe in males than females.</p></list-item><list-item><p id=\"P85\">ERRa and mitochondrial gene expression increased in females with myocarditis.</p></list-item><list-item><p id=\"P86\">ESRRA expression improves cardiac function in females with myocarditis.</p></list-item><list-item><p id=\"P87\">Low ESRRA is associated with increased myocarditis in males.</p></list-item></list></boxed-text>" ]

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[ "<fn-group><fn id=\"FN2\" fn-type=\"COI-statement\"><p id=\"P62\">Competing Interests</p><p id=\"P63\">The authors declare that they have no competing interests.</p></fn><fn id=\"FN3\"><p id=\"P64\">Ethics Approval and Consent to participate</p><p id=\"P65\">No humans were researched as a part of this study.</p></fn><fn id=\"FN4\"><p id=\"P66\">Animal Care and Ethics Statement</p><p id=\"P67\">Mice were used in accordance with the recommendations in the Guide for the Care and Use the Laboratory Animals of the National Institutes of Health (NIH) and approval obtained from the Animal Care and Use Committee at Mayo Clinic Florida for all procedures (IACUC# A00003984). Mice were maintained under pathogen-free conditions in the animal facility at Mayo Clinic Florida and mice were sacrificed according to the Guide for the Care and Use of Laboratory Animals of the NIH.</p></fn><fn id=\"FN5\"><p id=\"P68\">Consent for Publication</p><p id=\"P69\">Not applicable. We consent to publish the article.</p></fn><fn id=\"FN6\"><p id=\"P70\">Rights and permissions</p><p id=\"P71\">Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third-party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit <ext-link xlink:href=\"http://creativecommons.org/licenses/by/4.0/\" ext-link-type=\"uri\">http://creativecommons.org/licenses/by/4.0/</ext-link>. The Creative Commons Public Domain Dedication waiver (<ext-link xlink:href=\"http://creativecommons.org/publicdomain/zero/1.0/\" ext-link-type=\"uri\">http://creativecommons.org/publicdomain/zero/1.0/</ext-link>) applies to the data made available in this article, unless otherwise stated in a credit line to the data.</p></fn></fn-group>" ]

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[ "<title>Background</title>", "<p id=\"P8\">Medical students are essential to the development of a socially accountable medical school (##UREF##0##1##, ##UREF##1##2##). Social accountability for medical schools is defined as the obligation of the medical school to direct education, research, and service activities toward the most important needs of the community served by the medical school and its graduates (##UREF##2##3##). While medical students are expected to learn about social accountability and become socially accountable practitioners, little has been documented about their perceptions and experiences of social accountability.</p>", "<p id=\"P9\">Medical students have distinct and unique experiences during training, which may influence their future practice and choices following graduation (##UREF##3##4##, ##REF##25072172##5##). It is thus important to understand these experiences to improve the teaching and learning of social accountability. Previous studies have shown that medical students have a limited understanding of social accountability (##REF##25072172##5##–##UREF##4##7##). A survey of Deans in Korea reported that interactions between partners had the greatest influence on social accountability in medical education (##REF##35313788##8##). Medical students are thus key partners for social accountability; therefore, understanding their perceptions and experiences could be a starting point for improving their learning and adoption of social accountability (##UREF##0##1##).</p>", "<p id=\"P10\">The Students’ Toolkit on Social Accountability of Medical Schools was developed through a collaboration between the International Federation of Medical Students Associations (IFMSA) and the Training for Health Equity Network (THEnet) (##UREF##3##4##). The toolkit aims to provide a brief introduction to social accountability for medical students and empower them to make a difference in their schools in the area of social accountability. The toolkit is an evaluation tool for students to assess the progress made by their medical school in terms of social accountability and to create action plans to improve social accountability at the medical school (##UREF##0##1##). This kit has been applied in various settings, especially in high-income countries (##REF##36859255##9##, ##UREF##5##10##).</p>", "<p id=\"P11\">The perceptions and experiences of medical students regarding social accountability are unique because of contextual differences that may influence their learning and adoption of social accountability (##REF##32426393##11##, ##REF##21774646##12##). There is a dearth of published literature about these perceptions and experiences, particularly from sub-Saharan Africa. The purpose of this study therefore was to determine the perceptions and experiences of medical students regarding social accountability at the Makerere University School of Medicine.</p>" ]

[ "<title>Methods</title>", "<title>Study design and questionnaire:</title>", "<p id=\"P12\">A cross-sectional online questionnaire-based survey was conducted between September 2022 and October 2023 using Google Forms. This study collected responses from medical students from the Makerere University School of Medicine. The validated Students’ Toolkit on Social Accountability in Medical Schools, which has been applied widely to study perceptions of social accountability, was used for this study, with some modifications. The final study questionnaire had three sections: demographic information, perceptions, and experiences related to social accountability; and the evaluation section from the toolkit. The questionnaire was pretested on eight undergraduate medical students in their fourth or fifth year of study. The link to the questionnaire on Google Forms was sent to each student’s email. Each link was unique to each email address to avoid reuse and double enrollment.</p>", "<title>Study setting</title>", "<p id=\"P13\">The study was conducted at the Makerere University Medical School in Uganda. Makerere University is a government-owned university, and the Makerere University School of Medicine is the oldest medical school in East Africa. The Bachelor of Medicine and Bachelor of Surgery (MBCHB) program spans five years. Medical student recruitment is performed at the national level and is based mainly on academic performance (##REF##30873807##13##). Following graduation, students undertake one year of supervised internship. After successful completion of the internship period, graduates can obtain full registration status with the Uganda Medical and Dental Practitioners Council and thereafter practice as medical doctors. At this point, the doctor may decide to continue as a general practitioner or take on further postgraduate training in a field of choice, such as Family Medicine, Obstetrics and Gynecology, Internal Medicine, General Surgery or other fields.</p>", "<title>Characteristics of participants</title>", "<p id=\"P14\">All students in their fourth or fifth year of medical school were invited to participate in the study through trained research assistants. Fourth- and fifth-year students were selected because they were more likely to have had experiences related to social accountability in medical school. The research assistants were fourth- and fifth-year students who were trained in the study procedures. The research assistants obtained written informed consent from the participants and registered their email addresses. The links to the electronic survey were sent to the registered email addresses. Students who did not initially respond to the survey were reminded by the research assistant. The link was made available in two academic years, allowing two sets of fourth-year students to respond and one set of fifth-year students since these had been enrolled in the previous year.</p>", "<title>Statistical analysis</title>", "<p id=\"P15\">The data collected in Google Forms were exported to Microsoft Excel, checked for completeness and missing data, and then cleaned. The data were analyzed with R statistical software. The descriptive statistics are reported. The total score for the students’ toolkit 12 social accountability evaluation items was computed, and the means and standard deviations for the individual items are presented. The total scores were categorized according to the key provided in the student toolkit. The categories included 0–8, weak foundation; 9–17, some evidence of social accountability; 18–26, identify areas of improvement; and 27–36, strong foundation. The categories were divided into limited social accountability (weak and some social accountability) and strong social accountability (looking for areas of improvement and strong social accountability), and regression analysis was conducted to determine the associated factors. The chi-square test and Fisher’s exact test were used. A p value &lt; 0.05 indicated statistical significance.</p>" ]

[ "<title>Results</title>", "<p id=\"P16\">This study involved 426 medical students in their fourth or fifth year of study at the Makerere University School of Medicine. The mean age of the students was 25.24 ± 4.4 years. Most of the participants were in their fourth year of study, as the link was available over two academic years. The characteristics of the study participants are summarized in ##TAB##0##Table 1##.</p>", "<title>Student perceptions and experiences of social accountability</title>", "<p id=\"P17\">When asked if they had ever heard about social accountability, 165 (38.73%) students responded ‘yes’. However, only 6 (3.64%) of those who reported having heard about social accountability provided an accurate definition of the term. Of the 165 (39%) participants who reported hearing about social accountability before, 91 (55%) encountered the term in personal reading. The average time spent in community-based education research and service was 6.8 weeks, and most (40.14%) of the students reported feeling moderately prepared for their last COBERS experience. (##FIG##0##Fig. 1##)</p>", "<title>Students’ evaluation of social accountability at medical schools</title>", "<p id=\"P18\">Using the Students’ toolkit for social accountability in medical schools, 48.12% of the medical students evaluated the medical school as having good social accountability, with a total score between 18 and 26. In contrast, 1.41% of the students felt that the medical school had a weak foundation for social accountability.</p>", "<p id=\"P19\">Of the twelve items used to assess social accountability, seven items (1, 2, 3, 7, 8, 11 and 12) were related to perceptions, while five items (4, 5, 6, 9 and 10) were related to experiences. Item 10 (Does your school have community-based research?) 2.57 ± 0.62 and Item 12 (Does your school have a positive impact on the community?) 2.434 ± 0.67 had the highest mean scores. Item 4 (Do you learn about other cultures?) 1.37 ± 0.93 and Item 8 (Do your teachers reflect the sociodemographic characteristics of the reference population?) 1.60 ± 0.92 had the lowest mean scores. (##TAB##1##Table 2##)</p>", "<p id=\"P20\">The only factor that was significantly associated with evaluation of social accountability in medical school as strong was receiving career guidance in secondary school (p 0.003). (##TAB##2##Table 3##)</p>" ]

[ "<title>Discussion</title>", "<p id=\"P21\">We evaluated the perceptions and experiences of medical students at the Makerere University School of Medicine regarding social accountability. Our study findings suggest that medical students at the Makerere University School of Medicine have varied perceptions and experiences of social accountability; most of the students evaluated the medical school favorably, and receiving career guidance in secondary school was associated with a positive evaluation of social accountability. Most medical students felt that the medical school had a good level of social accountability and needed to look at areas of weakness and ways to advocate for improving social accountability. The highest mean scores for items in the students’ toolkit for social accountability in medical schools were for community-based research and for the positive impact of the medical school on the community. The least favorable assessment items include learning about other cultures and teachers reflecting the reference population. Medical students come from varied backgrounds and have different experiences during medical education; therefore, it is not surprising that the perceptions, experiences and evaluations of the medical school were diverse. Career guidance may have helped the students choose an appropriate career path, leading to more positive perceptions and experiences for students who had received career guidance prior to medical school. This study helps us to better understand medical students’ views regarding social accountability and the factors that may influence these views. These findings provide a starting point for improving student experiences of social accountability in medical education.</p>", "<p id=\"P22\">The high percentage of students who gave a good evaluation of social accountability at medical school reflects efforts by the medical school to achieve this goal. These efforts include community-based education, research and services and adopting a competency-based medical education curriculum to better meet the community’s needs (##REF##36242004##14##, ##UREF##6##15##). Our findings are comparable to those of a study conducted at a Saudi Arabian government-funded medical school where most students felt that the medical school was performing well in terms of social accountability (##UREF##3##4##).</p>", "<p id=\"P23\">There have been gains in pursuing social accountability goals, and more needs to be done to enable students to understand the concept and demonstrate its values. Our findings concur with a previous study that showed a poor understanding of social accountability among stakeholders. This previous study also provided evidence of social accountability in medical school activities (##REF##22240206##6##). Similarly, a qualitative study in the United Kingdom also showed that students did not understand the concept of social accountability or feel that it has implications for their medical education or future practice (##REF##25072172##5##).</p>", "<p id=\"P24\">Medical students are central to social accountability efforts in medical schools. The perceptions and experiences of the medical students in this study reflect exposure to the concept and practice of social accountability. Learning social accountability requires deliberate and meaningful efforts in which medical students are considered partners.</p>", "<p id=\"P25\">The major strength of this study lies in the relatively large sample size, as we tried to enroll all eligible participants. The limitation of this study is that it was conducted at one study site, which may limit the generalizability of the findings. However, the presence of similar findings in other settings supports the generalizability of our findings. Recall bias was minimized by enrolling students who were still in medical school.</p>" ]

[ "<title>Conclusions</title>", "<p id=\"P26\">Medical students at the Makerere University School of Medicine have varied perceptions and experiences of social accountability. Most students felt moderately prepared for COBERS and evaluated medical school positively for social accountability. Regarding individual items in the student toolkit, the presence of community-based research and the school having a positive impact on the community had higher mean evaluation scores. Receiving career guidance in secondary school was associated with a positive evaluation of the medical school.</p>", "<title>Recommendations</title>", "<p id=\"P27\">The medical school should provide students with more opportunities to learn about social accountability and routinely evaluate the perceptions and experiences of medical students regarding social accountability. Students should be better prepared for Community-Based Education Research and Service.</p>" ]

[ "<p id=\"P1\">Authors’ contributions</p>", "<p id=\"P2\">LO conception of the study, acquisition and analysis of data, drafting the paper and writing the final version.</p>", "<p id=\"P3\">IGM, SK and AGM-refined the study concept, reviewed the results, drafted the paper and approved its final version.</p>", "<title>Background</title>", "<p id=\"P4\">Medical schools are called to be socially accountable as a feature of excellent medical education. Medical students are essential to the development of socially accountable medical schools. Therefore, understanding the perceptions and experiences of medical students regarding social accountability is critical for efforts to improve social accountability practices and outcomes.</p>", "<title>Methods</title>", "<p id=\"P5\">This cross-sectional online questionnaire-based survey used Google Forms and involved medical students in their fourth and fifth years of study at the Makerere University School of Medicine. The survey was conducted between September 2022 and October 2023. We used a study questionnaire and the Students’ toolkit for social accountability in medical schools to collect data on demographics, perceptions and experiences and evaluate social accountability.</p>", "<title>Results</title>", "<p id=\"P6\">A total of 426 medical students responded to the online questionnaire. The mean age of the students was 25.24 ± 4.4 years. Most of the students were male (71.3%), and most were in their fourth year of study (65%). Most of the students (43.66%) evaluated the school as having a good level of social accountability. The evaluation items referring to community-based research and positive impact on the community had the highest mean scores. Only 6 (3.64%) students provided an accurate definition of social accountability. Students receiving career guidance in secondary school was associated with evaluating social accountability in the medical school as strong (p-0.003).</p>", "<title>Conclusions</title>", "<p id=\"P7\">Medical students evaluated the medical school favorably in terms of social accountability. Receiving career guidance in secondary school was significantly associated with a positive evaluation of social accountability.</p>" ]

[]

[ "<title>Funding</title>", "<p id=\"P28\">The research reported in this publication was supported by the Fogarty International Center of the National Institutes of Health, U.S. Department of State’s Office of the U.S. Global AIDS Coordinator and Health Diplomacy (S/GAC) and the President’s Emergency Plan for AIDS Relief (PEPFAR) under Award Number 1R25TW011213. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.</p>", "<title>Availability of data and materials</title>", "<p id=\"P29\">The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><p id=\"P36\">Level of Preparedness for Last Community-Based Education Research and Service Experience</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><p id=\"P37\">Evaluation categories of social accountability at the medical school</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\"><label>Table 1</label><caption><p id=\"P38\">Characteristics of the study participants</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Variable</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Characteristic</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">N = 426 (%)</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(95 CI)1–2</th></tr></thead><tbody><tr><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">Gender</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Female</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">122 (28.64)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(24.44–33.23)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Male</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">304 (71.36)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(66.77–75.56)</td></tr><tr><td rowspan=\"4\" align=\"left\" valign=\"top\" colspan=\"1\">Age</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">20-24</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">283 (66.43)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(61.70–70.86)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">25-30</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">86 (20.19)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(16.54–24.38)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">31-35</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">32 (7.51)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5.272–10.55)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">36-47</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">25 (5.87)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(3.341–7.833)</td></tr><tr><td rowspan=\"3\" align=\"left\" valign=\"top\" colspan=\"1\">Tribe</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Muganda</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">158 (37.26)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3.910–8.656)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Munyankole</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">51 (12.03)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(9.164–15.60)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Other tribe</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">215 (50.71)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(45.85–55.56)</td></tr><tr><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">Nationality</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Ugandan</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">410 (96.24)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(93.84–97.77)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Other Nationality</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">16 (3.76)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(2.235–6.156)</td></tr><tr><td rowspan=\"3\" align=\"left\" valign=\"top\" colspan=\"1\">Highest education level before medical school</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">A level and equivalent</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">317 (74.41)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(69.94–78.44)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Bachelor</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">36 (8.45)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(6.066–11.61)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Diploma</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">73 (17.14)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(13.75–21.13)</td></tr><tr><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">Year of study</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Year 4</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">276 (64.79)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(60.02–69.29)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Year 5</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">150 (35.21)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(30.71–39.98)</td></tr><tr><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">Received career guidance in secondary school</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Yes</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">343 (80.71)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(76.56–84.28)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">No</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">82 (19.29)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">(15.72, 23.44)</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T2\"><label>Table 2</label><caption><p id=\"P39\">Students' evaluation of social accountability at the Makerere University School of Medicine</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Question</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Excellent</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Good</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Somewhat</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">No</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mean ± SD</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Does your institution have a clear social mission statement around the communities that they serve?</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">91 (21.36)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">186 (43.66)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">109 (25.59)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">40 (9.39)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.77 ± 0.891</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Does your curriculum reflect the needs of the population you serve?</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">99 (23.24)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">216 (50.70)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">89 (20.89)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">22 (5.16)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.92 ± 0.802</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Does your school have community partners or stakeholders who shape your school?</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">87 (20.42)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">164 (38.50)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">120 (28.17)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">55 (12.91)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.664 ± 0.944</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Do you learn about other cultures and other social circumstances in medical context in your curriculum?</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">53 (12.44)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">133 (31.22)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">157 (36.85)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">83 (19.48)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.366 ± 0.934</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Do the places/locations you learn at in practice include the presence of the populations that you will serve?</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">161 (37.79)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">174 (40.85)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">74 (17.37)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">17 (3.99)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.124 ± 0.835</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Are you required to do community-based learning (opposed to only elective opportunities)?</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">211 (49.53)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">146 (34.27)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">44 (10.33)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">25 (5.87)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.275 ± 0.872</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Does your class reflect the sociodemographic characteristics of your reference population?</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">101 (23.71)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">175 (41.08)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">99 (23.24)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">51 (11.97)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.765 ± 0.946</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Do your teachers reflect the sociodemographic characteristics of your reference population?</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">71 (16.67)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">170 (39.91)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">129 (30.28)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">56 (13.15)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.601 ± 0.915</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Does your learning experience also provide an active service to your community?</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">145 (34.04)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">193 (45.31)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">76 (17.84)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12 (2.82)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.106 ± 0.789</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Does your school have community-based research?</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">270 (63.38)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">132 (30.99)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">21 (4.93)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (0.70)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.57 ± 0.622</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Does your school encourage you to undertake generalist specialties (eg. family medicine, general practice)?</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">153 (35.92)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">123 (28.87)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">94 (22.07)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">56 (13.15)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.876 ± 1.045</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Does your school have a positive impact on the community?</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">225 (52.82)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">164 (38.50)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">34 (7.98)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (0.70)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.434 ± 0.67</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T3\"><label>Table 3</label><caption><p id=\"P40\">Factors associated with students' evaluation of social accountability</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"top\" span=\"1\"/><col align=\"left\" valign=\"top\" span=\"1\"/><col align=\"left\" valign=\"top\" span=\"1\"/><col align=\"left\" valign=\"top\" span=\"1\"/><col align=\"left\" valign=\"top\" span=\"1\"/><col align=\"left\" valign=\"top\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Variable</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Characteristic</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Overall, N = 426</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Limited SA, N = 76</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Strong SA, N = 350</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">p value</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Gender</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Female</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">122</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">28 (22.95)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">94 (77.05)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.081</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Male</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">304</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">48 (15.79)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">256 (84.21)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Receiving career guidance in secondary school</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Yes</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">344</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">52 (15.12)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">292 (84.88)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.003</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">No</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">82</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">24 (29.27)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">58 (70.73)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">Year of study</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Year 4</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">276</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">47 (17.03)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">229 (82.97)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.6</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Year 5</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">150</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">29 (19.33)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">121 (80.67)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Age category</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">19–24</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">283</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">52 (18.37)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">231 (81.63)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.7</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">25–47</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">143</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">24 (16.78)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">119 (83.22)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Nationality</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Ugandan</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">410</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">75 (18.29)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">335 (81.71)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.3</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Other</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">16</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1 (6.25)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">15 (93.75)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Education level prior to medical school</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">A-level and equivalent</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">317</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">57 (17.98)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">260 (82.02)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.9</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Graduate</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">109</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">19 (17.43)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">90 (82.57)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CGPA category</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.36–3.99</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">352</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">61 (17.33)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">291 (82.67)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.5</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">4.00–5.00</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">67</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">14 (20.90)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">53 (79.10)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr></tbody></table></table-wrap>" ]

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[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN2\"><p id=\"P30\">Declarations</p><p id=\"P31\">Competing interests</p><p id=\"P32\">AGM is a member of the BMC Medical Education editorial board. The other authors declare that they have no competing interests.</p></fn><fn id=\"FN3\"><p id=\"P33\">Ethical approval and consent to participate</p><p id=\"P34\">Ethical approval for the study was obtained from the Makerere University School of Medicine Institutional Review Board (Mak-SOMREC-2021-77) and the Uganda National Council for Science and Technology. Written informed consent was obtained from the study participants.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"nihpp-rs3756902v1-f0001\" position=\"float\"/>", "<graphic xlink:href=\"nihpp-rs3756902v1-f0002\" position=\"float\"/>" ]

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{ "acronym": [ "COBERS" ], "definition": [ "Community-Based Education Research and Service" ] }

[ "<title>Introduction</title>", "<p id=\"P3\">Glaucoma is the second leading cause of irreversible blindness worldwide affecting about 70 million people^{##REF##16488940##1##–##REF##24974815##3##}. Primary open angle glaucoma (POAG), the most common form of glaucoma is associated with progressive loss of retinal ganglion cell (RGC) axons and optic nerve degeneration ^{##REF##10416758##4##–##REF##27544758##6##}. Elevated intraocular pressure (IOP), a major risk factor for glaucoma is caused by increased resistance to aqueous humor (AH) outflow through the trabecular meshwork (TM) ^{##REF##19279343##7##–##REF##24825645##9##}. Despite TM being the major site of glaucomatous pathology ^{##REF##12751655##10##}, mechanisms regulating outflow resistance in TM are poorly understood ^{##REF##25957840##11##}. Glaucoma is a multi-factorial disease associated with genetic and environmental factors ^{##REF##26315706##12##–##REF##28577860##15##}. <italic toggle=\"yes\">MYOC</italic> was the first glaucoma gene identified ^{##REF##9005853##16##–##REF##9280311##18##} and is responsible for approximately 4% of POAG and most cases of juvenile-onset glaucoma (JOAG) ^{##REF##10900113##19##–##REF##10196380##21##}. <italic toggle=\"yes\">MYOC</italic>-associated JOAG is often less-responsive to current medication since current treatments do not target the main pathology ^{##REF##12150989##22##–##REF##34497454##25##}. It is therefore critical to develop targeted therapies to prevent vision loss in young pediatric patients.</p>", "<p id=\"P4\"><italic toggle=\"yes\">MYOC</italic> is abundantly expressed in TM cells and other ocular and non-ocular tissues ^{##REF##10711688##26##–##REF##9675094##28##}. However, the exact function of <italic toggle=\"yes\">MYOC</italic> is still not clear, although there are suggestions that it may function as a matricellular protein ^{##REF##9548973##29##–##UREF##1##34##}. Various studies have demonstrated that WT <italic toggle=\"yes\">MYOC</italic> is not required for the regulation of IOP, however mutations in <italic toggle=\"yes\">MYOC</italic> lead to a gain-of-function phenotype ^{##REF##11152659##35##–##REF##11604506##40##}. Overexpression or knockout of WT <italic toggle=\"yes\">MYOC</italic> exhibited no ocular changes in mice ^{##REF##15456875##39##, ##REF##11604506##40##} indicating that the WT <italic toggle=\"yes\">MYOC</italic> is not required for homeostasis of IOP. This is further supported by the findings that homozygous or heterozygous deletion of myocilin in humans is not associated with glaucoma ^{##REF##11604506##40##–##REF##17317787##43##}. Mutant <italic toggle=\"yes\">MYOC</italic> forms detergent insoluble aggregates and accumulates in the endoplasmic reticulum (ER) causing ER stress ^{##REF##30483726##30##, ##REF##11152659##35##, ##REF##15069026##36##, ##REF##11401512##38##, ##REF##14680806##44##, ##REF##16466712##45##}. The insufficiency of TM cells to resolve chronic ER stress results in cell death, leading to IOP elevation ^{##REF##14680806##44##, ##REF##21821918##46##–##REF##17108164##48##}. Since myocilin is not required for IOP regulation and mutant myocilin acquires toxic gain-of-function phenotype leading to TM cell death, knocking out myocilin at the genomic level becomes an attractive strategy for developing a novel therapy for <italic toggle=\"yes\">MYOC</italic>-associated glaucoma.</p>", "<p id=\"P5\">The Clustered Regularly Interspaced Short Palindromic Repeats (CRISPR) in association with CRISPR-associated systems (Cas) is a powerful and widely used tool for genomic research ^{##REF##29968520##49##, ##REF##24906146##50##}. It has two major components: an endonucleases enzyme Cas9 that cuts DNA and a gRNA that guides Cas9 to specific DNA sites. Together, they form a ribonucleoprotein (RNP) complex that can identify and cut DNA at the specific site. Once bound, Cas9 introduces a double strand break in the DNA. Gene knockouts can be generated due to indels incorporated by non-homologous end joining (NHEJ) or a homologous sequence can be simultaneously introduced for homology-directed repair (HDR) ^{##REF##29968520##49##, ##REF##24906146##50##}.</p>", "<p id=\"P6\">Previously, our group has demonstrated the successful gene editing of <italic toggle=\"yes\">MYOC</italic> using the CRISPR/Cas9 system in mice and human donor eyes ^{##REF##28973933##51##}. In this study, the knockout of <italic toggle=\"yes\">MYOC</italic> was targeted by designing gRNA targeting exon 1. The Cas9 + guide RNA was delivered using the adenovirus (Ad)-5, which has specific tropism toward the TM ^{##REF##19959644##52##}. Although, Ad5 is a highly efficient system, Ad5 is inflammatory and induces a strong immune response in transduced tissues ^{##REF##37558962##53##}. Considering our goal of clinical development, we sought to investigate other viral vectors including adeno-associated viruses (AAVs) and lentiviral (LV) particles to deliver Cas9 targeting <italic toggle=\"yes\">MYOC</italic> to TM <italic toggle=\"yes\">in vitro</italic> and <italic toggle=\"yes\">in vivo</italic> models ^{##REF##33411993##54##, ##REF##31557521##55##}. These viruses hold potential for clinical application due to robust delivery with long-term transgene expression, efficient transduction in post-mitotic cells, low immunogenicity, and minimal toxicity ^{##REF##36312652##56##, ##REF##36074935##57##}. In the present study, we first explored whether various AAVs or LV particles have specific tropism to TM in <italic toggle=\"yes\">in vitro</italic> and <italic toggle=\"yes\">in vivo</italic> models. We further examined whether selected AAV or LV expressing Cas9 and gRNA targeting <italic toggle=\"yes\">MYOC</italic> (cr<italic toggle=\"yes\">MYOC</italic>) reduce myocilin misfolding and rescue glaucomatous phenotypes in <italic toggle=\"yes\">in vitro</italic> and <italic toggle=\"yes\">in vivo</italic> models.</p>" ]

[ "<title>Methods</title>", "<title>Viral Vector Constructs:</title>", "<p id=\"P7\">AAV 2, self-complementary AAV2 (scAAV2) and Trp-Mutant scAAV2 (scAAV2^Trp-Mut) were selected for the study based on previous studies that show tropism toward the trabecular outflow pathway ^{##REF##31611639##58##}. Ready to use AAV2, scAAV2 and ScAAV2^Trp-Mut expressing GFP under the control of the CMV promoter were purchased from the Viral Vector Core at the University of Florida, Gainesville, FL. LV expressing GFP under the control of the CMV promoter (LV_GFP) was purchased from Vector Builder, Inc (Product ID: LVMP-VB160109–10005).</p>", "<p id=\"P8\">Guide RNA (gRNA) targeting <italic toggle=\"yes\">MYOC</italic> (GGCCTGCCTGGTGTGGGATG) published in the previous study, had the highest efficiency and selectivity in targeting human <italic toggle=\"yes\">MYOC</italic>\n^{##REF##28973933##51##}. In our current study, this same gRNA was cloned with <italic toggle=\"yes\">spCas9</italic> in the shuttle vector for generating LV constructs. LV particles expressing Cas9 + g<italic toggle=\"yes\">MYOC</italic>, LV expressing GFP and LV expressing Cas9 + scrambled gRNA were manufactured by Vector Builder, Inc. The LV_<italic toggle=\"yes\">Cas9</italic> + scrambled gRNA expresses spCas9 with non-specific gRNA sequence that does not target any genomic DNA. A different gRNA (GACCAGCTGGAAACCCAAACCA) was designed for cloning into ssAAV2 vectors using saCas9 (AAV2_cr<italic toggle=\"yes\">MYOC</italic>; Product ID: AAV2 MP (VB 200728–1179 bqW)) as the packaging capacity of AAV is comparatively small. The efficiency of this gRNA to selectively target human <italic toggle=\"yes\">MYOC</italic> was found to be equivalently high. We have utilized AAV2 expressing an empty cassette as a control (AAV2_Null; Viral Gene Core, University of Iowa).</p>", "<title>Mouse Husbandry:</title>", "<p id=\"P9\">All mice were housed and bred in a research facility at the University of North Texas Health Science Center (UNTHSC, Fort Worth, TX, USA). Animals were fed standard chow <italic toggle=\"yes\">ad libitum</italic> and housed in cages with dry bedding. The animals were maintained in a 12 h light:12 h dark cycle (lights on at 0630hrs) under a controlled environment of 21–26°C with 40–70% humidity. C57BL/6J (male) mice were obtained from the Jackson Laboratories (Bar Harbor, ME, USA). We have utilized <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice that express mutant <italic toggle=\"yes\">MYOC</italic> and develop ocular hypertension by the age of 3-months as described previously ^{##REF##21821918##46##, ##REF##27820874##59##, ##REF##22328638##60##}. <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice on a pure C57BL/6J strain were utilized for this study. These mice were genotyped by PCR using primers specific to human <italic toggle=\"yes\">MYOC</italic> as described previously ^{##REF##21821918##46##, ##REF##27820874##59##, ##REF##22328638##60##}. Animal studies were executed in agreement with the guidelines and regulations of the UNTHSC Institutional Animal Care and Use Committee (IACUC) and the ARVO Statement for the Use of Animals in Ophthalmic and Vision Research. This study is reported in accordance with ARRIVE guidelines (<ext-link xlink:href=\"https://arriveguidelines.org/\" ext-link-type=\"uri\">https://arriveguidelines.org</ext-link>). Experimental protocols were approved by UNTHSC IACUC and Biosafety office under the approved protocol. At the end of experiment, mice will be sacrified by CO₂ inhalation followed by cervical dislocation.</p>", "<title>TM cell culture and <italic toggle=\"yes\">in vitro</italic> transduction:</title>", "<p id=\"P10\">TM3 cells were transfected with pDsRed2-<italic toggle=\"yes\">MYOC</italic> plasmids to generate stable cells expressing WT or mutant (Y437H or G364V) <italic toggle=\"yes\">MYOC</italic> using Lipofectamine 3000^™ transfection kit (Invitrogen, Life Technologies, Grand Island, NY, USA). These plasmids express <italic toggle=\"yes\">MYOC</italic> tagged with DsRed at the C-terminus. The confluent transfected cells were then treated with G418 antibiotic (0.6 mg/mL; Gibco, Life Technologies, Grand Island, NY, USA) for 7–10 days and individual colonies were selected and expanded. The cells stably expressing DsRed-tagged <italic toggle=\"yes\">MYOC</italic> (with or without mutations) were characterized as described previously ^{##REF##17325163##61##}, and maintained in DMEM media (Sigma-Aldrich Corp, St. Louis, MO, USA)) supplemented with G418 antibiotics, 10% FBS (Gibco), and streptomycin (Gibco). For viral transduction, TM3 cells were plated at 30–40% confluency. The following day, cells were incubated with AAV (5000 MOI/mL) or LV (10 MOI/mL) in antibiotic free and low serum (6%) media. 30 hours post viral treatment, cells were switched back to regular maintenance medium. Once confluent (at day 3 or 4 post-transduction), cells were later processed for DNA isolation, Western blotting, and immunostaining. Human primary TM cells (n = 2 strains) were grown to confluency in 12-well plates and treated with AAV2/2, AAV2/4, AAV2/5 and AAV2/8 at multiplicities of infection (MOI) of 2.5×10¹ to 2.5×10³ viral genomes (VG)/cell. GFP expression was examined by fluorescent microscopy after 72 hours of transduction.</p>", "<title>Intraocular Injections:</title>", "<p id=\"P11\">Viral deliveries were performed via intravitreal (IVT) and intracameral (IC) routes. Mouse eyes were anesthetized before injections by topical administration of proparacaine HCl drops (0.5%) (Akorn Inc., Lake Forest, IL, USA). Both IVT and IC bolus injections were performed on mice anesthetized intranasally with isoflurane (2.5%; with 0.8 L/min oxygen). However, in case of slow-IC infusion protocol, mice were anesthetized using xylazine/ketamine (10/100 mg/kg; Vetus; Butler Animal Health Supply, Westbury, NY/Fort Dodge Animal Health, Fort Dodge, IA, USA) cocktail administered intraperitoneally. As required, additional one-quarter to one-half of the initial dose was provided for continuous maintenance of the surgical anesthetic state. LV particles (2.5 × 10⁶ TU/eyes and 2.5 mL/eye) or various AAV2 (2 × 10¹⁰ GC/eye) were injected via IVT or IC route. Hamilton’s (Reno, NV, USA) glass micro-syringe (10 mL capacity) attached with a 33 gauge 1-inch-long needle was used for IVT injections as described previously ^{##UREF##2##62##}. For IC route, mouse eyes were treated topically with 1% cyclopentolate (Mydriacyl^®, Alcon Laboratories, Fort Worth, TX) to dilate the pupils. Using the same micro-syringe system, the 33-gauge needle was inserted through the cornea 1–2 mm from the limbus, positioned parallel to the iris, and pushed towards the chamber angle opposite to the cannulation point. Care was taken to not touch the iris, corneal endothelium, or the anterior lens capsule. The viral solution was slowly released into the anterior chamber over a period of 30s, after which the needle was kept inside for a further 1 min, before being rapidly withdrawn. For slow infusion, the glass micropipette system was loaded onto a micro-dialysis infusion pump (SP101I Syringe Pump; WPI) that delivered the viral solution at a flow rate of 0.083 mL/min over the course of 30 mins (total volume delivered, 2.5 mL). A drop of filtered saline was also applied through this procedure to prevent corneal drying.</p>", "<title>IOP measurements:</title>", "<p id=\"P12\">A TonoLab impact tonometer (Colonial Medical Supply, Londonderry, NH, USA) was used for IOP measurements on mice as previously described ^{##REF##16303957##63##}. Baseline IOPs for C57BL/6J and <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice were measured during dark conditions (between 6:00–8:00 AM). The mice were anesthetized via intranasal isoflurane (2.5%; 0.8 L/min oxygen) delivery and readings were noted within 3 min of isoflurane influence to avoid any of its side effects on IOP ^{##REF##33154371##64##}. Post-injections, IOPs were monitored weekly (daylight and dark) in a masked manner. The average value of six individual IOP readings were represented.</p>", "<title>Slit lamp imaging:</title>", "<p id=\"P13\">A slit lamp (SL-D7, Topcon Corporation, Tokyo, Japan) was used to determine inflammation and ocular abnormalities in the anterior segment, including corneal edema, and photo-documented with a digital camera (DC-4; Topcon) as described earlier ^{##REF##21821918##46##}.</p>", "<title>Histology and immunofluorescence staining:</title>", "<p id=\"P14\">Following viral transduction, mice were euthanized at specified timepoints, and eyes were carefully enucleated and placed in 4% paraformaldehyde (PFA, Electron Microscopy Sciences, Hatfield, PA, USA) overnight at 4°C. The next day, eyes were washed with 1x PBS (Sigma-Aldrich) and cryopreserved using increasing concentration of sucrose (10% and 20%), followed by OCT compound embedding and sectioning. For hematoxylin and eosin (H&amp;E) staining, the eyes were dehydrated in ethanol, and embedded in paraffin wax for sectioning. The paraffin-embedded mouse eyes were sectioned (sagittal) at 5 mm thickness, followed by deparaffinization in xylene, rehydration with gradual 5 min washes in each 100, 95, 70, and 50% ethanol solution and ending with a 10 min wash in 1x PBS. These sections were later stained with H&amp;E. The general morphology of the anterior segment was assessed including the TM structure at iridocorneal angle and corneal thickness by light microscopy. Images were captured using a Keyence microscope (Itasca, IL, USA).</p>", "<p id=\"P15\">The OCT-embedded sections from mouse eyes were incubated with 10% goat serum (EMD Millipore Corp) in 0.2% Triton X-100 (diluted in PBS; Fisher BioReagents, Fair Lawn, NJ, USA) for 2 hours. For <italic toggle=\"yes\">in vitro</italic> studies, TM cells were plated in 8-well chamber slides (Lab-Tek Nunc Brand Products, Rochester, NY, USA) and fixed with 4% PFA for 20 mins, followed by PBS washes. Fixed cells or sections were then incubated with 10% goat serum in 0.1% Triton X-100 for 2 hours. The slides were incubated with primary antibody (<italic toggle=\"yes\">MYOC</italic>, catalog # 60357: Proteintech Group Inc, Rosemont, IL, USA; or GRP78, Catalog# ab21685: Abcam, Cambridge, MA, USA). The slides were washed 4 times with 1x PBS before incubating with Alexa Fluor secondary antibody (1:500; Invitrogen, Life Technologies, Grand Island, NY, USA) at room temperature for 2 hours. The slides were washed again and mounted with DAPI antifade mounting medium (Vectashield, Vector Laboratories Inc., Burlingame, CA, USA) as described previously ^{##REF##28973933##51##, ##REF##27820874##59##, ##UREF##2##62##, ##UREF##3##65##}. For evaluating GFP expression in mice, the OCT sections were washed once with PBS and mounted with DAPI medium. Fluorescent images were captured, processed, and quantified using a Leica SP8 confocal microscope and LAS-X software (Leica Microsystems Inc., Buffalo Grove, IL, USA). Tissue sections and TM cells incubated without primary antibodies served as a negative control and were used to normalize the fluorescent intensities by background elimination. Sections of non-injected eyes served as a background control for GFP fluorescence. For quantifying staining specific to the mouse TM, a region of interest was drawn around the TM area and represented as the unit of fluorescence intensity per mm². MYOC fluorescent intensity in TM3 cells stably expressing mutant <italic toggle=\"yes\">MYOC</italic> was quantified by imaging thirteen to fifteen different non-overlapping areas of each treated wells. The fluorescent intensity was normalized using number of cells per image as determined by DAPI staining.</p>", "<title>Western Blot:</title>", "<p id=\"P16\">TM3 cells were lysed in 1x RIPA buffer containing protease inhibitors. Cellular lysates were loaded on denaturing 4%–12% gradient polyacrylamide readymade gels (NuPAGE Bis-Tris gels, Life Technologies). The proteins were separated using Invitrogen’s Mini Gel electrophoresis tank at constant voltage (150 V) and transferred onto a methanol-activated PVDF membrane (Immobilon-P, 0.45 mm pore size; Merk Millipore Ltd., St. Louis, MO, USA) as described previously ^{##UREF##2##62##}. The blots were blocked with 5% nonfat dry milk prepared in 1x PBS with Tween-20 (PBST), followed by overnight incubation at 4°C with respective primary antibodies (1:1000 dilutions). The primary antibodies used were KDEL (catalog# MBP1–97469, Novus Biologicals, Littleton, CO, USA); MYOC (catalog# ab41552, Abcam); ATF4 (catalog# 10835–1-AP, Proteintech); CHOP (catalog# 15204–1-AP, Proteintech; 6003–1395, Novus). GAPDH (catalog# 60004–1-Ig, Proteintech) was used as a loading control. After overnight primary antibody incubation, the blots were washed with 1x PBST and incubated with respective horseradish-peroxidase (HRP)-conjugated secondary antibodies (1:2500 dilution) and developed with enhanced chemiluminescence (ECL) detection reagent (SuperSignal West Femto Maximum Sensitivity Substrate; Life Technologies). Protein bands were visualized using an LI-COR Biosciences Odyssey-Fc image system (Lincoln, NE, USA) and quantified using ImageStudio software (LI-COR Biosciences) as previously explained ^{##UREF##3##65##, ##REF##24691439##66##}.</p>", "<title>Genomic endonuclease assay:</title>", "<p id=\"P17\">Genomic DNA was isolated using NucleoSpin^® Tissue (catalog# 740952, Macherey-Nagel, Allentown, PA, USA) from cells treated with LV_<italic toggle=\"yes\">crMYOC</italic>, AAV2_<italic toggle=\"yes\">crMYOC</italic>, LV_Null and AAV2_Null. Untreated cells were used as experimental control. <italic toggle=\"yes\">MYOC</italic>, which is a target of selected gRNA was amplified by PCR. PCR product was denatured and reannealed using the Alt-R^™ Genome Editing Detection Kit protocol (catalog# 1075932, Integrated DNA Technologies, Coralville, Iowa, USA). This generated mismatched heteroduplex DNA products containing strands with CRISPR/Cas9-induced indel reannealed to wild-type strands or different indel. The heteroduplexes were subsequently detected using T7 endonucleases (T7E1), that cleaved the mismatched DNA. The resulting cleaved products were analyzed by gel electrophoresis.</p>", "<title>CRISPR-Cas9 off-target effects by whole genome sequencing (WGS):</title>", "<p id=\"P18\">TM3 cells were transduced with lentivirus expressing Cas9 only (gScr), or Cas9 with gRNA against myocilin (g<italic toggle=\"yes\">MYOC</italic>). 48 hours after infection, genomic DNA was extracted from gScr, g<italic toggle=\"yes\">MYOC</italic>, and parental TM3 (NT) cells. Samples were sequenced on a Novaseq 6000 system at 30x coverage. The FASTQ files for all three samples (g<italic toggle=\"yes\">MYOC</italic>, gScr, NT) were aligned to the human reference genome (GRCh37) with BWA-mem and sorted with SAMtools ^{##REF##19505943##67##}. The resulting BAM files were processed to remove duplicate reads with Picard Tools ( <ext-link xlink:href=\"http://broadinstitute.github.io/picard/\" ext-link-type=\"uri\">http://broadinstitute.github.io/picard/</ext-link> ). Local realignment and base quality recalibration were performed with Genome Analysis Tollkit (GATK) ^{##REF##20644199##68##}. The most-likely off-target sites were determined using Cas-OFFinder ^{##REF##24463181##69##} based upon the human reference genome (GRCh37), allowing the alignment of the gRNA to the genome to have up to 3 mismatches, DNA bulge size less than or equal to 1, and an RNA bulge size less than or equal to 1. The resulting 1214 unique sites were prioritized using the crisprScore package in Bioconductor, with the CFD algorithm ^{##REF##36323688##70##}. The top 100 sites were selected based upon their crisprScore. Each site was inspected visually using the Integrated Genome Viewer ^{##REF##21221095##71##} with the analysis-ready BAM files for all three samples loaded. Sites were judged to be off target if indels were observed within 20 nt of the target in the g<italic toggle=\"yes\">MYOC</italic> sample and not in any of the other samples.</p>", "<title>Statistics:</title>", "<p id=\"P19\">Statistical analyses were performed using Prism 9.0 software (GraphPad, San Diego, CA, USA). A <italic toggle=\"yes\">P</italic> value of &lt;0.05 was considered significant. Data was represented as mean ± SEM. An unpaired Student’s <italic toggle=\"yes\">t</italic> test (two-tailed) was used for comparing data with two-groups. The IOP results that comprise more than two groups were analyzed by repeated-measures two-way ANOVA followed by a Bonferroni post-hoc correction.</p>" ]

[ "<title>Results</title>", "<title>Ocular transduction patterns of various AAV2 serotypes and lentiviral particles in mouse eyes:</title>", "<p id=\"P20\">Selective AAV serotypes were shown to have tropism towards the TM of mice, rats, and monkeys in previous studies ^{##REF##26052939##72##–##REF##28763501##74##}. These studies suggest that single-stranded (ss) ^{##REF##28763501##74##} and self-complimentary (sc) AAV2 efficiently transduces TM ^{##REF##26052939##72##, ##REF##16506246##75##}. However, AAV2 also exhibits strong tropism to other ocular tissues. Our recent study also showed robust tropism of LV to the mouse TM ^{##UREF##2##62##}. We therefore compared the TM specific tropism of various AAV2 capsid variants and LV particles expressing GFP. To select which AAV serotypes has the best tropism towards TM, we first screened several AAV2 serotypes in primary human TM cells (SI I). Human primary TM cells (n = 2 strains) were grown to confluency in 12-well plates and treated with AAV2/2, AAV2/4, AAV2/5 and AAV2/8 at multiplicities of infection (MOI) of 2.5×10¹ to 2.5×10³ viral genomes (VG)/cell as described previously ^{##REF##28763501##74##}. GFP expression was examined by fluorescent microscopy after 72 hours of transduction (SI I). No AAVs caused GFP expression at MOI of ×10¹ VG/cell (not shown), but we observed robust AAV2-GFP expression at MOI of 2.5×10³ VG/cell. Note that these high MOI are consistent with other cell types ^{##REF##26052939##72##, ##REF##28763501##74##, ##REF##23497173##76##}. Based on these data, we chose to further investigate whether various AAV2 capsids produce robust tropism in mouse TM. These viruses were injected via IVT or IC (bolus or slow perfusion) routes to determine GFP expression in ocular tissues (n = 3 for each vector per route of injections) and GFP was examined by confocal imaging 2-weeks post-injection. We chose to perform slow perfusion IC injection because bolus IC injections may wash out quickly through the outflow pathway, which can limit the viruses’ ability to transduce TM cells. None of the three capsid variants, ssAAV2_<italic toggle=\"yes\">GFP</italic>, scAAV2_<italic toggle=\"yes\">GFP</italic> and scAAV2^Trp-Mut_<italic toggle=\"yes\">GFP</italic> showed GFP expression in the TM region delivered by bolus IVT or IC injections (##FIG##0##Figure 1A##). Consistent with previous studies, we observed a robust GFP expression in the retina (data not shown). Slow IC infusion of scAAV2^Trp-Mut induced robust GFP expression in TM and other tissues at the iridocorneal angle compared to the ssAAV2 vector (##FIG##0##Figure 1A##). However, no GFP fluorescence was observed in the eyes of scAAV2 slow IC-infused eyes. Irrespective of their differences in TM transduction efficiency, these AAV2 variants were also found to transduce retina, optic nerve, and optic nerve head regions robustly in slow IC-treated eyes (##FIG##0##Figure 1B##). Since AAVs did not show selective and robust tropism to TM, we next evaluated the selective tropism of LV. Consistent with our previous study ^{##UREF##2##62##}, IVT bolus injections of LV_<italic toggle=\"yes\">GFP</italic> induced GFP expression in mouse TM (##FIG##1##Figure 2A##). This GFP fluorescence was seen throughout the TM. Minor expression was also observed in ciliary body region. Importantly, no GFP expression was observed in the retina confirming the specificity of our LV to the TM (##FIG##1##Figure 2B##). For comparative purposes, the efficiency of LV vectors was also examined via slow IC infusion. We observed robust and more efficient TM transduction via the slow-infused IC route (##FIG##1##Figure 2A##–##FIG##1##B##). However, significant LV_<italic toggle=\"yes\">GFP</italic> transduction was also observed in the inner corneal endothelium layer (##FIG##1##Figure 2B##). Since IVT delivery of LV demonstrated most selective and efficient tropism to TM, we utilized this approach to deliver <italic toggle=\"yes\">Cas9</italic> to TM in our subsequent studies.</p>", "<title>Comparison of AAV2 and LV-mediated <italic toggle=\"yes\">MYOC</italic> editing <italic toggle=\"yes\">in vitro</italic>:</title>", "<p id=\"P21\">We have previously demonstrated CRISPR/Cas9-mediated knockout of the <italic toggle=\"yes\">MYOC</italic> gene using the Ad5 delivery system ^{##REF##28973933##51##}. Here, we examined whether AAV2 and LV expressing cr<italic toggle=\"yes\">MYOC</italic> efficiently edit the <italic toggle=\"yes\">MYOC</italic> gene in human TM3 cells. TM3 cells stably expressing DsRed tagged human <italic toggle=\"yes\">MYOC</italic> with Y437H or G364V mutations (DsRed-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Mut</italic>}) exhibit reduced secretion and intracellular accumulation of mutant-<italic toggle=\"yes\">MYOC</italic>\n^{##REF##11152659##35##, ##REF##14680806##44##, ##REF##21821918##46##, ##REF##28973933##51##, ##REF##27820874##59##, ##UREF##3##65##}. Overall, a decrease in DsRed puncta was observed in TM3 cells transduced with AAV2_cr<italic toggle=\"yes\">MYOC</italic> and LV_cr<italic toggle=\"yes\">MYOC</italic> compared to TM3 cells transduced with controls viral particles (##FIG##2##Figure 3A##). We next quantified <italic toggle=\"yes\">MYOC</italic> accumulation using Image J, which revealed that LV-cr<italic toggle=\"yes\">MYOC</italic> reduced MYOC significantly by 62% while AAV2_cr<italic toggle=\"yes\">MYOC</italic>-mediated reduction was 34% (##FIG##2##Figure 3B##). The decrease in MYOC accumulation in LV_cr<italic toggle=\"yes\">MYOC</italic> treated cells was also reflected on GRP78 fluorescence, reduced significantly by 38% compared to the control cells.</p>", "<p id=\"P22\">We further determined genome editing efficiency of AAV_cr<italic toggle=\"yes\">MYOC</italic> or LV_cr<italic toggle=\"yes\">MYOC</italic> in TM3 cells stably expressing mutant MYOC using Western blot analysis (##FIG##3##Figure 4A##–##FIG##3##B##). Western blot and its densitometric analysis demonstrated significant reduction in MYOC and ER stress markers (GRP78, CHOP and GRP94) in LV_cr<italic toggle=\"yes\">MYOC</italic>-treated cells compared to cells transduced by LV-null. Although AAV2-cr<italic toggle=\"yes\">MYOC</italic> reduced MYOC and ER stress markers, this reduction was not statistically significant compared to cells treated with AAV2-null. Using the Alt-R^™ Genome Editing Detection Kit, we further confirmed T7 endonuclease (T7E1) induced cleaved product in both our LV_cr<italic toggle=\"yes\">MYOC</italic> and AAV2_cr<italic toggle=\"yes\">MYOC</italic> treated DNA samples (SI II). No cleaved product was observed in untreated control, LV_Null and AAV2_Null treated DNA samples. These data indicate that LV_cr<italic toggle=\"yes\">MYOC</italic> edits <italic toggle=\"yes\">MYOC</italic> and reduces its intracellular accumulation, relieving ER stress in human TM cells.</p>", "<title>LV_cr<italic toggle=\"yes\">MYOC</italic> decreases mutant myocilin in TM and reduces elevated IOP in mouse model of MYOC-associated glaucoma:</title>", "<p id=\"P23\">Since LV_cr<italic toggle=\"yes\">MYOC</italic> disrupts <italic toggle=\"yes\">MYOC</italic> efficiently <italic toggle=\"yes\">in vitro</italic>, we further determined whether LV_cr<italic toggle=\"yes\">MYOC</italic> rescues <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice, which expresses human <italic toggle=\"yes\">MYOC</italic> with the Y437H mutation. As shown in ##FIG##0##Figure 1##, intravitreal injection of LV targets mouse TM. We therefore performed a single intravitreal injection of LV_cr<italic toggle=\"yes\">MYOC</italic> in adult ocular hypertensive <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice. Before injections (0 day), we observed that <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice show IOP elevation compared to WT mice (##FIG##4##Figure 5A##). Ocular hypertensive <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice were injected intravitreally with LV_cr<italic toggle=\"yes\">MYOC</italic> or LV_<italic toggle=\"yes\">Cas9</italic>-Null (2.5 × 10⁶ TU/eyes; ##FIG##3##Figure 4##). While LV_Null treated <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice exhibited significant IOP elevation compared to age-matched C57BL/6J mice, LV_cr<italic toggle=\"yes\">MYOC</italic> mice demonstrated a significant reduction of IOP 3-weeks after injection and IOPs in LV_cr<italic toggle=\"yes\">MYOC</italic> injected mice were similar to WT mice 3-weeks after injection (##FIG##4##Figure 5A##). The mean dark-adapted IOP was ~17.75 mmHg in LV_cr<italic toggle=\"yes\">MYOC</italic>-injected <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice compared to ~21.07 mmHg in LV_Null treated <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice and ~18.4 mmHg in control WT mice. We next determined whether LV_cr<italic toggle=\"yes\">MYOC</italic> reduced mutant myocilin accumulation in <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice by immunolabeling of fixed anterior segments with MYOC antibody (##FIG##4##Figure 5B##). Immunostaining data revealed that LV_cr<italic toggle=\"yes\">MYOC</italic> treatment reduced MYOC labeling in the TM region of <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice compared to LV_null treated <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice. These data indicate that LV_cr<italic toggle=\"yes\">MYOC</italic> edits the <italic toggle=\"yes\">MYOC</italic> gene and prevents IOP elevation in <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice. Since viral vectors including Ad5 tend to cause ocular inflammation, we next examined ocular structures in LV_cr<italic toggle=\"yes\">MYOC</italic> injected <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice compared to Ad5_cr<italic toggle=\"yes\">MYOC</italic> injected <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice using slit lamp imaging and histological analysis of anterior segments (##FIG##4##Figure 5C## and SI III). The eyes injected with Ad5_cr<italic toggle=\"yes\">MYOC</italic> (2 × 10⁶ pfu/eyes) developed acute inflammation determined by an opaque white appearance of the anterior segments (##FIG##4##Figure 5C##). Moreover, H&amp;E staining of the anterior segment revealed increased corneal thickness in Ad5 injected eyes (SI III). In contrast, both slit lamp imaging and H&amp;E staining revealed that LV_cr<italic toggle=\"yes\">MYOC</italic> injected eyes showed no abnormalities in the anterior segments. These data indicate that an IVT injection of LV_cr<italic toggle=\"yes\">MYOC</italic> causes minimal ocular toxicity in mice.</p>", "<p id=\"P24\">One of the major concerns with CRISPR-Cas9 based genome editing is off-target effects. To determine the off-target effects due to LV_cr<italic toggle=\"yes\">MYOC</italic>, we performed WGS on TM3 cells transfected with LV_cr<italic toggle=\"yes\">MYOC</italic> or LV_crScrambled. The most obvious change is at the myocilin genome locus (SI IV and SI V ). Out of the top 100 predicted off-target sites based on crisprScore, 2 sites (MLLT3 with 7 reads, and FAM19A5 with one read) were identified with potential changes in TM3 cells treated with LV_cr<italic toggle=\"yes\">MYOC</italic>, but not in cells treated with LV_crScrambled gRNA or parental TM3 (NT) samples. These sites had crisprScores of 0.75 and 0.50 respectively. Both observed changes are in deep intronic regions, located more than 10,000 nt from the nearest exon. A total of nine off-target sites for LV_cr<italic toggle=\"yes\">MYOC</italic> were considered most likely (crisprScore ³ 0.5), of which only one falls in a coding region. That site is within SLC2A10 and has a crisprScore of 0.53 requiring 3 mismatches and a bulge. We detected the change at the MLLT3 site by the T7 endonuclease 1 assay (T7E1). We cannot detect the change at the FAM19A5 site by the T7E1 assay. This may be due to the lack of sensitivity of the T7E1 assay or sequencing error. Together, these data indicate that LV_cr<italic toggle=\"yes\">MYOC</italic> edits <italic toggle=\"yes\">MYOC</italic> with high efficiency with only limited off-target effects.</p>" ]

[ "<title>Discussion</title>", "<p id=\"P25\">Recent advances in genome editing technologies allow investigators to directly alter the genes associated with disease pathology. The gain-of-function mutation of the <italic toggle=\"yes\">MYOC</italic> gene serves as a direct target for gene editing without the need for gene replacement. Knocking down <italic toggle=\"yes\">MYOC</italic> expression in the eye does not compromise any normal ocular physiological function and it is relatively easy to knock out the gene compared to correcting its mutations ^{##REF##34497454##25##, ##REF##15456875##39##, ##REF##11604506##40##}. The eye is a favorable target to develop gene therapy attributed to its ease of accessibility for routine clinic-based applications and the fact that it is an isolated immune privileged compartment separated by the blood-retinal barrier ^{##REF##33411993##54##, ##REF##36312652##56##}. Importantly, long duration of efficacy can be obtained from a single dose of gene delivery, thus eliminating the requirement for patient compliance with routine eye drop application ^{##REF##28161916##77##}. We have previously demonstrated that Ad5_cr<italic toggle=\"yes\">MYOC</italic> decreases mutant MYOC in TM and rescues glaucoma in transgenic mice. Although Ad5 was used experimentally due to its tropism for the TM, Ad5 is not a suitable viral vector for clinical use due to its immunogenic response ^{##REF##36312652##56##}. Here, we show that lentiviral particles mediate optimum and efficient <italic toggle=\"yes\">MYOC</italic> editing in TM and prevent IOP elevation in a mouse model of <italic toggle=\"yes\">MYOC</italic>-associated POAG. For clinical application, the selectivity to transduce and target transgene expression in a specific cell region is important to avoid off-site gene editing ^{##REF##28161916##77##}. The modifications of viral serotypes or capsid can alter the cellular tropism of the viral vector ^{##REF##26052939##72##, ##REF##28872643##73##, ##REF##16101513##78##, ##REF##27131906##79##}. In addition, the route of vector delivery, the intraocular environment and proximity of the target tissue to the delivery site help determine the efficiency and selectivity of the transduction ^{##REF##27131906##79##, ##UREF##4##80##}. Based on the anatomy, the IC route provides the most efficient TM transduction in several studies using AAV or LV vectors ^{##REF##33411993##54##, ##REF##22052240##81##}. Most of the anterior segment aqueous humor flow exits via the TM, which is known for its phagocytic property ^{##REF##29526795##82##}. This further promotes the viral vectors to have high affinity for TM transduction compared to cornea, lens or ciliary body. However, due to the constant AH outflow, IC bolus injection leads to rapid washout of the viral vectors, with limited exposure to the target tissue, especially in the mouse which has a very small eye. Hence, we employed slow-IC infusing that delivers the virus for a extended period. In contrast, the IVT injection route provides a longer-lasting depot effect for sustained release of the injected vectors, proving to be an efficient route for gene therapy application with single dose administration. Our findings indicate that slow-IC infusion is the most efficient route for inducing robust transgene expression in the TM via both AAV2 capsid variants and LV vectors. However, LV vectors induced GFP expression in the corneal endothelium, which is consistent with a previous study ^{##REF##15988411##83##}. Nonetheless, the IVT route for LV_<italic toggle=\"yes\">GFP</italic> proved to be more specific and selective in transducing the mouse TM, with minor GFP expression noted in the ciliary body region. The slow and smaller release of the virus particles from the posterior vitreous, prevent their proximity and exposure to corneal endothelial, enough to reduce the propensity to transduce.</p>", "<p id=\"P26\">The AAV vectors are well known for their safety and efficacy in clinical application and are the preferred option for retinal gene therapy ^{##REF##36312652##56##, ##REF##28161916##77##}. This nonpathogenic ssDNA and replication deficient parvovirus provides long-term transgene expression, with only a mild immunogenic response. However, they are limited by their ability to transduce the tissues of the anterior segments. Several studies have emphasized the use of scAAV capsids or their mutant forms for efficient transduction of TM cells as they facilitate the generation of dsDNA ^{##REF##26052939##72##, ##REF##28872643##73##, ##REF##19684004##84##}. However, the size of the transgene cassette that can be inserted is very limited. In contrast, the capsid mutation of AAV serotype 2 (AAV2) have better TM transduction via the intracameral route in rodents, perfused anterior chamber, and cultured human TM cells, thus resolving the issue associated with transgene insertion size ^{##REF##28763501##74##}. While evaluating cellular tropism of AAV2 serotype capsid variants via GFP expression, the scAAV2^{Trp − Mut} induced prominent expression of GFP in TM via the slow-IC infusion route. TM transduction was also observed with our ssAAV2 capsid variant. However, we demonstrate that AAV2 is not selective to TM, with robust GFP expression observed in retina and ONH. The selectivity of transgene expression can also be determined by use of tissue specific promoters. The CMV promoters used in our vector constructs promotes ubiquitous transgene expression in a majority of ocular tissues including corneal endothelium, non-pigmentary epithelial cells and retinal tissues ^{##REF##15231389##85##}. A few studies have reported TM preferential promoters such as matrix Gla protein and chitinase-3-like-1 promoter ^{##REF##27131906##79##, ##REF##25540740##86##}. This non-specificity of AAVs to ocular tissues can increase Cas9-associated off-target effects, thus limiting its clinical applications for the treatment of glaucoma.</p>", "<p id=\"P27\">Lentiviruses are known for their capacity to induce sustained transgene expression with low immunogenic response. Both FIV and HIV based LV are used in ocular research ^{##REF##22052240##81##}. LV vector efficiency is currently being investigated in two macular degeneration clinical trials ^{##REF##28161916##77##, ##UREF##4##80##}. Our HIV based VSV-G pseudotyped vector proved to be selective towards the mouse TM via the IVT route. The ssRNA genome of lentivirus is reverse transcribed into dsDNA that becomes integrated into the host genome via integrase enzyme activity. This is one of the major limitations of using LV in clinical applications. Based on the recent advancement, our LV vectors are designed to avert insertional mutagenesis by inhibiting integrase. These integrase-deficient lentivirus vectors can be generated by introducing non-pleiotropic mutations within the open reading frame that specifically targets the integration function without affecting the life cycle of the virus^{##REF##32903507##87##}.</p>", "<p id=\"P28\">LVs are known for their high transgene loading capacity (7 kb), which is a major advantage over the AAV vectors (~ 4.6 kb). Therefore, they are more suitable for packaging gene editing constructs such as CRISPR/Cas9. Although scAAV vectors have higher TM transduction efficiency as reported by a previous study ^{##REF##19684004##84##}, they are limited by the capacity for packaging the cargo gene. Therefore, we used ssAAV2 variants to determine the efficiency of CRISPR/Cas9 based <italic toggle=\"yes\">MYOC</italic> gene editing. Moreover, we used SaCas9 for ssAAV2 based CRISPR assembly, as it is smaller in size compared to the SpCas9^{##REF##26524662##88##–##REF##33293588##90##}. Both LV and AAV2 expressing Cas9 were able to edit the <italic toggle=\"yes\">MYOC</italic> gene in human TM cells. However, the overall effect of <italic toggle=\"yes\">MYOC</italic> gene editing on MYOC protein levels and ER stress was more significantly pronounced in LV-treated cells compared to AAV2-treated cells. Comparable to our previous study ^{##REF##28973933##51##}, our LV_cr<italic toggle=\"yes\">MYOC</italic> was able to knock down <italic toggle=\"yes\">MYOC</italic> expression in transduced TM of <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice, resulting in significantly reduced IOP independent of any immunogenic response.</p>", "<p id=\"P29\">One serious concern with traditional Cas9 is off-target effects, which occur due to non-selectively of Cas9 to similar genomic regions. Traditional nuclease CRISPR/Cas9-based gene knockouts also introduce DNA double-strand breaks (DSBs), which pose serious risks such as large deletions, translocations, and chromosomal abnormalities. In addition, this effect can be more pronounced when Cas9 is expressed for longer period as in the case when delivered using viral vectors. WGS revealed that our LV_cr<italic toggle=\"yes\">MYOC</italic> targets <italic toggle=\"yes\">MYOC</italic> in TM cells with high efficiency but we have also observed limited off-target effects in LV_cr<italic toggle=\"yes\">MYOC</italic> treated TM cells. We utilized <italic toggle=\"yes\">in silico</italic> tools to select our gRNA targeting <italic toggle=\"yes\">MYOC</italic>. These in silico tools search for potential off-target sites in the whole genome and calculate the likelihood of off-target editing. Most off-target effects are often gRNA dependent and selecting another gRNA may reduce these off-target effects. In addition,viral vectors tend to cause prolonged expression of Cas9, which can increase off-target effects ^{##REF##25712100##91##}. To overcome these concerns, our future studies will be directed towards utilizing base editors and non-viral delivery approaches. Recent advances made in precision genome editing offers better promise in reducing these off-target effects ^{##REF##29160308##92##–##REF##36159595##94##}. Specially, adenine base editors, comprise a catalytically impaired Cas9 (nCas9) with adenosine deaminase (TadA) and enable the conversion of A•T to G•C or vice versa with high precision and efficiency without causing DNA double strand breaks ^{##REF##36122230##95##}. Base editors may exhibit some bystander effect in nearby regions with little or no off-target effects. Since we are knocking out <italic toggle=\"yes\">MYOC</italic>, this may not cause any serious issues. Our future experiments will be directed towards adapting precision genome editing for glaucoma. Several studies have recently utilized non-viral delivery platforms such as lipid nanoparticles to deliver Cas9 mRNA or protein for optimum gene editing with minimum off-target effects ^{##REF##36159595##94##, ##REF##34363036##96##–##REF##29490262##98##}. These non-viral deliveries of base editors provide a promising lead for efficient gene editing in ocular diseases with minimum off-target effects.</p>", "<p id=\"P30\">In conclusion, our studies show that LVs are highly efficient in delivering Cas9 to TM without any ocular toxicity and LV-mediated gene editing is highly efficient in reducing mutant myocilin and lowering elevated IOP in mouse model of glaucoma. Importantly, our studies lay the foundation for further development of gene editing methods to cure glaucoma.</p>" ]

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[ "<p id=\"P1\"><bold>Authors Contributions:</bold> GSZ, VCS and AFC designed research studies. SVP, BK, SR, YS, JCM, QZ, and CCS. performed experiments and analyzed data. SVP, and GSZ wrote the manuscript. VCS, AFC, TES, and QZ assisted in conducting key experiments, provided reagents, and assisted in manuscript preparation. All authors discussed the results and implications and commented on the manuscript at all stages.</p>", "<p id=\"P2\">Mutations in myocilin (<italic toggle=\"yes\">MYOC</italic>) are the leading known genetic cause of primary open-angle glaucoma, responsible for about 4% of all cases. Mutations in <italic toggle=\"yes\">MYOC</italic> cause a gain-of-function phenotype in which mutant myocilin accumulates in the endoplasmic reticulum (ER) leading to ER stress and trabecular meshwork (TM) cell death. Therefore, knocking out myocilin at the genome level is an ideal strategy to permanently cure the disease. We have previously utilized CRISPR/Cas9 genome editing successfully to target <italic toggle=\"yes\">MYOC</italic> using adenovirus 5 (Ad5). However, Ad5 is not a suitable vector for clinical use. Here, we sought to determine the efficacy of adeno-associated viruses (AAVs) and lentiviruses (LVs) to target the TM. First, we examined the TM tropism of single-stranded (ss) and self-complimentary (sc) AAV serotypes as well as LV expressing GFP via intravitreal (IVT) and intracameral (IC) injections. We observed that LV_<italic toggle=\"yes\">GFP</italic> expression was more specific to the TM injected via the IVT route. IC injections of Trp-mutant scAAV2 showed a prominent expression of GFP in the TM. However, robust GFP expression was also observed in the ciliary body and retina. We next constructed lentiviral particles expressing Cas9 and guide RNA (gRNA) targeting <italic toggle=\"yes\">MYOC</italic> (<italic toggle=\"yes\">crMYOC</italic>) and transduction of TM cells stably expressing mutant myocilin with LV_cr<italic toggle=\"yes\">MYOC</italic> significantly reduced myocilin accumulation and its associated chronic ER stress. A single IVT injection of LV_cr<italic toggle=\"yes\">MYOC</italic> in <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice decreased myocilin accumulation in TM and reduced elevated IOP significantly. Together, our data indicates, LV_cr<italic toggle=\"yes\">MYOC</italic> targets <italic toggle=\"yes\">MYOC</italic> gene editing in TM and rescues a mouse model of myocilin-associated glaucoma.</p>" ]

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[ "<title>Acknowledgments.</title>", "<p id=\"P31\">These studies were supported by the National Institutes of Health (EY026177 and EY030366) and facilitated by an NIH/NEI Center Support Grant to the University of Iowa (P30 EY025580). The authors acknowledge support from NIH grant P30EY034070 and from an unrestricted grant from Research to Prevent Blindness to the Gavin Herbert Eye Institute at the University of California, Irvine.</p>", "<title>Data Availability Statement:</title>", "<p id=\"P32\">All the datasets used and/or analysed in the present study is available from the corresponding author on reasonable request.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><title>AAV2-mediated GFP transduction in ocular tissues of C57BL/6J mice.</title><p id=\"P33\">ssAAV2, scAAV2 and scAAV2^Trp-Mut expressing GFP (2 × 10¹⁰ GC/eye) were injected in mouse eyes via IVT or IC bolus injections or slow IC infusion (n = 3 eyes each). GFP expression was examined by confocal imaging 2 weeks post-injections in anterior segment (<bold>A</bold>) and retina (<bold>B</bold>). Non-injected eyes serve as control for background fluorescent intensity. TM—trabecular meshwork; SC—Schlemm’s canal; CB—ciliary body; C—cornea; I—iris. White arrows show TM.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><title>LV-mediated GFP transduction in ocular tissues of C57BL/6J mice.</title><p id=\"P34\">LV particles expressing GFP (2.5 × 10⁶ TU/eyes) were injected in mouse eyes via IVT or IC bolus injections or slow IC infusion (n = 3 eyes each). GFP expression was examined by confocal imaging 2 weeks post-injections in anterior segment (<bold>A</bold>) and retina (<bold>B</bold>). Non-injected eyes served as control for background fluorescent intensity. TM—trabecular meshwork; SC—Schlemm’s canal; CB—ciliary body; C—cornea; I—iris. White arrows show TM.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3</label><caption><title>Comparison of AAV2- and LV-mediated <italic toggle=\"yes\">MYOC</italic>editing in TM cells.</title><p id=\"P35\"><bold>(A)</bold> Representative images showing AAV2_cr<italic toggle=\"yes\">MYOC</italic>or LV_cr<italic toggle=\"yes\">MYOC</italic> treatment of TM3 cells stably expressing DsRed tagged mutant <italic toggle=\"yes\">MYOC</italic>. AAV2_cr<italic toggle=\"yes\">MYOC</italic> or LV_cr<italic toggle=\"yes\">MYOC</italic>reduces intracellular <italic toggle=\"yes\">MYOC</italic> and ER stress marker GRP78 (cyan) (scale bar = 50 mm; n=3). <bold>(B)</bold> Quantitative analysis of fluorescent intensities demonstrates a significant reduction of MYOC fluorescence in both LV and AAV2-Cr<italic toggle=\"yes\">MYOC</italic> treated TM cells. For GRP78 immunostaining, only LV_cr<italic toggle=\"yes\">MYOC</italic> cells showed significant decrease. Unpaired (two-tailed) student <italic toggle=\"yes\">t</italic> test, with *<italic toggle=\"yes\">p</italic> &lt; 0.05, **<italic toggle=\"yes\">p</italic> &lt; 0.01, ***<italic toggle=\"yes\">p</italic>&lt; 0.001 and ****<italic toggle=\"yes\">p</italic> &lt; 0.0001. Quantitative data represented as mean ± SEM.</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4</label><caption><title>Effect of AAV2- and LV-mediated <italic toggle=\"yes\">MYOC</italic>editing on ER stress markers in TM cells.</title><p id=\"P36\"><bold>A)</bold> Representative Western blot showing decreased protein levels of MYOC, GRP78, GRP94, and CHOP predominantly in LV_cr<italic toggle=\"yes\">MYOC</italic> treated cells compared to AAV2_cr<italic toggle=\"yes\">MYOC</italic>(n = 3). <bold>B)</bold> The densitometric analysis confirms significant decrease in MYOC and associated ER stress markers with LV_cr<italic toggle=\"yes\">MYOC</italic> treatment only, with no significant effect observed in AAV2_cr<italic toggle=\"yes\">MYOC</italic> treated cells. Unpaired (two-tailed) student <italic toggle=\"yes\">t</italic> test, with *<italic toggle=\"yes\">p</italic> &lt; 0.05, **<italic toggle=\"yes\">p</italic>&lt; 0.01, ***<italic toggle=\"yes\">p</italic> &lt; 0.001 and ****<italic toggle=\"yes\">p</italic> &lt; 0.0001. Quantitative data represented as mean ± SEM.</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5</label><caption><title>LV-Cr<italic toggle=\"yes\">MYOC</italic> knockout of human <italic toggle=\"yes\">MYOC</italic> reduces mutant myocilin in TM and lowers elevated IOP in <italic toggle=\"yes\">Tg</italic>-<italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice.</title><p id=\"P37\"><bold>(A)</bold> IOP measurements in <italic toggle=\"yes\">Tg</italic>- <italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice injected with LV_Cr<italic toggle=\"yes\">MYOC</italic>. Ocular hypertensive <italic toggle=\"yes\">Tg</italic>- <italic toggle=\"yes\">MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice were injected with LV_Cas9-Null or LV_Cr<italic toggle=\"yes\">MYOC</italic> (2.5 × 10⁶ TU/eyes) and IOPs were measured weekly (n = 6 mice each; &gt;9 months old). LV_Cr<italic toggle=\"yes\">MYOC</italic> reduced elevated IOP significantly compared to ocular hypertensive LV_CrNull injected <italic toggle=\"yes\">Tg- MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice. <bold>(B)</bold> Representative images showing decreased MYOC in TM of <italic toggle=\"yes\">Tg-MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice transduced with LV_cr<italic toggle=\"yes\">MYOC</italic> (n=2). (<bold>C</bold>) Representative slit-lamp images revealed no ocular inflammation in <italic toggle=\"yes\">Tg-MYOC</italic>^{<italic toggle=\"yes\">Y437H</italic>} mice injected intravitreally with LV_cr<italic toggle=\"yes\">MYOC</italic> (2.5 × 10⁶ TU/eyes) compared to eyes transduced with Ad5_cr<italic toggle=\"yes\">MYOC</italic> (2 × 10⁶ pfu/eyes; n = 3 each). Data represented as mean ± SEM; *<italic toggle=\"yes\">p</italic> &lt; 0.05, **<italic toggle=\"yes\">p</italic> &lt; 0.01, ***<italic toggle=\"yes\">p</italic> &lt; 0.001 and ****<italic toggle=\"yes\">p</italic> &lt; 0.0001. Two-way ANOVA with repeated measures and Bonferroni post-hoc analysis.</p></caption></fig>" ]

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{ "acronym": [], "definition": [] }

[ "<title>INTRODUCTION</title>", "<p id=\"P2\">Post-acute sequelae of SARS-CoV-2 infection (i.e., long COVID) is a growing public health crisis globally.^{##UREF##0##1##} There have been over 769 million confirmed COVID-19 cases worldwide, and more than 200 million individuals infected have had or will develop persistent symptoms of COVID-19.^{##UREF##0##1##,##UREF##1##2##} Previous studies have highlighted the presence of a broad spectrum of clinical, social, and psychological consequences following primary infection with the SARS-CoV-2 virus.^{##REF##35589549##3##} Despite the public health importance, long COVID remains largely unexplained and understudied.</p>", "<p id=\"P3\">Given the complexity and variety of its presentations, one of the persistent challenges hindering long COVID research has been the successful identification of cases. As such, the National COVID Cohort Collaborative (N3C) collaboratively collected a large electronic health record repository of SARS-CoV-2-infected individuals. The group developed an advanced machine learning (ML) based phenotype for long COVID and tested it using the NIH’s All of Us (AoU) data.^{##REF##35589549##3##,##REF##37218289##4##} While an important step in identifying long COVID patients, the model relies entirely on patients’ EHR data, which is only representative of a patient’s recorded history in a medical setting. This potentially limits the algorithm’s ability to detect cases.</p>", "<p id=\"P4\">COVID research is rapidly evolving; indeed, since the original N3C model was published in 2022, researchers have identified several factors that are indicative of the presence of long COVID. Multiple studies have shown that long COVID has a significant association to a person’s genetic. For example, a recent large genome-wide association analysis demonstrated that the Forkhead Box P4 gene (<italic toggle=\"yes\">FOXP4</italic>) has a significant association to long COVID.^{##UREF##2##5##} Further, there have been over 20 genetic variants identified with significant associations to COVID-19 contraction and hospitalization.^{##UREF##3##6##} Given these findings, it is plausible that these genetic variants may also have an effect on a person’s risk to have long lasting side effects after contracting COVID-19.</p>", "<p id=\"P5\">Along with genetic factors, researchers have also identified that social and lifestyle factors captured in survey and mobile device data may be associated with COVID-19 (and thus could be used to improve models). For example, findings among AoU research participants showed a statistically significant decrease in step count before and after the COVID-19 pandemic and vulnerable populations, including individuals at a lower socioeconomic status were at the highest risk of reduced activity.^{##UREF##4##7##} Further, studies demonstrated that external factors (e.g., socioeconomic indicators) are associated with the diagnosis and treatment of long COVID.^{##REF##37056649##8##} Because social determinants of health (SDOH) contribute to a persons’ susceptibility to long COVID, our team aims to leverage survey and mobile device data available in the AoU platform, which include features from four of the five domains of social determinates of health. ^{##REF##31412182##9##}</p>", "<p id=\"P6\">Given the plausibility of genetic, social, and lifestyle factors’ association with long COVID, we hypothesized that the integration of genotypes, mobile device information, and health survey data into a predictive model could improve the performance of the existing EHR-based algorithm. One of the challenges facing algorithm development has been the perceived requirement of extensive data sharing or repeatedly developing algorithms with different datasets. We sought to expedite this process by utilizing and further developing an existing algorithm. Our objective was to validate and further enhance the predictability of long COVID by leveraging the genetic, mobile device, and survey data available through AoU Workbench by building from the existing machine learning infrastructure provided from the N3C algorithm.</p>" ]

[ "<title>METHODS</title>", "<title>Data Source and Cohort</title>", "<p id=\"P23\">Data for this study were derived from the AoU data enclave (version 7) available in the preproduction environment,^{##REF##31412182##9##} which consist of EHRs, survey-based information, genotyping data, and lifestyle data (collected through mobile devices) of non-deceased adults (18 years or older) living in the United States. EHR data encompass demographics, health-care visit details, medical conditions, and prescription drug orders for each patient.</p>", "<p id=\"P24\">Our cohort inclusion criteria largely paralleled that used for the original N3C study.^{##REF##35589549##3##} Specifically, cohort members were required to have either 1) an International Classification of Diseases, Tenth Revision, Clinical Modification (ICD-10-CM) COVID-19 diagnosis code (i.e., U07.1) from an inpatient or emergency health-care visit, or 2) a positive SARS-CoV-2 PCR or antigen test.</p>", "<p id=\"P25\">We required patients to have EHR data available in AoU before and after the diagnosis date or positive test result. We excluded patients with &lt; 90 days between the positive COVID-19 test or diagnosis date and the end of the study (May 2023) to ensure sufficient follow-up to assess long COVID.^{##REF##35589549##3##} To accommodate the existing EHR data model, we gathered EHR data from AoU participants. We only gathered variables that were used in the pre-existing models and made only the exact same transformations given from the previous literature’s data cleaning protocol.^{##REF##35589549##3##} We used EHR data in the AoU preproduction environment, which follows the common OMOP standard data model. The only additional inclusion criterion was that patients must have valid data (described below) for least one other data type (i.e., genetic, survey, or mobile device data).</p>", "<p id=\"P26\">From the genetic data, we used discrete variables showing the minor allele counts (0, 1, and 2) for each genetic variant. We extracted 25 genetic variants from AoU whole genome sequencing (WGS) data version 7 (<bold>Supplemental Table 1</bold>). These variants were effect alleles identified by the COVID-19 Genetic Consortium and were significantly associated to COVID-19 infection or hospitalization due to COVID-19.^{##UREF##3##6##} Participants with genetic data were included only if they had non-null values for the genetic expression of each of our pre-specified 25 genetic variants (<bold>Supplemental Table 1</bold>).</p>", "<p id=\"P27\">From the health survey information available in AoU, we selected 32 questions which gave information that was distinctively different from the data available through our EHR data (<bold>Supplemental Table 2</bold>). These questionnaire outputs were transformed into scalar variables for model testing and training. Participants considered to have valid survey data must have completed the AoU core survey questionnaire (the basics, lifestyle, and overall health) within the time span of their EHR history in AoU. Further, participants who skipped 10% or more of the questions used for our algorithms were excluded.</p>", "<p id=\"P28\">The mobile device data collected in AoU derives from patients’ Fitbits or other smart devices, which provide measures such as heart rate, steps, and sleeping status. Several factors contributed to variability in data availability. First, patients varied in the consistency of their monitoring. Second, the timing of the COVID-19 diagnosis relative to the span of available data differed. For instance, if a patient received a COVID-19 diagnosis 3 months into a year of data, most of the that participant’s data would be after their COVID-19 diagnosis, while others who received a COVID-19 diagnosis later would have proportionally more data before their diagnosis.</p>", "<p id=\"P29\">Given this variability, our goal was to select a set of features which gave interpretative value to the mobile device data for our models and worked regardless of observation period. We narrowed our inclusion criteria down to only patients with a positive COVID-19 test or diagnosis result available in their EHR history. Long COVID is normally defined as persisting symptoms more than 28 days after a positive COVID-19 test, and tests are relatively accurate up to a week of COVID-19 contraction. Because of this, we excluded all mobile device measurements between 7 days before the COVID-19 test to 28 days after the COVID-19 test. We then created averages for the total measurements before and after this excluded time frame. For example, step counts from a patient’s mobile device data were extracted into two features for our machine learning model. The first feature measures the patients average step count per day from the start of the survey to 7 days prior to their first COVID-19 positive test. The second feature measures the patient’s average step count per day starting 28 days after the positive COVID-19 test (<bold>Supplemental Table 3</bold>).</p>", "<p id=\"P30\">Participants with mobile device data were considered only if they had sufficient data points both before and after a COVID-19 test or diagnosis date in their EHR. We required patients to have two or more data points earlier than seven days prior to their COVID-19 test or diagnosis, and two or more data points following 28 days after their positive COVID-19 test or diagnosis date. Further, participants’ measures had to fall within a clinically reasonable average range for heart rate, step count, and average minutes asleep for us to consider their mobile device data reasonable. Specifically, we applied exclusion criteria based on the reasonability of a participants’ average health metrics, by giving each variable a range and excluding data that fell over that range. The reasonable ranges for each metric are as follows: average heart rate &lt; = 100bmp, minimum heart rate (for a single day) &lt; = 125bpm, median minutes asleep &lt; = 600, and median steps &lt; = 100000.</p>", "<p id=\"P31\">Data cleaning, feature engineering, and model training were completed using AoU Researcher Workbench cloud computing.</p>", "<title>Training, Testing, and Validating Sets</title>", "<p id=\"P32\">We first reserved the patients with all data sources (i.e., EHR, survey and/or mobile device, and genetic data) for validation. We named this cohort the gold standard testing set, and this set was used to evaluate the combined prediction given from all the model’s combined predictive output. From the remaining participants, we separated patients based on available data, and trained each model with 75% of the patients with that data type available. This data set is referenced later as the training data set. The corresponding 25% of remaining patients with that data were designated as a preliminary testing set. The preliminary testing set was used to test the output of a single model iteratively in the model training process (##FIG##0##Fig. 1##).</p>", "<title>Transformation of Training Data</title>", "<p id=\"P33\">While long COVID is a common disease, the associated ICD code U09.9 is quite rare and under-utilized in clinical practice, and it was not available before October 2021. Because long COVID is a rare diagnostic billing codes, the rarity of true cases makes training machine learning models difficult. Given the limited scope of our data, there is a restrictively small number of true long COVID cases to train our models. However, the N3C model was created using a more robust dataset. Therefore, to increase the number of cases in our training set, we also counted patients that the N3C model gave a predictive probability value of &gt; 0.9 as a long COVID case. Thus, we defined long COVID cases for our training set as participants with either (1) the U09.9 diagnostic code or (2) a predicted probability for having long COVID of &gt; 0.9 based on the original N3C EHR algorithm.</p>", "<p id=\"P34\">Although this definition is helpful for training new models on limited data sets, it is still important to maintain some level of accuracy for our total evaluation set. For this reason, the definition for long COVID in our testing sets remained unchanged, and we evaluated the performance of our models only using the U09.9 diagnostic code for long COVID.</p>", "<title>Model Design and Machine Learning Infrastructure Selection</title>", "<p id=\"P35\">We implemented the N3C model to determine baseline performance for all data sets. We then trained two models independently: (1) genetic data alone, and (2) survey and mobile device data. We grouped mobile device data and survey data due to their high correlation and aptitude for missing elements. Following the N3C approach, we used the existing XGBoost algorithm for EHR data. We then selected the most appropriate ML infrastructure for the remaining data sources among Logistic Regression, Polynomial Regression, Random Forests, Convolutional Neural Networks, K Nearest Neighbors, Support Vector Machines, and Extreme Gradient Boosting Trees (##FIG##0##Fig. 1##). We first ran preliminary tests to find the most effective ML infrastructure to represent each data type individually. Our preliminary tests consisted of calculating the AUROC score for each model using preliminary testing data and comparing the base performance of different ML infrastructures. Then a different model was created for each ML infrastructure, and the best infrastructure was picked based on the leading AUROC score for each data type. Given the missingness of the survey and mobile device data, extreme gradient boosting (XGBoost) was the best performing model type for this data. The genetic data alone was found to have the highest AUROC score when using a convolutional neural network (CNN) machine learning infrastructure.</p>", "<title>Predictive Method</title>", "<p id=\"P36\">The individual predictive outcomes from our independent models were combined using a weighted average function to give a single overall outcome for our combined methodology. Our weighted average determined a person’s likelihood of long COVID by taking each of models results and multiplying it by a given proportional weight. The weights were combined in a sum for overall confidence using the equation below:\n</p>", "<p id=\"P37\"> represents the participant’s data, with subsets of the participant’s data represented by a subscript. represents the probabilistic outcome of a machine learning model with , and representing the outcomes of the EHR, genetic, and survey models, respectively. The non-negative constants , and are proportional weights given to the predictive outcome of each model. The total output is given as a percentage that represents the overall predictive method described as a probability. For this reason, , and are fitted using the equation below.\n</p>", "<p id=\"P38\">These proportional weights were used as a fine-tuning parameter for our predictive method. By taking the sum of these products, the result is a single probability percentage for a patient’s risk for long COVID.</p>", "<p id=\"P39\">To calculate the optimal weights for these parameters, we designed a function to iterate through possible weights for , and . The function found the combination of weights that gave the best AUROC score for the preliminary testing set, accurate to the nearest hundredth.</p>", "<title>Cross Validation</title>", "<p id=\"P40\">We created a set of parameters for training models and validated our results using an original cross validation method. We trained each model separately using a 5-fold cross validation method. Then, we divided our Gold-Standard Evaluation cohort into 4 groups and cycled through portions of the Gold-Standard Evaluation set to evaluate the prediction given by each combination of models. Because of this, our final validation used 100 unique iterations of our data to train, test, and validate the predictive method (##FIG##1##Fig. 2##). We assessed AUROC, positive predictive values, specificity, and sensitivity for each iteration and then the average of the 100 iterations for each model. We used a two-sample t-test for difference in sample means to assess the significance of difference between the AUROC.</p>", "<p id=\"P41\">Further, to assess the difference in the specificity for our predictive models, we created an optimization function to choose a threshold for each model so that the results would have a resulting sensitivity closest 0.7. Then, we did a two-sample t-test for difference in sample means for the specificity of each predictive model’s outcome.</p>", "<title>Assessing Individual Predictive Features</title>", "<p id=\"P42\">Using Shapely as an analysis tool for ML methods, we determined the most important features in the final predictive method based on Shapley values and the relative direction for each new feature. To do so, we transformed each model’s Shapely values by dividing each value by the sum off all Shapely values for the model. The resulting proportional feature contribution for each model were then combined by multiplying each model’s set of Shapely values, by the model’s relative contribution to the overall predictive method.</p>" ]

[ "<title>RESULTS</title>", "<title>Study population</title>", "<p id=\"P7\">Our final cohort included 17,755 AoU participants out of which 976 had long COVID. The data availability for the cohort is represented in ##FIG##2##Fig. 3##. Our genetic model was trained on a total of 3,145 participants with 55 long COVID cases. The survey and mobile device model was trained with a total of 14,368 participants, 907 of which were long COVID cases. The gold standard test cohort included 944 participants who had both genetic and survey-based data. Of these participants, &lt; 20 participants were long COVID cases.</p>", "<title>Machine Learning Infrastructure Selection and Combined Model Weighting</title>", "<p id=\"P8\">Our findings showed that XGBoost was the most effective method for representing survey-based data, and CNN was the most effective method for modeling genetic data to predict a long COVID diagnosis.</p>", "<p id=\"P9\">When we calculated optimal weights for the combined model parameters, the function found the combination of weights giving the best AUROC score for the preliminary testing set:\n</p>", "<title>Model Results</title>", "<p id=\"P10\">We used the 100 predictions generated in cross validation to create average AUROC Curves (##FIG##3##Fig. 4##). We also assessed average positive predictive value, specificity, and sensitivity for each of the models (##TAB##0##Table 1##).</p>", "<p id=\"P11\">The AUROC average improved when new data type was added to the algorithm’s predictive method. The addition of survey and mobile device data to the prediction produced the largest increase, raising the AUROC score from an average of 0.721 to an average of 0.842. This increase was significant (p = 0.01) in a two-sample t-test. The combined addition of survey, mobile, and genetic data to the prediction further raised the average AUROC score to 0.855, but this increase was not found to be a statistically significant difference.</p>", "<p id=\"P12\">While Sensitivity was held as a constant, the specificity also increased for predictive methods that used more data sources. The N3C originally had an average specificity of 0.700. Our final model using EHR, genetic, survey, and mobile device data has a corresponding specificity of 0.867. From this, we see that the predictive method grew in specificity with the addition of new data sources, and that our final predictive model grew drastically in its ability to specify between long COVID cases and controls.</p>", "<title>Fairness Analysis</title>", "<p id=\"P13\">Access to healthcare and social determinates can often contribute not only to a person’s likelihood to receive a diagnosis, but further their likelihood to be accurately categorized by a machine learning based predictive algorithm. For this reason, we tested our machine learning model’s performance across a few social determinants that were available in the AoU survey questionnaire to see if the model performed better on certain cohorts of participants. We tested performance of our model across sexual orientation, annual income, and education level.</p>", "<p id=\"P14\">We found significant differences in the model’s performance in participants whose annual income is greater than 50,000 US dollars (average AUROC = 0.841) and participants whose annual income is less than or equal to 50,000 US dollars (average AUROC = 0.947). Similarly, we found that our model performed better on participants who have not attended college for at least one year of their life (average AUROC = 0.908) than participants who have attended a full year of college (average AUROC = 0.840). We found a small but not significant change in model performance between straight participants (average AUROC = 0.845), and sexual minority participants (average AUROC = 0.925).</p>", "<title>Features</title>", "<p id=\"P15\">Along with assessing models with additional information, we also identified the twenty features that contributed the most predictive capacity to the models (##FIG##4##Fig. 5##). Of the twenty features with the most relative proportionate contribution to the prediction, ten of the features were existing features from the N3C XGBoost model using EHR data.^{##REF##35589549##3##} These features also comprised all of the top three features in relative proportionate contribution which were post covid outpatient utilization, age, and post covid inpatient utilization. The other ten features with high feature importance all came from the survey and mobile device data model. The survey and mobile device features with the strongest positive correlation of this set were how many years a participant has been living in their current home, how many people under the age of 18 live with the participant, and the participant’s frequency of alcohol consumption in the last year. Features with a negative correlation included Average Pain in the last 7 days, median steps after a positive COVID19 diagnosis, general health rating, general social health rating, and average heart rate after a positive COVID19 diagnosis.</p>" ]

[ "<title>DISCUSSION</title>", "<p id=\"P16\">Our findings show that combining multiple data sources using a weighted averaging method can increase the overall accuracy of the existing predictive N3C predictive algorithm. The inclusion of different data types in AoU led to a direct increase in the AUC ROC score of our model. The approach (i.e., using an existing model, but adding new data sources by creating independently run models) allowed us to incorporate the existing predictive strength of the N3C model, which had a larger and more robust training group for EHR data. Additionally, the functional interpretation of our algorithm’s features shows that predictive models incorporating characteristics captured in AoU survey/mobile data improve performance most. When isolating the factors contributing most to the combined predictive model, the N3C components remained the most significant, but the survey and mobile data also played a substantive role. Notably, the genetic variants were not among the top contributors.</p>", "<p id=\"P17\">The inclusion of different data types led to improved accuracy of our overall prediction, even with data types being trained and modeled separately. These results show that well-crafted existing models can be improved by adding predictors from a smaller, more selective dataset in certain circumstances. The success of this method (i.e., starting with an original model created using a powerful dataset, and adding new independently trained models incorporating more recently available new datatypes) demonstrates that we can leverage existing Machine Learning algorithms and improve on the predictive outcome using new data types without requiring extensive data sharing or repeated algorithmic development.</p>", "<p id=\"P18\">The models exhibited fairness for predicting long COVID risk across various metrics by sexual orientation, annual income, and education level. For instance, both the AUROC and PPV were both higher among sexual minority participants compared to straight participants and the overall sample. This may be attributed to the deliberate oversampling of individuals from underrepresented communities in biomedical research by AoU, highlighting the significance of representation from marginalized communities in developing prediction algorithms to enhance model fairness. However, it is important to recognize that evaluating fairness based on binary demographic characteristics is an oversimplification. Health inequities often result from complex systems of intersecting power (e.g., heterosexism, racism) and necessitate an intersectional approach to evaluating model fairness when deploying algorithms to ensure equitable outcomes in mitigating the risk of long COVID.^{##UREF##5##10##–##REF##37600144##13##}</p>", "<p id=\"P19\">Many phenotypes have utilized Health Survey Data in AoU to increase the scope of data utilized for predictive models. Health survey data in AoU is collected on enrollment in the study and covers a wide range of information including overall health, living situation, education, income, alcohol and drug usage, home ownership, marital status, sex, insurance, and birthplace.</p>", "<p id=\"P20\">Although long COVID has considerable health concerns for many patients, there are striking differences between the diagnosis and treatment of long COVID.^{##REF##37217471##14##} Due to this challenge, an application of a machine learning algorithm could be a valuable clinical aid to identifying patients with long COVID and assessing susceptibility among a wide range of patients. Further, the interpretation of feature contributions in our predictive model highlights the relative high importance of survey data and a lesser observable impact of known genetic factors to a person’s risk for long COVID. By implementing similar surveys in hospitals, our algorithm could be used as a screening tool for clinicians to anticipate long term health complications for patients at risk for long COVID.</p>", "<p id=\"P21\">This study has some limitations. Our original models were trained in part to make use of the output from the N3C teams machine learning model. Because of this, the biases and limitations present in the N3C model likely carry over to our predictive method. Our algorithm based on AoU data has not been validated with any external datasets. Further there are many limitations on the data used to create our algorithm. We only used preidentified COVID-19 related genetic variants instead of whole genome sequencing data. As such, the genetic contribution is limited known genetic indicators of COVID-19 hospitalization and contraction. This validation of this predictive method represents a particular challenge given that the survey data required for assessment limits the overall study cohort; however, these findings may also encourage the collection of this type of data on a broader scale. AoU data contains its own limitations as well, including an age restriction of &gt; = 18 years, non-universal observational periods for each participant, and occasional missing data points (more prominent in the survey data). Finally, because long COVID is a relatively new disease, diagnosis in a patient’s medical history is uncertain, potentially causing variability in the accuracy of our data.</p>", "<p id=\"P22\">Despite these limitations, this algorithm can improve the power of predicting long COVID risk for patients by leveraging genetic, survey, mobile device, and EHR data. Further, the method for our algorithm’s development reached some success in improving the performance of an existing model using new data. This means others could use the same method to improve the performance of an existing model by leveraging the power of new data sources as they become available.</p>" ]

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[ "<p id=\"P1\">Over 200 million SARS-CoV-2 patients have or will develop persistent symptoms (long COVID). Given this pressing research priority, the National COVID Cohort Collaborative (N3C) developed a machine learning model using only electronic health record data to identify potential patients with long COVID. We hypothesized that additional data from health surveys, mobile devices, and genotypes could improve prediction ability. In a cohort of SARS-CoV-2 infected individuals (n=17,755) in the All of Us program, we applied and expanded upon the N3C long COVID prediction model, testing machine learning infrastructures, assessing model performance, and identifying factors that contributed most to the prediction models. For the survey/mobile device information and genetic data, extreme gradient boosting and a convolutional neural network delivered the best performance for predicting long COVID, respectively. Combined survey, genetic, and mobile data increased specificity and the Area Under Curve the Receiver Operating Characteristic score versus the original N3C model.</p>" ]

[]

[ "<title>Acknowledgements:</title>", "<p id=\"P43\">This study was supported under award numbers R01 GM139891, R01AG069900, and U01 HG011181.</p>", "<title>Data Availability</title>", "<p id=\"P44\">The All of Us dataset can be accessed through the Researcher Workbench by following the detailed data application process outlined at <ext-link xlink:href=\"https://www.researchallofus.org/\" ext-link-type=\"uri\">https://www.researchallofus.org</ext-link>.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><title>Training, Testing, and Validation Set Parameters.</title><p id=\"P49\">Flowchart describing how the AoU data was divided into sets based on data availability.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><title>Model Output Sources and Predictive Method.</title><p id=\"P50\">Flowchart showing how the predictive components of separate models are combined into a single predictive outcome.</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3</label><caption><title>Cohort Data Availability.</title><p id=\"P51\">Venn diagram showing the data available for AoU participants.</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4</label><caption><title>Average ROC of Models.</title><p id=\"P52\">Shows the Average ROC curve for each combination of models from cross validation results. Shaded area represents the standard deviation of the ROC curve, and color represents models used.</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5</label><caption><title>Top Twenty Predictive Features in the Combined Model.</title><p id=\"P53\">The relative contribution of the 20 most powerful features in our predictive method. Color of each bar represents the data source of the feature.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\"><label>Table 1</label><caption><p id=\"P54\">Predictive Model Results</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">Model</th><th align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">AUC ROC Score</th><th align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">Specificity</th><th align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">Sensitivity</th><th align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">PPV</th></tr></thead><tbody><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">EHR, Genetic, and Survey Data</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.855 +/− 0.060</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.867 +/− 0.130</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.692 +/− 0.021</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.149 +/− 0.082</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">EHR and Survey Data</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.842 +/− 0.064</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.869 +/− 0.115</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.698 +/− 0.026</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.145 +/− 0.077</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">EHR and Genetic Data</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.745 +/− 0.052</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.636 +/− 0.122</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.686 +/− 0.011</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.038+/− 0.015</td></tr><tr><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">EHR data alone</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.721 +/− 0.053</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.700 +/− 0.078</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.686 +/− 0.011</td><td align=\"left\" valign=\"middle\" rowspan=\"1\" colspan=\"1\">0.043 +/− 0.010</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN2\"><p id=\"P45\">Declarations</p><p id=\"P46\">The authors declared no competing interests for this work.</p></fn><fn id=\"FN3\"><p id=\"P47\">Code Availability</p><p id=\"P48\">The source code associated with this study is publicly available at: <ext-link xlink:href=\"https://github.com/The-Wei-Lab/AoU-COVID-Machine-Learning\" ext-link-type=\"uri\">https://github.com/The-Wei-Lab/AoU-COVID-Machine-Learning</ext-link></p></fn></fn-group>", "<table-wrap-foot><fn id=\"TFN1\"><p id=\"P55\">EHR = electronic health record; AUC = area under the curve; ROC = receiver operating characteristics; PPV = positive predictive value</p></fn></table-wrap-foot>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Background</title>", "<p id=\"P5\">Ensuring access to high-quality maternal health care across each stage of the perinatal continuum of care is critical for reducing maternal morbidity and mortality and improving pregnancy and postpartum experiences. India has made dramatic progress in reducing maternal mortality over the past three decades with an 81% reduction from 556 per 100,000 live births in 1990 to 103 per 100,000 live births in 2017–2019.^{##UREF##0##1##–##UREF##2##3##} Supported by a series of Indian-government-initiated maternal and neonatal health schemes,^{##REF##34011316##4##–##UREF##3##6##} including an extensive community health workers program,^{##REF##25025872##7##–##UREF##5##9##} national data illustrate continued improvement in access to key reproductive health and perinatal care points, with current estimates as follows: early antenatal care receipt (70.0%), a minimum of four antenatal care visits (58.1%), institutional births (88.6%), maternal and child postnatal check-up by skilled health personnel within two days of childbirth (78.0% and 79.1%, respectively) and decreases in unmet need for contraception (9%).^{##UREF##6##10##} While basic measures are up, indicators of high-quality care, such as maternal consumption of iron and folic acid (26.0% for 180 days or more), anemia among pregnant women (52.2%), and postnatal checks beyond the first week of birth remain below recommendations.^{##UREF##6##10##} Additionally, notable deficits persist in equity across socioeconomic status and rurality.^{##REF##34289804##11##} Thus, intensified efforts are required to address quality and equity.</p>", "<p id=\"P6\">Significant disruptions in the perinatal care continuum occur postpartum, where limited care is provided beyond the first two days after childbirth. High-quality postpartum care and social support has been associated with reductions in maternal and neonatal mortality,^{##UREF##7##12##,##REF##26479476##13##} and increased engagement throughout the postpartum period in behaviors promoting maternal (e.g., appropriate care-seeking for maternal postpartum health concerns, postpartum contraceptive adoption) and newborn health (e.g., exclusive breastfeeding and vaccination acceptance).^{##UREF##8##14##,##REF##29683911##15##}</p>", "<p id=\"P7\">Innovations in postpartum care provision and linkage are needed to overcome persisting barriers to postpartum care access and improve postnatal health knowledge and evidence-based behaviors. Prevalent barriers include poverty, low education, social norms around health-related decision-making, gender norms around mobility, lack of health insurance, costs, and perceptions of poor quality or lack of benefit of services.^{##UREF##8##14##–##UREF##9##16##} Group-oriented mHealth (or digital health) approaches may help mitigate the dual impact of the limited care access and lack of social support that characterize this period. Furthermore, group mHealth approaches are promising logistically, particularly given current high mobile phone penetration and low cost. Their potential builds on the evidence bases of group participatory learning and action models and mHealth, including interventions which have successfully increased antenatal care, infant vaccination, and perinatal social support for improved mental health,^{##REF##15308962##23##–##REF##33485310##30##} improved health and social support outcomes when applied to other health topics,^{##REF##33750340##31##,##REF##31178636##32##} and acceptability and costing of mHealth interventions.^{##UREF##13##33##,##UREF##14##34##}</p>", "<p id=\"P8\">To address these needs, we developed a group intervention called <italic toggle=\"yes\">MeSSSSage</italic> (Maa Shishu Swasthya Sahayak Samooh meaning maternal and child health support group) to increase perinatal women’s knowledge, refer them to in-person care as needed, and to connect them with a virtual social support group of other mothers with similarly-aged infants through weekly calls and text chat. Informed by the capabilities and motivation constructs of the capabilities, opportunities, motivation and behavior (COM-B) framework,^{##UREF##9##16##,##REF##19184385##17##} the intervention targeted knowledge of health-promoting behaviors and parental self-efficacy and empowerment to impact health knowledge, behaviors, and maternal and child health outcomes (##FIG##0##Fig. 1##). Our development process included two iterative rounds of pilot testing, the first to inform key platform and design factors,^{##REF##35767348##18##} and the second to understand feasibility and acceptability of our revised model.^{##UREF##10##19##}</p>", "<p id=\"P9\">In the current analysis, we sought to describe the preliminary effectiveness of the <italic toggle=\"yes\">MeSSSSage m</italic>Health education and social support intervention on maternal health knowledge, behaviors and outcomes at six months postpartum. In this analysis, we focus on changes in maternal health knowledge from pre to post-intervention; postpartum health care seeking and receipt; postpartum physical health, mental health, and functional mobility status; and initiation of postpartum contraception to space or limit births.</p>" ]

[ "<title>Methods</title>", "<p id=\"P10\">The <italic toggle=\"yes\">MeSSSSage</italic> mHealth educational and social support intervention pilot study occurred within Boothgarh district, Punjab state, northern India. Eligibility criteria for participation included age 18 or above, 28–32 weeks pregnant, living in the study area, and having no major maternal complications. This open-label pilot study employed a pretest-posttest nonrandomized control group design, with quantitative survey data collected at study enrollment and intervention completion (~ 6 months postpartum) across different combinations of intervention modalities (##TAB##0##Table 1##).</p>", "<p id=\"P11\">Our study team pre-screened potential participants using antenatal clinic registry data maintained by community health workers. Potential participants were then contacted over the phone, screened, and if eligible, invited to participate and led through an informed consent process which included a discussion of the study procedures, risks and benefits. Verbal consent was obtained from all participants. Of 397 potential participants with whom our study team attempted to contact, we successfully recruited 201 participants. Reasons for non-enrollment included the following: unable to reach the potential participant over cell phone (switched off or out of service), wrong number, no longer pregnant (miscarriage or preterm birth), inconsistent access to mobile phone, and not interested.</p>", "<p id=\"P12\">Participant engagement in the intervention lasted a total of eight months; participants were recruited, and the eight groups formed in a staggered fashion, resulting in a study timeline of August 2021 – November 2022. Participant enrollment and baseline quantitative survey administration took place between August - December 2021. Intervention implementation occurred from August 2021 through July 2022, and our endline quantitative survey was administered between May and November 2022. Participant retention across the eight-months of the study was 67% (n = 135 participants in endline survey) and ranged from 65%–78% by intervention modality group (78% for group call, 70% in the IVR and WhatsApp arm, and 65% in the control arm).</p>", "<title>MeSSSage intervention</title>", "<p id=\"P13\">A detailed description of the pilot intervention is provided elsewhere;^{##UREF##10##19##} briefly, <italic toggle=\"yes\">MeSSSage</italic> intervention modalities comprised weekly audio or audio-video group calls, group text chats, and audio educational content provided via automated IVR or application (##TAB##0##Table 1## and ##FIG##1##Fig. 2##). Weekly educational content and group calls began prenatally at 32 weeks of gestation (2 sessions) and continued weekly for six months postpartum. This pilot study assessed eight intervention groups utilizing four different intervention modality combinations: 1) app only (1 group; n = 20), 2) IVR only (1 group; n = 20), 3) Group call + Whatsapp + App (3 groups, n = 60), 4) Group call + Whatsapp + IVR (3 groups, n = 60). Data were collected from 20 control participants. 21 individuals originally enrolled in the study but withdrew before intervention start and were replaced. If in group call or text chat a physical examination for mother or infant was needed, they were referred to a health provider. All participants, including the control arm, received the standard of care which in this setting, comprised of community health worker-led home visits, counseling, and immunization services.</p>", "<p id=\"P14\">For the purposes of analysis, we collapsed study participants into two intervention groups because of our primary interest in understanding the differences in asynchronous (groups 1 and 2 described above) vs. synchronous communication (groups 3 and 4) modalities and intervention engagement. We expected group call participants to experience greater social connectedness than those receiving just the educational app or IVR who received asynchronous information without connecting directly with others.</p>", "<title>Study measures</title>", "<p id=\"P15\">Pre and post-intervention data were collected through interviewer administration over the phone.1 Participant sociodemographic characteristics collected at pre-intervention included age, relationship status, educational attainment, religion, caste, ration card and type, parity, and mobile phone ownership. Maternal health knowledge assessed both pre- and post-intervention focused on assessment of nine key maternal health danger signs during pregnancy and childbirth (prolonged labor; excessive bleeding; convulsions; swelling of hands, body or face; high fever; foul-smelling vaginal discharge; severe pain in lower abdomen; uterine water discharge; or other), twelve key maternal health danger signs during postpartum (excessive bleeding; convulsions; swelling of hands, body or face; high fever or severe headache; foul-smelling vaginal discharge; severe pain in lower abdomen; chest pain or shortness of breath; calf pain, redness or swelling; increased perineal pain; problems urinating or leaking; swollen, red or tender breasts or nipples; severe depression or suicidal behavior; or other), and seven key preparations for institutional birth (hospital selection, contact for transportation vehicle, clothes and pads for mother and infant, identify people to accompany mother, save or arrange for money, prepare documents required for birth, identify someone to take care of house during absence, or other). Endline assessment of maternal health behaviors and status focused on postpartum health care use, overall postpartum health and functional mobility, postpartum mental health (i.e., level of depressive and anxiety symptoms, assessed using the Edinburgh Postnatal Depression Scale and the Generalized Anxiety Disorder-7 scale),^{##REF##33177069##20##,##REF##16717171##21##} and current and intended postpartum contraceptive use.</p>", "<title>Ethical approvals</title>", "<p id=\"P16\">This study received approval from the Health Ministry Screening Committee of the Indian Council of Medical Research and Health Department of Government of Punjab. The study protocol was approved by the University of California, San Francisco Institutional Review Board (19–299723); the Ethics Committee of the Post Graduate Institute of Medical Education and Research (IEC-03/2020 – 1567); the Collaborative Research Committee of the Post Graduate Institute of Medical Education and Research (79/30-Edu-13/1089–90); and the Indian Council of Medical Research (ID 2020–9576). Informed consent was obtained by all study participants.</p>", "<title>Data analysis</title>", "<p id=\"P17\">Given our primary focus on change across time, we limited the analytic sample for this paper to participants with both baseline and endline data (n = 135). We first compared the sociodemographic characteristics of the three analysis arms (group call, other modes (IVR/WhatsApp), and control) through identification of standardized differences.^{##REF##19757444##22##} Because we identified significant differences across groups in the distribution of age, age at marriage, household composition, educational attainment, household income and ration card, mobile phone ownership and smart phone access, we employed inverse probability weighting to make participants comparable across arms. This approach is akin to direct standardization and considers multiple differences in underlying demographics between the arms at baseline.^{##REF##15308962##23##,##REF##25030033##24##}</p>", "<p id=\"P18\">We summarized sociodemographic characteristics using proportions and means of the matched, reweighted study population surveyed pre- and post- implementation of the <italic toggle=\"yes\">MeSSSage</italic> intervention stratified by three arms (group call, other intervention modes (IVR/App), and control). We then assessed the effect of being in each intervention arm (either group call, other, or control arm) on primary outcomes (changes in knowledge of maternal danger signs during pregnancy or postpartum, planning for facility birth planning, and knowledge of family planning methods) using mixed effects linear regression including a random intercept for participant with robust standard errors to adjust within individual clustering due to the longitudinal structure of the data. The difference-in-difference (<italic toggle=\"yes\">Beta</italic>) is the interaction term between a categorical variable denoting the time (before vs. after the intervention was implemented) and the intervention arm (group call vs. other modes; group call vs. control; other modes vs. control). We interpreted this term differential change over time in each intervention arm compared to the reference group. For outcomes collected only at endline (postpartum health check, postpartum general health, postpartum depression or anxiety, and postpartum contraceptive method use) we analyzed the differences between the arms (group call vs other, group call vs control, and other vs control) using logistic regression. Differences where p &lt; 0.05 were considered statistically significant. All analyses are presented using weighted estimates. Data entry was done through REDCap, and all statistical analyses were conducted using Stata 15.^{##UREF##11##25##}</p>", "<p id=\"P19\">[1] For individuals who were not able to be reached for endline survey, local ASHAs were engaged to facilitate in-person quantitative survey administration.</p>" ]

[ "<title>Results</title>", "<title>Sociodemographic characteristics</title>", "<p id=\"P20\">At enrollment, study participants were mean age 26.8 years (mean 26.8, SD 3.9) and almost all were married (99.3%; ##TAB##1##Table 2##). Most women had either completed a high school education (44.8%) or higher (44.3%). Nearly two-thirds of the sample belonged to the Sikh religion (65.3%) and one-third to marginalized caste (scheduled caste and scheduled tribe; 36.4%). Approximately half of participants (48.0%) possessed a ration card, an official government document given to eligible poor families to get subsidized food grains from government fair price shops. Parity was one (53.3%) or more (46.8%). Mobile phone ownership at the household-level was near-universal (99.2%), and most women owned their own phone (92.5%). On average, participants achieved 5.5 antenatal care visits, 65% achieving the Indian national guideline of four or more antenatal care visits (not shown), and most (83%) of participants reported initiating antenatal care in the first trimester of pregnancy.</p>", "<title>Knowledge of maternal danger signs</title>", "<p id=\"P21\">Despite increases noted across time, maternal danger signs during pregnancy/at childbirth and in the postpartum period remained relatively low (##TAB##2##Table 3##). Of eight pregnancy/childbirth and twelve postpartum risk factors, the mean number known ranged from mean 1.13 to 2.05 at baseline and 0.79 to 2.10 at endline across groups (##TAB##1##Table 2##; Table S1). Assignment to the group call intervention was associated with a small but significantly greater increase in the mean number of danger signs known during pregnancy/at childbirth when compared to those in the other intervention group (mean difference 0.94, 95% CI 0.15–1.73) and those in the control group (mean increase 0.89, 95% CI 0.25–1.52), and in the mean number of postpartum maternal danger signs known (mean increase 0.63, 95% CI 0.02–1.24). No differences were identified between other intervention group versus control participants.</p>", "<title>Knowledge of planning for institutional delivery steps</title>", "<p id=\"P22\">Knowledge related to planning for institutional delivery (steps known) was similarly low but increased over time (##TAB##2##Table 3##). Of seven steps total, the mean steps known ranged from 0.89 to 1.20 at baseline and 1.31 to 2.07 at endline. Differences in change over time among group call participants were noted only in comparison to the control group, with a statistically significant mean increase of 1.14 (95% CI 0.46–1.82) when compared to the control group; no other group differences were identified.</p>", "<title>Knowledge of family planning methods</title>", "<p id=\"P23\">Family planning method knowledge was low and increase over time only for group call participants (##TAB##2##Table 3##). Of eight family planning methods, the mean number of methods known at baseline ranged from 1.39 to 1.93 at baseline and 1.34 to 2.36 at endline. Group call participants had a statistically significant mean increase of 0.51 (95% CI 0.02–1.00) when compared to those in the other intervention group.</p>", "<title>Postpartum health care, physical and mental health</title>", "<p id=\"P24\">Achievement of a maternal postpartum health check with a clinical provider within six weeks of giving birth was reported by half of group call participants (50.0%), one-quarter of other intervention participants (25.7%) and one-fifth of control participants (21.0%; ##TAB##3##Table 4##)). Group call participants had nearly three-fold increased odds of postpartum health check compared to other intervention participants (OR 2.88, 95% CI 1.07–7.74). Differences observed between group call and control participants did not meet statistical significance.</p>", "<p id=\"P25\">Few participants experienced postpartum health concerns (3%, 5%, and 0%, respectively across group call, other and control groups). Self-rated excellent or very good health varied across groups at 63%, 43% and 30%, respectively across group call, other intervention, and control groups, but these differences were not statistically significant.</p>", "<p id=\"P26\">No differences across groups were noted in postpartum mental health. The proportion of individuals with potential depressive symptoms was similar across group call, other intervention, and control groups, at 46%, 41%, and 41%, respectively. Some differences were noted in anxiety symptoms which were reported by 9%, 12%, and 0% of respondents, but differences did not meet statistical significance or were untestable.</p>", "<title>Postpartum contraceptive use</title>", "<p id=\"P27\">The proportion of individuals reporting using a postpartum contraceptive method ranged from 66% in the other intervention group to 90% in the control group, with no statistically significant differences identified across groups (##TAB##3##Table 4##). Plan to use contraceptive in future was relatively high, ranging from 73% in other group to 90% in both group call and control groups, and no differences noted across groups.</p>" ]

[ "<title>Discussion</title>", "<p id=\"P28\">Our preliminary effectiveness assessment of the <italic toggle=\"yes\">MeSSSSage</italic> mHealth education and social support intervention found our group call modality had a positive impact on change in knowledge of maternal danger signs at pregnancy and childbirth, institutional delivery preparatory steps, and family planning methods as well as increased postpartum health check achievement. However, no differences were noted in prevalence of mental health symptoms. While further robust effectiveness testing will be required to determine the true effect of the <italic toggle=\"yes\">MeSSSSage</italic> intervention compared to the standard of care on key maternal and infant health outcomes, these preliminary results are promising, especially in conjunction with confirmation of feasibility and acceptability of <italic toggle=\"yes\">MeSSSSage</italic>.^{##UREF##10##19##}</p>", "<p id=\"P29\">Our findings that <italic toggle=\"yes\">MeSSSSage</italic>’s synchronous educational and social support sessions significantly increased maternal health knowledge are consistent with other literature which has identified that both mHealth interventions^{##REF##37296420##26##,##REF##27144393##27##} and interventions incorporating health within a group self-help context improve knowledge outcomes.^{##UREF##12##28##–##REF##33485310##30##} Social support has been identified as an important mechanism influencing health behaviors within both mHealth interventions and participatory group approaches, incorporates a similar emphasis on development of social support.^{##REF##33750340##31##} Our other intervention modality, which included audio educational messages only, was less effective, and this is similar to other research that identified no difference in knowledge simply through the distribution of written educational materials.^{##REF##31178636##32##} These findings and their contextualization provide further support for health education strategies employing social support, and in particular, the potential for integration of mHealth delivery into combined social support and health education interventions.</p>", "<p id=\"P30\">A major concern identified within this study was the relatively low knowledge of maternal danger signs during pregnancy/childbirth and postpartum, institutional delivery preparedness, and contraceptive methods. While knowledge across domains was higher at follow-up within our group call intervention modality, the mean number of danger signs and institutional delivery preparatory steps remained only at two and family planning methods just over one. This low level of knowledge is inconsistent with most study participants having initiated antenatal care in the first trimester and met Indian national guidelines for achievement of four or more antenatal care visits. However, other research on levels of maternal health knowledge in India supports low knowledge and educational quality on maternal danger signs even in the context of antenatal care;^{##UREF##13##33##–##UREF##15##35##} however, results are mixed.^{##UREF##16##36##} Studies have suggested knowledge deficits despite adequate care access may be due to quality concerns, including limited time with providers and lack of formal antenatal education classes, among others.^{##UREF##13##33##}</p>", "<p id=\"P31\">The largest impact associated with the group call intervention compared to other intervention participants was the three-fold increased odds of postpartum health check with a clinical provider. Given significant drops in the perinatal continuum of care achievement occurring postpartum for Indian women,^{##UREF##17##37##,##REF##22308538##38##} and the important role of postpartum care in optimizing maternal health and well-being,^{##REF##26479476##13##,##REF##25326202##39##} this is an encouraging and finding and may be a way to combat the low rates of postpartum care visits for Indian women. Future research should explore the timing and mechanism of this change, including the timing of increased postpartum visits, and mechanism of impact (e.g., whether this was due to increased knowledge about the importance of postpartum visits, changes in social norms, pressure from group dynamics, or increased accessibility).</p>", "<p id=\"P32\">Our preliminary effectiveness findings identified no impact of the group call modality on postpartum mental health status. We had hypothesized that this modality would have a mental health impact given the strong relationship between social support and maternal mental health^{##REF##36705738##40##} and the relatively high prevalence of postpartum depression reported in India.^{##REF##29147043##41##} Further inquiry into the intervention impact on mental health will be integrated into subsequent research where more robust research design and sample size will allow for assessment of effectiveness and potential mechanisms of impact.</p>", "<p id=\"P33\">In combination with our team’s findings confirming the feasibility and acceptability of multiple combinations of <italic toggle=\"yes\">MeSSSSage</italic> modalities, these findings support preliminary effectiveness of this mHealth education and social support intervention.^{##UREF##10##19##} However, since our study was designed for feasibility and acceptability assessment, a number of limitations exist relating to estimation of preliminary effectiveness. We combined multiple group modalities to estimate the impact of synchronous versus asynchronous intervention engagement and control, and due to our interest in the social support of a group call, this was our largest group. Our sample size was relatively small and characterized by baseline imbalance across groups in participant sociodemographic characteristics. Weighting was employed to overcome this imbalance; however, future research using more appropriate study designs and sample sizes for estimating intervention effectiveness will provide more robust conclusions. Given the ongoing importance of social and structural factors in perinatal health care continuity in India,^{##REF##32514389##42##} subgroup evaluation for identifying interventions capable of reducing health inequity will be important.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"P34\">Expansion of the evidence base for interventions to improve postpartum care and support are needed to overcome deficits in postpartum care seen across multiple settings. MHealth interventions such as <italic toggle=\"yes\">MeSSSSage</italic> which combine education, social support, and referral may be an important strategy for efficiently reaching this target population, particularly in locations where mobile penetration is high, as a supplement to strategic health systems improvements. Continued advancements in supportive care models may broadly contribute to reducing maternal and neonatal mortality through increasing knowledge and supporting health-promoting behaviors.</p>" ]

[ "<title>Background.</title>", "<p id=\"P1\">Significant disruptions in the perinatal continuum of care occur postpartum in India, despite it being a critical time to optimize maternal health and wellbeing. Group-oriented mHealth approaches may help mitigate the impact of limited access to care and the lack of social support that characterize this period. Our team developed and pilot tested a provider-moderated group intervention to increase education, communication with providers, to refer participants to in-person care, and to connect them with a virtual social support group of other mothers with similarly aged infants through weekly calls and text chat.</p>", "<title>Methods.</title>", "<p id=\"P2\">We analyzed the preliminary effectiveness of the pilot intervention on maternal health knowledge through 6 months postpartum among 135 participants in Punjab, India who responded to baseline and endline surveys. We described change in knowledge of maternal danger signs, birth preparedness, postpartum care use, postpartum physical and mental health, and family planning use over time between individuals in group call (synchronous), other intervention (asynchronous), and control groups.</p>", "<title>Results.</title>", "<p id=\"P3\">Participant knowledge regarding danger signs was low overall regarding pregnancy, childbirth and the postpartum period (mean range of 1.13 to 2.05 at baseline and 0.79 to 2.10 at endline). Group call participants had a significantly higher increase over time in knowledge of danger signs than other intervention and control group participants. Birth preparedness knowledge ranged from mean 0.89–1.20 at baseline to 1.31–2.07 at baseline, with group call participants having significantly greater increases in comparison to the control group. Group call participants had nearly three-fold increased odds of postpartum health check with a clinical provider than other intervention participants (OR 2.88, 95% CI 1.07–7.74). No differences were noted in postpartum depressive and anxiety symptoms.</p>", "<title>Conclusions.</title>", "<p id=\"P4\">Preliminary effectiveness results are promising, yet further robust testing of the MeSSSSage intervention effectiveness is needed. Further development of strategies to support health knowledge and behaviors and overcoming barriers to postpartum care access can improve maternal health among this population.</p>" ]

[]

[ "<title>Acknowledgements:</title>", "<p id=\"P35\">We greatly appreciate our research participants who shared their time and experiences with us. We also appreciate our intervention moderators, field teams, and community health professionals who facilitated our research. This paper is dedicated to Dr. Vijay Kumar, our close colleague, a tireless advocate for high-quality community-based maternal and child health care, who passed away in August 2023.</p>", "<title>Funding:</title>", "<p id=\"P36\">This work was supported by the National Institute of Child Health and Development grant R21HD101786 and the Government of India Ministry of Education’s Scheme for Promotion of Academic and Research Collaboration (SPARC; Project ID 66). The funders held no role in study design, data collection, and analysis, decision to publish, or preparation of the manuscript.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><title>Conceptual Framework of Intervention Context, <italic toggle=\"yes\">MeSSSSage</italic> Intervention Targets, Outcomes and Anticipated Long-Term Impacts</title><p id=\"P40\">MeSSSSage intervention targets include knowledge, social support, and postnatal care. Primary behavioral and health outcomes are italicized (postpartum depression, exclusive breastfeeding, and postpartum contraceptive uptake).</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><title>Weekly maternal and neonatal content of MeSSSage intervention</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\"><label>Table 1</label><caption><p id=\"P41\"><italic toggle=\"yes\">MeSSSSage</italic> intervention modalities</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Intervention Modality</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Description</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>Audio or video or audio group call sessions (“group call”)</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Trained moderator-led weekly group educational and social support group sessions included icebreakers/group-building activities, facilitated discussions on weekly themes (Fig. 2), and open question/discussion sessions. A gynecologist participated in one call per month prenatally, and a gynecologist and neonatologist participated in one call per month postnatally. Groups were audio only (TATA platform) or video (Zoom platform) per participant preference.</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>WhatsApp-based group text Chat</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Trained moderator-facilitated WhatsApp group text chats where educational audio and visual messages were shared weekly and group members were encouraged to ask questions for moderator and other group member response.</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>MeSSSage mobile application (app)</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Weekly educational audio messages focused on key information regarding weekly themes (Figure X). The mobile app organized sections for weekly audio messages, providing women with the flexibility to access health education content at their convenience.</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>Interactive Voice Response (ivR)</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">IVR calls were dispatched to participants once per week at designated days and times to share weekly educational audio messages.</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T2\"><label>Table 2</label><caption><p id=\"P42\">Sociodemographic characteristics of <italic toggle=\"yes\">MeSSSSage</italic> intervention participants (n = 135)</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Group Call (n = 94)</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Other(n = 28)</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Control (n = 13)</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Total(n = 135)</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Age^{<xref rid=\"TFN2\" ref-type=\"table-fn\">a</xref>}</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">26.7 (0.38)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">26.8 (0.61)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">26.7 (1.57)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">26.7 (0.33)</td></tr><tr><td colspan=\"5\" align=\"left\" valign=\"top\" rowspan=\"1\">Relationship status</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Married or domestic partnership</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">94 (100%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">27 (98%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">13 (100%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">134 (99.8%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Separated</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0 (0%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1 (1.2%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0 (0%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1 (0.2%)</td></tr><tr><td colspan=\"5\" align=\"left\" valign=\"top\" rowspan=\"1\">Educational attainment</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">None</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0 (0%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2 (7.2%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1 (6%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (1.5%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Up to secondary</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9 (9.6%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1 (3.1%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (22.2%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">13 (9.4%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Higher secondary</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">41 (43.6%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">15 (50.1%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6 (47.1%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">62 (44.8%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Diploma or higher</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">44 (46.8%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">10 (39.6%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (24.7%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">57 (44.3%)</td></tr><tr><td colspan=\"5\" align=\"left\" valign=\"top\" rowspan=\"1\">Religion</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Hindu</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">23 (24.5%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">11 (32.2%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">4 (26.3%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">38 (25.7%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Muslim</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9 (9.6%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0 (0%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (21.8%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12 (9%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Sikh</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">62 (66%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">17 (67.8%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6 (51.9%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">85 (65.3%)</td></tr><tr><td colspan=\"5\" align=\"left\" valign=\"top\" rowspan=\"1\">Caste</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">General</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">41 (43.6%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">16 (55.7%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5 (39.9%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">62 (45.2%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Schedule caste/tribe</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">36 (38.3%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6 (21.2%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6 (48.3%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">48 (36.4%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Other backward class</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">15 (16%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (9.8%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2 (11.8%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">20 (14.8%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Other</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2 (2.1%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (13.3%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0 (0%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5 (3.6%)</td></tr><tr><td colspan=\"5\" align=\"left\" valign=\"top\" rowspan=\"1\">Ration card</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Yes</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">45 (47.9%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">16 (45.7%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5 (35.9%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">66 (48.0%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">No</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">49 (52.1 %)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12 (54.3%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">8 (64.1%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">69 (52.0%)</td></tr><tr><td colspan=\"5\" align=\"left\" valign=\"top\" rowspan=\"1\">Parity</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">50 (53.2%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">14 (51.6%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">8 (60.2%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">72 (53.4%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">&gt;1</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">44 (46.8%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">14 (48.4%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5 (39.8%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">63 (46.6%)</td></tr><tr><td colspan=\"5\" align=\"left\" valign=\"top\" rowspan=\"1\">Mobile phone ownership</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Individual</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">93 (98.9%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">25 (91.6%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9 (80.6%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">122 (92.5%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Household</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">88 (93.6%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">28 (100%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">13 (100%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">134 (99.2%)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Total antenatal care visits^{<xref rid=\"TFN2\" ref-type=\"table-fn\">a</xref>}</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5.3 (4.8–5.7)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6.5 (5.77.3)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6.5 (5.1 –7.8)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5.5 (5.15.9)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Antenatal care initiated first trimester</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">78 (83.0%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">15 (83.8%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6 (80.5)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">99 (82.9%)</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T3\"><label>Table 3</label><caption><p id=\"P45\">Comparisons between pre and post-intervention maternal health-related knowledge by intervention group (N = 135)</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">Knowledge area</th><th rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">T</th><th colspan=\"3\" align=\"left\" valign=\"top\" rowspan=\"1\">Mean (95% CI)</th><th colspan=\"3\" align=\"left\" valign=\"top\" rowspan=\"1\">Group*Time Parameter (95% CI)</th></tr><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Group Call</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Other</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Control</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Group call vs. other</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Group call vs. control</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Other vs. control</th></tr></thead><tbody><tr><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">Number of pregnancy/childbirth maternal danger signs known (n = 8)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>BL</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.13 (1.01, 1.24)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.79 (1.32, 2.26)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.23 (0.81, 1.65)</td><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">0.94<xref rid=\"TFN5\" ref-type=\"table-fn\">**</xref> (0.15, 1.73)</td><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">0.89<xref rid=\"TFN5\" ref-type=\"table-fn\">**</xref> (0.25, 1.52)</td><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">−0.05 (−1.00, 0.90)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>EL</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.10 (1.87, 2.33)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.81 (1.29, 2.35)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.31 (0.91, 1.71)</td></tr><tr><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">Number of postpartum maternal danger signs known (n = 12)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>BL</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.87 (1.75, 1.99)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.05 (1.69, 2.42)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.92 (1.49, 2.35)</td><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">0.26 (−0.40, 0.93)</td><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">0.63<xref rid=\"TFN6\" ref-type=\"table-fn\">*</xref> (0.02, 1.24)</td><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">0.36 (−0.48, 1.21)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>EL</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.37 (1.16, 1.59)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.30 (0.78, 1.80)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.79 (0.36, 1.23)</td></tr><tr><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">Number of planning for institutional delivery steps known (n = 7)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>BL</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.89 (0.66, 0.98)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.01 (0.63, 1.39)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.20 (0.61, 1.79)</td><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">0.48 (−0.17, 1.12)</td><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">1.14<xref rid=\"TFN5\" ref-type=\"table-fn\">**</xref> (0.46, 1.82)</td><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">0.66 (−0.19, −1.52)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>EL</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.07 (1.84, 2.31)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.79 (1.38, 2.20)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.31 (0.87, 1.76)</td></tr><tr><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">Number of family planning methods known (n = 8)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>BL</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.93 (1.76, 2.10)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.71 (1.28, 2.14)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.39 (1.10, 1.67)</td><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">0.51<xref rid=\"TFN6\" ref-type=\"table-fn\">*</xref> (0.02, 1.00)</td><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">0.47 (−0.24, 1.18)</td><td rowspan=\"2\" align=\"left\" valign=\"top\" colspan=\"1\">−0.04 (−0.82, 0.74)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>EL</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.36 (2.12, 2.60)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.63 (1.27, 1.99)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.34 (0.80, 1.88)</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T4\"><label>Table 4</label><caption><p id=\"P51\">Postpartum health care, physical and mental health, and contraceptive use at endline by intervention group (N = 135)</p></caption><table frame=\"box\" rules=\"rows\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Group call (n = 94)</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Other (n = 28)</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Control (n = 13)</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Group call vs. other</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Group call vs. control</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Other vs. control</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>Postpartum health care</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">n (%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">n (%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">n (%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">OR 95%CI</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">OR 95% CI</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">OR 95%CI</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Postpartum health check within six weeks</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">48 (51.6%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">7 (32.4%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (31.9%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.22 (0.79, 6.25)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.27 (0.51, 9.99)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.02 (0.17, 6.04)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Postpartum health check with a clinical provider</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">47 (50%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">7 (25.7%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2 (21.0%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.88<xref rid=\"TFN10\" ref-type=\"table-fn\">*</xref> (1.07, 7.74)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3.75 (0.74, 18.96)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.30 (0.20, 8.33)</td></tr><tr><td colspan=\"5\" align=\"left\" valign=\"top\" rowspan=\"1\">\n<bold>Postpartum physical health</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Experienced postpartum health concern</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (3.3%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1 (4.5%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0 (0%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.71 (0.07, 7.52)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td colspan=\"5\" align=\"left\" valign=\"top\" rowspan=\"1\">Self-rated health</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Excellent/very good</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">59 (62.7%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9 (42.8%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (30%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.25 (0.83, 6.09)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3.92 (0.84, 18.22)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.73 (0.28, 10.46)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Good/Fair/Poor</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">35 (37.3%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12 (57.3%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">8 (70%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td colspan=\"5\" align=\"left\" valign=\"top\" rowspan=\"1\">Functional mobility</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">High difficulty (More than 1.16)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">24 (25.5%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5 (26.5%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5 (50%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.95 (0.30, 2.97)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.34 (0.09, 1.36)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.35 (0.06, 2.03)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Low difficulty (&lt;1.16 tasks, median)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">70 (74.5%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">16 (73.5%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6 (50%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td colspan=\"5\" align=\"left\" valign=\"top\" rowspan=\"1\">\n<bold>Postpartum mental health</bold>\n</td></tr><tr><td colspan=\"5\" align=\"left\" valign=\"top\" rowspan=\"1\">Postpartum depression</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Not likely</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">51 (54.3%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">17 (59.2%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">8 (58.9%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.22 (0.50, 3.00)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.21 (0.34, 4.31)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.98 (0.22, 4.35)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Possible/Likely/Probable</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">43 (45.8%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">10 (40.8%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">5 (41.1%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Postpartum anxiety</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Minimal</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">86 (91.5%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">25 (87.9%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">13 (100%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mild/Moderate/Severe</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">8 (8.6%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (12.1%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0 (0%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.67 (0.16, 2.87)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>Postpartum contraception use</bold>\n</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Currently using contraceptive method^{<xref rid=\"TFN12\" ref-type=\"table-fn\">a</xref>}</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1.27 (0.44, 3.60)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.26 (0.03, 2.29)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.20 (0.02, 2.29)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Yes</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">67 (71.3%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">14 (66.1%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">10 (90.3%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">No</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">27 (28.7%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">7 (33.9%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1 (9.7%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Would like to be using a contraceptive method ^{<xref rid=\"TFN13\" ref-type=\"table-fn\">b</xref>}</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3.67 (0.34, 39.62)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Yes</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">11 (40.7%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1 (15.8%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0 (0%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">No</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">16 (59.3%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6 (84.2%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1 (100%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\"/></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Planning to use contraceptive in future</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">82 (89.1%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">15 (73.4%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">10 (90.3%)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2.97 (0.90, 9.77)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.87 (0.09, 8.00)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0.29 (0.03, 3.30)</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group><fn fn-type=\"COI-statement\" id=\"FN1\"><p id=\"P37\"><bold>Competing interests:</bold> No potential competing interest was reported by the authors.</p></fn><fn id=\"FN2\"><p id=\"P38\"><bold>Ethics approval and consent to participate</bold>: This study received ethical approval from Institute Ethics committee of Postgraduate Institute of Medical Education and Research (PGIMER), Chandigarh and the University of California, San Francisco.</p></fn><fn id=\"FN3\"><p id=\"P39\"><bold>Availability of data and materials</bold>: The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.</p></fn></fn-group>", "<table-wrap-foot><fn id=\"TFN1\"><p id=\"P43\">Notes:</p></fn><fn id=\"TFN2\"><label>a</label><p id=\"P44\">Mean(SD)</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"TFN3\"><p id=\"P46\">Notes: Time (T): baseline (BL) vs. endline (EL)</p></fn><fn id=\"TFN4\"><label>***</label><p id=\"P47\">p &lt; 0.001</p></fn><fn id=\"TFN5\"><label>**</label><p id=\"P48\">p &lt; 0.01</p></fn><fn id=\"TFN6\"><label>*</label><p id=\"P49\">p &lt; 0.05</p></fn><fn id=\"TFN7\"><p id=\"P50\">Full model output for these analyses is presented in Table S1.</p></fn></table-wrap-foot>", "<table-wrap-foot><fn id=\"TFN8\"><label>***</label><p id=\"P52\">p &lt; 0.001</p></fn><fn id=\"TFN9\"><label>**</label><p id=\"P53\">p &lt; 0.01</p></fn><fn id=\"TFN10\"><label>*</label><p id=\"P54\">p &lt; 0.05</p></fn><fn id=\"TFN11\"><p id=\"P55\">Notes:</p></fn><fn id=\"TFN12\"><label>a</label><p id=\"P56\">among those individuals not pregnant</p></fn><fn id=\"TFN13\"><label>b</label><p id=\"P57\">among those individuals not currently using a contraceptive method</p></fn><fn id=\"TFN14\"><label>c</label><p id=\"P58\">among individuals using a contraceptive method.</p></fn></table-wrap-foot>" ]

[ "<graphic xlink:href=\"nihpp-rs3746241v1-f0001\" position=\"float\"/>", "<graphic xlink:href=\"nihpp-rs3746241v1-f0002\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"P3\">Chronic kidney disease (CKD) is a major health epidemic that affects 37 million adults in the United States and many more worldwide.^{##REF##17986697##1##,##REF##27383068##2##} Patients with CKD face significantly higher risks of cardiovascular disease and subsequent mortality compared to patients without CKD.^{##REF##15385656##3##} Left ventricular hypertrophy (LVH), which affects 50–70% of patients with early and intermediate stages of CKD and up to 90% by the time they reach dialysis, is a major cause of heart failure and in particular heart failure with preserved ejection fraction (HFpEF) in patients with CKD.^{##REF##15100371##4##} Among many factors involved in the pathogenesis of LVH and heart failure, alterations in mineral homeostasis may uniquely contribute in patients with CKD.^{##REF##27573728##5##,##REF##36104509##6##}</p>", "<p id=\"P4\">Fibroblast growth factor (FGF) 23 regulates calcium and phosphate homeostasis.^{##REF##29892265##7##} In CKD, FGF23 levels progressively rise as kidney function declines. Elevated FGF23 initially helps maintain normal serum phosphate in the setting of impaired kidney function,^{##REF##21389978##8##} but higher FGF23 is dose-dependently associated with increased risks of LVH, heart failure and mortality in patients with CKD.^{##REF##21985788##9##–##REF##27434583##11##} Mechanistically, FGF23 directly induces hypertrophic growth of cultured cardiac myocytes and contributes to development of LVH in rodents via activation of FGF receptor (FGFR) 4 and the phospholipase Cγ (PLCγ) – calcineurin – nuclear factor of activated T-cells (NFAT) signaling pathway,^{##REF##21985788##9##,##REF##26437603##12##,##REF##31758962##13##} which is a potent inducer of structural cardiac remodeling.^{##REF##14656927##14##} Human genetic data support the link between FGF23 and development of LVH and heart failure, particularly in patients at risk for developing CKD.^{##REF##35902130##15##}</p>", "<p id=\"P5\">In the pathogenesis of cardiac remodeling that culminates in LVH, mounting evidence suggests that metabolic remodeling and mitochondrial dysfunction precede most, if not all other pathological alterations.^{##REF##30124471##16##} Metabolic inefficiency and loss of coordinated anabolic activity have emerged as proximal causes of cardiac structural remodeling.^{##REF##29915254##17##} Over the past decades, numerous metabolic pathways have been implicated in cardiac hypertrophic growth and heart failure.^{##REF##36064969##18##} However, the role of cardiac metabolism in heart failure specifically in CKD has not been studied extensively, and whether CKD-specific mechanisms, including excess activation of the FGF23–FGFR4 axis, contribute to cardiac metabolic remodeling and mitochondrial dysfunction remains unknown.^{##REF##37053280##19##} In this report, we analyzed cardiac metabolism, mitochondrial composition and function in bioengineered cardio-bundles, cultured cardiomyocytes and multiple rodent models to test the hypotheses that cardiac mitochondrial dysfunction and metabolic remodeling occur in animal models of CKD; that these changes predate the development of structural cardiac remodeling; and that FGF23–FGFR4 signaling are mechanisms of cardiac mitochondrial dysfunction and metabolic remodeling.</p>" ]

[ "<title>Methods</title>", "<title>Antibodies, recombinant proteins and heparin</title>", "<p id=\"P6\">Carrier-free recombinant mouse FGF23 (2629FG025/CF) from R&amp;D systems was used at 25 ng/mL and 100 ng/mL. The isoform specific FGFR4 small molecule inhibitor BLU9931 from Selleck Chemicals (Cat. No S7819, TX, USA) was used at 10 ng/mL. Heparin solution (McKesson Corporation, NC, USA) was used at 0.2 USP/mL. Primary antibodies include sarcomeric a-actinin (EA-53, Sigma-Aldrich, USA). Secondary antibodies are Cy3-conjugated goat–anti-mouse (Cat. No. 115165166, Jackson ImmunoResearch Laboratories, PA, USA).</p>", "<title>Isolation and cultivation of neonatal rat ventricular myocytes</title>", "<p id=\"P7\">Neonatal rat ventricular myocytes were isolated from Sprague-Dawley rat pups (postnatal day 0–3) using a commercially available kit (Cat.No. LK003300,Worthington, NJ, USA) as described previously and in the supplemental methods.^{##UREF##0##20##}</p>", "<title>Fabrication of bio-engineered cardio-bundles</title>", "<p id=\"P8\">Bioengineered cardio-bundles were generated as described previously and as described in the supplemental methods.^{##REF##32857373##21##} In selected studies, FGF23 (25 ng/mL) and BLU9931 (10 ng/mL) were added to culture medium on day 7 and replenished with each media change during the additional 7 days.</p>", "<title>Measurement of contractile force and action potential propagation</title>", "<p id=\"P9\">Isometric contractile forces were measured as described previously.^{##REF##32857373##21##} In brief, cardio-bundles were immersed in Tyrode’s solution containing 1.8 mM CaCl2 and connected to a force transducer. Contractions were elicited by electric field stimulus from parallel platinum electrodes. Optical mapping of action potentials was performed as described previously.^{##REF##27723557##22##} Briefly, cardio-bundles were stained with a transmembrane voltage-sensitive dye (di-4-ANEPPS) and paced at different rates by suprathreshold point stimulus to map propagation of action potentials.</p>", "<title>Immunofluorescence and morphometry of cultured myocytes and cardio-bundles</title>", "<p id=\"P10\">Hypertrophic growth of isolated NRVM was analyzed on laminin-coated glass coverslips after 48 hours of treatment as done previously and as described in the supplemental methods.^{##REF##26437603##12##,##REF##32857373##21##,##REF##28512310##23##}</p>", "<title>Live-cell metabolic analysis</title>", "<p id=\"P11\">Mitochondrial oxygen consumption rate and glycolytic rate were determined in NRVM using the Seahorse XF Mito Stress Test Kit and the Seahorse XF Glycolytic Rate Assay Kit according to manufacturer’s protocols. All live-cell metabolic assays were performed in collaboration with Duke’s Cardiovascular Physiology Core using the metabolic flux analyzer Agilent Seahorse XF96 (Agilent Technologies), see supplemental methods for details and references.</p>", "<title>RNA isolation and quantification</title>", "<p id=\"P12\">Total RNA was extracted from hearts and cultured cardiomyocytes using a RNeasy Plus Mini Kit (Qiagen) following the manufactures’ instructions. 0.5 – 2 μg of RNA was reverse transcribed to cDNA using Applied Biosystems High-Capacity cDNA Reverse Transcription Kit (cat. No 4368813, ThermoFisher). Quantitative real-time PCR was performed with SSoAdvanced Universal Probe Supermix (Bio-rad) and sequence specific Taqman probes (ThermoFisher) as indicated in ##TAB##0##Table 1##. Samples were run in duplicates on a Quantstudio 3 (Applied Biosystems, ThermoFisher) Real Time detection instrument. Relative gene expression was normalized to expression levels of housekeeping genes β2-microglobulin (for in vitro studies) or 18S rRNA (for in vivo studies). Results were evaluated using the 2^−ΔΔCt method and expressed as mean ± standard error of the mean (SEM).</p>", "<title>Mice</title>", "<p id=\"P13\">Constitutive FGF receptor four null mice (FGFR4^−/−),^{##REF##9716527##24##} constitutive Col4a3^−/− null mice (Alport)^{##REF##8956999##25##} and constitutive FGFR4 knock in mice (FGFR4-Arg385)^{##REF##20068154##26##} were maintained on a C57Bl/6 background. Mice with inducible cardiomyocyte-specific deletion of FGFR4 (α-MHC^MerCreMer-FGFR4^flox) were generated by crossing α-MHC^MerCreMer mice^{##REF##11440973##27##} with FGFR4 floxed mice.^{##REF##28094278##28##} α-MHC^MerCreMer-FGFR4^flox mice were maintained on a C57Bl/6 background. Cre-recombination was induced by tamoxifen injections (30 mg/kg bodyweight, i.p., every 48 hours for a total of 3 injections). Dietary interventions were started 10 days after the last tamoxifen injection. Male and female mice were used in the distribution indicated in the supplemental methods.</p>", "<title>Adenine model of chronic kidney disease</title>", "<p id=\"P14\">As done before, chronic kidney disease was induced by feeding 12–16-week-old mice an adenine diet (0.15%, TD.170304 to 0.2% adenine, TD.140290; control diet TD.170303, Envigo, IN, USA)^{##REF##31575945##29##}. All mice were put on control diet for one week before study start and then mice were randomized to receive either adenine diet or control diet. After 8, 12 or 16 weeks respectively, mice were sacrificed, plasma collected and hearts were prepared for molecular and histopathologic analyses.</p>", "<title>Non-invasive assessment of kidney function</title>", "<p id=\"P15\">Glomerular filtration rate was determined non-invasively in mice using NIC kidney devices as described previously and in the supplemental methods.^{##REF##25388805##30##,##UREF##1##31##}</p>", "<title>Non-invasive and invasive assessment of cardiac function</title>", "<p id=\"P16\">All echocardiographic analyses were performed by Duke’s Cardiovascular Physiology Core using a Vevo 3100 imaging system (FUJIFILM VisualSonics, Toronto, Canada) please see supplemental methods for details.</p>", "<title>Serum chemistry</title>", "<p id=\"P17\">At study end, blood was collected from mice at the time of killing via cardiac puncture, transferred into Microvette heparin plasma tubes (Sarstedt, Germany) and centrifuged at 21,000g for 10 minutes. Plasma supernatants were collected and stored at −80 °C. Blood urea nitrogen (BUN), phosphate and hemoglobin were measured at the University of North Carolina Animal Histopathology and Lab Medicine Core as done previously with an Alfa Wassermann Vet Axcel^® Chemistry Analyzer (Alfa Wassermann Diagnostic Technologies, LLC, NJ, USA). Intact and C-terminal FGF23 levels were determined by ELISA (Immutopics, CA, USA) according to the manufacturers protocol. PTH levels were measured with the mouse PTH 1–84 ELISA kit (Immutopics, CA, USA) according to the manufactures protocol.</p>", "<title>Mitochondrial respiration</title>", "<p id=\"P18\">Mitochondrial isolation from frozen mouse hearts was performed as previously described and in the supplemental methods.^{##REF##35756609##32##} High-resolution oxygen consumption rate (<italic toggle=\"yes\">J</italic>O₂) was assessed via the Oroboros Oxygraph-2K (Oroboros Instruments, Innsbruck, Austria) as previously described,^{##REF##35756609##32##} with minor adjustments, see supplemental methods for details.</p>", "<title>RNA sequencing</title>", "<p id=\"P19\">RNA sequencing was performed in collaboration with Duke’s Center for Genomic and Computational Biology Core Facility. In brief, RNA-seq data was processed using the TrimGalore toolkit^{##UREF##2##33##} which employs Cutadapt^{##UREF##3##34##} to trim low-quality bases and Illumina sequencing adapters from the 3’ end of the reads. Only reads that were 20nt or longer after trimming were kept for further analysis. Reads were mapped to the GRCm38v73 version of the mouse genome and transcriptome^{##REF##22067447##35##} using the STAR RNA-seq alignment tool^{##REF##23104886##36##}. Please see supplemental methods for further details.</p>", "<title>Proteomics</title>", "<p id=\"P20\">Proteomics of isolated cardiac mitochondria was performed in collaboration with Duke’s Proteomics and Metabolomics Core Facility. Please see supplemental methods for detailed description and references.</p>", "<title>Metabolomics</title>", "<p id=\"P21\">Metabolomic measurements were performed at the Metabolomics Core Laboratory at Duke Molecular Physiology Institute. Please see supplemental methods for detailed description and references.</p>", "<title>Transmission electron microscopy</title>", "<p id=\"P22\">Transmission electron microscopy was performed by Duke’s Center for Electron Microscopy. Please see supplemental methods for detailed description.</p>", "<title>Study approval</title>", "<p id=\"P23\">All animal protocols and experimental procedures for adenine diet in mice, and primary cardiomyocyte isolations from neonatal Sprague Dawley rats were approved by the Institutional Animal Care and Use Committees (IACUC) at Duke University. All animals were maintained in a ventilated rodent-housing system with temperature-controlled environments (22–23°C) with a 12-hour light/dark cycle and allowed ad libitum access to food and water. All protocols adhered to the Guide for Care and Use of Laboratory Animals to minimize pain and suffering. No animals were excluded from analysis.</p>", "<title>Statistical Analysis</title>", "<p id=\"P24\">All data are presented as means ± SEM. Identification of possible statistical outliers was performed by ROUT method (Q=1%) in GraphPad Prism. P &lt; 0.05 was considered statistically significant. All data were analyzed using GraphPad Prism9 (Graphpad Software, CA, USA) followed by student’s T-tests or when appropriate by 2-way ANOVA.</p>" ]

[ "<title>Results</title>", "<title>CKD alters cardiac mitochondrial structure prior to onset of LVH</title>", "<p id=\"P25\">To investigate cardiac metabolic remodeling in heart failure due to CKD, we analyzed cardiac structure and function in wild-type mice fed 0.2% adenine diet to induce CKD.^{##REF##31575945##29##} Adenine-induced CKD caused progressive LVH over 16 weeks as indicated by increased LV mass index (LVMI), increased posterior and intra-septal wall thickness, and reduced systolic LV diameter (##FIG##0##Figure 1A##); LV function, marked by fractional shortening, was unchanged (Supplemental Figure S1).</p>", "<p id=\"P26\">After 12 weeks of consuming the adenine diet when glomerular filtration rate (GFR) was significantly reduced but no cardiac structural remodeling could be detected (##FIG##0##Figure 1B##), cardiac expression of <italic toggle=\"yes\">Nppa</italic> and <italic toggle=\"yes\">Timp1</italic> mRNA were increased, indicating that hypertrophic and fibrotic remodeling processes had already begun (##FIG##0##Figure 1C##). Interestingly, expression of the transcription factors PGC-1α and FOXO1, which are key regulators of myocardial metabolism and cardiac mitochondrial function, were also significantly increased in CKD versus controls (##FIG##0##Figure 1C##). At that time point, electron microscopy revealed grossly normal myofibrillar structures in the CKD myocardium, but mitochondria were misaligned, swollen and the cristae were disorganized, suggesting possible mitochondrial dysfunction (##FIG##0##Figure 1D##).</p>", "<p id=\"P27\">To characterize myocardial mitochondrial function, we assessed respiratory capacity across the electron transport system complexes. Respiration by both respiratory chain complex I (NADH supported) and complex II (succinate supported) was significantly increased in CKD hearts versus controls, suggesting that changes in cardiac mitochondrial function precede hypertrophic and fibrotic structural remodeling (##FIG##0##Figure 1E##). While many studies of heart failure describe reduced cardiac oxidative phosphorylation capacity, increases in mitochondrial respiration have also been reported in a rat model of right ventricular heart failure induced by pulmonary hypertension.^{##REF##17582388##37##}</p>", "<p id=\"P28\">In a second genetic model of progressive CKD that also develops LVH,^{##UREF##4##38##} we confirmed the presence of pathologic mitochondria and increased expression of profibrotic and hypertrophic markers in the hearts of C57BL6J/Col4a3^−/− mice with severe CKD (Supplemental Figure S2). These data support adverse effects of CKD on mitochondria across different models.</p>", "<title>CKD changes the cardiac mitoproteome and metabolome</title>", "<p id=\"P29\">To investigate cardiac metabolic remodeling in response to CKD, we isolated mitochondria from the hearts of 12-week adenine fed CKD mice and assessed the cardiac mitoproteome using tandem mass spectrometry (##FIG##1##Figure 2A##). Using a previously reported mitochondrial enrichment factor,^{##REF##33077793##39##} we achieved mitochondrial enrichment of approximately 75% (data not shown). Of the 781 mitochondrial genes identified across CKD and control mice, 56 were upregulated and 62 were downregulated in CKD hearts when compared to controls without CKD (##FIG##1##Figure 2A##). The downregulated proteins were significantly enriched in six different KEGG (Kyoto Encyclopedia of Genes and Genomes) pathways, fatty acid oxidation, acetyl-CoA metabolism, regulation of biosynthetic processes from pyruvate and NADPH, and anti-oxidant activity (##FIG##1##Figure 2B##, top). Upregulated proteins were involved in mitochondrial ribosomes and translation (##FIG##1##Figure 2B##, bottom). These results suggest that CKD induces functional changes in cardiac mitochondria.</p>", "<p id=\"P30\">To assess the cardiac metabolome in CKD, we performed targeted liquid chromatography-mass spectrometry (LC-MS) analysis of serum and heart tissue (##FIG##1##Figure 2C##). Several medium and long chain acylcarnitines (MLAC) were significantly altered in CKD (##FIG##1##Figure 2C##). In serum, MLAC were mostly upregulated whereas in cardiac tissue some MLACs were up and downregulated in when compared to controls. Several amino acids including arginine, phenylalanine and citrulline were increased in hearts and serum of CKD mice, whereas cardiac concentrations of branch chained amino acids including valine, leucine/isoleucine tended to be lower. In line, serum levels of branch chained keto acids were also significantly reduced in CKD. Significant changes were detected in cardiac organic acids and TCA intermediates. Pyruvate was significantly higher whereas lactate trended lower in CKD versus controls (##FIG##1##Figure 2C##). TCA cycle intermediates, including citrate, also trended to higher levels in CKD hearts (##FIG##1##Figure 2C##). Taken together, these data suggest that CKD alters fatty acid, amino acid and glucose metabolism in in the heart.</p>", "<title>FGF23-FGFR4 induce hypertrophic growth of bio-engineered cardio-bundles</title>", "<p id=\"P31\">To investigate whether elevated FGF23-FGFR4 signaling might directly contribute to the cardiac metabolic remodeling observed in CKD, we studied cardio-bundles, 3-dimensional multicellular cylindrical tissues bio-engineered from neonatal rat ventricular myocytes (NRVM) and fibroblasts. The cardio-bundles spontaneously contract and exhibit mature functional properties similar to postnatal rat myocardium.^{##REF##27723557##22##} Acute treatment of cardio-bundles with FGF23 (20 minutes) significantly increased contractility compared to vehicle, as reported previously (##FIG##2##Figure 3A##).^{##REF##28512310##23##} In contrast, chronic FGF23 treatment (7 days) significantly decreased contractility; this effect was blocked by co-treatment with BLU9931, a small molecule isoform-specific inhibitor of FGFR4 (##FIG##2##Figure 3A##), confirming the specific involvement of the FGF23-FGFR4 axis.</p>", "<p id=\"P32\">To assess electrophysiological function, cardio-bundles were paced with a voltage-sensitive dye, followed by optical mapping of the action potential propagation. Chronic FGF23 treatment prolonged action potential duration (APD) compared to controls (##FIG##2##Figure 3B##). Conduction velocity was ~32% slower in FGF23-treated versus vehicle-treated cardio-bundles, an effect that was attenuated by BLU9931 (##FIG##2##Figure 3C##). The selected dose of BLU9931 had no effects on contractile force or conduction velocity of control cardio-bundles (##FIG##2##Figure 3A##,##FIG##2##C##), ensuring specificity for FGF23 induced, FGFR4-mediated effects.</p>", "<p id=\"P33\">Seven days of FGF23 treatment stimulated hypertrophic growth of cardio-bundles as indicated by increased cross-sectional area of individual myocytes and elevated mRNA expression of the hypertrophic markers TRPC6 and RCAN1 (##FIG##2##Figure 3D##,##FIG##2##E##,##FIG##2##G##); co-treatment with BLU9931 blocked these effects. Similar to the results from CKD mice, mRNA expression of metabolic transcription factors PGC-1α and FOXO1 increased in FGF23-treated cardio-bundles (##FIG##2##Figure 3F##).</p>", "<p id=\"P34\">To investigate the mechanism of FGF23-FGFR4 induced cardiac remodeling, we profiled changes in gene expression of cardio-bundles subjected to 7 days of FGF23 treatment using RNA sequencing. Metabolic pathways were highly enriched upon FGF23 treatment, including fatty acid metabolism, adipogenesis and cholesterol homeostasis (##FIG##2##Figure 3H##). In addition, FGF23-treated cardio-bundles showed strong enrichment in molecular processes related to mitochondrial structure and function including oxidative phosphorylation, respiratory chain, organelle fission and organelle inner membrane (##FIG##2##Figure 3I##). Downregulated genes involved molecular processes related to angiogenesis, vascular development, TNFα signaling, and the P53 pathway (##FIG##2##Figure 3I##). Taken together, these results suggest that FGF23 can induce <italic toggle=\"yes\">in vitro</italic> changes in cardiac tissue remodeling that parallel those observed in mice with CKD and that these effects are mediated by FGFR4.</p>", "<title>FGF23-FGFR4 alters mitochondrial function in cultured cardiomyocytes</title>", "<p id=\"P35\">Next, we determined if FGF23-FGFR4 directly modulates substrate utilization and mitochondrial respiration in cultured cardiomyocytes. NRVM were treated with FGF23 with and without BLU9931 for 48 hours. FGF23 induced hypertrophy of NRVM as determined by significant increases in the area of immunolabeled cells and elevated mRNA expression of the hypertrophic markers, TRPC6 and RCAN1 (##FIG##3##Figure 4A##,##FIG##3##B##), as previously reported.^{##REF##26437603##12##} Co-treatment with BLU9931 blocked hypertrophic growth of cardiomyocytes, while treatment with BLU9931 by itself had no effect on NRVM (##FIG##3##Figure 4A##,##FIG##3##B##).</p>", "<p id=\"P36\">Activation of glycolysis is involved in cell growth, including cardiac remodeling.^{##REF##29929976##40##} To determine if glycolysis is directly stimulated by FGF23 or increases indirectly in response to cellular hypertrophy, we treated NRVM with FGF23 for 1 hour, before cellular hypertrophy was present (Supplemental Figure S3). Using the Seahorse XF analyzer, we evaluated the extracellular acidification rate (ECAR) in NRVM as an indirect measure of glycolysis (##FIG##3##Figure 4C##). ECAR was significantly higher in FGF23-treated than control cells; this effect was abolished by BLU9931 (##FIG##3##Figure 4C##), which had no effect on its own (data not shown). Next, we assessed the glycolytic rate of NRVM, which removes the contribution of mitochondrial CO2 to ECAR and allows more accurate measurement of glycolysis. FGF23 significantly increased basal and compensatory glycolysis, as determined by elevated total proton efflux rates (PER) and glycolysis-specific proton efflux rates, whereas BLU9931 blocked these effects (glycoPER; ##FIG##3##Figure 4C##).</p>", "<p id=\"P37\">To directly examine mitochondrial function in response to FGF23, we applied the Seahorse mitochondrial stress test assay to cultured cardiomyocytes. FGF23 significantly increased basal and maximum mitochondrial respiration in NRVM (##FIG##3##Figure 4D##). Similarly, ATP production-linked spare respiratory capacity and non-mitochondrial oxygen consumption rate (OCR) was higher in FGF23-treated cells (##FIG##3##Figure 4D##). In contrast, FGF23 significantly decreased coupling efficiency, indicating uncoupling of substrate oxidation and ATP synthesis (##FIG##3##Figure 4D##). The observed reduction in coupling efficiency was attributable to significantly increased proton leak (##FIG##3##Figure 4D##), which is the predominant mechanism for incomplete coupling.^{##REF##11053672##41##} Pharmacologic inhibition of FGFR4 with BLU9931 and inhibition of calcineurin using cyclosporin A prevented the effects of FGF23 on mitochondrial respiration, implicating the FGFR4–PLCγ–calcineurin axis in the metabolic effects of FGF23 on glycolysis (##FIG##3##Figure 4D##).</p>", "<title>Activation of FGFR4 causes cardiac metabolic remodeling independently of CKD</title>", "<p id=\"P38\">To test the hypothesis that FGFR4 activation contributes to cardiac metabolic remodeling, we investigated cardiac mitochondria in knock-in mice that express FGFR4-Arg385, which is a gain-of-function mutation of FGFR4.^{##REF##26437603##12##} As we reported previously,^{##REF##26437603##12##} there were no differences in kidney function or mineral metabolism between wild-type and FGFR4-Arg385 mice (##FIG##4##Figure 5A##, Supplemental Figure S4). Six-month-old FGFR4-Arg385 mice developed mild LVH, characterized by increased wall thickness and mRNA expression of hypertrophic and profibrotic markers, but overall LV mass, LV diameters and systolic and diastolic function were unchanged until 24 months of age when FGFR4-Arg385 mice developed overt LVH (##FIG##4##Figure 5A##,##FIG##4##B##, Supplemental Figure S4). Transmission electron microscopy revealed cardiac mitochondrial morphological changes in 6-month-old FGFR4-Arg385 mice that were similar to the changes observed in wild-type mice with CKD, including disorganized alignment and swollen mitochondrial cristae (##FIG##4##Figure 5C##). Analysis of the cardiac and serum metabolome in 6-month-old FGFR4-Arg385 mice demonstrated significant changes in organic acids, several MLAC, branch chained amino acids and branch chained keto acids in a similar pattern as observed in the CKD mice (##FIG##4##Figure 5D##–##FIG##4##F##). Taken together, these morphologic and metabolomic data suggest that expression of a constitutively active FGFR4 is sufficient to induce cardiac metabolic remodeling, which occurs prior to the onset of overt LVH and in a pattern that is similar to that observed in mice with CKD and wild-type FGFR4 expression.</p>", "<title>Global deletion of FGFR4 prevents remodeling of the cardiac mitoproteome in CKD</title>", "<p id=\"P39\">Global deletion of FGFR4 (FGFR4^−/−) protects mice from LVH caused by chronic high phosphate diet.^{##REF##28512310##23##} To test if FGFR4 deletion also protects against LVH in CKD, we subjected FGFR4^−/− mice to 16 weeks of adenine diet. As reported previously,^{##REF##31575945##29##} all mice developed CKD with elevations in serum BUN and FGF23 levels (##FIG##5##Figure 6A##, Supplemental Figure S5). Control but not FGFR4^−/− mice developed pathological cardiac remodeling (##FIG##5##Figure 6A##) and elevated cardiac expression of pro-hypertrophic (<italic toggle=\"yes\">Nppb</italic>) and pro-fibrotic (<italic toggle=\"yes\">Fn1</italic>) markers (Supplemental Figure S5). Cardiac mRNA expression of metabolic transcription factor PGC-1α was upregulated in control but not FGFR4^−/− mice (Supplemental Figure S5).</p>", "<p id=\"P40\">We isolated mitochondria from the hearts of wild-type mice and FGFR4^−/− littermates with CKD and assessed the cardiac mitoproteome with LC-MS. The 9 proteins that were significantly downregulated and the 13 that were significantly upregulated in wild-type CKD mice were unchanged in FGFR4^−/− CKD mice, including proteins of mitochondrial respiration and function (##FIG##5##Figure 6B##,##FIG##5##C##). In addition, 83 proteins were downregulated and 57 were upregulated only in FGFR4^−/− CKD mice; enrichment analysis of these differentially expressed proteins suggests that deletion of FGFR4 upregulates pathways related to NADH activity, mitochondrial ribosomes, and mitochondrial translation and downregulates pathways linked to ATP transport, protein channel and fatty acid activity. (##FIG##5##Figure 6D##,##FIG##5##E##). Taken together, these results indicate that deletion of FGFR4 attenuates pathologic changes in cardiac mitochondrial composition in CKD.</p>", "<title>FGF23-FGFR4 signaling mediates cardiac metabolic remodeling in adenine-induced CKD</title>", "<p id=\"P41\">Mice with cardiomyocyte-specific deletion of FGFR4 do not develop LVH in response to repeated short-term (5 days) FGF23 injections.^{##REF##31758962##13##} To determine if FGF23 and cardiac FGFR4 contribute to cardiac metabolic remodeling in CKD, we created mice with inducible cardiomyocyte-specific deletion of FGFR4 (α-MHC^MerCreMer-FGFR4^flox). α-MHC^MerCreMer-FGFR4^flox mice were injected with tamoxifen and cardiac FGFR4 expression was evaluated 10 days later. FGFR4 mRNA expression was significantly reduced in the hearts but not the kidneys of α-MHC^MerCreMer-FGFR4^flox mice compared to controls, indicating specific deletion of FGFR4 from cardiomyocytes (Supplemental Figure S6). After 16 weeks of adenine diet, all mice developed kidney injury to the same degree (##FIG##6##Figure 7A##, Supplemental Figure S6). Echocardiography revealed significantly lower LV mass, wall thickness and heart weight to tibia length ratio in α-MHC^MerCreMer-FGFR4^flox mice versus controls (##FIG##6##Figure 7A##,##FIG##6##C##). In addition, cardiac mRNA expression of pro-hypertrophic (<italic toggle=\"yes\">Nppb</italic>) and pro-fibrotic markers (<italic toggle=\"yes\">Timp1</italic>) were significantly decreased in experimental mice versus controls (##FIG##6##Figure 7B##). Control but not α-MHC^MerCreMer-FGFR4^flox mice developed diastolic dysfunction (##FIG##6##Figure 7C##, Supplemental Figure S6).</p>", "<p id=\"P42\">We analyzed the cardiac metabolome of α-MHC^MerCreMer-FGFR4^flox mice and littermate controls (##FIG##6##Figure 7D##). Cardiomyocyte-specific deletion of FGFR4 elevated cardiac levels of MLAC. While amino acids and branch chained amino acids remained mostly unchanged, expression levels of organic acids suggested normalization of cardiac pyruvate, succinate and lactate concentrations. Taken together, these results indicate that blocking cardiac FGFR4 attenuates cardiac metabolic remodeling in CKD.</p>" ]

[ "<title>Discussion</title>", "<p id=\"P43\">In this report, we demonstrate that cardiac mitochondrial dysfunction and metabolic remodeling are complications of CKD that precede structural remodeling of the heart that eventually culminates in LVH. Using bioengineered cardio-bundles, primary culture of NRVM, gain-of-function and loss-of function genetic mouse models, we further show that FGF23-mediated activation of FGFR4 is one potential underlying mechanism of cardiac mitochondrial dysfunction and metabolic remodeling in mice with CKD.</p>", "<p id=\"P44\">Studies that assessed cardiac metabolism in CKD are scarce and mostly addressed single metabolic pathways.^{##REF##18191738##42##–##REF##20515820##46##} In contrast, we used an agnostic approach to report on global changes in cardiac amino acids, acylcarnitines and organic acid metabolism in wild-type and FGFR4-deficient mice. Our results align with recent reports that compared cardiac metabolites from patients with heart failure with preserved ejection fraction (HFpEF) or reduced ejection fraction (HFrEF) and reports of different animal models.^{##REF##36856044##47##,##REF##30971818##48##} This suggests that HFpEF and cardiac metabolic remodeling with or without CKD likely share common mechanisms.</p>", "<p id=\"P45\">Genetic activation of FGFR4 caused HFpEF in the absence of kidney injury or elevated serum FGF23 levels in our study. Similar to CKD, metabolic changes manifested before overt LVH. Since the cardiac metabolome of CKD mice and FGFR4-Arg385 knock in mice were similar, we hypothesized that FGF23-FGFR4 signaling contributes to regulation of cardiac metabolism. This hypothesis is supported by our finding that LVH and HFpEF were attenuated and cardiac metabolism was normalized in mice with CKD overlaying global or cardiomyocyte-specific deletion of FGFR4 compared to mice with CKD and intact cardiac FGFR4. Our in vitro data in cardio-bundles and NRVM in which FGF23 drove contractile, electrical, and metabolic dysfunction that were prevented by a specific small molecule inhibitor of FGFR4 lend further support to our hypothesis.</p>", "<p id=\"P46\">Cardiac mitochondria play a major role in regulating cardiac energy homeostasis. Our analysis of cardiac mitochondria using transmission electron microscopy, proteomics and respirometry indicate that CKD causes substantial changes to cardiac mitochondrial structure and function before structural changes of the heart are evident. Interestingly, FGF23-mediated mitochondrial dysfunction is characterized by increased mitochondrial respiration in-vitro. These results are in line with our respirometry findings from CKD mice. Similar results have also been observed in right ventricular heart failure,^{##REF##17582388##37##} in contrast to other heart failure models. Our results are supported by previous studies that described pathologic mitochondria in a rat model of CKD using high resolution imaging and increased respiration and proton leak in aged mitochondria.^{##REF##33319746##49##,##REF##29889834##50##} The pathological augmented basal respiration coupled with sustained proton leak likely augments bioenergetic stress, burdens the mitochondrial work load, ultimately resulting in a decline in respiratory efficiency. In addition, changes in mitochondrial oxidative catabolism have been described to drive adaptive changes in metabolic properties.^{##REF##35150906##51##} Our results indicate that FGF23-mediated increases in mitochondrial respiration function as a retrograde signaling mechanism that ultimately increases glycolysis.</p>", "<p id=\"P47\">Limitations of this report include the lack of metabolic flux studies. Our static metabolomic results only provide a snapshot on cardiac metabolic pathways and do not allow a full interpretation of cardiac glycolysis and fatty acid metabolism in CKD. Moreover, we currently do not know how FGFR4 mediates its downstream metabolic effects. Enriched pathways in cardio-bundles treated with FGF23 included processes related to peroxisomal physiology and transferrin receptor biology, highlighting a possible mechanistic link between FGF23 and iron homeostasis, as has been reported previously.^{##REF##23505057##52##,##REF##37053547##53##} Increased respiration with elevated proton leak could indicate that increased production of reactive oxygen species plays a role in this mechanism. Additional experiments will be needed to elucidate the pathway from FGFR4 to mitochondrial dysfunction and whether it includes the PLCγ-calcineurin signaling cascade as indicated by the rise in TRPC6 and RCAN1 expression in cardio-bundles that we observed. Recently, renal glycolysis has been identified as a mammalian phosphate sensor and thus energy metabolism serves as a critical regulator of phosphate homeostasis that controls osseous FGF23 secretion via glycerol-3-phosphate.^{##REF##32065590##54##,##UREF##5##55##} Our work now closes this “metabolic loop” by demonstrating that FGF23 itself exerts direct metabolic effects. HFpEF increases mortality in patients with CKD, but to date, we do not know enough about the early pathogenesis of HFpEF in CKD to support treat patients before cardiac structural changes become evident. Our findings that FGF23 mediates cardiac metabolic remodeling and mitochondrial dysfunction strongly supports the need to develop and test small molecule inhibitors of FGFR4 or its downstream effectors in preclinical models and early clinical studies for the treatment of cardiac metabolic remodeling to prevent subsequent development of LVH and HFpEF.</p>" ]

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[ "<p id=\"P1\">Chronic kidney disease (CKD) is a global health epidemic that significantly increases mortality due to cardiovascular disease. Left ventricular hypertrophy (LVH) is an important mechanism of cardiac injury in CKD. High serum levels of fibroblast growth factor (FGF) 23 in patients with CKD may contribute mechanistically to the pathogenesis of LVH by activating FGF receptor (FGFR) 4 signaling in cardiac myocytes. Mitochondrial dysfunction and cardiac metabolic remodeling are early features of cardiac injury that predate development of hypertrophy, but these mechanisms of disease have been insufficiently studied in models of CKD. Wild-type mice with CKD induced by adenine diet developed LVH that was preceded by morphological changes in mitochondrial structure and evidence of cardiac mitochondrial and metabolic dysfunction. In bioengineered cardio-bundles and neonatal rat ventricular myocytes grown in vitro, FGF23-mediated activation of FGFR4 caused a mitochondrial pathology, characterized by increased bioenergetic stress and increased glycolysis, that preceded the development of cellular hypertrophy. The cardiac metabolic changes and associated mitochondrial alterations in mice with CKD were prevented by global or cardiac-specific deletion of FGFR4. These findings indicate that metabolic remodeling and eventually mitochondrial dysfunction are early cardiac complications of CKD that precede structural remodeling of the heart. Mechanistically, FGF23-mediated activation of FGFR4 causes mitochondrial dysfunction, suggesting that early pharmacologic inhibition of FGFR4 might serve as novel therapeutic intervention to prevent development of LVH and heart failure in patients with CKD.</p>", "<title>Graphical Abstract</title>", "<p id=\"P2\">\n\n</p>" ]

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[ "<title>Acknowledgment</title>", "<p id=\"P48\">We would like to acknowledge the excellent and expert technical support provided to this project by: The Animal Clinical Laboratory Services Core of the University on North Carolina (Chapel Hill), Duke Cardiovascular Physiology Core, Duke Center for Genomics and Computational Biology, Duke Molecular Physiology Institute - Metabolomics Core, Duke Proteomics and Metabolomics Core, Duke Center for Electron Microscopy and Nanoscale Technology, Duke Substrate Services Core Research Support, Duke Light Microscopy Core Facility.</p>", "<p id=\"P49\">Graphical abstract was designed using <ext-link xlink:href=\"http://biorender.com\" ext-link-type=\"uri\">biorender.com</ext-link></p>", "<title>Funding:</title>", "<p id=\"P50\">American Heart Association AHA center grant 15SFDRN25080048 to Myles Wolf, American Society of Nephrology Karl W. Gottschalk Award to Alexander Grabner, NIH U01HL134764 to Nenad Bursac. Sonja Hänzelmann was supported by SFB1192 B8 of the German Research Foundation, Fabian Hausmann received a postdoctoral stipend of the University Hospital Hamburg.</p>", "<title>Data sharing statement</title>", "<p id=\"P51\">All data presented in this manuscript will be made available to other researchers upon request to the corresponding author.</p>" ]

[ "<fig position=\"float\" id=\"F1\"><label>Figure 1</label><caption><title>Cardiac function, remodeling and changes to cardiac mitochondria in CKD.</title><p id=\"P71\">Cardiac function of control and adenine fed mice was evaluated at the outset of the experiment at 8 weeks and at study termination at 16 weeks.</p><p id=\"P72\">Significant LVH was detectable at the final timepoint after 16 weeks CKD as indicated by left ventricular mass index (LVMI), posterior wall thickness (PW) and left ventricular end-systolic diameter (LVD;s) (A). After 12 weeks adenine diet, significant decrease in glomerular filtration rate (GFR) indicated kidney damage but no clear manifestation of structural LVH as indicated by the unchanged posterior wall thickness and overall wall thickness after 12 weeks on the adenine diet (B). Measurement of remodeling parameters <italic toggle=\"yes\">Nppa, Timp1, Foxo1</italic> and <italic toggle=\"yes\">Pgc-1α</italic> indicate that hypertrophic and fibrotic remodeling had been initiated at 12 weeks CKD (C). Electron microscopy revealed clear changes in mitochondrial morphology with misalignment of mitochondria and changes in cristae appearance (D). Mitochondrial respiration after 12 weeks of adenine diet indicated that respiration through complex I and II was significantly increased before structural cardiac remodeling was detectable (E).</p><p id=\"P73\">Bar graphs represent mean with SEM and individual values included in the graph. * indicate p&lt; 0.05. Scale bar in D is 500nm.</p></caption></fig>", "<fig position=\"float\" id=\"F2\"><label>Figure 2</label><caption><title>Changes to mitochondrial proteome and cardiac metabolome in CKD.</title><p id=\"P74\">CKD leads to significant changes in the mitochondrial proteome of wildtype after 12 weeks of adenine diet, before structural remodeling becomes detectable (A,B). A total of 118 mitochondrial genes were significantly regulated in the mitoproteome of mice with CKD compared to controls (A). KEGG pathway analysis showed that, downregulated proteins represented fatty acid metabolism, and amino acid degradation (B, top). The up-regulated genes were enriched in ribosomal processes and translation (B, bottom).</p><p id=\"P75\">Analysis of metabolomics also showed significant changes in serum acylcarnitines, amino acids and keto acids (C, left). Among upregulated serum amino acids were phenylalanine and citrulline. These were also upregulated in cardiac tissue (C, right). Here, more amino acids were significantly downregulated, leucine and isoleucine showed a downward trend (p = 0.07). In contrast to serum, more cardiac acylcarnitines were significantly downregulated. For organic acids from cardiac tissue of CKD mice only pyruvate reached significance but lactate and citrate showed trends (p= 0.07).</p></caption></fig>", "<fig position=\"float\" id=\"F3\"><label>Figure 3</label><caption><title>FGFR4 regulates metabolic transcription and hypertrophy in bio-engineered cardio bundles.</title><p id=\"P76\">Treatment of neonatal rat ventricular myocyte cardio-bundles with FGF23 for 20 minutes significantly increased contractile force, while 7 days of chronic treatment led to a significant reduction in contractile force that could be rescued by co-application of BLU9931, a selective FGFR4 inhibitor (A). Electrophysiological function was evaluated by pacing of cardio-bundles and application of Di-4-ANEPPS as voltage sensitive dye. Chronic exposure of cardio-bundles to FGF23 lead to significantly longer action potential durations (B). FGF23-treated bundles exhibited significantly lower conduction velocity that was normalized after co-application of BLU9931 (C). Beside functional changes, chronic FGF23 treatment also led to cardio-bundle hypertrophy indicated by the significant rise in cross-section (D,G) and increased expression of hypertrophic mRNA markers <italic toggle=\"yes\">Rcan1</italic> and <italic toggle=\"yes\">Trpc6</italic> (E). Increased expression of <italic toggle=\"yes\">Rcan1</italic> and <italic toggle=\"yes\">Trpc6</italic> was blocked by parallel treatment with BLU9931. Metabolic transcription factors that were increased in CKD mice, also increased in cardio-bundles after FGF23 treatment (F). Representative images of cardio-bundles indicate cellular hypertrophy after FGF23 treatment by increased myocyte cross-sections (G). Gene set enrichment analysis of control and FGF23 treated cardio-bundles showed an enrichment of metabolic pathways, particularly fatty acid metabolism, adipogenesis and cholesterol homeostasis (H). Additional enrichment was detected in pathways related to mitochondrial function, such as oxidative phosphorylation, respiratory chain, organelle fission and organelle inner membrane (I). Downregulated pathways after FGF23 treatment include angiogenesis, vascular development TNFα signaling and P53.</p><p id=\"P77\">Bar graphs represent mean with SEM and individual values included in the graph. * indicate p&lt;0.05. Scale bars in G are 10 μm.</p></caption></fig>", "<fig position=\"float\" id=\"F4\"><label>Figure 4</label><caption><title>FGFR4 mediates metabolic remodeling in cultured cardiomyocytes</title><p id=\"P78\">Cultured neonatal rat ventricular myocytes (NRVM) esponded to 48h of FGF23 treatment with significant hypertrophy indicated by increased cross-sectional area and expression of pro-hypertrophic markers (A and B). Pro-hypertrophic mRNA expression and cellular hypertrophy could be mitigated by parallel treatment with the FGFR4-specific inhibitor BLU9931. Cardiac mitochondria isolated from NRVM treated with FGF23 for 1h, before observable hypertrophy takes place, were analyzed in a Seahorse XF analyzer for extracellular acidification rate (ECAR), elevated total proton efflux rates (PER) and glycolysis specific proton efflux rates (glycoPER) (C). ECAR was significantly higher in mitochondria from FGF23-treated cells, which could be reduced to control levels by BLU9931. PER showed elevated basal and compensatory glycolysis in mitochondria with FGF23 treatment; glycolysis-specific proton efflux was also increased. These FGF23 mediated effects were blocked by BLU9931 application. Seahorse mitochondrial stress test assay showed increased basal and maximal mitochondrial respiration after FGF23 treatment of NRVM (D). ATP production-linked, spare respiratory capacity and non-mitochondrial oxygen consumption rate increased in parallel after FGF23 treatment. The significant decrease in coupling efficiency and the increased proton leak indicate uncoupling of substrate oxidation and ATP synthesis after 1h of FGF23 treatment. Application of BLU9931 or the calcineurin inhibitor, cyclosporin A, prevented the changes to mitochondrial function caused by FGF23.</p><p id=\"P79\">Bar graphs represent mean with SEM and individual values included in the graph. * indicate p&lt;0.05. Graphs in C represent 3 independent experiments.</p></caption></fig>", "<fig position=\"float\" id=\"F5\"><label>Figure 5</label><caption><title>Cardiac function, remodeling and metabolomics of FGFR4-Arg385 mice in the absence of CKD.</title><p id=\"P80\">FGFR4-Arg385 mice did not have impaired kidney function as indicated by BUN values, but beginning LVH is detectable by increased wall thickness at 6 months of age. By 24 month of age, renal function remained unchanged and significant LVH/HFpEF was detected in FGFR4-Arg385 mice indicated by robust structural remodeling and increased fractional shortening (A). mRNA expression levels of remodeling, pro-fibrotic and pro-hypertrophic markers support initiation of cardiac remodeling at 6 months of age (B). Transmission electron microscopy showed similar changes in the mitochondria of six-month-old FGFR4-Arg385 mice as observed in mice with adenine induced CKD (C).</p><p id=\"P81\">Metabolomic analysis of FGFR4-Arg385 mice at 6 month of age showed significant increase in some serum acylcarnitines and reduction in cardiac MLACs (D). Similar to wildtype CKD animals, cardiac citrulline was upregulated while a number of other amino acids, including leucin and isoleucine, were downregulated (E). Reduction of serum keto acids was also in line with results obtained from the adenine CKD model. Organic acids also showed similar changes with a significant upregulation of pyruvate and a downregulation of lactate (F).</p><p id=\"P82\">Bar graphs represent mean with SEM and individual values included in the graph. * indicate p&lt;0.05.</p></caption></fig>", "<fig position=\"float\" id=\"F6\"><label>Figure 6</label><caption><title>Global deletion of FGFR4 prevents LHV and changes to cardiac mitoproteome in CKD.</title><p id=\"P83\">Mice with global deletion of FGFR4 develop CKD to the same degree as control mice after 16 weeks of adenine diet as indicated by the rise in FGF23 (A). Wildtype animals developed LVH at 16 weeks with increased LV mass, wall thickness and ratio of heart weight/tibia length. These changes were absent in FGFR4−/− mice (B). Cardiac mitoproteome of wildtype and FGFR4−/− mice was evaluated after 12 weeks adenine feeding, before overt remodeling is observed. 22 proteins were significantly regulated in wildtype CKD mice, but were not changed in FGFR4−/− CKD mice, with 9 proteins downregulated and 13 upregulated (B). Analysis showed enrichment in pathways connected to mitochondrial respiration and function (C). Additionally, 163 proteins were identified that were only regulated in FGFR4−/− CKD mice with 83 downregulated and 57 upregulated proteins (D). Enrichment analysis showed a partial normalization of mitochondrial proteins in FGFR4−/− mice (E).</p><p id=\"P84\">Bar graphs represent mean with SEM and individual values included in the graph. * indicate p&lt;0.05.</p></caption></fig>", "<fig position=\"float\" id=\"F7\"><label>Figure 7</label><caption><title>Cardiomyocyte expression of FGFR4 mediates metabolic remodeling in adenine-induced CKD.</title><p id=\"P85\">After 16 weeks of adenine diet, control animals and α-MHCMerCreMer-FGFR4flox mice developed kidney damage to a similar degree with elevated FGF23. LV mass, wall thickness and ratio of heart weight/tibia length were significantly lower in α-MHCMerCreMer-FGFR4flox mice than control animals (A). Cardiac-specific deletion of FGFR4 also significantly reduced the cardiac expression of pro-hypertrophic and pro-fibrotic markers (B). Echocardiography showed no structural remodeling or abnormalizes in the hearts of α-MHCMerCreMer-FGFR4flox, whereas hearts of control animals showed significant wall thickening and remodeling (C). Analysis of the cardiac metabolome showed elevation of a greater number of MLAC when compared to control animals. Expression levels of organic acids indicate a normalization of glucose utilization in α-MHCMerCreMer-FGFR4flox mice and reduction in cardiac pyruvate and citrate concentrations (D). Amino acids were unchanged between groups.</p><p id=\"P86\">Bar graphs represent mean with SEM and individual values included in the graph. * indicate p&lt;0.05.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\"><label>Table 1 :</label><caption><p id=\"P87\">Taq-man probes used for RNA quantification</p></caption><table frame=\"box\" rules=\"all\"><colgroup span=\"1\"><col align=\"left\" valign=\"middle\" span=\"1\"/><col align=\"left\" valign=\"middle\" span=\"1\"/></colgroup><thead><tr><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Target</th><th align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Assay ID</th></tr></thead><tbody><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Eukaryotic 18S rRNA</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">4352930E</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">FGFR4</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mm01341851_g1</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Fib</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mm01256744_m1</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Foxol</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mm00490672_m1</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Myh6</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mm00440359_m1</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Myh7</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mm00600555_m1</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Nppa</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mm01255747_g1</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Pgc-1α</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mm01208835_m1</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Rcanl</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Rn00596606_m1</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Timpl</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mm00441818_m1</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Trpc6</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mm01176083_m1</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">β2-microglobulin</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Mm00437762_m1</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group><fn id=\"FN1\"><p id=\"P52\">Disclosure statement</p><p id=\"P53\">MW has equity interests in Akebia, Unicycive, and Walden and has served as a consultant for Bayer, Enyo, Jnana, Kissei, Launch, Pharmacosmos and Reata.</p></fn><fn id=\"FN2\"><p id=\"P54\">Supplemental figure S1: cardiac functional parameters of wildtype mice with adenine induced CKD.</p><p id=\"P55\">Additional functional parameters were measured for mice with different degrees of CKD. Left ventricular diastolic diameter (LVD;d) and fractional shortening were not changed at any of the investigated timepoints. Measurement of BUN indicated significant renal damage in mice that had been fed the adenine diet for 8 weeks and at 16 weeks on the diet. Functional parameters of mice that were fed the adenine diet for 12 weeks, did not indicate impaired function or structural remodeling, as indicated by fractional shortening and left ventricular (LV) mass. In line with other profibrotic parameters, expression of fibronectin mRNA was also increased after 12 weeks on the adenine diet.</p><p id=\"P56\">Bar graphs represent mean with SEM and individual values included in the graph. * indicate p&lt; 0.05.</p><p id=\"P57\">Supplemental figure S2: Cardiac function of Col4a3^−/− mice.</p><p id=\"P58\">Changes in kidney and cardiac function of C57BLJ/6-Col4a3^−/− mice, a genetic model for CKD, and changes in relevant mRNA parameters. Col4a3^−/− mice and their respective controls were evaluated at 20 weeks of age, when CKD and beginning LVH were manifest as indicated by significant increase of BUN, iFGF23, heart weight/tibia length and heart weight/bodyweight. Measurement of pro-hypertrophic, pro-fibrotic and metabolic transcription factors are further evidence of cardiac remodeling (A). Electron microscopy revealed clear changes in mitochondrial morphology similar to that observed in mice with CKD due to the feeding of an adenine containing diet (B). Changes observed in Col4a3^−/− mice were in line with those observed in the adenine model, indicating a generalized mechanism directing the development of LVH, not model specific particularities.</p><p id=\"P59\">Bar graphs represent mean with SEM and individual values included in the graph. * indicate p&lt;0.05. Scale bar in B is 500nm.</p><p id=\"P60\">Supplemental figure S3: Hypertrophy makers in NRVM treated for 1h with FGF23.</p><p id=\"P61\">To determine if glycolysis is directly stimulated by FGF23 or indirectly a response to cellular hypertrophy, we treated NRVM with FGF23 for 1 hour, when no signs of cellular hypertrophy were present. At 1h after FGF23 treatment, markers of cellular hypertrophy were not significantly induced compared to vehicle treated cells.</p><p id=\"P62\">Bar graphs represent mean with SEM and individual values included in the graph.</p><p id=\"P63\">Supplemental figure S4: Additional renal and cardiac function parameters of FGFR4-Arg385 mice.</p><p id=\"P64\">At 6 month of age FGFR4-Arg385 mice showed no impaired cardiac or renal function. Expression of <italic toggle=\"yes\">Myh7</italic> and <italic toggle=\"yes\">FN</italic> mRNA however was already increased, indicating initiation of structural remodeling, but not detectable structural changes (top). Measurement of serum phosphate in FGFR4-Arg385 mice at 24 month of age indicates no changes to mineral metabolism or renal function, but significant impairment of cardiac function, consistent with LVH/HFpEF.</p><p id=\"P65\">Supplemental figure S5: renal and cardiac function in FGFR4^−/− mice with CKD</p><p id=\"P66\">Measurement of BUN after 16 weeks on the adenine diet indicated FGFR4^−/− mice develop CKD to the same degree as the control animals. Cardiac function, evaluated by fractional shortening was not changed between groups. Deletion of FGFR4 also normalized expression of cardiac pro-hypertrophic and pro-fibrotic markers.</p><p id=\"P67\">Bar graphs represent mean with SEM and individual values included in the graph. * indicate p&lt;0.05.</p><p id=\"P68\">Supplemental figure S6: Tissue specific deletion of FGFR4 in cardiomyocytes and functional parameters.</p><p id=\"P69\">Deletion of FGFR4 under the α-MHC promotor was induced with a total of 3 injection of 30 mg/kg bodyweight tamoxifen, i.p. every 48 hours. Expression of FGFR4 in cardiomyocytes of α-MHC^MerCreMer-FGFR4^flox was measured 10 days after the last injection by qPCR. Abundance of kidney <italic toggle=\"yes\">Fgfr4</italic> mRNA was measured to verify tissue specificity of the deletion. Serum levels of BUN, PTH and Phosphate indicate the same degree of renal damage after 16 weeks on the adenine diet in control animals and α-MHC^MerCreMer-FGFR4^flox mice. Control but not α-MHC^MerCreMer-FGFR4^flox mice developed diastolic dysfunction marked by changes in E/A ratio with preserved systolic function.</p><p id=\"P70\">Bar graphs represent mean with SEM and individual values included in the graph. * indicate p&lt;0.05</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>The service sectors are growing faster as compared to other sectors in developing nations (\n##UREF##60##Mukherjee, 2015##;\n##UREF##47##Latha, 2016##;\n##UREF##76##Service sectors in India, 2021##). The services industry not only accounts for the bulk of India’s Gross Domestic Product (GDP), but it also attracts significant foreign investment, contributes significantly to export, and employs a vast majority of the population. In the financial year 2021, the services sector in India contributed 54 percent of the total Gross Value Added. Hence the service sector is an integral component of economic growth in India and necessitates investigation on various economic and behavioral dimensions.</p>", "<p>The service industry comprises routine, positive, and negative services. “Routine or neutral services” raise familiar experiences that they navigate frequently. For example, the housekeeping and beauty services. “Positive services” are those associated with travel and entertainment. Tourism and hospitality services are examples of positive services. “Negative services” are those associated with unpleasant circumstances. Law and healthcare services are examples of negative services (\n##UREF##59##Morgan &amp; Rao, 2006##).</p>", "<p>The healthcare industry is one of the fastest-growing service industries in recent years. In the last few decades, extensive attempts have been made to incorporate “information and communication technology” (ICT) into healthcare operations (\n##REF##29893224##Fullman\net al., 2018##;\n##UREF##1##Aayog, 2019##). It is projected that India’s healthcare industry is expected to grow up to 372 billion dollars by 2022. The country’s healthcare market had grown rapidly to become one of the largest sectors in terms of income and jobs (\n##UREF##82##Statista Research Department, 2021##). This necessitates studies on the healthcare sector.</p>", "<p>The services offered by law and healthcare are considered to be unpleasant (\n##UREF##33##Hellén &amp; Sääksjärvi, 2011##). These services are referred to as “adverse services” or “negative services” since the clients are exposed to uncomfortable situations such as overcrowding, long waiting times, anxiety, and risk of infections. In adverse services customers are exposed to these situations which cause an element of uneasiness which further influences their service experience (\n##UREF##59##Morgan &amp; Rao, 2006##;\n##UREF##75##Schwartz, 2015##;\n##UREF##8##Bahadori, Teymourzadeh, Ravangard, &amp; Raadabadi, 2017##). Although healthcare services are hostile, many people are forced to use them at some point in their lives (\n##UREF##33##Hellén &amp; Sääksjärvi, 2011##). Negative emotional states such as worry, uncertainty, and unease are frequently created in hospital settings (\n##UREF##59##Morgan &amp; Rao, 2006##;\n##UREF##11##Berry &amp; Bendapudi, 2007##;\n##UREF##58##Miller, Luce, Kahn, &amp; Conant, 2009##). Researchers have examined the association between emotions and service quality evaluations (\n##UREF##57##Mattila &amp; Enz, 2002##;\n##UREF##80##Slåtten, 2011##;\n##UREF##61##Naami, &amp; Hezarkhani, 2018##).\n##UREF##10##Ben-Ze’ev (2000)## opined that positive emotions such as joy or happiness express satisfaction, while negative emotions such as rage or guilt express dissatisfaction.</p>", "<p>Scholars in the field of “positive psychology” have defined happiness as “a summary appraisal of one’s life” (\n##UREF##24##Diener, Scollon, &amp; Lucas, 2009##). Happiness is described as a feeling, sentiment, or transitory form of joy (\n##UREF##45##Labroo &amp; Patrick, 2008##;\n##UREF##44##Labroo &amp; Mukhopadhyay, 2009##). An individual’s happiness level impacts how happy and unhappy people acquire, understand, and evaluate the same situation. The happiness level of the consumer predicts their ability to cope in difficult conditions (\n##UREF##12##Boehm, Ruberton, &amp; Lyubomirsky, 2017##). According to past research, consumers’ attitudes vary depending on the services they have received (positive or negative). Happiness, on the other hand, seems to remain consistent over time and in various situations (\n##UREF##33##Hellén &amp; Sääksjärvi, 2011##). During service encounters, consumers have a strong tendency to pursue psychological needs such as hedonistic needs, inner joy, and happiness (\n##UREF##12##Boehm\net al., 2017##). Researchers revealed that consumers with positive emotional states are more likely to evaluate services in a positive way (\n##UREF##5##Ali, Amin, &amp; Cobanoglu, 2016##).</p>", "<p>Service quality (SQ) evaluation is a cognitive process in which clients compare service quality expectations to the actual services obtained (\n##UREF##48##Lee, Lee, &amp; Yoo, 2000##;\n##UREF##2##Abedniya, &amp; Zaeim, 2011##). Therefore, hospitals must ensure that good quality services are provided to meet their client’s expectations (\n##REF##21456497##Suki, Lian, &amp; Suki, 2011##;\n##UREF##74##Rechel, Doyle, Grundy, &amp; McKee, 2009##). Marketers need to explore the precursors of service quality evaluation. In service marketing literature, precursors of service quality evaluation by the customers have received considerable attention over the recent decade (\n##UREF##39##Kiran &amp; Diljit, 2017##;\n##UREF##84##Tan, Benbasat, &amp; Cenfetelli, 2013##;\n##UREF##83##Sultan &amp; Wong, 2011##;\n##UREF##78##Shamdasani, Mukherjee, &amp; Malhotra, 2008##).</p>", "<p>Service quality is critical for healthcare organizations, and it has a significant impact on patient satisfaction (\n##UREF##19##Dagger, Sweeney, &amp; Johnson, 2007##;\n##UREF##94##Zaim, Bayyurt, &amp; Zaim, 2010##). The major outcomes of a business are SQ and customer satisfaction (CS), in this case, patient satisfaction. Patients are treated as guests who are seeking positive outcomes as well as quality service experiences (\n##REF##20210071##Otani, Waterman, Faulkner, Boslaugh, &amp; Dunagan, 2010##;\n##UREF##56##Marzo\net al., 2021##). Delivering exceptional services leads to a high CS, which leads to customer retention (\n##REF##20210071##Otani\net al., 2010##;\n##UREF##51##Loureiro, Miranda, &amp; Breazeale, 2014##;\n##UREF##65##Oluwafemi &amp; Dastane, 2016##).</p>", "<p>Patient satisfaction (PS) is an individual opinion of the standard of care received (\n##REF##16268410##Otani, Kurz, Harris, &amp; Byrne, 2005##). PS is a critical analysis of patients’ happiness with the quality of health care they receive both in and out of the doctor’s office. PS is a major determinant of the quality of healthcare outcomes (\n##UREF##56##Marzo\net al., 2021##). CS is essential to every industry since satisfied customers are loyal and bring in new business. The healthcare industry is no exception to this. Many studies believe that happy patients are more likely to tell their friends about their doctors and return when they need help again (\n##REF##20210071##Otani\net al., 2010##). CS reflects the feelings of healthcare patients about the quality of service they expect in comparison to what they currently experience. It’s also possible to presume that the satisfaction level of the patient is determined by the number of expectations and realities learned from the health services he/she has received (\n##UREF##41##Kotler &amp; Keller, 2016##).</p>", "<p>In the field of psychology, the importance of emotions has acquired a lot of attention, but it is lacking in the marketing literature (\n##UREF##32##Hellén, 2010##). Positive services are well researched when compared to negative services (\n##UREF##59##Morgan &amp; Rao, 2006##;\n##UREF##11##Berry &amp; Bendapudi, 2007##;\n##UREF##58##Miller\net al., 2009##). Thus, the goal of this study is to determine the relationship between patients’ happiness and satisfaction through patients’ mood and perceived service quality at a healthcare setup. This research contributes to the service marketing literature by demonstrating how patient satisfaction is enhanced in an unpleasant service context by mood and perceived service quality. Healthcare consumers are susceptible to deal with the adverse features of the service differently and to evaluate the quality of service encounters through emotions to develop satisfaction. The hospital management can support healthcare consumers to improve satisfaction by changing the servicescape of the hospital.</p>", "<p>This research article begins with a review of literature on the constructs of the study. In the subsequent sections the methodology adopted and the data analysis is presented followed by the major findings, limitations of the study, and directions for future research.</p>" ]

[ "<title>Methods</title>", "<title>Study design</title>", "<p>This research work is highly focused on patient satisfaction in a healthcare setting to empirically validate the hypotheses framed. This study has adopted a quantitative approach. A cross-sectional research design was applied in the study. The relationship between the dependent, independent, and mediating variables is examined in this research. A structural model was created to examine the impact of patient’s happiness on PS through mood and PSQ.</p>", "<title>Setting</title>", "<p>The top four districts of Karnataka, India (Bangalore Urban, Dakshina Kannada, Udupi, and Mysore) were identified based on the Human Development Index (HDI) details made available by the Government of Karnataka (\n##UREF##37##HDI report, 2015##). High HDI was considered a significant inclusion criterion as it strongly captures the three dimensions of health, literacy, and standard of living. Research evidence suggests a strong association between HDI and happiness and a stronger positive relationship between HDI and life satisfaction. There is a significant positive association between HDI and the happiness index of a region as reported by\n##UREF##50##Leigh and Wolfers (2006)##; United Nations Development Programme’s (UNDP) Human Development Report authored by\n##UREF##31##Hall, Helliwell, and Helliwell (2014)##. A simple random sampling technique was adopted to select the district and the hospital. Among the four districts mentioned above, Udupi district was selected for this study. As per the inclusion criteria, this study was conducted in a tertiary care hospital located in this district. The data was gathered during the period from November to December 2021.</p>", "<title>Participants</title>", "<p>The participants included in the study were outpatients who had more than two visits to the tertiary care hospital, were aged between 18-65 years, and spoke English or Kannada. The outpatient departments (OPDs) considered for the study were medicine and medical specialties and surgery and surgical specialties. Pediatric and psychiatric OPDs were excluded. Participants were approached at the pharmacy, which is their final point of outpatient service encounter at the hospital.</p>", "<p>The sample size was calculated based on the number of items on the rating scale which is multiplied by 10 (\n##UREF##29##Hair, Sarstedt, Ringle, &amp; Gudergan, 2017##) i.e. 17*10 = 170. Accounting for a non-response rate of 20%, 170+34=204. So, it was approximated to 210. The total sample size of the study was 227.</p>", "<title>Ethics and consent</title>", "<p>Ethical approval was obtained from the Institutional Ethics Committee (IEC) of Kasturba Medical College and Kasturba Hospital Manipal, Karnataka, India (IEC: 868/2020). Total confidentiality of the data is maintained by not using participant identifiers. This has been included in the participant information sheet (Clause No.12).</p>", "<title>Data collection</title>", "<p>Primary data was collected through a structured questionnaire. The questionnaire contained a set of scales evaluating the level of happiness of the participants (\n##UREF##52##Lyubomirsky &amp; Lepper, 1999##). This was rated on a seven-point Likert scale. The set of scales evaluating participants’ mood (\n##UREF##70##Peterson &amp; Sauber, 1983##), service quality perceptions (\n##UREF##13##Brady &amp; Cronin, 2001##;\n##UREF##68##Parasuraman, Zeithaml, &amp; Berry, 1988##), and patient satisfaction (\n##UREF##27##Greenfield &amp; Attkisson, 1989##;\n##UREF##63##Oliver, 1997##) were rated on a five-point Likert scale. The questionnaire had a total of 23 questions of which 6 captured demographic details of the participants, and 17 were related to the constructs of the study. After the questionnaire was finalized, it was translated into Kannada by a language expert. Both the English and Kannada versions of the questionnaire were created using Microsoft Word 2013, then printed for participant use. This hardcopy is used to collect data from the participants. Data collection was carried out at the hospital pharmacy since it is the last point of contact in outpatient services. The participants were selected purposively. They were informed about the purpose and procedures of the study through a participant information sheet. Written consent was obtained from the participants and then the researcher distributed the questionnaire to participants. The researcher instructed the participants to tick the appropriate response on the questionnaire.</p>", "<title>Pilot testing</title>", "<p>Before collecting the data, a pretest procedure was carried out which involved pre-testing of a survey questionnaire to evaluate the complete questionnaire using validity and reliability checks. Validity is the accuracy with which an instrument measures what it is supposed to measure. In this research endeavor, the following validity checks have been implemented.</p>", "<p>Face validity: This was checked to see whether at face value the questions/items appeared to be measuring the construct or what is intended to measure.</p>", "<p>Content validity: A panel of judges who were experts in healthcare management, marketing, and operations evaluated the draft questionnaire. The survey items were rated based on the clarity, relevance, appropriateness, and redundancy of the items. Out of the 10 experts approached, six of them responded with their comments and suggestions. Suggestions given were incorporated.</p>", "<p>Construct validity: This involved convergent and divergent validity checked after receiving data from the final sample.</p>", "<p>The data was collected from 47 respondents during the pilot study. Data was collected through a structured questionnaire. A copy of the questionnaire can be found in the\n<italic toggle=\"yes\">Extended data</italic> (\n##UREF##4##Akthar\net al., 2022##). The questionnaire was administered personally. The written consent was obtained from the participants before giving the questionnaire. The respondents were approached in September 2021.</p>", "<p>Reliability is the consistency or repeatability of the measure. The pilot study helped to determine the construct reliability.</p>", "<p>Internal consistency: The reliability within a scale was checked to see whether all the items were designed to measure a particular construct. The reliability scores, Cronbach’s α value of the constructs are as follows: Happiness (0.812), Mood (0.643), Perceived service quality (0.756), and patient satisfaction (0.862) and all the values are well above the threshold limit (\n##UREF##29##Hair\net al., 2017##).</p>", "<p>While conducting the pilot study it was observed that a few participants perceived difficulty in responding to two questions (Items H3: Some people are generally very happy. They enjoy life regardless of what is going on, getting the most out of everything. To what extent does this characterization describe you? and H4: Some people are generally not very happy. Although they are not depressed, they never seem as happy as they might be. To what extent does this characterization describe you?) pertaining to the construct happiness.</p>", "<p>These items were reworded and the cognitive interviewing technique (involving two methods i.e. think-aloud interviewing and probing) was adopted (\n##UREF##90##Willis, 2004##) to check the validity of the statements. In order to check the accuracy of the reworded statements, it was subjected to 10 respondents. The paraphrased statements are as follows- H3: People are mostly happy and enjoy their life no matter what is going on to make the most out of everything. Does this describe you?; H4: People are mostly not very happy and not once appear happy as they may be. Does this describe you?</p>", "<title>Analysis</title>", "<p>Descriptive statistics were calculated using IBM SPSS statistics 27 (IBM SPSS Statistics, RRID: SCR_016479; Armonk, NY: IBM Corp). The proposed hypotheses were tested and the mediation analysis was performed using the SmartPLS 3 (SmartPLS, RRID: SCR_022040)\n<bold>.</bold> The results are represented in the form of tables and figures in the subsequent section. SmartPLS 3.0 software was used to analyze the data of this research endeavor. “Partial Least Squares Regression-Structural Equation Modeling” (PLS-SEM) adopts the SEM technique and has many similarities to regression. In addition, PLS also models the theoretical association between the latent variables and also the relationship between the latent variable and its indicators (\n##UREF##16##Chin, Marcolin, &amp; Newsted, 1996##). PLS was also preferred to other covariance-based techniques, like LISREL, as it can be run on smaller sample sizes.</p>" ]

[ "<title>Results</title>", "<p>Descriptive statistics are estimated and the output is presented in\n##TAB##0##Tables 1## &amp;\n##TAB##1##2##. The full dataset can be found in the\n<italic toggle=\"yes\">Underlying data</italic> (\n##UREF##4##Akthar\net al., 2022##).</p>", "<p>The measurement model was estimated using the data and it is presented below (\n##FIG##1##Figure 2##).</p>", "<p>The construct reliability was established by estimating Cronbach’s alpha, factor loadings, and composite reliability (\n##TAB##2##Table 3##). Composite reliability is said to be a better accurate measure of internal consistency as the measure of composite reliability doesn’t tend to increase with the addition of every new item. The threshold value of internal consistency reliability is 0.8 (\n##REF##18487745##Daskalakis &amp; Mantas, 2008##), which is established in this research endeavor. In a reflective model, the outer loadings of all indicators have to be above 0.7 (\n##UREF##34##Henseler\net al., 2014##), which is evident in this case. Further, the “average variance extracted (AVE)” of each construct must be above 0.5 (\n##UREF##88##Wasko &amp; Faraj, 2005##;\n##UREF##91##Wixom &amp; Watson, 2001##) indicating at least 50 percent variance of each construct could be explained by the indicator. These criteria have been fulfilled and presented in\n##TAB##2##Table 3##.</p>", "<p>“Discriminant validity” is verified by comparing the AVEs with the squared multiple correlations of each latent variable (\n##UREF##15##Chin, 1998##). In this analysis, Fornell and Larcker criterion is adopted (\n##TAB##3##Table 4##).</p>", "<p>As the AVE of each construct is higher than the squared multiple correlations, it is concluded that the constructs of this research endeavor exhibit discriminant validity. Collinearity among the constructs was tested using the “Variance Inflated Factor (VIF)” guidelines. The predictor variables displayed VIF values below 5 (\n##TAB##2##Table 3##). This implied that collinearity is not a constraint in this structural model.</p>", "<p>Hypotheses proposed in this study were examined by comparing the direct effect of patients’ happiness level on PSQ and the indirect effect of the level of patients’ happiness on PS. The results are displayed in\n##TAB##4##Table 5##. H1 proposed that the patient’s happiness level positively influences mood and it is supported (β=0.26, t=3.770, p&lt;0.01). H2 proposed that mood positively influenced the PSQ and is supported (β=0.552, t=10.957, p&lt;0.01). H3 proposed that happiness positively influenced PSQ and this hypothesis is also supported (β=0.212, t=3.958, p≤0.01). H4 proposed a direct positive effect of PSQ on PS (β=0.791, t=29.516, p≤0.01). The path values (β values) and the empirical t values of all the hypotheses are above the cutoff value of 0.2 and 1.96 respectively, which substantiates the proposed hypotheses of this research endeavor. The influence of all the exogenous latent variables namely, happiness, mood, perceived service quality, on the endogenous latent variable of patient satisfaction is estimated to be moderate (R\n²=62.5%) (\n##UREF##29##Hair\net al., 2017##).</p>", "<p>The effect size, f\n² of all the exogenous latent variables was calculated (\n##TAB##4##Table 5##). The effect size measures the extent of influence of the variables independent of the scope of the sample analyzed (\n##UREF##17##Cohen, 1988##).\n##UREF##17##Cohen (1988)## proposes a threshold to gauge the extent of the effect of the constructs. Effect size above 0.35 is reported as a large effect; value in the range of 0.15 to 0.35 is reported as moderate effect and values below 0.15 is reported as a low effect. In our research endeavor, the effect size of mood on PSQ (f\n²=0.484) and the effect size of PSQ on PS (f\n²=1.669) are estimated to be large. Model Fitness is assessed with the help of the value of “Standard Root Mean square Residual (SRMR)” as proposed by\n##UREF##34##Henseler\net al. (2014)##. The threshold value of model fitness is 0.8 (\n##UREF##36##Hu &amp; Bentler, 1998##). The SRMR value of this model is reported as 0.073 which indicates a good model fit.</p>", "<p>A mediation analysis was undertaken to assess the mediating effect of the construct ‘mood’ between the constructs of happiness and PSQ (\n##FIG##2##Figure 3##).</p>", "<p>The direct effect of happiness on perceived service quality (0.163) is significant and the indirect effect of happiness on PSQ through mood (0.116) is also significant. The VAF (Variance Accounted for) value of 41.58 percent indicates a partial mediation between happiness and PSQ (\n##TAB##5##Table 6##).</p>", "<title>Importance performance matrix analysis (IPMA)</title>", "<p>Importance Performance Matrix Analysis (IPMA) provides researchers with an insight into the relative importance of the performance of the exogenous latent variables in their association with the endogenous latent variables and was first proposed by\n##UREF##55##Martilla and James (1977)##. This method enables the researchers to examine the importance of an item in addition to its performance. The rationale of this analysis is to identify the total effect of the predecessor constructs (mood, perceived service quality, and happiness) in forestalling the target endogenous construct (patient satisfaction) (\n##UREF##28##Hair, Hult, Ringle, &amp; Sarstedt, 2016##, p. 276;\n##UREF##30##Hair, Sarstedt, &amp; Ringle, 2018##, p. 105). The total effect establishes the importance of the constructs while the mean value of their scores reflects their performance (ranging from 0, which is the lowest, to 100, which is the highest) (\n##UREF##35##Höck, Ringle, &amp; Sarstedt, 2010##, p. 201).</p>", "<p>The results of IPMA are presented in\n##FIG##3##Figure 4## and\n##TAB##6##Table 7##. Analyses demonstrate that PSQ is ranked high on performance (81.48) in comparison to the other exogenous constructs. In addition, the total effect of PSQ on PS is 0.791 which is also high. Thus, a unit increment in the performance of PSQ from 81.448 to 82.448 will result in an increase in the performance of PS from 77.662 to 78.453. The total effect and performance of the exogenous construct mood are 0.437 and 77.313 respectively. Thus one unit increment of mood from 77.313 to 78.313 would increase the performance of PS from 77.662 to 78.099. Similarly, the total effect and performance of the exogenous construct happiness are 0.281 and 74.553. Thus, an increment of one unit of happiness from 74.553 to 75.553 would yield an increment in patient satisfaction to 77.943. This study found that the total effect of PSQ has the strongest and most significant effect on patient satisfaction followed by Mood and then Happiness. This is an important implication to healthcare service providers.</p>", "<p>The Importance Performance Map is presented below (\n##FIG##4##Figure 5##).</p>" ]

[ "<title>Conclusions/Discussion</title>", "<p>In this research endeavor, we intended to explore the role of emotions in adverse services. We proposed to explore if mood mediates the relationship between happiness and service quality perception which had a direct bearing on patient satisfaction. Happiness was explored as a significant antecedent to perceived service quality of adverse services, such as hospitals, and it provides a significant foundation to determine patient satisfaction. These findings enable us to conclude that happy people are most likely to experience better service quality and patient satisfaction. We can also conclude that patients’ emotions (mood) at the hospital play an instrumental role in developing service quality perceptions and indirectly strengthening patient satisfaction.</p>", "<p>The results of this research endeavor to uphold the results of previous research from positive psychology and service marketing literature showing that the happiness of people significantly influences their mood which directly or indirectly influences their service quality perception, especially in adverse services such as hospitals, legal services, etc (\n##UREF##59##Morgan &amp; Rao, 2006##;\n##UREF##24##Diener\net al., 2009##;\n##UREF##33##Hellén &amp; Sääksjärvi, 2011##;\n##REF##19725210##Badri\net al., 2009##;\n##UREF##92##Yesilada &amp; Direktör, 2010##;\n##UREF##26##Fatima\net al., 2018##). However, this research endeavor extends the literature by displaying that happiness is a significant predictor of mood and mood mediates the association between happiness and perceived service quality in adverse services. This study contributes to the body of knowledge by highlighting the role of patients’ moods in predicting service quality and thereby patient satisfaction.</p>", "<p>These results could be used to deduce the following. Firstly, happy people would experience positive affective states, and consequently, a person who scores high on happiness is more likely to report a positive mood. Secondly, the mood is a reflection of their happiness and hence the results could be biased (\n##UREF##21##Diener, 2009##). This study contributes to the body of knowledge by highlighting the role of patients’ moods in predicting service quality and thereby patient satisfaction. This result does not echo the outcomes of past research such as\n##UREF##43##Kumar and Oliver (1997)## and\n##UREF##62##Oliver (1993)##, who proposed that there is no significant relationship between consumers’ affective response and service quality. However,\n##UREF##89##White (2006)## has proposed a positive association between mood and service quality, and our findings are in line with this research output.</p>", "<p>This research also presents significant managerial implications to industry, especially to the services of adverse nature. We recommend that marketers of adverse service must design strategies to enhance the mood of their patients or customers. Every element of the servicescape in adverse services must be designed such that it enhances the mood of customers. From service providers (doctors and nurses) and support staff to peripheral service encounters, there should be effective management that contributes to elevating the customer’s mood. Doctors and nurses can be trained to handle customers’ queries about the line of treatment and medication effectively. Health care providers or medical teams, environmental conditions, and hospital completeness are all elements that influence patient satisfaction. To enhance patient satisfaction, the quality of care provided by health services, human resources, and infrastructure must be improved. As a result, the entire service encounter can be made more enjoyable for the customers by reducing the distress caused by adverse services.</p>", "<p>This study is also subject to limitations. First, this study was conducted in a tertiary hospital of a high HDI district. There is a significant positive association between HDI and the happiness index of a region as reported by\n##UREF##50##Leigh and Wolfers (2006)##. Future research would benefit from conducting a comparative study amongst high HDI and low HDI districts. Second, this study adopted a quantitative approach. Future researchers would consider improving patient satisfaction by exploring the elements of servicescape in adverse services through an experimental approach. The study design adopted in positive psychology research endeavors consists of a two-step process. In the preliminary stage, happy and unhappy subjects are identified and are then subject to the experimental setting. This paves the way for effective comparison in both groups. In this scenario, service quality perceptions could have been effectively captured in the controlled group and the experimental group. Thirdly, this study utilized the scale developed by\n##UREF##70##Peterson and Sauber (1983)## to measure the construct mood which might be a cause of concern. This scale fails to capture the extent of influence of elements of servicescape on a patient’s general mood. Thus there could be a possibility of an element of error in capturing the patient’s mood. However, it can also be argued that a patient’s mood captured at the hospital is attributed to elements of servicescape.</p>", "<p>In conclusion, the concept of consumer emotions and its implication on service quality evaluation had gained momentum. Taking a step in this direction, this research endeavor explored the impact of consumers’ happiness on service quality perception at a hospital, which is considered to be an adverse service by nature. This research outcome indicated that consumers’ mood partially mediated the association between happiness and service quality perception. This outcome provides significant evidence that goes against the theoretical underpinnings of positive psychology theories which suggest that happy people significantly experience situations more positively. However, outcomes of this research endeavor contribute to service marketing literature which highlights the role of servicescape to modify the mood of consumers thereby influencing their service quality perception. Hospitals must design their servicescape effectively, to trigger positive emotions (mood) among patients that will have a direct bearing on service quality evaluation and thereby patient satisfaction. This is of paramount importance, especially during the current COVID-19 pandemic. COVID-19 has created significant distress economically and emotionally across the globe. Mental health has become the focal point of discussion and concern. Hence, hospitals must ensure that all their marketing strategies revolve around creating a positive affective state (mood) among their patients which will enable them to perceive adverse services in hospitals in a positive way.</p>" ]

[ "<title>Conclusions/Discussion</title>", "<p>In this research endeavor, we intended to explore the role of emotions in adverse services. We proposed to explore if mood mediates the relationship between happiness and service quality perception which had a direct bearing on patient satisfaction. Happiness was explored as a significant antecedent to perceived service quality of adverse services, such as hospitals, and it provides a significant foundation to determine patient satisfaction. These findings enable us to conclude that happy people are most likely to experience better service quality and patient satisfaction. We can also conclude that patients’ emotions (mood) at the hospital play an instrumental role in developing service quality perceptions and indirectly strengthening patient satisfaction.</p>", "<p>The results of this research endeavor to uphold the results of previous research from positive psychology and service marketing literature showing that the happiness of people significantly influences their mood which directly or indirectly influences their service quality perception, especially in adverse services such as hospitals, legal services, etc (\n##UREF##59##Morgan &amp; Rao, 2006##;\n##UREF##24##Diener\net al., 2009##;\n##UREF##33##Hellén &amp; Sääksjärvi, 2011##;\n##REF##19725210##Badri\net al., 2009##;\n##UREF##92##Yesilada &amp; Direktör, 2010##;\n##UREF##26##Fatima\net al., 2018##). However, this research endeavor extends the literature by displaying that happiness is a significant predictor of mood and mood mediates the association between happiness and perceived service quality in adverse services. This study contributes to the body of knowledge by highlighting the role of patients’ moods in predicting service quality and thereby patient satisfaction.</p>", "<p>These results could be used to deduce the following. Firstly, happy people would experience positive affective states, and consequently, a person who scores high on happiness is more likely to report a positive mood. Secondly, the mood is a reflection of their happiness and hence the results could be biased (\n##UREF##21##Diener, 2009##). This study contributes to the body of knowledge by highlighting the role of patients’ moods in predicting service quality and thereby patient satisfaction. This result does not echo the outcomes of past research such as\n##UREF##43##Kumar and Oliver (1997)## and\n##UREF##62##Oliver (1993)##, who proposed that there is no significant relationship between consumers’ affective response and service quality. However,\n##UREF##89##White (2006)## has proposed a positive association between mood and service quality, and our findings are in line with this research output.</p>", "<p>This research also presents significant managerial implications to industry, especially to the services of adverse nature. We recommend that marketers of adverse service must design strategies to enhance the mood of their patients or customers. Every element of the servicescape in adverse services must be designed such that it enhances the mood of customers. From service providers (doctors and nurses) and support staff to peripheral service encounters, there should be effective management that contributes to elevating the customer’s mood. Doctors and nurses can be trained to handle customers’ queries about the line of treatment and medication effectively. Health care providers or medical teams, environmental conditions, and hospital completeness are all elements that influence patient satisfaction. To enhance patient satisfaction, the quality of care provided by health services, human resources, and infrastructure must be improved. As a result, the entire service encounter can be made more enjoyable for the customers by reducing the distress caused by adverse services.</p>", "<p>This study is also subject to limitations. First, this study was conducted in a tertiary hospital of a high HDI district. There is a significant positive association between HDI and the happiness index of a region as reported by\n##UREF##50##Leigh and Wolfers (2006)##. Future research would benefit from conducting a comparative study amongst high HDI and low HDI districts. Second, this study adopted a quantitative approach. Future researchers would consider improving patient satisfaction by exploring the elements of servicescape in adverse services through an experimental approach. The study design adopted in positive psychology research endeavors consists of a two-step process. In the preliminary stage, happy and unhappy subjects are identified and are then subject to the experimental setting. This paves the way for effective comparison in both groups. In this scenario, service quality perceptions could have been effectively captured in the controlled group and the experimental group. Thirdly, this study utilized the scale developed by\n##UREF##70##Peterson and Sauber (1983)## to measure the construct mood which might be a cause of concern. This scale fails to capture the extent of influence of elements of servicescape on a patient’s general mood. Thus there could be a possibility of an element of error in capturing the patient’s mood. However, it can also be argued that a patient’s mood captured at the hospital is attributed to elements of servicescape.</p>", "<p>In conclusion, the concept of consumer emotions and its implication on service quality evaluation had gained momentum. Taking a step in this direction, this research endeavor explored the impact of consumers’ happiness on service quality perception at a hospital, which is considered to be an adverse service by nature. This research outcome indicated that consumers’ mood partially mediated the association between happiness and service quality perception. This outcome provides significant evidence that goes against the theoretical underpinnings of positive psychology theories which suggest that happy people significantly experience situations more positively. However, outcomes of this research endeavor contribute to service marketing literature which highlights the role of servicescape to modify the mood of consumers thereby influencing their service quality perception. Hospitals must design their servicescape effectively, to trigger positive emotions (mood) among patients that will have a direct bearing on service quality evaluation and thereby patient satisfaction. This is of paramount importance, especially during the current COVID-19 pandemic. COVID-19 has created significant distress economically and emotionally across the globe. Mental health has become the focal point of discussion and concern. Hence, hospitals must ensure that all their marketing strategies revolve around creating a positive affective state (mood) among their patients which will enable them to perceive adverse services in hospitals in a positive way.</p>" ]

[ "<p>No competing interests were disclosed.</p>", "<title>Background</title>", "<p>Managing emotions during hospital visits is important to enhance patient satisfaction. The purpose of this paper is to explore the relationship between patients’ happiness and satisfaction through patients’ mood and perceived service quality at a healthcare setup.</p>", "<title>Methods</title>", "<p>This study was conducted in a tertiary care hospital located in coastal Karnataka during the period from November to December 2021. Primary data was collected through a structured questionnaire from 227 respondents. “Statistical Package for the Social Sciences (SPSS) 27.0” and “SmartPLS 3.0” software was used for data analysis.</p>", "<title>Results</title>", "<p>Hypotheses proposed in this study were examined by comparing the direct effect of patients’ happiness level on perceived service quality and the indirect effect of the level of patients’ happiness on patient satisfaction. The influence of all the exogenous latent variables namely, happiness, mood, perceived service quality, on the endogenous latent variable of patient satisfaction is estimated to be moderate (R\n²=62.5%).</p>", "<title>Conclusion</title>", "<p>This study empowers hospital managers to recognize how patient satisfaction is dependent on patients’ happiness. In order to enhance patient satisfaction, the quality of care provided by health services, human resources, and infrastructure must be improved. As a result, the entire service encounter can be made enjoyable to the customers by reducing the distress caused by adverse services. Managers can utilize the outcomes of the study to develop marketing strategies to influence patients’ emotions in the healthcare setup by modifying the servicescape elements.</p>", "<title>Amendments from Version 1</title>", "<p>The changes as suggested by reviewer 2, have been incorporated in the revised version. The major difference between the previous version and this version is as follows: 1. The methodology section has been revised as suggested.  Study Design: This study has adopted a quantitative approach. A cross-sectional research design was applied in the study. The relationship between the dependent, independent, and mediating variables is examined in this research. A structural model was created to examine the impact of patient’s happiness on PS through mood and PSQ. Study Setting: A simple random sampling technique was adopted to select the district and the hospital. Among the four districts mentioned above, Udupi district was selected for this study. As per the inclusion criteria, this study was conducted in a tertiary care hospital located in this district. The data was gathered during the period from November to December 2021. 2. Full form of UNDP: United Nations Development Programme 3. When we look at the constructs happiness and mood, they look similar. However, as per positive psychology literature, happiness and mood are different.  4. A positive environment does impact patient's happiness and mood. In this research, the hospital is selected through simple random sampling.</p>" ]

[ "<title>Literature review</title>", "<title>Theoretical background</title>", "<p>\n##REF##10467897##Ashby and Isen (1999)## proposed that positive affect has a consistent impact on a variety of cognitive activities. Many of these effects are explained by the neuropsychological theory, which claims that the pleasant effect is linked to higher levels of dopamine in the brain. They have suggested that “positive affect influences olfaction, the consolidation of long-term or episodic memories, working memory, and creative problem-solving”. For example, the idea claims that higher dopamine release in the “anterior cingulate” promotes “cognitive flexibility” and allows the choice of “cognitive perspectives”, which helps with creative problem-solving. The resulting theory has several advantages over other methods to the study of positive affect now in use. First and foremost, it offers a neuropsychological explanation for a variety of well-known positive affect occurrences. Second, it predicts “positive affect” influences on tasks that have never been studied by the researchers. Third, it lists several tasks in which positive emotion is not expected to have an impact on performance. Fourth, it connects positive emotion research findings to previously unconnected neuropsychological studies. For example, it compares cognitive processing in healthy people with cognitive processing in some neuropsychological patient groups.</p>", "<title>Happiness</title>", "<p>Subjective wellbeing (SWB) is a synonym for happiness (\n##REF##11577847##Lu, Gilmour, &amp; Kao, 2001##;\n##UREF##14##Chang &amp; Nayga, 2010##). In the positive psychology literature, happiness is defined as “a summary appraisal of one’s life” (\n##UREF##24##Diener\net al., 2009##;\n##UREF##87##Veenhoven, 2010##). Diener stated that “the term happy in common English usage refers to a transient, positive state of mind brought on by a specific experience, such as a nice social engagement” (\n##UREF##24##Diener\net al., 2009##). One of the most important aspects of a human being’s life is their mental state of mind (\n##UREF##23##Diener, Lucas, &amp; Oishi, 2002##).</p>", "<p>Various factors influence happiness levels, such as individual feelings of joy, positive well-being, and a sense of a good and meaningful life (\n##REF##9418274##Lyubomirsky &amp; Ross, 1997##;\n##UREF##53##Lyubomirsky &amp; Tucker, 1998##;\n##UREF##54##Lyubomirsky, Tucker, &amp; Kasri, 2001##).\n##UREF##54##Lyubomirsky\net al. (2001)## and\n##REF##16351326##Lyubomirsky, King, and Diener (2005)## have found that happy people make more positive decisions than unhappy people. As a result, happiness plays a crucial role in determining outcomes. Happy customers are more likely to be pleased when they make a specific decision. Happy people have a more consistent reaction to life situations, as they are considerably better at dealing with stressful conditions than unhappy people (\n##UREF##33##Hellén &amp; Sääksjärvi, 2011##). Happy people are more likely to be connected by more positive life circumstances, and as a result, they have more positive outcomes in their lives (\n##REF##16809528##Kahneman, Krueger, Schkade, Schwarz, &amp; Stone, 2006##).</p>", "<title>Mood</title>", "<p>Customers’ moods are transient states of emotion that prompt them to evaluate services precisely (\n##UREF##72##Pieters &amp; Raaij, 1988##). When compared to other emotional states that endure longer, mood has a lower intensity, is more distributed, and is unintentional (\n##UREF##7##Bagozzi\net al., 1999##).\n##UREF##46##Lane and Terry (2000)## have defined mood as “a collection of experiences that are fleeting in character, vary in intensity and length, and frequently involve multiple emotions”. This definition states that mood is a chain of expressive emotions that forms a frame of mind to change incoming events gradually through day-to-day activities, as happy customers are in a better mood, they have a greater opinion of service quality than disgruntled customers (\n##UREF##33##Hellén &amp; Sääksjärvi, 2011##). Therefore, happiness is defined as an emotional state of well-being that is stored in the memory as a mood rather than an emotion (\n##UREF##9##Belanche, Casaló, &amp; Guinalu, 2013##).</p>", "<title>Perceived service quality (PSQ)</title>", "<p>“The customer’s assessment of an entity’s total excellence or superiority can be defined as service quality” (\n##UREF##95##Zeithaml, 1988##). Consumers evaluate the service quality of an organization by comparing their perceptions to their expectations (\n##UREF##79##Sivakumar, Li, &amp; Dong, 2014##). Service providers must ensure that service recipients have pleasant service interactions, as negative experiences will be shared with others (\n##UREF##71##Petzer, Meyer, Svari, &amp; Svensson, 2012##). According to recent research in the service industry, negative emotions experienced by service recipients during the service contact have an impact on their loyalty levels (\n##UREF##39##Kiran &amp; Diljit, 2017##;\n##UREF##5##Ali\net al., 2016##;\n##UREF##69##Park, Lee, Kwon, &amp; Del Pobil, 2015##;\n##UREF##81##Son, Jung, &amp; Lee, 2015##).</p>", "<p>Service quality is a critical factor in determining whether or not a service provider is favored, hence it must be carefully measured and improved (\n##REF##29954278##Javed &amp; Ilyas, 2018##;\n##UREF##56##Marzo\net al., 2021##). According to researchers, hospitals must now meet their criteria and deliver the greatest health care services to patients as a result of growing expectations for common facilities (\n##UREF##66##Padma, Rajendran, &amp; Lokachari, 2010##;\n##UREF##67##Pai &amp; Chary, 2014##). PSQ in the healthcare sector has received a lot of attention. It should be mentioned that in both public and private institutions, patients’ perceptions of healthcare services are influenced by the quality of care they receive (\n##UREF##77##Shabbir, Malik, Malik, &amp; Wiele, 2016##).</p>", "<title>Patient satisfaction (PS)</title>", "<p>One of the most widely researched topics in literature is satisfaction (\n##REF##23324161##Sawyer\net al., 2013##;\n##REF##23669268##Barnett\net al., 2013##). According to\n##UREF##64##Oliver (2000)##, satisfaction is “a post-consumption judgment by the consumer that a service provides a pleasant level of consumption-related fulfillment, including under or over-fulfillment.” Patient satisfaction is one of the most often reported outcome indicators for quality of care in the healthcare sector, and it can be referred to as consumer satisfaction (\n##UREF##56##Marzo\net al., 2021##). Patient satisfaction is described as “meeting or exceeding the requests and expectations of the patient” (\n##UREF##3##Akbulut, 2016##;\n##UREF##93##Yilmaz, 2011##). This situation may arise as a result of patients’ inability to assess the medical element of the treatments provided. During the examination, physicians’ compassion, empathy, and other related abilities have a beneficial impact on patient satisfaction (\n##REF##34259619##Akbolat, Sezer, Ünal, &amp; Amarat, 2021##). Patients who are happy with the service they receive will share their experience among people they know (\n##UREF##38##Juhana, Manik, Febrinella, &amp; Sidharta, 2015##).</p>", "<p>The hypotheses proposed in this study are shown in\n##FIG##0##Figure 1##. The model proposes that patients’ happiness influences their satisfaction via mood and PSQ.</p>", "<p>Happiness causes recurrent pleasant moods because happy persons are more likely to have optimistic ideas (\n##REF##16351326##Lyubomirsky\net al., 2005##;\n##UREF##22##Diener &amp; Biswas-Diener, 2005##). According to psychologists, a cheerful individual might experience a bad mood. Moods fluctuate to some extent as a result of positive and negative events. Even when their moods vary, the happy person adapts to events while maintaining a positive attitude (\n##UREF##24##Diener\net al., 2009##). Happiness is important in healthcare services because it protects consumers from the harmful consequences of those services. Individuals who are happy cope with stress better and can maintain a positive attitude in adverse services (\n##UREF##33##Hellén &amp; Sääksjärvi, 2011##). As a result, the association between happiness and mood is likely to be substantial in healthcare services.\n<statement id=\"state1\"><label>\n<italic toggle=\"yes\">H1</italic>:</label><p>\n<italic toggle=\"yes\">Happiness is positively related to mood in adverse services.</italic>\n</p></statement>\n</p>", "<p>Previous researchers have discovered an association between mood and PSQ (\n##UREF##40##Kocabulut &amp; Albayrak, 2019##;\n##UREF##73##Pornpitakpan, Yuan, &amp; Han, 2017##;\n##UREF##89##White, 2006##). Experts have documented that the impact of mood is greatest when the buyer is uninformed of the goods or services. Individuals are more likely to rely on their feelings in uncertain situations (\n##UREF##25##Dubé, &amp; Morgan, 1996##). According to a comprehensive understanding, patients find it difficult to measure service quality in unpleasant services, thus they lean on their moods for assessment (\n##UREF##18##Collier, 1994##). Similarly,\n##UREF##20##Darke, Chattopadhyay, and Ashworth (2006)## have demonstrated that when data is lacking, mood serves as a clue, and this conclusion is consistent with customers’ tendency to grasp actual opinions. These findings show that when additional evidence is absent, people rely on their mood for assessment. According to previous research, evaluating adverse services is difficult for clients (\n##UREF##59##Morgan &amp; Rao, 2006##). As a result, it is hypothesized that mood and perceived service quality in healthcare services have a substantial link.\n<statement id=\"state2\"><label>\n<italic toggle=\"yes\">H2</italic>:</label><p>\n<italic toggle=\"yes\">Mood is positively connected to perceived service quality in adverse services.</italic>\n</p></statement>\n</p>", "<p>Customer’s emotion is a vital factor in evaluating services that they received (\n##UREF##86##Tsaur, Luoh, &amp; Syue, 2015##). Patients depend on their moods when evaluating healthcare services. Researchers believe that happiness affects PSQ through mood since happy people are more likely to be in a positive frame of mind, and hence are more likely to rate service quality positively (\n##UREF##33##Hellén &amp; Sääksjärvi, 2011##;\n##UREF##89##White, 2006##), as they are bound to encounter positive emotional states while in adverse circumstances, a happy customer may be more likely to experience enhanced SQ (\n##UREF##33##Hellén &amp; Sääksjärvi, 2011##). As a result, it is hypothesized that happiness and service quality perceptions in healthcare services are mediated by mood.\n<statement id=\"state3\"><label>\n<italic toggle=\"yes\">H3</italic>:</label><p>\n<italic toggle=\"yes\">The association between happiness and perceived service quality is mediated by mood.</italic>\n</p></statement>\n</p>", "<p>The impact of healthcare quality on PS has been thoroughly researched by scientists (\n##UREF##56##Marzo\net al., 2021##;\n##UREF##3##Akbulut, 2016##;\n##REF##19725210##Badri, Attia, &amp; Ustadi, 2009##;\n##UREF##96##Zineldin, 2006##;\n##UREF##6##Amin, &amp; Nasharuddin, 2013##). Various researchers in the field of marketing have found hypothetical as well as practical links between SQ and other user behaviors such as satisfaction, value, purchase/revisit intentions, and so on (\n##REF##34259619##Akbolat\net al., 2021##;\n##UREF##49##Lee &amp; Carter, 2011##;\n##UREF##85##Theodorakis &amp; Alexandris, 2008##;\n##UREF##42##Kouthouris, &amp; Alexandris, 2005##).</p>", "<p>The disparity between clients’ thoughts and expectations of the services is referred to as hospital service quality (\n##UREF##0##Aagja &amp; Garg, 2010##). Patients are the hospital’s most valuable asset in a healthcare setting. Hospital service quality has grown increasingly important as a means of satisfying and sustaining patients (\n##REF##21938970##Alhashem, Alquraini, &amp; Chowdhury, 2011##;\n##REF##18437935##Arasli, Haktan, &amp; Turan, 2008##). Scholars have discovered a link between PSQ and PS, demonstrating that if healthcare SQ is higher, patient satisfaction will be higher (\n##UREF##92##Yesilada &amp; Direktör, 2010##;\n##UREF##26##Fatima, Malik, &amp; Shabbir, 2018##). PS is utilized to define SQ in a healthcare context. SQ and satisfaction are found to have a substantial relationship (\n##UREF##77##Shabbir\net al., 2016##). Furthermore, it is assumed that greater services are required to satisfy customers (\n##UREF##26##Fatima\net al., 2018##;\n##UREF##56##Marzo\net al., 2021##). As a result, a considerable relationship between PSQ and PS in healthcare services is envisaged.\n<statement id=\"state4\"><label>\n<italic toggle=\"yes\">H4</italic>:</label><p>\n<italic toggle=\"yes\">Perceived service quality is positively related to patient satisfaction.</italic>\n</p></statement>\n</p>", "<title>Data availability</title>", "<title>Underlying data</title>", "<p>Figshare: Underlying data: Can positive emotions predict consumer satisfaction in adverse services? An empirical investigation.\n<ext-link xlink:href=\"https://doi.org/10.6084/m9.figshare.19360625.v3\" ext-link-type=\"uri\">https://doi.org/10.6084/m9.figshare.19360625.v3</ext-link> (\n##UREF##4##Akthar\net al., 2022##)</p>", "<p>This project contains the following underlying data:\n<list list-type=\"simple\"><list-item><label>-</label><p>Dataset.csv</p></list-item></list>\n</p>", "<title>Extended data</title>", "<p>This project contains the following extended data:\n<list list-type=\"simple\"><list-item><label>-</label><p>Questionnaire.docx</p></list-item><list-item><label>-</label><p>Informed Consent.docx</p></list-item><list-item><label>-</label><p>Participant Information Sheet.docx</p></list-item></list>\n</p>", "<p>Data are available under the terms of the\n<ext-link xlink:href=\"http://creativecommons.org/publicdomain/zero/1.0/\" ext-link-type=\"uri\">Creative Commons Zero “No rights reserved” data waiver</ext-link> (CC0 1.0 Public domain dedication).</p>" ]

[]

[ "<fig position=\"float\" fig-type=\"figure\" id=\"f1\"><label>Figure 1. </label><caption><title>Proposed hypotheses.</title></caption></fig>", "<fig position=\"float\" fig-type=\"figure\" id=\"f2\"><label>Figure 2. </label><caption><title>Structural model.</title></caption></fig>", "<fig position=\"float\" fig-type=\"figure\" id=\"f3\"><label>Figure 3. </label><caption><title>Mediation analysis.</title></caption></fig>", "<fig position=\"float\" fig-type=\"figure\" id=\"f4\"><label>Figure 4. </label><caption><title>Importance performance matrix analysis.</title></caption></fig>", "<fig position=\"float\" fig-type=\"figure\" id=\"f5\"><label>Figure 5. </label><caption><title>Importance performance map.</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\"><label>Table 1. </label><caption><title>Demographic characteristics (N=227).</title></caption><table frame=\"hsides\" rules=\"groups\" content-type=\"article-table\"><thead><tr><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Characteristics</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Components</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">N</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">%</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Gender</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Male</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">97</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">42.7</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Female</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">130</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">57.3</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Marital status</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Single</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">51</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">22.5</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Married</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">175</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">77.1</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Divorced/Widowed</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">.4</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Age</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">18-25</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">35</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">15.4</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">26-40</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">109</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">48.0</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">41-55</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">67</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">29.5</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">56-65</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">16</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">7.0</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Education level</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Up to 12\n^th\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">86</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">37.9</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Graduate</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">113</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">49.8</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Postgraduate</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">28</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">12.3</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Occupation</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Unemployed</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">85</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">37.4</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Employed</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">73</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">32.2</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Professional</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">43</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">18.9</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Business</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">26</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">11.5</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Monthly income</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">25000 and below</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">108</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">47.6</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">25001-75000</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">91</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">40.1</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">75001-125000</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">20</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">8.8</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">125001-200000</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">8</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">3.5</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Above 200000</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T2\"><label>Table 2. </label><caption><title>Outpatient departments (OPDs).</title></caption><table frame=\"hsides\" rules=\"groups\" content-type=\"article-table\"><thead><tr><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Category</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">OPDs</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">N</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">%</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Medicine</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Medicine</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">47</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">20.7</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Medical Specialties</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Dental</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">19</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">8.4</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Dermatology</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">18</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">8.0</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Endocrinology</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">9</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">4.0</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">ENT</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">4</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1.8</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Eye</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">2</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">.9</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Gastroenterology</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">9</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">4.0</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Nephrology</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">14</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">6.2</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Neurology</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">16</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">7.0</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">OBG</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">14</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">6.2</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Oncology</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">5</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">2.2</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Ophthalmology</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">10</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">4.4</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Cardiology</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">9</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">4.0</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Pulmonology</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">17</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">7.5</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Urology</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">11</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">4.8</td></tr><tr><td colspan=\"1\" rowspan=\"1\"/><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Ortho</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">20</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">8.8</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Surgery</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Surgery</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">2</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">.9</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Surgical specialties</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Cardiothoracic</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">.4</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T3\"><label>Table 3. </label><caption><title>Measurement model evaluation.</title></caption><table frame=\"hsides\" rules=\"groups\" content-type=\"article-table\"><thead><tr><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Construct</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Indicators</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Outer loading</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Composite reliability</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">AVE</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Cronbach’s alpha</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Outer weight</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">VIF</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"1\" rowspan=\"3\" valign=\"top\">Happiness</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">H1</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.951\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"3\" valign=\"top\">0.948</td><td align=\"left\" colspan=\"1\" rowspan=\"3\" valign=\"top\">0.859</td><td align=\"left\" colspan=\"1\" rowspan=\"3\" valign=\"top\">0.918</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.357\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">3.886</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">H2</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.942\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.401\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">3.119</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">H3</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.886\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.319\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">2.569</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"2\" valign=\"top\">Mood</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">M1</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.890\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"2\" valign=\"top\">0.978</td><td align=\"left\" colspan=\"1\" rowspan=\"2\" valign=\"top\">0.782</td><td align=\"left\" colspan=\"1\" rowspan=\"2\" valign=\"top\">0.721</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.578\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1.461</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">M2</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.879\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.553\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1.467</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"5\" valign=\"top\">Patient satisfaction</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">PS1</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.761\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"5\" valign=\"top\">0.854</td><td align=\"left\" colspan=\"1\" rowspan=\"5\" valign=\"top\">0.594</td><td align=\"left\" colspan=\"1\" rowspan=\"5\" valign=\"top\">0.771</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.249\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1.699</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">PS2</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.788\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.272\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1.839</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">PS3</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.799\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.258\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1.943</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">PS4</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.838\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.247\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">2.303</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">PS5</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.762\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.240\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1.726</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"4\" valign=\"top\">Perceived service quality</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">SQ1</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.716\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"4\" valign=\"top\">0.892</td><td align=\"left\" colspan=\"1\" rowspan=\"4\" valign=\"top\">0.624</td><td align=\"left\" colspan=\"1\" rowspan=\"4\" valign=\"top\">0.849</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.278\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1.420</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">SQ2</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.781\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.347\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1.506</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">SQ3</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.843\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.356\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1.842</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">SQ4</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.738\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.311\n<xref rid=\"tfn1\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1.476</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T4\"><label>Table 4. </label><caption><title>Squared multiple correlations (SMC).</title></caption><table frame=\"hsides\" rules=\"groups\" content-type=\"article-table\"><thead><tr><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\"/><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Happiness</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Mood</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Perceived service quality</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Patient satisfaction</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Happiness</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.927</td><td colspan=\"1\" rowspan=\"1\"/><td colspan=\"1\" rowspan=\"1\"/><td colspan=\"1\" rowspan=\"1\"/></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Mood</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.260</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.884</td><td colspan=\"1\" rowspan=\"1\"/><td colspan=\"1\" rowspan=\"1\"/></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Perceived service quality</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.355</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.608</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.791</td><td colspan=\"1\" rowspan=\"1\"/></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Patient satisfaction</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.297</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.593</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.771</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.790</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T5\"><label>Table 5. </label><caption><title>Hypothesis testing, f\n².</title></caption><table frame=\"hsides\" rules=\"groups\" content-type=\"article-table\"><thead><tr><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Relationship</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Path coefficient</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">t-Value</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Bias Corrected 95% Confidence Interval</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">f\n²\n</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Happiness - mood</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.260\n<xref rid=\"tfn4\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">3.3770</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">(0.111,0.409)</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.072</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Happiness - perceived service quality</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.212\n<xref rid=\"tfn4\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">3.958</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">(0.106, 0.313)</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.071</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Mood - perceived service quality</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.553\n<xref rid=\"tfn4\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">10.957</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">(0.736, 0.651)</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.484</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Perceived service quality - patient satisfaction</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.791\n<xref rid=\"tfn4\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">29.516</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">(0.736, 0.846)</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1.669</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T6\"><label>Table 6. </label><caption><title>Mediating effect of Mood.</title></caption><table frame=\"hsides\" rules=\"groups\" content-type=\"article-table\"><thead><tr><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\"/><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Direct effect</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Indirect effect</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Total effect</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">VAF</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Mediation</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">\n##FIG##2##Figure 3##\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.163\n<xref rid=\"tfn7\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.116\n<xref rid=\"tfn7\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.882\n<xref rid=\"tfn7\" ref-type=\"table-fn\">***</xref>\n</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">41.58%</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Partial</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"T7\"><label>Table 7. </label><caption><title>Importance performance matrix analysis of institutional effectiveness.</title></caption><table frame=\"hsides\" rules=\"groups\" content-type=\"article-table\"><thead><tr><th align=\"left\" colspan=\"1\" rowspan=\"2\" valign=\"bottom\">Latent constructs</th><th align=\"left\" colspan=\"2\" rowspan=\"1\" valign=\"top\">Patient satisfaction</th></tr><tr><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Importance (total effects)</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Performance (index values)</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Happiness</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.281</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">74.553</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Mood</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.437</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">77.313</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Perceived service quality (PSQ)</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">0.791</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">81.488</td></tr></tbody></table></table-wrap>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Biliary tract cancer (BTC) is a malignant tumor originating from the differentiation of bile ducts or gallbladder epithelial cells. It encompasses intrahepatic cholangiocarcinoma (ICC), hilar cholangiocarcinoma (HCCA), extrahepatic cholangiocarcinoma (eCCA) and gallbladder carcinoma (GBC).##REF##33516341##\n1\n## Radical surgery remains the mainstay of treatment for BTC, although only a limited percentage of patients (30–35%) are eligible for this approach.##REF##33592563##\n2\n## Moreover, the recurrence rate following surgical intervention is alarmingly high. Patients with ICC who undergo radical surgical resection experience a recurrence rate of 53.3%, with the meagre 5‐year disease‐free survival rate (DFS) of 32%.##REF##32438491##\n3\n##\n</p>", "<p>Gemcitabine combined with cisplatin is the established first‐line treatment for patients with advanced BTC. However, the median overall survival (mOS) achieved with this treatment is only 11.6–13.4 months.##UREF##0##\n4\n## In the recent phase III TOPAZ‐1 study, the combination of durvalumab with gemcitabine and cisplatin demonstrated an objective response rate (ORR) of 26.7%, a mOS of 12.8 months, and a median progression‐free survival (mPFS) of 7.2 months for patients with advanced BTC.##UREF##1##\n5\n## This combination has been approved by the US Food and Drug Administration (FDA) as a first‐line treatment for advanced BTC, but the trial results indicated that OS remains limited.</p>", "<p>Limited treatment options are available for patients who progress after receiving first‐line standard chemotherapy. As a result, extensive research has been conducted to explore the potential of targeted therapy, immunotherapy and combination therapy. However, despite the adoption of the FOLFOX regimen as the standard second‐line treatment for BTC, its mOS rate of only 6.2 months remains low.##REF##33798493##\n6\n## Genetic sequencing of BTC has revealed that approximately 7% of cases exhibit genetic mutations, with fibroblast growth factor receptor 2 (FGFR2) mutations accounting for 6.1%.##REF##26373574##\n7\n## Consequently, FGFR has emerged as the primary target for targeted therapy in BTC. A phase II trial of pemigatinib, a small molecule inhibitor of FGFR1‐3, was conducted to treat 38 patients (35.5%) with FGFR2 fusion or rearrangement, resulting in complete remission in 3 patients and partial remission in 35 patients.##REF##32203698##\n8\n## This study led to the approval of Pemigatinib for advanced BTC. Furthermore, another FGFR inhibitor, futibatinib, has demonstrated an ORR of 41.7%, a mPFS of 9.0 months and a mOS of 21.7 months in patients with FGFR2 fusion/rearrangement in ICC.##UREF##2##\n9\n##\n</p>", "<p>The results of tumor tissue immunohistochemistry revealed that 7.3% of ICC and 5.2% of hilar or distal cholangiocarcinoma tested positive of PD‐L1,##REF##31949927##\n10\n## suggesting the potential benefit of immunotherapy in BTC. However, clinical studies have not demonstrated significant efficacy of immune monotherapy for advanced BTC. For instance, the KEYNOTE‐028 study included 24 PD‐L1‐positive patients with advanced BTC who received pembrolizumab, resulting in an ORR of 13%, PFS of 1.8 months, and mOS of 5.7 months.##REF##32359091##\n11\n## Similarly, in the KEYNOTE‐158 study, pembrolizumab demonstrated an ORR of 5.8%, mOS of 7.4 months, and mPFS of 2.0 months in patients with advanced BTC.##REF##32359091##\n11\n##\n</p>", "<p>The efficacy of targeted monotherapy and immune monotherapy in the treatment of patients with advanced BTC is limited. However, the combination of lenvatinib and pembrolizumab has demonstrated impressive results, with an ORR of 25%, mPFS of 4.9 months (95% CI: 4.7–5.2 months) and mOS of 11.0 months (95% CI: 9.6–12.3 months) in patients with BTC of standard who have progressed after first‐line therapy.##REF##32832493##\n12\n## Additionally, the combination of durvalumab and tremelimumab had an ORR of 10.8% in BTC patients.##REF##35611499##\n13\n## These findings indicate that while targeted monotherapy and immunotherapy may have limited efficacy in treating advanced BTC, combination regimens of PD‐1 monoclonal antibody demonstrate greater therapeutic potential.</p>", "<p>Angiogenesis is a fundamental characteristic of cancer and plays a critical role in tumor growth and progression. It has been observed that anti‐angiogenic therapy can upregulate the expression of immune checkpoints.##REF##35842983##\n14\n## Anlotinib, a novel multi‐target tyrosine kinase inhibitor, has demonstrated significant anti‐tumor effects. In ICC, anlotinib exerts its inhibitory effects on tumor cell proliferation and invasion by interfering with the VEGF/PI3K/AKT signalling pathway.##REF##32709873##\n15\n## Moreover, anlotinib has shown the potential to enhance the potency of immunotherapy by normalising the vasculature and inducing T‐cell inflammation in the tumor microenvironment (TME).##REF##34844980##\n16\n## Toripalimab, a humanised monoclonal antibody against programmed death protein 1 (PD‐1), blocks the interaction between PD‐1 and its ligands.##REF##35095833##\n17\n## Therefore, the objective of this study was to assess the safety and efficacy of anlotinib hydrochloride in combination with toripalimab for patients with unresectable BTC. Additionally, considering the complex nature of the BTC TME, we will also explore potential markers that may be associated with treatment efficacy.</p>" ]

[ "<title>Methods</title>", "<title>Experimental procedure</title>", "<title>Study design and patients</title>", "<p>This is a single‐centre, prospective, single‐arm, exploratory clinical study to assess the safety and efficacy of anlotinib hydrochloride in combination with toripalimab for the treatment of unresectable BTC.</p>", "<p>Patient inclusion criteria were as follows: (1) Age ≥ 18 years, patients volunteered to participate in the study and signed an informed consent form (ICF). (2) Histologically confirmed locally advanced or metastatic biliary tract tumors (including ICC, HCCA, eCCA and GBC). (3) Patients who failed first‐line chemotherapy or who chose not to undergo first‐line chemotherapy. (4) Disease progression within 14 months prior to enrolment (RECIST1.1 criteria must be used as the basis for assessing disease progression). (5) Patients who had normal function of major organs. (6) ECOG PS: 0–2 score. (7) Expected survival ≥ 3 months.</p>", "<p>Exclusion criteria were as follows: (1) Recent use (within 6 months) of VEGFR‐TKI small molecule drugs. (2) Patients who had been treated with any antibodies/drugs targeting T‐cell co‐regulatory proteins (immune checkpoints). (3) Confirmed allergy to the investigational drug and/or its excipients. (4) Patients who could not take oral medication. (5) People with high blood pressure that could not be reduced to within the normal range with antihypertensive medication. Detailed study criteria can be found in Supplementary table ##SUPPL##0##1##.</p>", "<title>Procedures</title>", "<p>All patients enrolled were treated with anlotinib hydrochloride in combination with toripalimab. For the treatment regimen, patients were administered Anlotinib Hydrochloride Capsules at a dose of 12 mg orally once daily before breakfast for 2 weeks, followed by a 1‐week break period, making up a 21‐day cycle. Toripalimab was given intravenously at a dose of 240 mg on the first day, also for a 21‐day cycle. Patients achieving complete remission (CR), partial remission (PR) and stable disease (SD) in response to the treatment continued to receive medication until disease progression, intolerable toxicity or patient request to discontinue the treatment. If drug toxicity caused a suspension of treatment, the cumulative duration of suspension should not exceed 2 weeks per dosing cycle, and no more than two pauses per cycle to ensure the proper intensity of treatment. The trial will be terminated for patients whose treatment cycle is delayed by more than 2 weeks, as well as those who exceed the above criteria.</p>", "<title>Efficacy evaluation and statistical analysis</title>", "<p>Primary study endpoints included the objective response rate (ORR), which was assessed every 6 weeks to determine the efficacy. Secondary study endpoints included progression‐free survival (PFS), overall survival (OS) and disease control rate (DCR). The efficacy was assessed based on the RECIST version 1.1 criteria. Point estimates and 95% confidence intervals were provided for efficacy endpoints such as ORR and DCR, and intervals were estimated using the Clopper‐Pearson method. The Kaplan–Meier method was utilised to estimate PFS and OS and to calculate 95% confidence intervals.</p>", "<title>Gene sequencing</title>", "<p>Paraffin sections from 11 patients involved in this clinical trial for NGS (next‐generation sequencing) analysis. NGS analysis was performed at a CLIA/CAP‐compliant Molecular Diagnostics Service laboratory of OrigiMed Co., Ltd (Shanghai, China). The 706 tumor‐related gene panel sequencing method was used. To begin, at least 50 ng DNA was extracted from each 40 mM formalin‐fixed paraffin‐embedded (FFPE) tumor sample using a DNA extraction kit (QIAamp DNA FFPE Tissue Kit). Subsequently, genes were captured and sequenced using the Illumina NextSeq 500 platform (Illumina Incorporated, San Diego, CA, USA).</p>", "<title>Immunohistochemistry</title>", "<p>The expression of CD4, CD8, Foxp3 and PD‐L1 was evaluated in pathological specimens of 11 patients at our institution. The tissue sections were stained following the steps of section preparation, antigen repair, antibody hybridisation, colour development and sealing. The results were interpreted by two experienced pathologists in a double‐blind method. Tumor cells and lymphocytes expressing PD‐L1 were considered positive, while CD4 and CD8 with a value of ≥ 10 were considered to be high expression. A Foxp3 value greater than 1 was also considered to be high expression.</p>" ]

[ "<title>Results</title>", "<title>Effective</title>", "<p>From May 2020 to September 2021, a total of 15 eligible patients were enrolled to a combination therapy of anlotinib and toripalimab. Among these patients, there were nine with ICC, three with HCCA and three with GBC. Out of the 15 patients, four (26.7%) who were unable to tolerate chemotherapy received anlotinib in combination with toripalimab as first‐line therapy, while 11 (73.3%) were undergoing second‐line therapy. The baseline characteristics of the patients are presented in Table ##TAB##0##1##.</p>", "<p>Each patient received a minimum of two treatment cycles, and as of the data cut‐off date on 6 September 2022, all patients were evaluated. The efficacy of all patients, assessed according to RECIST v1.1, is presented in Table ##TAB##1##2##. The evaluation demonstrated a PR in 4 patients, SD in 9 patients, PD in two patients. The ORR was 26.7% (4/15), and the DCR was 86.7% (13/15). The mPFS was 8.6 months (95% CI: 2.1–15.2) for all patients (Figure ##FIG##0##1a##) and 5.1 months (95% CI: 0.9–9.2) for patients who received second‐line treatment (Figure ##FIG##0##1b##). The mOS was 14.53 months (95% CI: 0.8–28.2) for all patients (Figure ##FIG##0##1c##) and 6.6 months (95% CI: 0–18.2) for patients who received second‐line treatment (Figure ##FIG##0##1d##). Two patients achieved PR after 2 cycles of treatment, and the corresponding computed tomography images are presented in Figure ##FIG##0##1f##.</p>", "<p>Furthermore, one patient (Patient No. 2) in this study demonstrated successful conversion following treatment with the proposed protocol. This patient, a 68‐year‐old man diagnosed with HCCA in stage IIIB in July 2020, underwent three cycles of anlotinib and toripalimab treatment. The imaging evaluation of this patient is presented in Figure ##FIG##0##1e##. Following a multidisciplinary treatment (MDT) discussion, it was determined that the patient was eligible for radical surgical intervention. On 1 December 2020, the patient underwent surgical treatment. Subsequently, he received oral tegafur for 6 months, resulting in tumor‐free survival.</p>", "<title>Safety</title>", "<p>The majority of patients (14 out of 15, 93.3%) experienced adverse reactions, which were predominantly mild in severity (Grades 1 and 2). Notably, no Grade 4 adverse reactions were observed. As presented in Table ##TAB##2##3##, these adverse reactions included thrombocytopaenia (60.0%), leucopoenia (53.3%), hypertension (20.0%), malaise (13.3%), hypothyroidism (6.7%) and fever (6.7%). In addition, Grade 3 adverse reactions observed were hypertension and reduced platelet count.</p>", "<title>Results of patients NGS assessment</title>", "<p>Tumor tissues from 11 patients in this study were collected for next‐generation sequencing (NGS) analysis. The identified mutated genes were primarily TP53, KRAS, CDKN2A, STK11 and CDK4 (Figure ##FIG##1##2a##). Additionally, a comparative analysis was conducted between patients with a favourable efficacy evaluation (PR) as the good efficacy group (<italic toggle=\"yes\">N</italic> = 4) and those with an unfavourable efficacy evaluation (PD and SD) as the poor efficacy group (<italic toggle=\"yes\">N</italic> = 7). TP53, KRAS and CDKN2A were commonly gene mutated in the good group (Figure ##FIG##1##2b##). In contrast, TP53, KRAS and STK11 were commonly mutated genes in the poor efficacy group (Figure ##FIG##1##2c##). Notably, the <italic toggle=\"yes\">STK11</italic> gene exhibited a mutation frequency of 43% in the poor efficacy group, whereas no mutations in this gene were identified in the good efficacy group.</p>", "<title>Tumor markers expression analysis by immunohistochemistry (IHC)</title>", "<p>To further characterise the molecular profile of patients who responded to the treatment, immunohistochemistry (IHC) analysis was performed to examine the expression of CD8, CD4, Foxp3 and PD‐L1 in 11 patients with efficacy evaluations including PR, SD and PD. The expression levels of these biomarkers in all patients are shown in Supplementary table ##SUPPL##0##2##. Figure ##FIG##2##3## illustrates that patients with PR showed a higher prevalence of low expression of Foxp3 and high expression of PD‐L1. To consolidate these findings in terms of OS, we established the parameter CD8/Foxp3. X‐tile analysis the optimal cut‐off values of CD8/Foxp3 ratio is 3.0 (Figure ##FIG##3##4e##). Patients in the CD8/Foxp3 ratio &gt; 3 group exhibited prolonged survival compared with those in the CD8/Foxp3 ratio ≤ 3 group (<italic toggle=\"yes\">P</italic> = 0.0397) (Figure ##FIG##3##4f##). As mentioned earlier, patient NO.2 was a successfully converted patient. Analysis of their immunohistochemical background revealed significant expression of CD8 (Figure ##FIG##3##4a##), CD4 (Figure ##FIG##3##4b##), Foxp3 (Figure ##FIG##3##4c##) and PD‐L1 (Figure ##FIG##3##4d##).</p>" ]

[ "<title>Discussion</title>", "<p>Biliary tract cancer constitutes approximately 3% of all gastrointestinal malignancies and ranks as the second most prevalent hepatobiliary cancer.##REF##34626563##\n18\n## Because of its high malignancy and aggressive nature, effective systemic treatment options for advanced BTC are limited. Currently, chemotherapy stands as the mainstay treatment for advanced BTC. However, there exist limited data to assess the potential advantages of combining targeted therapy with immunotherapy in the management of BTC. This phase II clinical trial aimed to evaluate the safety and efficacy of anlotinib in combination with toripalimab was assessed in patients with advanced BTC.</p>", "<p>In this study, the first‐line treatment achieved an ORR of 50% and a DCR of 100%. In the second‐line treatment, the ORR was 18.2% and the DCR was 81.8%. The majority of observed adverse events were Grade 1 or 2, with no reports of serious immune‐related adverse events. Notably, one patient with ICC achieved successful conversion after receiving three cycles of anlotinib in combination with toripalimab as first‐line treatment. These findings highlight the potential of targeted combination immunotherapy as a viable treatment approach for advanced BTC.</p>", "<p>LKB1(STK11) is an oncogene implicated in the proliferation and migration of cancer cells.##REF##34479873##\n19\n## The downregulation of LKB1 has been found to significantly enhance the Wnt/β‐catenin signalling pathway in ICC cells.##REF##26056085##\n20\n## Additionally, studies have reported that silencing LKB1 can lead to upregulation of PD‐L1 surface expression in ICC cells.##REF##32653521##\n21\n## Our study's gene sequencing analysis revealed a higher prevalence of STK11 (LKB1) mutations in patients with unfavourable outcomes. This suggests that individuals with a high frequency of SKT11 mutations may exhibit lower PD‐L1 expression, which could potentially compromise the effectiveness of their treatment.</p>", "<p>The efficacy of immunotherapy is often influenced by the interactions between tumor cells and the TME.##REF##32559423##\n22\n## CD8, CD4 and Foxp3 are markers of T cells, which are important components of the TME.##REF##34303196##\n23\n##, ##REF##34495808##\n24\n## Despite the high PD‐L1 expression of PD‐L1, tumor‐infiltrating lymphocytes (TILs) and tumor mutational burden (TMB) in BTC, the response to immunotherapy has been reported to be unsatisfactory.##REF##32359091##\n11\n##, ##REF##34124390##\n25\n##, ##REF##35121803##\n26\n## Our analysis of these markers yielded similar results, showing no correlation between the expression of CD8, CD4, PD‐L1 and Foxp3 and the efficacy of immunotherapy, although the sample size was limited. However, a multicentre randomised phase II trial of atezolizumab with or without cobimetinib for BTC revealed increased expression of antigen processing and presentation gene and higher CD8/FoxP3 ratios following combination therapy, indicating a potential correlation between the CD8/Foxp3 ratio and patient overall survival.##REF##34907910##\n27\n## Therefore, the findings of this study suggest that the CD8/Foxp3 ratio may serve as a critical factor in determining patient survival.</p>", "<p>This phase II study investigating the combination of anlotinib and toripalimab in patients with BTC has shown promising results. Nevertheless, it is important to note that the study had limitations in terms of sample size and the heterogeneity of the patient population. Hence, further research is warranted to validate these findings, employing larger sample sizes and focusing on more refined tumor subtypes. In conclusion, this study suggests that combination therapy could represent a novel and effective treatment approach for advanced BTC. Moreover, certain markers such as STK11 and the CD8/Foxp3 ratio may indicate potential benefit from this treatment.</p>" ]

[]

[ "<title>Abstract</title>", "<title>Objectives</title>", "<p>To assess the safety and efficacy of anlotinib (a multi‐targeted tyrosine kinase inhibitor) combined with toripalimab (a PD‐1 monoclonal antibody) in the treatment of unresectable biliary tract cancer (BTC).</p>", "<title>Methods</title>", "<p>In this prospective, single‐arm, single‐centre exploratory clinical study, patients with locally progressed or metastatic BTC were included. Patients were treated with anlotinib (12 mg, PO, QD, for 2 weeks and then stopped for a week, 21 days for a cycle) and toripalimab (240 mg, IV, Q3W). The primary endpoint of the study was the objective response rate (ORR), as defined in RECIST version 1.1 criteria.</p>", "<title>Results</title>", "<p>In this study, 15 BTC patients who met the criteria were enrolled. The ORR was 26.7%, the median progression‐free survival (mPFS) was 8.6 months (95% CI: 2.1–15.2), the median overall survival (mOS) was 14.53 months (95% CI: 0.8–28.2) and the disease control rate (DCR) was 87.6%. A patient with hilar cholangiocarcinoma was successfully converted after three cycles of treatment and underwent surgical resection. Furthermore, patient gene sequencing revealed that STK11 was mutated more frequently in patients with poor outcomes. In addition, patients with a CD8/Foxp3 ratio &gt; 3 had a longer survival than those with a CD8/Foxp3 ratio ≤ 3 (<italic toggle=\"no\">P</italic> = 0.0397).</p>", "<title>Conclusions</title>", "<p>In patients with advanced BTC, the combination of anlotinib and toripalimab demonstrated remarkable anti‐tumor potential, with increased objective response rates (ORR), longer overall survival (OS) and progression‐free survival (PFS). Moreover, STK11 and CD8/Foxp3 may be as biomarkers that can predict the effectiveness of targeted therapy in combination with immunotherapy.</p>", "<p>In this study, we found the combination of anlotinib (a multi‐targeted tyrosine kinase inhibitor) combined with toripalimab (a PD‐1 monoclonal antibody) demonstrated remarkable anti‐tumor potential in the treatment of unresectable biliary tract cancer (BTC). Moreover, STK11 and CD8/Foxp3 may be as biomarkers that can predict the effectiveness of targeted therapy in combination with immunotherapy.\n</p>" ]

[ "<title>Ethical approval</title>", "<p>This study protocol was reviewed and approved by ethics committee of Comprehensive Cancer Center of Drum Tower Hospital of Nanjing University and Drum Tower Hospital of Nanjing University. The patients/participants provided their written informed consent to participate in this study.</p>", "<title>Author contributions</title>", "<p>\n<bold>Mingzhen Zhou:</bold> Data curation; formal analysis; writing – original draft. <bold>Yuncheng Jin:</bold> Data curation; investigation. <bold>Sihui Zhu:</bold> Data curation; investigation. <bold>Chen Xu:</bold> Data curation; investigation. <bold>Lin Li:</bold> Data curation; investigation. <bold>Baorui Liu:</bold> Conceptualization; resources; supervision; writing – review and editing. <bold>Jie Shen:</bold> Conceptualization; funding acquisition; resources; supervision; writing – original draft; writing – review and editing.</p>", "<title>Conflict of interest</title>", "<p>The authors declare that they have no conflict of interests.</p>", "<title>Funding information</title>", "<p>The study was supported by the National Natural Science Foundation of Nanjing University of Chinese Medicine (No. XZR2023075); the Hospital Management Research of Jiangsu Province (No. JSYGY‐3‐2023‐618); and the Medical Science and Technology Development Foundation of Nanjing (No. YKK22095).</p>", "<title>Consent for publication</title>", "<p>The patients/participants provided their written informed consent to participate in this study.</p>", "<title>Supporting information</title>" ]

[ "<title>Data availability statement</title>", "<p>All data generated or analysed during this study are included in this published article.</p>" ]

[ "<fig position=\"float\" fig-type=\"Figure\" id=\"cti21483-fig-0001\"><label>Figure 1</label><caption><p>Kaplan–Meier progression‐free survival curves of all patients <bold>(a)</bold> and second‐line treatment patients <bold>(b)</bold>; Kaplan–Meier overall survival curves of all patients <bold>(c)</bold> and the second‐line treatment patients <bold>(d)</bold>. <bold>(e)</bold> The CT and MR of the patient with successfully converted. <bold>(f)</bold> The CT of two patients with the evaluated of PR.</p></caption></fig>", "<fig position=\"float\" fig-type=\"Figure\" id=\"cti21483-fig-0002\"><label>Figure 2</label><caption><p>Next‐generation sequencing (NGS) gene sequencing results. <bold>(a)</bold> Mutations in overall patients. <bold>(b)</bold> Efficacy evaluation for PR patients with mutations. <bold>(c)</bold> Efficacy evaluation for SD and PD patients with mutations.</p></caption></fig>", "<fig position=\"float\" fig-type=\"Figure\" id=\"cti21483-fig-0003\"><label>Figure 3</label><caption><p>Number of people in each efficacy evaluation among the different expressed tumor markers.</p></caption></fig>", "<fig position=\"float\" fig-type=\"Figure\" id=\"cti21483-fig-0004\"><label>Figure 4</label><caption><p>Immunohistochemical straining image of CD8 <bold>(a)</bold>, CD4 <bold>(b)</bold>, Foxp3 <bold>(c)</bold> and PD‐L1 <bold>(d)</bold> of a patient with successfully converted <bold>(e)</bold>. X‐tile analysis of the CD8/Foxp3. The optimal cut‐off value of CD8/Foxp3 is 3.0 <bold>(f)</bold>. Kaplan–Meier overall survival curves of the CD8/Foxp3.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"cti21483-tbl-0001\" content-type=\"Table\"><label>Table 1</label><caption><p>Demographic and disease characteristics at baseline</p></caption><table frame=\"hsides\" rules=\"all\"><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><thead valign=\"bottom\"><tr style=\"border-bottom:solid 1px #000000\"><th align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Eligible patients' characteristics</th><th align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Eligible patients (<italic toggle=\"yes\">n</italic> = 15)</th></tr></thead><tbody valign=\"top\"><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Median age (IQR)</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">59 (38–74)</td></tr><tr><td colspan=\"2\" align=\"left\" valign=\"top\" rowspan=\"1\">Gender, <italic toggle=\"yes\">n</italic> (%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Male</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9 (60.0%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Female</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6 (40.0%)</td></tr><tr><td colspan=\"2\" align=\"left\" valign=\"top\" rowspan=\"1\">ECOG PS, <italic toggle=\"yes\">n</italic> (%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2 (13.3%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">11 (73.4%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2 (13.3%)</td></tr><tr><td colspan=\"2\" align=\"left\" valign=\"top\" rowspan=\"1\">Prior line of treatment, <italic toggle=\"yes\">n</italic> (%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">4 (26.7%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">11 (73.3%)</td></tr><tr><td colspan=\"2\" align=\"left\" valign=\"top\" rowspan=\"1\">Radical surgery</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Yes</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9 (60%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">No</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6 (40%)</td></tr><tr><td colspan=\"2\" align=\"left\" valign=\"top\" rowspan=\"1\">Primary tumor site, <italic toggle=\"yes\">n</italic> (%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Intrahepatic cholangiocarcinoma</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9 (60.0%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Hilar cholangiocarcinoma</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (20.0%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Gallbladder carcinoma</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (20.0%)</td></tr><tr><td colspan=\"2\" align=\"left\" valign=\"top\" rowspan=\"1\">Disease stage<xref rid=\"cti21483-note-0004\" ref-type=\"table-fn\">\n^a\n</xref>\n</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">II</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2 (13.3%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">III</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">4 (26.7%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">IV</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">9 (60%)</td></tr><tr><td colspan=\"2\" align=\"left\" valign=\"top\" rowspan=\"1\">Extrahepatic metastasis</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Lung</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">2 (13.3%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Lymph nodes</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">12 (80%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Bone</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">3 (20%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Other</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1 (6.7%)</td></tr><tr><td colspan=\"2\" align=\"left\" valign=\"top\" rowspan=\"1\">Number of metastatic sites</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">1 (6.7%)</td></tr><tr><td align=\"left\" style=\"padding-left:10%\" valign=\"top\" rowspan=\"1\" colspan=\"1\">≥ 2</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">14 (93.3%)</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"cti21483-tbl-0002\" content-type=\"Table\"><label>Table 2</label><caption><p>Best overall response and disease control rate</p></caption><table frame=\"hsides\" rules=\"all\"><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><thead valign=\"bottom\"><tr style=\"border-bottom:solid 1px #000000\"><th align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Response % (<italic toggle=\"yes\">n</italic>)</th><th align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">First‐line treatment eligible patients (<italic toggle=\"yes\">n</italic> = 4)</th><th align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Second‐line treatment eligible patients (<italic toggle=\"yes\">n</italic> = 11)</th></tr></thead><tbody valign=\"top\"><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">CR</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">PR</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">50.0% (2)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">18.2% (2)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">SD</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">50.0% (2)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">63.6% (7)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">PD</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">18.2% (2)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">ORR</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">50.0% (2)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">18.2% (2)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>ORR‐Total</bold>\n</td><td colspan=\"2\" align=\"left\" valign=\"top\" rowspan=\"1\">\n<bold>26.7% (4)</bold>\n</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">DCR</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">100.0% (4)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">81.8% (9)</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">\n<bold>DCR‐Total</bold>\n</td><td colspan=\"2\" align=\"left\" valign=\"top\" rowspan=\"1\">\n<bold>86.7% (13)</bold>\n</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"cti21483-tbl-0003\" content-type=\"Table\"><label>Table 3</label><caption><p>Patients with treatment‐related AEs</p></caption><table frame=\"hsides\" rules=\"all\"><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><col align=\"left\" span=\"1\"/><thead valign=\"bottom\"><tr style=\"border-bottom:solid 1px #000000\"><th align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">TRAEs</th><th align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Grade 1–2% (<italic toggle=\"yes\">n</italic>)</th><th align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Grade 3% (<italic toggle=\"yes\">n</italic>)</th><th align=\"left\" valign=\"bottom\" rowspan=\"1\" colspan=\"1\">Grade 4% (<italic toggle=\"yes\">n</italic>)</th></tr></thead><tbody valign=\"top\"><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Platelet count decreased</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">60.0% (9)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Leukocyte count decreased</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">53.3% (8)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Hypertension</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">20.0% (3)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Fatigue</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">13.3% (2)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">13.3% (2)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Fever</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6.7% (1)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td></tr><tr><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">Hypothyroidism</td><td align=\"char\" char=\"(\" valign=\"top\" rowspan=\"1\" colspan=\"1\">6.7% (1)</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td><td align=\"left\" valign=\"top\" rowspan=\"1\" colspan=\"1\">0</td></tr></tbody></table></table-wrap>" ]

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[ "<boxed-text position=\"anchor\" content-type=\"graphic\" id=\"cti21483-blkfxd-0001\"></boxed-text>" ]

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[ "<supplementary-material id=\"cti21483-supitem-0001\" position=\"float\" content-type=\"local-data\"><caption><p>\nSupplementary table 1\n</p><p>\nSupplementary table 2\n</p></caption></supplementary-material>" ]

[ "<fn-group id=\"cti21483-ntgp-0001\"><fn id=\"cti21483-note-0001\"><p>Trial registration: ChiCTR2000037847.</p></fn></fn-group>", "<table-wrap-foot id=\"cti21483-ntgp-0002\"><fn id=\"cti21483-note-0002\"><p>Data are median (IQR) or <italic toggle=\"yes\">N</italic> (%).</p></fn><fn id=\"cti21483-note-0003\"><p>ECOG, Eastern Cooperative Oncology Group.</p></fn><fn id=\"cti21483-note-0004\"><label>\n^a\n</label><p>Modified staging classification for intrahepatic cholangiocarcinoma based on the sixth and seventh editions of the American Joint Committee on Cancer and the Union for International Cancer Control TNM staging systems.</p></fn></table-wrap-foot>" ]

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[ "<media xlink:href=\"CTI2-13-e1483-s001.docx\"><caption><p>Click here for additional data file.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Myasthenia gravis (MG) is an autoimmune disease that results in impaired signal transmission due to the presence of pathogenic autoantibodies to several target antigens on the postsynaptic membrane of the neuromuscular junction. Antibodies against acetylcholine receptor (AChR) and muscle-specific kinase (MuSK) induce myasthenic weakness by inhibiting normal neuromuscular signaling. Although symptoms and response to treatment vary depending on the age of onset and antibodies involved, it typically begins with ocular symptoms such as ptosis and diplopia [##REF##28029925##1##]. Within MG, the anti-muscle-specific kinase antibody-positive subtype (MuSK-positive MG) is rare, accounting for 5-8% of all cases [##REF##29266255##2##]. Unlike AChR-MG, which often affects females in their 20s and males aged 60 or older, MuSK-positive MG often affects younger individuals, particularly females under 40 years, and rarely presents after the age of 70 years [##UREF##0##3##]. In addition, MuSK-MG is clearly predominant in females (78-100%) [##REF##23914322##4##].</p>", "<p>There are subtypes of MG defined by age of onset, including late-onset (50-64 years) MG and elderly-onset (≥65 years) MG, whose prevalence has increased recently [##REF##21074862##5##]. A nationwide survey in Japan revealed a 1.5-fold increase in the incidence of late-onset MG over 19 years, with a 2.3-fold increase among elderly patients [##REF##21440910##6##]. MG may be underdiagnosed in the elderly population, or the incidence of late-onset MG may be increasing, or possibly both [##REF##21074862##5##]. Elderly-onset MG is more likely to be severe with life-threatening events at its onset. However, appropriate diagnosis and treatment lead to a good prognosis even in the elderly, as many factors, including comorbidities, medications, and age-related changes, can delay diagnosis [##UREF##1##7##]. In this report, we present a case of an elderly patient with MuSK-positive MG, who presented with isolated type 2 respiratory failure.</p>" ]

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[ "<title>Discussion</title>", "<p>MG is the most common neuromuscular disorder, characterized by fluctuating muscle weakness that worsens with use and improves with rest [##REF##28029925##1##]. Its symptoms vary depending on the age of onset and the antibodies involved [##REF##28029925##1##].</p>", "<p>MG can present with the following signs and symptoms: ptosis in 71.9% of cases, diplopia in 47.3%, weakness of facial muscles in 5.3%, bulbar symptoms in 14.9%, weakness of extremities in 23.1%, and dyspnea in 2.3% [##REF##21440910##6##]. Ocular symptoms are the most common initial presenting complaint. However, the patient did not exhibit typical ocular symptoms. Although MuSK-positive MG tends to manifest atypical symptoms, there are three distinct clinical patterns of MuSK-positive MG as follows: oculopharyngeal weakness, with occasional profound tongue and facial atrophy; neck, shoulder, and respiratory weakness without ocular involvement; and a phenotype indistinguishable from AChR-MG [##REF##23914322##4##]. Our patient exhibited a type of MuSK-positive MG characterized by neck, shoulder, and respiratory weakness without ocular involvement.</p>", "<p>Respiratory muscle weakness can be a life-threatening symptom as it may lead to pneumonia and type 2 respiratory failure [##REF##24532751##8##]. Myasthenic crisis, defined as any exacerbation of MG necessitating mechanical ventilation, occurred in 7.7% of cases within one year after the onset of initial symptoms. The overall frequency of crisis was 13.3% throughout the whole period [##REF##21440910##6##]. Previous studies have reported that in 2% of MuSK-positive MG cases, the initial symptom was respiratory failure, and MuSK-positive MG patients can present with selective respiratory muscle weakness without the typical ocular and limb symptoms [##REF##23914322##4##,##REF##21674519##9##]. Our elderly female patient also exhibited type 2 respiratory failure at the initial onset, but there were no other typical symptoms except for iliopsoas weakness, the clinical significance of which was challenging to determine due to complicating factors.</p>", "<p>Given that MuSK-positive MG is more common in younger individuals, especially females under 40 years of age, and rarely occurs after 70 years of age, our case is considered even rarer [##UREF##0##3##]. In the elderly, physical findings are difficult to recognize due to age-related muscle weakness and the effects of medications, such as cholinesterase inhibitors for dementia. Consequently, without MG being considered as a differential diagnosis, the diagnosis can be easily delayed. Previous studies have reported an increasing incidence of late-onset and elderly-onset MG [##REF##21074862##5##,##REF##21440910##6##]. So it is important to be aware that we may encounter elderly patients who are initially noted to have MG.</p>", "<p>In terms of treatment and prognosis, control agents such as pyridostigmine have been found to be less effective in managing MuSK-positive MG compared to other MG subgroups [##REF##28029925##1##]. The primary treatment approach for MuSK-positive MG treatment is immunosuppression [##REF##29266255##2##]. Initially, this subtype was associated with more severe symptoms, including myasthenic crisis, and a worse prognosis than other subtypes. However, recent reports suggest that MuSK-positive MG can respond well to treatment suggesting that it might not always imply a poor prognostic factor [##REF##31811563##10##]. Furthermore, patients with late-onset MG, including the elderly onset, tend to have better outcomes in terms of drug requirements, drug responsiveness, and the time required to wean off ventilation when experiencing myasthenic crisis. In our case, the patient was able to wean off NPPV with acute treatment and improved the activities of daily living. It is important to be fully aware of late-onset MG and atypical cases that exhibit type 2 respiratory failure, as proper diagnosis and treatment can improve prognosis.</p>" ]

[ "<title>Conclusions</title>", "<p>In conclusion, this case underscores the atypical presentation of MuSK-positive MG in an elderly patient, initially manifested as type 2 respiratory failure. Some patients with MG do not exhibit the typical ocular or limb symptoms and, instead, develop respiratory failure. In the elderly, factors, such as comorbidities, medications, and age-related muscle weakness, can delay the diagnosis of MG. In elderly patients with type 2 respiratory failure, MG should be suspected even in the absence of typical symptoms.</p>" ]

[ "<p>Myasthenia gravis (MG) is an autoimmune disease and represents one of the most common disorders associated with neuromuscular transmission defects. Within MG, the anti-muscle-specific kinase antibody-positive subtype (MuSK-positive MG) is rare. While it shares similarities with the common form of MG by presenting with ocular weakness, MuSK-positive MG typically presents with more atypical symptoms. Although MuSK-positive MG can lead to type 2 respiratory failure due to respiratory weakness, there have been limited reports where initial presentation involves only respiratory compromise. This study details a case of MuSK-positive MG presenting dyspnea. An 84-year-old female presented to the emergency department due to a three-day history of progressive respiratory distress, characterized by increased respiratory effort and shallow breathing, resulting in a diagnosis of type 2 respiratory failure. Despite the absence of neurological abnormalities, she tested positive for anti-muscle-specific kinase antibodies, confirming a diagnosis of MuSK-positive MG. This case highlights the significance of considering MG in the context of type 2 respiratory failure, even in the absence of typical neurological symptoms, especially in elderly patients.</p>" ]

[ "<title>Case presentation</title>", "<p>An 84-year-old female presented to the emergency department with a three-day history of worsening respiratory distress. Her medical problems included reflux esophagitis, insomnia, and Alzheimer's disease. She had no history of obstructive lung pathology and no history of tobacco use. Her home medications consisted of lansoprazole, mosapride citrate hydrate, donepezil hydrochloride, memantine hydrochloride, brotizolam, and albumin tannate. There was no documented history of autoimmune diseases or significant family medical history related to these conditions. Additionally, no obvious triggers, such as recent infections or heightened stress levels, were identified.</p>", "<p>On initial evaluation in the emergency department, her vital signs were as follows: respiratory rate was 25 breaths/min, SpO₂ 90% (on room air), blood pressure 189/109 mmHg, pulse rate 109 beats/min, and her body temperature was 36.5°C. Her Glasgow Coma Scale score was 15 (E4V5M6). Chest, cardiovascular, and abdominal examinations were unremarkable. No obvious neurological abnormalities could be noted. Arterial blood gas analysis on room air showed pH of 7.35, PaCO₂ 64.9 mmHg, PO₂ 51.0 mmHg, and HCO₃- 35.4 mmol/L. Blood test showed hypokalemia (serum potassium level: 2.9 mEq/L); however, it was unclear whether this was related to her respiratory distress. The other laboratory evaluation showed no remarkable findings (Table ##TAB##0##1##). Her chest radiography findings were unremarkable (Figure ##FIG##0##1##).</p>", "<p>Contrast-enhanced computed tomography (CT) was performed to evaluate for pulmonary thromboembolism, with no evidence of pulmonary thromboembolism or thymoma observed. Echocardiography showed normal heart function, with an ejection fraction of 55% using the Pombo method.</p>", "<p>Her level of consciousness worsened shortly after entering the inpatient ward, with PaCO₂ retention increasing to a level of 87.3 mmHg. Her respiratory status and level of consciousness improved approximately 30 minutes after receiving non-invasive positive pressure ventilation (NPPV) with the following settings: inspiratory positive airway pressure set at 10 cmH₂O, expiratory positive airway pressure set at 4 cmH₂O, respiratory rate at 14 breaths/min, fraction of inspired oxygen at 0.21. During the initial days of admission, she required continuous NPPV because her respiratory status worsened shortly after attempts were made to discontinue NPPV.</p>", "<p>We considered the possibility of medication side effects or neuromuscular diseases in our diagnosis, as there was no medical history or imaging evidence indicating a respiratory condition. Other than dyspnea, there were no symptoms suggestive of cholinergic crisis; however, we considered cholinergic crisis, caused by a cholinesterase inhibitor administered for dementia a month before the visit, as a differential diagnosis. Memantine hydrochloride and donepezil hydrochloride were discontinued, but her respiratory status did not improve. Neurological findings showed mild bilateral iliopsoas muscle weakness (manual muscle testing {MMT} score decreased to 4), but It was challenging for us to determine whether the decrease in MMT was due to aging or an underlying medical condition. Her limb muscle strength, reflexes, and sensation were intact. No ptosis, diplopia, or bulbar symptoms were observed. Head CT and magnetic resonance imaging for the evaluation of intracranial disease showed no obvious abnormalities. A lumbar puncture revealed no evidence of Guillain Barre syndrome. While her hypokalemia corrected to within the reference range, respiratory distress did not improve. Nerve conduction and repetitive stimulation tests showed no abnormal findings. On day 13 of admission, she tested positive for anti-MuSK antibody, leading to a diagnosis of MG. Negative findings were observed for anti-nuclear antibody, anti-Sjogren’s syndrome-B antibody, anti-Jo-1 antibody, and anti-aquaporin 4 antibody. The anti-Sjogren’s syndrome-A antibody level was 21.2 U/mL (reference range: 0.0-9.9 U/mL). After her respiratory status improved with intravenous immunoglobulin and tacrolimus for MG, NPPV was discontinued. She was able to wean off NPPV on day 25 of admission. Prednisolone was introduced in addition to acute immunotherapy and pyridostigmine was also initiated as symptomatic therapy. With these acute treatments, the activities of daily living improved to the extent that the patient was able to walk with supervision and take oral intake. Although she had no complications of note during her hospitalization, as progressive disuse and dementia made it difficult for her to be discharged home, she was transferred to a hospital for rehabilitation on day 77 of admission.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Chest x-ray showed no remarkable findings.</title></caption></fig>" ]