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World Economics Association
World Social and Economic Review
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Capital Accumulation, Production and Employment:
Can We Bend the Arc of Global Capital Toward Justice?
A conference from the WEA
15th May to 15th July 2016
Theoretical Background
Employment in a Just Economy
John Komlos
PhD in History (1978) and Economics (1990) by the University of Chicago. Professor Emeritus, University of Munich
Please cite the paper as:
John Komlos, (2016), Employment in a Just Economy, World Economics Association (WEA) Conferences, No. 1 2016, Capital Accumulation, Production and Employment:, 15th May to 15th July 2016
This paper has been included in the publication
"Capital and Justice" Edited by Gerson P. Lima and Maria Alejandra Madi
Download from WEA books Purchase from Amazon UK
This paper is based on the ideas of political philosopher John Rawls and suggests that in a just economy full employment would have to go beyond the implications of NAIRU. Rawls argues that the just economy is one which would be created behind a veil of ignorance, that is to say, without knowing where we would end up in the society's distribution of talent and other attributes. If we were to create it from scratch we would not create the kind of labor markets that exist currently, because we would be too concerned about ending up among the excluded, i.e., those without full time jobs which today in the U.S. is still 10% of the labor force or some 15 million adults. This is substantial but it does not even include at least 7 million additional people who have dropped out of the labor force altogether or the 2.3 million in jail!
Hence, the "natural" in the natural rate of unemployment is a misnomer, insofar as unemployment does not occur in nature. The concept is especially misleading, because many economists and media commentators inappropriately but invariably equate it with "full" employment as Martin Feldstein did in the January meeting of the American Economic Association. As a consequence, endemic un- and underemployment is accepted as an inevitable attribute of the labor market. This is insidious inasmuch as the concept assumes that the institutional structure of the labor market is held constant. However, with creative restructuring of that market our aim should be to bring unemployment down to the minimum feasible rate which in the U.S. is most likely around 1.2%,–the rate which prevailed in 1944 and which probably represents the lower bound attainable. Instead of the prevailing system, the right to work needs to be recognized as a natural right, because the right to life depends upon it. Several ways are proposed to create an inclusive labor market that distributes the available work in a more equitable fashion than the current system and envisions a just economy that risk-averse people would be willing to enter at random, i.e., without knowing where they would end up in it.
Keywords: full employment, justice, Labor Market, NAIRU, natural rate of unemployment, Rawls
1 Comment ↓
Stephen I. Ternyik says:
Thank you for attacking this very important socio-economic problem of `making a living and preserving self-esteem`, Prof. Komlos! The NAIRU is indeed the ACRU (artificially created rate of unemployment) and the monetary theory behind this coining has already proven to be economically not sustainable. Economic systems evolution empirically points now to a more just economy as envisioned by John Rawls or to a historical tragedy of social explosion, in thermo-economic terms. The binary introduction of a basic income guarantee (envisioned e.g. by T.Paine) and the right to labor (to earn an income) could restore human dignity under the current conditions, i.e. computer power is destroying `jobs` in an accelerated manner. We will have economically to be very creative to institutionalize an income for all citizens in a just economy, to balance the ongoing `creative destruction` of our socio-economic fabric.
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[124] Myeloid cells have also been shown to regulate susceptibili
Posted on February 28, 2019 by syks1232
[124] Myeloid cells have also been shown to regulate susceptibility to EAE following activation of type I NKT cells by αGalCer.[134] Hence, depletion of immunosuppressive myeloid-derived suppressor cells from the spleen results in the loss of αGalCer-induced protection from EAE.
These reports suggest that activation of NKT cell subsets BGJ398 nmr in different tissues may not only lead to their interaction with professional APCs but also with other immune regulatory cells, including myeloid-derived suppressor cells and Treg cells, and that they can cooperate to provide protection from autoimmune pathology. In this review, we have attempted to identify key outstanding issues related to the role of NKT cell subsets in health and disease, and how some of these issues may be addressed experimentally and clinically. Based on current evidence, we have proposed a hypothesis that states that while type I NKT cells have pathogenic and protective roles in autoimmune disease susceptibility, type II NKT cells possess mainly a protective role. We have discussed how new experimental mouse models coupled with the application of novel techniques, namely intravital cellular imaging in vivo and mass cytometry, may test this hypothesis and also
provide important insights into the role of NKT–DC interactions and cytokine/chemokine secretion profiles in determining the outcome of health versus disease. As the CD1d-dependent
antigen recognition pathway is highly conserved from mice to humans, several key features of NKT cell MAPK Inhibitor Library subsets are shared between them. Although most studies in mice have analysed NKT cells from the thymus, spleen or liver, the systemic results of their manipulation indicate that follow-up clinical studies are warranted. Therefore, discoveries in experimental models can be translated into the clinical setting,[1, 128] and allow the application of murine studies to humans. Fortunately, type II NKT cells occur more frequently than type I NKT cells Alectinib nmr in humans, which facilitates their further characterization using appropriate lipid ligands. A detailed characterization of type II NKT cells and their ligands in humans is important for their appropriate manipulation in disease conditions. Phase I/II clinical trials of the anti-tumour effects of human type I NKT cells stimulated by αGalCer have yielded promising results.[129, 130, 71, 131] Other analogues of αGalCer that skew conventional CD4+ T-cell responses towards either a Th1- or a Th2-like profile remain to be tested in similar trials. In the near future, it may be possible to differentially activate or inhibit type I and type II NKT cells for the development of novel immunotherapeutic protocols in the treatment and prevention of autoimmune disease.
02 × [serum] (CI 0 99, 1 02) across the range 0–30,000 pg/mL R-s
02 × [serum] (CI 0.99, 1.02) across the range 0–30,000 pg/mL. R-squared for the model was 0.99. The co-variates age, gender, weight, smoking status, SBP, DBP, CKD stage, GFR or diagnostic category had no significant
influence of the regression relationship. Conclusion: There is excellent correlation of Midkine levels between serum and plasma, confirming either specimen type may be used to accurately assay Midkine levels. 162 DEFECTIVE MITOPHAGY ACTIVITY IN EXPERIMENTAL DIABETIC NEPHROPATHY GC HIGGINS1,2, TV NGUYEN1, SA PENFOLD1, V THALLAS-BONKE1, BE HARCOURT3, PM ROBB1, G RAMM4, G JERUMS5, A SKENE6, R. MACISAAC7, EI EKINCI5,8, DA POWER9, KE WHITE10, RW BILOUS11, ME COOPER1,12, JM FORBES3, Neratinib nmr MT COUGHLAN1,12 1Glycation, Nutrition and Metabolism Laboratory, Diabetic Complications Division, Baker IDI Heart & Diabetes Institute,
Melbourne, Victoria; 2Department of Biochemistry & Molecular Biology, Monash University, Clayton, Victoria; 3Glycation & Diabetes, Mater Medical Research Institute, South Brisbane, Queensland; 4Membrane Biology Group, Department of Biochemistry and Molecular Biology, Monash University, Clayton, Victoria; 5Austin Health and the University of Melbourne, Melbourne, Victoria; 6Department of Anatomical Pathology, Austin Health, Melbourne, Victoria; 7Department of Endocrinology & Diabetes, St Vincent's Hospital, Melbourne, Victoria; 8Menzies School of Health Research, Charles Darwin University; 9Department of Nephrology, Austin buy Vemurafenib Health and the University of Melbourne, Melbourne, Victoria, Australia;
10EM Research Services, Faculty of Medical Sciences, Newcastle University, Newcastle upon Tyne; 11James Cook University Hospital, Middlesbrough, www.selleck.co.jp/products/Gefitinib.html United Kingdom; 12Department of Medicine, Central Clinical School, Monash University, Alfred Medical Research & Education Precinct, Melbourne, Victoria, Australia Aim: Here, we aimed to determine if there was an impairment in mitophagy and changes in mitochondrial dynamics in the kidney in Diabetic Nephropathy (DN). Background: DN is the major cause of end stage renal disease in the Western world. Defects in mitochondrial bioenergetics are evident in DN and are thought to initiate renal impairment. Accumulation of fragmented mitochondria are found in the renal cortex in experimental diabetes, suggesting that in tandem with a shift in dynamics, mitochondrial clearance mechanisms may be impaired. The process of mitophagy is the selective targeting of damaged or dysfunctional mitochondria to autophagosomes for degradation through the autophagy pathway. Methods: Markers of mitophagy and mitochondrial bioenergetics and dynamics were followed in the renal cortex from rodents rendered diabetic with the beta cell toxin streptozotocin (STZ).
Currently, the only approved vaccine against TB is the attenuated
Currently, the only approved vaccine against TB is the attenuated Mycobacterium bovis strain
Bacillus Calmette–Guerin (BCG). BCG is highly variable in efficacy (from 0 to 80%), as evidenced by reports showing that it is efficacious in protecting children, but not adults, from TB [7, 8]. Also, emerging multidrug-resistant strains have contributed to the increase in the rate of mortality caused by TB [9]. Thus, the development of a new and more effective vaccine is needed to control TB. As a consequence, the search for a new vaccine has intensified, especially in regard to the study of using immunodominant M. tuberculosis antigens such as Ag85A, Ag85B, ESAT-6, CFP-10 and TB10.4 (along with fusion proteins that combine these antigens) as vaccines. Such formulations have provided effective protection against M. tuberculosis in animal models [10–14]. In addition, studies have demonstrated that T cell-mediated selleckchem immune responses are required to control TB disease. Nevertheless, the evidence suggests that the adjuvants play an important role in stimulating these cells. Many adjuvants have been used with vaccines, including the classical adjuvanted subunit vaccines, BGC, the aluminium salts and synthetic cationic adjuvants like IC31 [15–18]. However, the recent progress in the development of novel
delivery systems has allowed the fusion of M. tuberculosis antigens to biological molecules to couple the adjuvant with the antigen [19]. In this regard, calreticulin this website has been of particular interest because it allows fused antigens to be directly targeted for MHC class I presentation because it can associate with peptides delivered to the endoplasmic reticulum by transporters associated with antigen processing (TAP-1 and TAP-2) and with MHC class I β2-microglobulin molecules [20–23]. In fact, tumour antigen linked to calreticulin can generate tumour-specific immunity and eradicate established tumours [24, 25]. Others have demonstrated
that calreticulin linked to the protective antigen domain IV from Bacillus anthracis enhances antibody responses [26]. Here, we describe the development and characterization of a recombinant replication-deficient adenoviral vector that expresses immunogenic M. tuberculosis Ag ESAT-6 fused to calreticulin. Additionally, we evaluated its ability to induce the production of tumour necrosis factor (TNF)-α Clomifene and interferon (IFN)-γ, two cytokines required for protective immunity, and its capacity to protect against a M. tuberculosis challenge. Our data demonstrate that the calreticulin–ESAT-6 and calreticulin–ESAT-6–CFP10 fusion proteins generate a specific immune response, but this response does not confer protection against pulmonary M. tuberculosis infection. Construction and characterization of the recombinant replication-deficient adenoviruses. The gene fusions ESAT-6–CFP10 and ESAT-6 were purchased from Invitrogen (Carlsbad, CA, USA) already cloned into pUC plasmids (pESAT-6–CFP10 and pESAT-6, respectively).
3) Furthermore, the same treatment regimen reduced significantly
3). Furthermore, the same treatment regimen reduced significantly the levels of IFN-γ, IL-1β, IL-2, IL-17 and TNF-α both in the spleen and pancreatic lymph nodes compared to control mice (Fig. 4a,b). The same differences, with the exception of TNF-α being undetectable, were also observed in murine sera in the same experimental conditions (Fig. 4c). Finally, prolonged treatment with apoTf did not change significantly the proportion of splenic CD4+ regulatory T cells (Treg) (CD4+/CD25+/FoxP3+) cells compared to control mice (Fig. 5). ApoTf plasma levels were significantly lower in patients with selleckchem ND-type 1 diabetes compared to matched
controls, while this difference was not observed comparing patients with CR or LS disease (Fig. 6).When biochemical and clinical features of ND-type 1
diabetes were correlated with apoTf levels we found a significant association with HbA1c determined at disease onset using both laboratory methods (r = −0·452, P = 0·045 with RID; r = −0·564, P = 0·01 with nephelometry) but not with basal or stimulated C peptide levels, ICG-001 in vitro GADA and IA2 antibodies, weight loss prior to diagnosis or symptom duration (data not shown). No correlation with any of the analysed clinical and biochemical features was encountered in patients with LS or CR type 1 diabetes (data not shown). The data presented herein were obtained from different murine and cellular models as well as human samples to demonstrate for the first time that recombinant human apoTf or human-derived apoTf acts to inhibit significantly the inflammatory many pathways leading to diabetes. The affected pathways included cytokine-induced beta cell death in
vitro and disease onset in well-established models. In particular, apoTf was associated with milder signs of insulitis and profound modulation of cytokine secretory profile in NOD mice. Several findings may prove significant in our understanding of type 1 diabetes pathogenesis and the role of apoTf. First, the prolonged ex-vivo treatment with apoTf leads to down-modulation of the destructive Th1 and Th17 autoimmune responses [17,21,22] that produce IL-1β, IL-2, TNF-α, IFN-γ, IL-17 and IL-18 [23], which are crucial to diabetes development in the NOD mouse. Th1, Th17 and Treg are thought to be regulated reciprocally and, therefore, changes in Treg could be expected in the immune modulating activity we observed during apoTf treatment in NOD mice [24]. Nevertheless, we could not observe significant changes in the prevalence of Treg (CD4+/CD25+/FoxP3+) cells in the spleen of animals treated for 12 weeks. Further studies are being carried out to demonstrate whether ApoTf exerts its anti-diabetogenic effect by up-regulating Treg function without modifying their numbers or whether it acts via Treg-independent pathways.
One reason for this might be a decreased bone marrow output Afte
One reason for this might be a decreased bone marrow output. After these changes within the first years of life the absolute number of B cells remain stable while the shift from naive to memory B cells
continues. It has been suggested that the molecular pathways underlying the generation of memory B cells differ between CD27+IgD+ and CD27+IgD- memory B cells. Whereas CD27+IgD- memory B cells seem to represent post-germinal centre B cells, the development of CD27+IgD+ memory B cells (including the acquisition of somatic hypermutation) might be independent of germinal centre Tofacitinib price reactions [8,24]. It has been suggested that CD27+IgD+ memory B cells represent a cellular surrogate of T cell-independent humoral immunity. Humoral immunity against encapsulated bacteria has been attributed to the presence of these memory B cells [25]. However, it is interesting to note that the age-dependent frequencies of both memory B cell subsets indicate comparable developmental stimuli (Figs 2 and 3). Recently, a B cell population lacking surface expression of CD27 but harbouring signs of memory
B cells (somatic hypermutation and immunoglobulin class switch) could be demonstrated in peripheral blood as well as in tonsils [9,10]. These memory B cells seem to be expanded in systemic autoimmunity (e.g. systemic lupus erythematosus) and chronic infectious diseases (e.g. human immunodeficiency Pazopanib virus, malaria) Temozolomide chemical structure [10,26,27]. The role of these B cell subsets in a physiological context is not elucidated well. Although the frequency of CD27-IgD- B cells increased during the first 5 years of age, the frequency of these B cells remained stable afterwards (Fig. 2). This is in contrast to the other memory B cell subsets, which increased gradually during age. Whether the differentiation and expansion of this particular memory B cell subset underlies different molecular and cellular pathways is a matter of research. In most individuals CD24-CD38++ B cells, representing circulating plasmablasts, could be detected in low
frequencies. Frequencies of plasmablasts almost never exceeded 5% of total B cells and did not seem to show significant changes between age groups (Fig. 2). This observation seems to be worth mentioning, as expansion of plasmablasts in the peripheral blood seems to be a characteristic pattern in distinct systemic autoimmune diseases [18]. Therefore, sustained expansion of plasmablasts above this defined cut-off might be an indicator of systemic autoimmune diseases (e.g. systemic lupus erythematosus), and seems to correlate with disease activity in this disease [18]. As well as disturbed B cell homeostasis in autoimmune diseases, B cell development and differentiation is impaired in several immune deficiencies.
3D) After 4 wk, three to five times more CD34+ cells were presen
3D). After 4 wk, three to five times more CD34+ cells were present in those cultures using IL-32 than in control samples (p<0.018, Table 2). These differences were
in part accompanied by a higher number of 2-wk cobblestones formed by cells cultured in IL-32 plus SCF (p<0.015) than those formed by cells cultured in SCF alone. The highest numbers of 5-wk cobblestones, an indicator for more primitive HPCs, were achieved in cultures supplemented with 100 ng/mL IL-32 (compared with intra-assay control p=0.014). After 2 wk in culture, the frequency of CD34+ cells ranged from 5 to 39%. The IL-32 expanded cells continued to be positive for CD34 until the end of the culture period; they also increasingly expressed CD45, indicating FK506 manufacturer leukocyte differentiation (Fig. 4A and B). The cells' colony-forming capacity, especially the total number of burst-forming unit erythrocyte and the plating efficiency were significantly better than in control
cultures consisting of medium only (Fig. 4C). The total numbers of colonies of cells cultured with IL-32 were equivalent to those cultured in SCF alone, while they led to a significantly higher plating efficiency (11±1.3% versus 4.9±0.43%, p<0.001). The other potential growth factors we tested led to significantly fewer numbers of colonies than SCF (Supporting Information Fig.). Injections of 5-fluorouracil (FU) produce profound myelosuppression in Balb/c mice within 7 days, and regeneration usually begins around day 10 24. In our study, myelosuppression was attenuated when BYL719 supplier human recombinant IL-32 was applied after 5-FU treatment. Both white blood cell (WBC) and platelet counts were significantly higher in mice treated with IL-32 on day 7 (Fig. 5A and B). On day 4, WBC counts were 30% higher, if 5 μg IL-32 had been administered (97.5±15×108/L versus normal saline 68.6±5.5×108/L, p<0.03). On day 7, the difference was even more prominent (53±6.6×108/L versus normal saline 33.6±3.1×108/L, p=0.011), which paralleled significantly higher monocyte counts (191.2±41.8×106 versus normal saline 34.5±10.1×106, p=0.002).
On this day, platelet counts of mice treated with 5 μg IL-32 were also significantly higher than in the control group (169.4±11×109/L versus normal saline 130.2±10.3×109/L, p=0.013), and they were surpassed by platelet counts in Inositol oxygenase mice, which had received the high dosage of 50 μg IL-32 (216.9±22.4×109/L, p=0.038). Though the number of thrombocytes seemed to be higher in IL-32 treated mice on days 10 and 14, differences discontinued to be significant (p>0.1). On day 14, twice the number of granulocytes was present in mice treated with 50 μg IL-32 compared with the normal saline group (1315.6±344×106 versus 670.3±290.8×106, p=0.04). No differences between the three different treatment groups were found in the hemoglobin contents, hematocrits, lymphocyte and red blood cell counts.
Several cytokines have been exploited for their immunostimulatory
Several cytokines have been exploited for their immunostimulatory properties, either as single agents or in combination
therapies 10. The first was type I IFN, in particular IFN-α, which strongly activates both the innate and adaptive arms of the immune response 11 (Fig. 1). Interleukin (IL)-2 was introduced Enzalutamide supplier in the 1980s as a T-cell stimulatory agent and has been approved since then for therapeutic use in renal cell carcinoma and melanoma 12; however, it is also a growth and survival factor for Treg cells, and was used in a recent study to dampen the inflammatory response in hepatitis C-induced vasculitis 13. GM-CSF is a myeloid differentiation factor and mostly activates phagocytes; however, recent evidence shows that it can also promote IL-10-producing T cells through pDC activation 14. Other studies pointed at a potential role of GM-CSF in tolerance induction 15, 16, illustrating the pleiotropic effects of this cytokine (Fig. 1). IL-12 is an interesting candidate to promote immunity to intra-cellular pathogens, such as mycobacteria and viruses 17. Based on their specific biology, the cytokines discussed in this paragraph have been studied as adjuvants in vaccine formulations that are currently under clinical development 1, 10. The results of clinical studies have produced mixed results 18 and, to the best of our knowledge, none of them has reached the stage of FDA approval
in this context. In recent years, a resurgence of interest in cytokines as therapeutic agents has emerged following the discovery of a see more number of interesting cytokines involved in various physiopathological processes, including infection, allergy, and auto-immunity. These include IL-17, IL-21, IL-22, IL-23, IL-27, and thymic stromal lymphopoietin Methane monooxygenase (TSLP). TSLP is an IL-7-like short-chain hematopoietic cytokine that was initially cloned in the mouse as a B-cell growth and differentiation factor 19. In the human, it mostly acts on DCs and mast
cells 19. Its direct effect on T cells remains controversial 20. No effect on B cells has been reported to date. A large number of studies have implicated TSLP in the physiopathology of allergic inflammation through its ability to induce the production of pro-allergic chemokines by DCs, together with a pro-inflammatory Th2-cell response 21, 22. In this issue of the European Journal of Immunology, Van Roey et al. 23 explore different possible vaccine adjuvants with regard to protection of HIV infection in an experimental setting, where there is a strong need for adjuvants to shape a protective immune response 24. The mucosal intranasal route is chosen in order to preferentially induce mucosal immunity through sIgA and infiltrating T cells. This route is known to provide protection not only in the upper respiratory tract but also in the vaginal mucosa, potentially interfering with the sexual transmission of HIV.
A C has received personal compensation
A. C. has received personal compensation Osimertinib mouse for activities with Almirall Hermal GmbH, Bayer Schering, Biogen Idec, Merck Serono, Novartis and Teva Neuroscience, research support from Bayer Schering, Biogen Idec, Merck Serono and Novartis and research grants from the German Ministry for Education and Research [BMBF, 'German Competence Network Multiple Sclerosis'
(KKNMS), CONTROL MS, 01GI0914]. "
"Analysis of the molecular mechanisms governing the ability of IL-10 to keep inflammation under control has highlighted the existence of a great degree of plasticity and specificity with regard to innate immune cells. In this respect, neutrophils represent a perfect example of innate immune cells conditioned by external signals (for instance, by LPS), as well as by intracellular regulatory pathways, that render them optimally responsive to IL-10 only
when required. The focus of this review are the recent experimental findings that have uncovered the sophisticated and complex molecular mechanisms responsible for the modulation of pro- and anti-inflammatory cytokine production learn more by IL-10 in neutrophils and other innate immune cells. Understanding how IL-10 exerts its anti-inflammatory response, particularly in the case of neutrophils, will provide novel clues leading, hopefully, to Resveratrol the therapeutic control of neutrophil-driven inflammatory reactions, such as septic infections, rheumatoid arthritis, osteoarthritis and chronic obstructive pulmonary disease. The biological activities of IL-10 function, in part, as key homeostatic mechanisms that control the degree and duration of the inflammatory response. IL-10, in fact, acts as a potent anti-inflammatory cytokine by conditioning the activation and function of innate and Ag-specific immune cells. Accordingly, a crucial effect of IL-10 is its ability to
selectively block the expression of pro-inflammatory genes encoding cytokines and chemokines in myeloid cells activated by PRR ligands, such as LPS, lipoteichoic acid or the TLR9 agonist, CpG, thereby dampening inflammation 1. By contrast, IL-10 simultaneously enhances the expression and production of anti-inflammatory molecules 1. The molecular mechanisms whereby IL-10 modulates the production of pro- and anti-inflammatory cytokines by target cells are, however, incompletely understood. Nonetheless, a general consensus exists regarding the following IL-10-triggered signaling steps, occurring both in murine and in human systems.
[18, 31] Studies have demonstrated
dynamic changes in the
dynamic changes in the ultrastructural features of the cell wall during morphogenic transformation to germ tubes, and have shown that the cell wall of germ tubes possesses selleck kinase inhibitor stratification comparable to that of the blastospore wall.[32] Other studies have shown internally collapsed cells with an intact cell wall leaving 'ghosts-like' cells and deflated Candida cells following exposure to subcidal concentrations of nystatin.[23] Therefore, it is not surprising that nystatin-induced changes in the cell wall structure of C. dubliniensis isolates would affect active budding and multiplication, thus suppressing its growth resulting in a PAFE of nearly 2 h in addition to subduing its adhesion to BEC as well as GT formation even after a brief exposure to nystatin. Microbial structures that contribute to the CSH include outer membrane protein, selleck chemical lipoprotein, phospholipid, lipopolysaccharide and fimbriae.[28] Thus, drugs that modify these structural features have shown to reduce the CSH of microbes.[29] In the case of C. albicans, it has been shown that the CSH correlates with the concentration of fibrils in the exterior layer of the cell wall. As C. dubliniensis isolates are phenotypically similar to C. albicans isolates, the observed suppression of CSH elicited by nystatin (approximately 35% reduction) on
C. dubliniensis though very much less than the other two adhesion attributes observed in the current investigation (mean reduction of 34.81% for CSH vs. 74.45% for adhesion to BEC and 95.92% for GT formation), it too may be related to the aforementioned pharmacodynamics of the nystatin on the cell wall of C. dubliniensis.[18, 31] Therefore, it is reasonable to speculate that by affecting the cell wall structure, nystatin may be capable of suppressing the CSH of this Candida species. Analysis of the variation between the impact that nystatin had on the three pathogenic attributes revealed
that there was a significant positive relationship between the Phosphoprotein phosphatase suppressive effect of the drug on adhesion to BEC and GT formation by C. dublinienis isolates (P = 0.046), whereas the effect elicited by nystatin on CSH did not have a positive relationship with the clampdown of adhesion to BEC and GT formation. Relative CSH is considered as a non-biological physical force related to adhesion whereas adhesion to BEC and GT formation by Candida are direct biological traits pertaining to adhesion. Hence, this difference in the relationship on the impact of nystatin on adhesion attributes of C. dubliniensis, which has not been documented hitherto, may be due to the difference in biological and non-biological forces involved in the adhesion process. Notwithstanding these differences, nystain was capable of suppressing both biological and non-biological adhesion attributes of C. dubliniensis as seen in this study.
Table 4 shows the presence of genes encoding Hox orthologs in the
Table 4 shows the presence of genes encoding Hox orthologs in the genomes of Hymenolepis and Echinococcus spp., S. mansoni, polyopisthocotylean 'monogeneans', and the planarian S. mediterranea. From these representatives, it appears that flatworms have a core set of one anterior gene (Hox1/Lab) and three central
genes (Hox3, Hox4/Dfd, Lox4/Abd-A). see more In addition, both characteristic lophotrochozoan posterior Hox genes (Post-1/2) are found, although those were initially thought to be missing from flatworms (128,142). Planarians also show the presence of Hox5 orthologs and larger numbers of central and posterior paralogs than found in parasitic flatworms, although it must be noted that whereas some of the homeobox sequences (e.g. Hox1, Hox4/Dfd and Hox8/Abda) show high levels of similarity to cognates outside the group, other flatworm homeoboxes are divergent and difficult to classify. Nevertheless, compared with other major lophotrochozoan groups such as annelids and molluscs, both free-living and parasitic flatworms show reductions in the numbers of Hox gene classes, and this may relate to their lack of axial elaboration. Hymenolepis is also oddly missing www.selleckchem.com/products/nutlin-3a.html an ortholog of the central Hox3 gene found in all other flatworms examined. In all cases, flatworm Hox genes are found to be widely dispersed in the genome and have been
shown previously to reside on at least two different chromosomes in S. mansoni (139). RNA-seq data indicate the presence of multiple non-Hox coding regions flanking the Hox genes in the Hymenolepis genome and thus further confirm the complete lack of clustering of flatworm Hox genes. The genomic structure of Hymenolepis orthologs appears normal, and full-length transcripts Sulfite dehydrogenase range in size between ∼1500 (HmHox1)–2600 (HmPost-2) bp and are
made up of 2–4 exons separated by introns 81–8946 bp in length. The abdominal-B ortholog HmPost-2 shows a characteristic intron interrupting the homeobox region. In contrast, typically structured Post-1 orthologs have not been described in flatworms, and the one (possibly two) Hymenolepis Post-1 orthologs appear as pseudogenes, and full-length exons cannot be deduced from present data. Expression of Hox genes in parasitic flatworms is so far known only from quantitative PCR and RNA-seq data that indicate dynamic patterns throughout their complex life cycles. Stage-specific expression has been demonstrated in S. mansoni (139), the 'monogenean'Polystoma gallieni (143), and in Hymenolepis and RNA-seq data in Hymenolepis also indicate at least minimal expression levels during both adult and larval development, with peaks of expression seen in central and posterior genes. How the dispersed structure of their genes affects the principal of colinearity is not known, and only a few studies of Hox spatial expression have been conducted in free-living flatworms, with somewhat inconsistent results (144), and none in parasitic flatworms.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: What is the formula for Nichrome Having no knowledge of electricals, I am interested in learning the applications of Nichrome.
I understand it has swg ratings. But what I am looking for is a formula that can calculate a particular thing if other certain inputs are provided.
Like the V = I/R triangle, is there a formula for Nichrome containing the following propreties:
Voltage, Amperage, Temperature of Nichrome, Length of Nichrome wire and Rating of Nichrome wire (swg) - I believe these are all the variables applicable, please correct me if I am wrong.
Example 1:
So for instance, if I input the Temperature of the Nichrome I want to achieve, say 50C celius, the length of the Nichrome wire to be 2 Inches and the 32 swg rating, using these, can the formula calculate the voltage and amperage requirements?
Example 2:
Using the inputs, voltage, amperage, length of wire and swg rating, can it calculate the temperature?
Thanks

A: The nichrome wire is essentially a resistor. It varies its resistance slightly over temperature, but the basic idea is it's just a resistor dissipating power. It's nothing magical, just conservation of energy. A vast majority of the electrical energy being dissipated by the nichrome wire will be as heat.
This site gives a tabulated chart for relating size of the nichrome, temperature, and amperage required.

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}

Fractional calculus is a natural extension of ordinary calculus, where integrals and derivatives are defined for arbitrary real orders.  Since $17th$ century, when fractional calculus was born, several different derivatives have been introduced: Riemann-Liouville, Hadamard, Grunwald--Letnikov, Caputo, just to mention a few \cite{Kilbas,Podlubny,Samko}, each of them with its own advantages and disadvantages. The choice of an appropriate fractional derivative (or integral) depends on the considered system, and for this reason we find a large number of works devoted to different fractional operators. On the other hand, it is natural to look for and study generalized fractional operators, for which the known ones are particular cases. Recently, U. Katugampola presented new types of fractional operators, which generalize both the Riemann--Liouville and Hadamard fractional operators \cite{Katugampola1,Katugampola2,Katugampola3}.

In this paper we present a new fractional operator, called the Caputo--Katugampola derivative, which generalizes the concept of Caputo and Caputo--Hadamard fractional derivatives (Section~\ref{sec:FC}). The new operator is the left inverse of the Katugampola fractional integral and keeps some of the fundamental properties of the Caputo and Caputo--Hadamard fractional derivatives. In Section~\ref{sec:FDE} we study a fractional differential equation (FDE), where the differential operator is the Caputo--Katugampola derivative, showing an existence and uniqueness theorem for the Cauchy type problem. Section~\ref{sec:approx} provides an approximated expansion of the Caputo--Katugampola derivative, that involves only the first-order derivative. Applying this formula to a given Caputo--Katugampola fractional Cauchy problem we obtain a sequence of systems of $N+1$ ordinary differential equations with $N+1$ initial conditions, where $N\in \mathbb{N}$. A sequence $(x_N)$ of the solutions to those systems of ODE's converges to the solution of the fractional problem. The numerical procedure to solve the fractional differential equation is presented in Section~\ref{Sec:examples} throughout an example. Finally, Section~\ref{con} contains some conclusions.

\section{Caputo--Katugampola fractional derivative}\label{sec:FC}

In \cite{Katugampola1}, the author introduced new fractional integral operators, called the Katugampola fractional integrals, in the following way:
$${I_{a+}^{\alpha,\rho}}x(t)=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_a^t \frac{\tau^{\rho-1}}{(t^\rho-\tau^\rho)^{1-\alpha}}x(\tau) d\tau,$$
$${I_{b-}^{\alpha,\rho}}x(r)=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_t^b \frac{\tau^{\rho-1}}{(\tau^\rho-t^\rho)^{1-\alpha}}x(\tau) d\tau,$$
where $0<a<b<\infty$, $x:[a,b]\rightarrow\mathbb{R}$ is an integrable function, and $\alpha\in(0,1)$ and $\rho>0$ two fixed real numbers.
Then, in \cite{Katugampola2}, it appeared a generalization to { the} Riemann--Liouville and Hadamard fractional derivatives, called the Katugampola fractional derivatives:
$${D_{a+}^{\alpha,\rho}}x(t)=\frac{\rho^\alpha}{\Gamma(1-\alpha)}t^{1-\rho}\frac{d}{dt}\int_a^t \frac{\tau^{\rho-1}}{(t^\rho-\tau^\rho)^\alpha}x(\tau) d\tau,$$
and
$${D_{b-}^{\alpha,\rho}}x(t)=\frac{-\rho^\alpha}{\Gamma(1-\alpha)}t^{1-\rho}\frac{d}{dt}\int_t^b \frac{\tau^{\rho-1}}{(\tau^\rho-t^\rho)^\alpha}x(\tau) d\tau.$$
The relation between these two fractional operators is the following:
$${D_{a+}^{\alpha,\rho}}x(t)=t^{1-\rho}\frac{d}{dt}{I_{a+}^{1-\alpha,\rho}}x(t) \quad \mbox{and} \quad {D_{b-}^{\alpha,\rho}}x(r)=-t^{1-\rho}\frac{d}{dt}{I_{b-}^{1-\alpha,\rho}}x(r).$$
Also, when $\alpha>0$ is an integer, these fractional integrals are generalizations of the n-fold integrals
$$\int_a^t \tau_1^{\rho-1}\, d\tau_1\,\int_a^{\tau_1} \tau_2^{\rho-1}\, d\tau_2\,\ldots \int_a^{\tau_{n-1}} \tau_n^{\rho-1}x(\tau_n)\, d\tau_n$$
and
$$\int_t^b \tau_1^{\rho-1}\, d\tau_1\,\int_{\tau_1}^b \tau_2^{\rho-1}\, d\tau_2\,\ldots \int_{\tau_{n-1}}^b \tau_n^{\rho-1}x(\tau_n)\, d\tau_n,$$
respectively.

Motivated by those definitions, we introduce a new fractional derivative as follows.

\begin{definition}\label{CKFD}
Let $0<a<b<\infty$, $x:[a,b]\rightarrow\mathbb{R}$ be an integrable function, and $\alpha\in(0,1)$ and $\rho>0$ two fixed reals. The left and right Caputo--Katugampola fractional derivatives of order $\alpha$ are defined by
$${^CD_{a+}^{\a,\r}} x(t) ={D_{a+}^{\alpha,\rho}}[x(t)-x(a)]=\frac{\rho^\alpha}{\Gamma(1-\alpha)}t^{1-\rho}\frac{d}{dt}\int_a^t \frac{\tau^{\rho-1}}{(t^\rho-\tau^\rho)^\alpha}[x(\tau)-x(a)] d\tau,$$
and
$${^CD_{b-}^{\a,\r}} x(t)={D_{b-}^{\alpha,\rho}}[x(t)-x(b)]=\frac{-\rho^\alpha}{\Gamma(1-\alpha)}t^{1-\rho}\frac{d}{dt}\int_t^b \frac{\tau^{\rho-1}}{(\tau^\rho-t^\rho)^\alpha}[x(\tau)-x(b)] d\tau,$$
respectively.
\end{definition}

Observe that, when $x(a)=0$ (resp. $x(b)=0$), the left (resp. right) Caputo--Katugampola and the Katugampola fractional derivatives coincide.
Moreover, there exists a relation between these { two types} of fractional derivatives. Namely, if both types of fractional derivative exist, then
\begin{equation}\label{eq:derivat}
{^CD_{a+}^{\a,\r}} x(t)={D_{a+}^{\alpha,\rho}}[x(t)-x(a)]={D_{a+}^{\alpha,\rho}}x(t)-{D_{a+}^{\alpha,\rho}}x(a)={D_{a+}^{\alpha,\rho}}x(t)-\frac{x(a)\rho^\alpha}{\Gamma(1-\alpha)}(t^\rho-a^\rho)^{-\alpha},
\end{equation}
and
$${^CD_{b-}^{\a,\r}} x(t)={D_{b-}^{\alpha,\rho}}[x(t)-x(b)]={D_{b-}^{\alpha,\rho}}x(t)-{D_{b-}^{\alpha,\rho}}x(b)={D_{b-}^{\alpha,\rho}}x(t)-\frac{x(b)\rho^\alpha}{\Gamma(1-\alpha)}(b^\rho-t^\rho)^{-\alpha}.$$

\begin{theorem}\label{teo:equiv} Let $x\in C^1([a,b]).$ Then,
$${^CD_{a+}^{\a,\r}} x(t) =\frac{\rho^\alpha}{\Gamma(1-\alpha)}\int_a^t (t^\rho-\tau^\rho)^{-\alpha}x'(\tau) d\tau,$$
and
$${^CD_{b-}^{\a,\r}} x(t)=\frac{-\rho^\alpha}{\Gamma(1-\alpha)}\int_t^b (\tau^\rho-t^\rho)^{-\alpha}x'(\tau) d\tau.$$
\end{theorem}

\begin{proof}
Taking
$$u'(\tau)=\tau^{\rho-1}(t^\rho-\tau^\rho)^{-\alpha} \quad\mbox{and} \quad v(\tau)=x(\tau)-x(a),$$
and then integrating by parts we obtain
$${^CD_{a+}^{\a,\r}} x(t) =\frac{\rho^\alpha}{\Gamma(1-\alpha)}t^{1-\rho}\frac{d}{dt}\int_a^t u'(\tau)v(\tau)d\tau.$$

We may now differentiate the integral, with respect to variable $t$, to conclude that
$${^CD_{a+}^{\a,\r}} x(t) =\frac{\rho^\alpha}{\Gamma(1-\alpha)}\int_a^t (t^\rho-\tau^\rho)^{-\alpha}x'(\tau) d\tau.$$
The second formula can be proven in a similar way.
\end{proof}

\begin{remark}
Observe that, from Theorem~\ref{teo:equiv}, we immediately obtain the following:
\begin{equation}\label{Derivative_Integral}{^CD_{a+}^{\a,\r}} x(t) ={I_{a+}^{1-\alpha,\rho}}\left(\tau^{1-\rho}\frac{d}{d\tau}x\right)(t) \quad \mbox{and} \quad {^CD_{b-}^{\a,\r}} x(t) ={I_{b-}^{1-\alpha,\rho}}\left(-\tau^{1-\rho}\frac{d}{d\tau}x\right)(t),\end{equation}
where ${I_{a+}^{1-\alpha,\rho}}$ and ${I_{b-}^{1-\alpha,\rho}}$ denote the left and right Katugampola fractional integrals.
\end{remark}

\begin{remark} Although in Definition~\ref{CKFD} we put $\alpha\in(0,1)$, it is possible to define the Caputo--Katugampola fractional derivatives for an arbitrary $\alpha>0$.  Let $n$ be the smallest integer greater than $\alpha$. Then, the left and right Caputo--Katugampola fractional derivatives of order $\alpha>0$ are defined by
$${^CD_{a+}^{\a,\r}} x(t) =\frac{\rho^{\alpha-n+1}}{\Gamma(n-\alpha)}\int_a^t \frac{\tau^{(\rho-1)(1-n)}}{(t^\rho-\tau^\rho)^{\alpha-n+1}}x^{(n)}(\tau)d\tau,$$
and
$${^CD_{b-}^{\a,\r}} x(t)=\frac{(-1)^n\rho^{\alpha-n+1}}{\Gamma(n-\alpha)}\int_t^b \frac{\tau^{(\rho-1)(1-n)}}{(\tau^\rho-t^\rho)^{\alpha-n+1}}x^{(n)}(\tau)d\tau,$$
respectively. In this case, when $\rho=1$, we obtain the left and right Caputo fractional derivatives (cf. \cite{Kilbas}), and for $\rho\to0^+$, attending that
$\lim_{\rho\to 0^+} (t^\rho-\tau^\rho)/\rho=\ln(t/\tau)$,
we obtain the left and right  Caputo--Hadamard fractional derivatives as defined in \cite{Baleanu1,Baleanu2}.
\end{remark}

Now let us prove that the left and the right Katugampola fractional integrals are bounded in the space of continuous functions.
\begin{theorem}\label{lem:contInt}
The operators ${I_{a+}^{\a,\r}}$ and ${I_{b-}^{\a,\r}}$ are linear and bounded from $C([a,b])$ to $C([a,b])$, i.e.,
\begin{equation}\label{eq:boundedL}
\left\|{I_{a+}^{\a,\r}} x\right\|_C\leq K_{\alpha,\rho}\left\|x\right\|_C,~\left\|{I_{b-}^{\a,\r}} x\right\|_C\leq K_{\alpha,\rho}\left\|x\right\|_C,
~\textnormal{with } K_{\alpha,\rho}=\frac{\rho^{-\alpha}}{\Gamma(\alpha+1)}(b^{\rho}-a^{\rho})^{\alpha},
\end{equation}
where $\left\|x\right\|_C=\max\limits_{t\in [a,b]} \left|x(t)\right|$.
\end{theorem}
\begin{proof}
For any $x\in C([a,b])$, one has
$$\begin{array}{ll}
\displaystyle\left|\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_a^t\frac{\tau^{\rho-1}}{(t^{\rho}-\tau^{\rho})^{1-\alpha}}x(\tau)\; d\tau\right|&
\displaystyle\leq\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\left\|x\right\|_C\int_a^t\frac{\tau^{\rho-1}}{(t^{\rho}-\tau^{\rho})^{1-\alpha}}\; d\tau\\
&\displaystyle\leq\frac{\rho^{-\alpha}}{\Gamma(\alpha+1)}(b^{\rho}-a^{\rho})^{\alpha}\left\|x\right\|_C.
\end{array}$$
In the case of the right Katugampola fractional integral, the proof is similar.
\end{proof}
Using the above lemma, one can show the following theorem concerning continuity of the Caputo--Katugampola fractional derivatives.

\begin{theorem}\label{teo:bounds} Let $x\in C^1([a,b]).$ Then, the Caputo--Katugampola fractional derivatives ${^CD_{a+}^{\a,\r}} x$ and ${^CD_{b-}^{\a,\r}} x$ exist and are continuous on the interval $[a,b]$. Moreover,  ${^CD_{a+}^{\a,\r}} x(t)=0$ at $t=a$, and ${^CD_{b-}^{\a,\r}} x(t)=0$ at $t=b$.
\end{theorem}

\begin{proof} Knowing that
$${^CD_{a+}^{\a,\r}} x(t) ={I_{a+}^{1-\alpha,\rho}}\left(\tau^{1-\rho}\frac{d}{d\tau}x\right)(t),$$
by Theorem~\ref{lem:contInt}, we get a continuity of ${^CD_{a+}^{\a,\r}} x$. In order to prove the second part, we start with the formula
$${^CD_{a+}^{\a,\r}} x(t) =\frac{\rho^\alpha}{\Gamma(1-\alpha)}\int_a^t (t^\rho-\tau^\rho)^{-\alpha}x'(\tau) d\tau,$$
that holds for all $t\in [a,b]$. Now, letting $M=\max_{t\in[a,b]}|t^{1-\rho}x'(t)|$, we obtain
$$\left|{^CD_{a+}^{\a,\r}} x(t)\right|\leq M\frac{\rho^{\alpha-1}}{\Gamma(2-\alpha)}(t^\rho-a^\rho)^{1-\alpha}.$$ Therefore, ${^CD_{a+}^{\a,\r}} x(t)=0$ at $t=a$.
One can obtain the results for the right Caputo-Katugampola fractional derivative in a similar way.
\end{proof}

In the space $C^1([a,b])$ we consider the norm
$$\|x\|_{C^1}:=\max_{t\in[a,b]}|x(t)|+\max_{t\in[a,b]}|x'(t)|.$$
Also, we define the two subspaces $C_a([a,b])$  and $C_b([a,b])$  of the space $C([a,b])$ by
$$C_a([a,b]):=\left\{x\in C([a,b])\, : \, x(a)=0\right\}, $$
and
$$C_b([a,b]):=\left\{x\in C([a,b])\, : \, x(b)=0\right\},$$
endowed with the norm
$$\|x\|_{C}:=\max_{t\in[a,b]}|x(t)|.$$

\begin{theorem} The Caputo--Katugampola fractional derivatives ${^CD_{a+}^{\a,\r}}$ and ${^CD_{b-}^{\a,\r}}$ are bounded operators from $C^1([a,b])$ to $C_a([a,b])$ and $C_b([a,b])$, respectively. Moreover,
$$\left\|{^CD_{a+}^{\a,\r}} x\right\|_{C}\leq M \|x\|_{C^1} \quad \mbox{and} \quad \left\|{^CD_{b-}^{\a,\r}} x\right\|_{C}\leq M \|x\|_{C^1},$$
where
$$M= \frac{\rho^{\alpha-1}(b^\rho-a^\rho)^{1-\alpha}}{\Gamma(2-\alpha)}\max\{a^{1-\rho},b^{1-\rho}\}.$$
\end{theorem}

\begin{proof} It is an immediate consequence of \eqref{Derivative_Integral} and Theorem~\ref{lem:contInt}.
\end{proof}

Under appropriate assumptions, the Caputo--Katugampola fractional derivatives and Katugampola fractional integrals are inverse operations of each other. In fact, we have the following.

\begin{theorem}\label{thm:DerInt} Let $x\in C([a,b]).$ Then,
$${^CD_{a+}^{\a,\r}} {I_{a+}^{\alpha,\rho}} x(t)=x(t).$$
\end{theorem}

\begin{proof} Using formula \eqref{eq:derivat}, by Theorem 3.2 in \cite{Katugampola2}, we have
$$\begin{array}{ll}
\displaystyle {^CD_{a+}^{\a,\r}} {I_{a+}^{\alpha,\rho}} x(t)&=\displaystyle {D_{a+}^{\alpha,\rho}} {I_{a+}^{\alpha,\rho}} x(t)-\frac{\rho^\alpha}{\Gamma(1-\alpha)}(t^\rho-a^\rho)^{-\alpha}\left[{I_{a+}^{\alpha,\rho}} x(t)\right]|_{t=a}\\
&=\displaystyle  x(t)-\frac{\rho^\alpha}{\Gamma(1-\alpha)}(t^\rho-a^\rho)^{-\alpha}\left[{I_{a+}^{\alpha,\rho}} x(t)\right]|_{t=a}.\\
\end{array}$$
On the other hand, since $x$ is a continuous function, there exists $M=\max_{t\in[a,b]}|x(t)|$. Then,
$$\begin{array}{ll}
\displaystyle \left|{I_{a+}^{\alpha,\rho}} x(t)\right|&\displaystyle\leq M\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_a^t \frac{\tau^{\rho-1}}{(t^\rho-\tau^\rho)^{1-\alpha}} d\tau\\
&\displaystyle =M\frac{\rho^{-\alpha}}{\Gamma(1+\alpha)}(t^\rho-a^\rho)^{\alpha},\\
\end{array}$$
and ${I_{a+}^{\alpha,\rho}} x(t)=0$ at $t=a$. Thus, ${^CD_{a+}^{\a,\r}} {I_{a+}^{\alpha,\rho}} x(t)=x(t)$.
\end{proof}

\begin{theorem}\label{teo:ID} Let $x\in C^1([a,b]).$ Then,
$${I_{a+}^{\alpha,\rho}}\, {^CD_{a+}^{\a,\r}} x(t)=x(t)-x(a).$$
\end{theorem}

\begin{proof} By Theorem 4.1 in \cite{Katugampola1} and by the definition of the Katugampola fractional integral, we have
$${I_{a+}^{\alpha,\rho}}\, {^CD_{a+}^{\a,\r}} x(t)={I_{a+}^{\alpha,\rho}}{I_{a+}^{1-\alpha,\rho}}(t^{1-\rho}x'(t))={I_{a+}^{1,\rho}}(t^{1-\rho}x'(t))$$
$$=\int_a^t \tau^{\rho-1}\tau^{1-\rho}x'(\tau)d\tau=x(t)-x(a).$$
\end{proof}

\begin{lemma}\label{lemma:power} For $v>0$, define
$$x(t)=\left(\frac{t^\rho-a^\rho}{\rho}\right)^v \quad \mbox{and} \quad y(t)=\left(\frac{b^\rho-t^\rho}{\rho}\right)^v.$$
Then,
$${^CD_{a+}^{\a,\r}} x(t)=\frac{\rho^{\alpha-v}\Gamma(1+v)}{\Gamma(1-\alpha+v)}(t^\rho-a^\rho)^{v-\alpha}  \quad \mbox{and} \quad
{^CD_{b-}^{\a,\r}} y(t)=\frac{\rho^{\alpha-v}\Gamma(1+v)}{\Gamma(1-\alpha+v)}(b^\rho-t^\rho)^{v-\alpha}.$$
\end{lemma}

\begin{proof} By Theorem \ref{teo:equiv},
$${^CD_{a+}^{\a,\r}} x(t) =\frac{\rho^\alpha}{\Gamma(1-\alpha)}\int_a^t (t^\rho-\tau^\rho)^{-\alpha}v\tau^{\rho-1}\left(\frac{\tau^\rho-a^\rho}{\rho}\right)^{v-1} d\tau.$$
Performing the change of variables $u=(\tau^\rho-a^\rho)/(t^\rho-a^\rho)$, and using the Beta function $B(\cdot,\cdot)$, we get
$$\begin{array}{ll}
\displaystyle{^CD_{a+}^{\a,\r}} x(t) &=\displaystyle \frac{\rho^{\alpha-v}v}{\Gamma(1-\alpha)}(t^\rho-a^\rho)^{v-\alpha}\int_0^1 (1-u)^{-\alpha}u^{v-1} du\\
&=\displaystyle \frac{\rho^{\alpha-v}v}{\Gamma(1-\alpha)}(t^\rho-a^\rho)^{v-\alpha}B(1-\alpha,v)\\
&=\displaystyle \frac{\rho^{\alpha-v}v}{\Gamma(1-\alpha)}(t^\rho-a^\rho)^{v-\alpha}\frac{\Gamma(1-\alpha)\Gamma(v)}{\Gamma(1-\alpha+v)}\\
&=\displaystyle \frac{\rho^{\alpha-v}\Gamma(1+v)}{\Gamma(1-\alpha+v)}(t^\rho-a^\rho)^{v-\alpha}.
\end{array}$$
The second formula can be proven in a similar way.
\end{proof}

\begin{remark} If $\rho=1$, then we obtain the power function $x(t)=(t-a)^v$, and as $\rho\to0^+$, we  get $x(t)=(\ln(t/a))^v$. The expressions obtained in Lemma \ref{lemma:power} coincide with the ones given in Property~2.16 \cite{Kilbas} and Property~2.6 \cite{Baleanu2}.
\end{remark}

Let us recall the definition of the Mittag--Leffler function.

\begin{definition} The Mittag--Leffler function with dependence on two parameters $\mu$ and $\beta$ is defined by the series
$$E_{\mu,\beta}(x)=\sum_{k=0}^\infty\frac{x^k}{\Gamma(\mu k+\beta)},\quad \mu>0 \, \mbox{and} \, \beta>0.$$
where $\mu, \beta \in \mathbb{C}$, $Re(\mu)>0$, $Re(\beta)>0$. When $\mu, \beta \in \mathbb{R}$ and $\mu, \beta>0$, the series is convergent.
As a particular case, when $\beta=1$, we obtain
$$E_{\mu}(x)=\sum_{k=0}^\infty\frac{x^k}{\Gamma(\mu k+1)},\quad \mu>0.$$
\end{definition}

When $\mu=1$, the Mittag--Leffler function reduces to the exponential function, $E_{1}(x)=\exp(x)$.

\begin{lemma}\label{lemma:exp} For $\mu>0$, $\beta\geq1$ and $\lambda\in\mathbb R$, we have
$$\displaystyle{^CD_{a+}^{\a,\r}} \left((t^\rho-a^\rho)^{\beta-1}E_{\mu,\beta}\left[\lambda(t^\rho-a^\rho)^\mu\right]\right)=\displaystyle\left\{
\begin{array}{ll}
\displaystyle\rho^\alpha(t^\rho-a^\rho)^{\beta-\alpha-1}E_{\mu,\beta-\alpha}\left[\lambda(t^\rho-a^\rho)^\mu\right]& \mbox{ if } \quad \beta>1\\
\displaystyle\lambda\rho^\alpha(t^\rho-a^\rho)^{\mu-\alpha}E_{\mu,\mu-\alpha+1}\left[\lambda(t^\rho-a^\rho)^\mu\right]& \mbox{ if } \quad \beta=1\\
\end{array} \right. $$
and
$$
\displaystyle{^CD_{b-}^{\a,\r}} \left((b^\rho-t^\rho)^{\beta-1}E_{\mu,\beta}\left[\lambda(b^\rho-t^\rho)^\mu\right]\right)=\displaystyle\left\{
\begin{array}{ll}
\displaystyle\rho^\alpha(b^\rho-t^\rho)^{\beta-\alpha-1}E_{\mu,\beta-\alpha}\left[\lambda(b^\rho-t^\rho)^\mu\right]& \mbox{ if } \, \beta>1\\
\displaystyle\lambda\rho^\alpha(b^\rho-t^\rho)^{\mu-\alpha}E_{\mu,\mu-\alpha+1}\left[\lambda(b^\rho-t^\rho)^\mu\right]& \mbox{ if } \, \beta=1\\
\end{array} \right.. $$
\end{lemma}

\begin{proof} Let $\beta>1$. Using Lemma \ref{lemma:power}, we have the following
$$\begin{array}{ll}
\displaystyle{^CD_{a+}^{\a,\r}} \left((t^\rho-a^\rho)^{\beta-1}E_{\mu,\beta}\left[\lambda(t^\rho-a^\rho)^\mu\right]\right)&
 = \displaystyle\sum_{k=0}^\infty\frac{\lambda^k}{\Gamma(\mu k+\beta)}{^CD_{a+}^{\a,\r}} (t^\rho-a^\rho)^{\mu k+\beta-1}\\
& = \displaystyle\sum_{k=0}^\infty\frac{\lambda^k}{\Gamma(\mu k+\beta)}\frac{\rho^\alpha\Gamma(\mu k+\beta)}{\Gamma(\mu k+\beta-\alpha)}(t^\rho-a^\rho)^{\mu k+\beta-\alpha-1}\\
&=\displaystyle\rho^\alpha(t^\rho-a^\rho)^{\beta-\alpha-1}E_{\mu,\beta-\alpha}\left[\lambda(t^\rho-a^\rho)^\mu\right].\\
\end{array}$$
For $\beta=1$, we have that
$$\begin{array}{ll}
\displaystyle{^CD_{a+}^{\a,\r}} \left(E_{\mu,1}\left[\lambda(t^\rho-a^\rho)^\mu\right]\right)&
 = \displaystyle{^CD_{a+}^{\a,\r}} 1+\sum_{k=1}^\infty\frac{\lambda^k}{\Gamma(\mu k+1)}{^CD_{a+}^{\a,\r}} (t^\rho-a^\rho)^{\mu k}\\
& = \displaystyle\sum_{k=1}^\infty\frac{\lambda^k}{\Gamma(\mu k+1)}\frac{\rho^\alpha\Gamma(\mu k+1)}{\Gamma(\mu k-\alpha+1)}(t^\rho-a^\rho)^{\mu k-\alpha}\\
&=\displaystyle\lambda\rho^\alpha(t^\rho-a^\rho)^{\mu-\alpha}\sum_{k=0}^\infty\frac{\lambda^k}{\Gamma(\mu k+\mu-\alpha+1)}(t^\rho-a^\rho)^{\mu k}\\
&=\displaystyle\lambda\rho^\alpha(t^\rho-a^\rho)^{\mu-\alpha}E_{\mu,\mu-\alpha+1}\left[\lambda(t^\rho-a^\rho)^\mu\right].
\end{array}$$
The formula for the right fractional derivatives can be proven in a similar way.
\end{proof}

In particular, setting $\mu=\beta=1$, we get the formulas for the fractional derivative of the exponential functions:
$${^CD_{a+}^{\a,\r}} \left(\exp\left[\lambda(t^\rho-a^\rho)\right]\right)=\lambda\rho^\alpha(t^\rho-a^\rho)^{1-\alpha}E_{1,2-\alpha}\left[\lambda(t^\rho-a^\rho)\right]$$
and
$${^CD_{b-}^{\a,\r}} \left(\exp\left[\lambda(b^\rho-t^\rho)\right]\right)=\lambda\rho^\alpha(b^\rho-t^\rho)^{1-\alpha}E_{1,2-\alpha}\left[\lambda(b^\rho-t^\rho)\right].$$
Observe that, for $\rho=1$ and $\alpha\to1^-$, we obtain the usual derivative of the exponential function:
$$\frac{d}{dt} \left(\exp\left[\lambda(t-a)\right]\right)=\lambda\exp\left[\lambda(t-a)\right].$$
For $\alpha=\mu>0$ and $\lambda\in\mathbb R$, we have
$$\displaystyle{^CD_{a+}^{\a,\r}} \left(E_\alpha\left[\lambda(t^\rho-a^\rho)^\alpha\right]\right)=\lambda\rho^\alpha E_\alpha\left[\lambda(t^\rho-a^\rho)^\alpha\right]$$
and
$${^CD_{b-}^{\a,\r}} \left(E_\alpha\left[\lambda(b^\rho-t^\rho)^\alpha\right]\right)=\lambda\rho^\alpha E_\alpha\left[\lambda(b^\rho-t^\rho)^\alpha\right].$$

\section{Caputo--Katugampola fractional Cauchy problem}\label{sec:FDE}

Consider the nonlinear fractional differential equation of order $\alpha\in(0,1)$,
\begin{equation}\label{eq:CauchyEq}
{^CD_{a+}^{\a,\r}} x(t)=f(t,x(t)), \quad t\in[a,b],
\end{equation}
with the initial condition
\begin{equation}\label{eq:BoundCond}
x(a)=x_a, \quad x_a\in\mathbb R,
\end{equation}
where $f:[a,b]\times\mathbb R\to \mathbb R$ is a continuous function with respect to all its arguments.
We seek conditions that guarantee the existence and uniqueness of solution to problem \eqref{eq:CauchyEq}--\eqref{eq:BoundCond}
in the following set of functions
$$\mathcal{U}:=\left\{x\in C([a,b]):{^CD_{a+}^{\a,\r}} x\in C([a,b])\right\}.$$

First, let us observe that, for $x\in C([a,b])$, the Cauchy type problem \eqref{eq:CauchyEq}--\eqref{eq:BoundCond} is equivalent to the problem
of finding solutions to the following Volterra integral equation
\begin{equation}\label{eq:Volterra}
x(t)=x_a+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_a^t\frac{\tau^{\rho-1}}{(t^{\rho}-\tau^{\rho})^{1-\alpha}}f(\tau,x(\tau))\; d\tau.
\end{equation}

Indeed, if $x\in C([a,b])$ satisfies \eqref{eq:CauchyEq}--\eqref{eq:BoundCond}, then applying operator ${I_{a+}^{\a,\r}}$ to the both
sides of \eqref{eq:CauchyEq}, using the initial condition \eqref{eq:BoundCond} and applying Theorem~\ref{teo:ID}
we obtain equation \eqref{eq:Volterra}. Conversely, taking { $t\rightarrow a^+$}, we see that the initial condition
\eqref{eq:BoundCond} is satisfied and applying operator ${^CD_{a+}^{\a,\r}}$ to both sides of \eqref{eq:Volterra}, by
Theorem~\ref{thm:DerInt}, we arrive to problem \eqref{eq:CauchyEq}--\eqref{eq:BoundCond}.

Now we will use the above results to prove the following theorem, ensuring the existence and uniqueness of solution
to the Cauchy type problem \eqref{eq:CauchyEq}--\eqref{eq:BoundCond}.

\begin{theorem}
Let $f:[a,b]\times\mathbb R\to \mathbb R$ be a continuous function and Lipschitz with respect to the second variable, i.e.,
\begin{equation}\label{eq:LipType}
\left|f(t,x_1)-f(t,x_2)\right|<L\left|x_1-x_2\right|,~(L>0),
\end{equation}
for all $t\in [a,b]$ and all $x_1,x_2\in\mathbb{R}$. Then, the Cauchy type problem \eqref{eq:CauchyEq}--\eqref{eq:BoundCond}
possesses a unique solution in the space $\mathcal{U}$.
\end{theorem}

\begin{proof}
The proof is similar in spirit to \cite{Kilbas2,Kilbas}. We start by showing that, for \eqref{eq:CauchyEq}--\eqref{eq:BoundCond}, there exists a unique
solution $x\in C([a,b])$. Let us recall that
the Cauchy type problem \eqref{eq:CauchyEq}--\eqref{eq:BoundCond} is equivalent to the problem
of finding solutions to the Volterra integral equation \eqref{eq:Volterra}.
This allows us to use the well known method, for nonlinear Volterra integral equations,
where first we prove existence and uniqueness of solutions on a subinterval of $[a,b]$.\\
Let us choose $a<t_1<b$ to be such that the following condition is satisfied
\begin{equation}\label{eq:t1}
0<L\frac{\rho^{-\alpha}}{\Gamma(\alpha+1)}(t_1^{\rho}-a^{\rho})^{\alpha}<1.
\end{equation}
We shall prove the existence of a unique solution $x\in C([a,t_1])$ to \eqref{eq:Volterra} on the subinterval $[a,t_1]\subset[a,b]$.
Let us define the following integral operator:
\begin{equation}
\fonction{\mathcal{S}}{C([a,b])}{C([a,b])}{x(t)}{\mathcal{S}x(t):=\displaystyle x_a+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_a^t\frac{\tau^{\rho-1}}{(t^{\rho}-\tau^{\rho})^{1-\alpha}}f(\tau,x(\tau))\; d\tau,}
\end{equation}
and rewrite \eqref{eq:Volterra} in the form $x(t)=\mathcal{S}x(t)$. Note that $\mathcal{S}$ is well defined and is a  bounded operator on the complete metric space $C([a,b])$ (by Theorem~\ref{lem:contInt}).
Using Theorem~\ref{lem:contInt} and the condition \eqref{eq:LipType} one has
\begin{multline*}
\left\|\mathcal{S}x_1-\mathcal{S}x_2\right\|_{C([a,t_1])}
\leq\left\|{I_{a+}^{\a,\r}}\left( f(t,x_1)-f(t,x_2)\right)\right\|_{C([a,t_1])}\\
\leq L\left\|{I_{a+}^{\a,\r}}\left(x_1-x_2\right)\right\|_{C([a,t_1])}
\leq L\frac{\rho^{-\alpha}}{\Gamma(\alpha+1)}(t_1^{\rho}-a^{\rho})^{\alpha}\left\|x_1-x_2\right\|_{C([a,t_1])}
\end{multline*}
and because condition \eqref{eq:t1} is satisfied, by the Banach fixed point theorem, there exists a unique solution $x^{*1}\in C([a,t_1])$ to the equation \eqref{eq:Volterra}
on the interval $[a,t_1]$. Moreover, if we define the sequence $x_m^1(t):=\mathcal{S}^m x_0(t)$ where, for $x_0(t)=x_a$,
\begin{equation*}
\mathcal{S}^m x_0(t):=x_0(t)+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_a^t\frac{\tau^{\rho-1}}{(t^{\rho}-\tau^{\rho})^{1-\alpha}}f(\tau,\mathcal{S}^{m-1} x_0(\tau))\; d\tau,~m=1,2,\dots,
\end{equation*}
then, again, by the Banach fixed point theorem, we obtain the solution $x^{*1}$ as a limit of the sequence $x_m^1$, i.e.,
\begin{equation*}
\lim\limits_{m\rightarrow\infty}\left\|x_m^1-x^{*1}\right\|_{C([a,t_1])}=0.
\end{equation*}
Now, let us choose $t_2=t_1+h_1$, with $h_1>0$ such that $t_2<b$ and
\begin{equation}\label{eq:t2}
0<L\frac{\rho^{-\alpha}}{\Gamma(\alpha+1)}(t_2^{\rho}-t_1^{\rho})^{\alpha}<1.
\end{equation}
Consider the interval $[t_1,t_2]$ and write equation \eqref{eq:Volterra} in the form
\begin{equation}\label{eq:Volterra2}
x(t)=x_0(t)+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{t_1}^t\frac{\tau^{\rho-1}}{(t^{\rho}-\tau^{\rho})^{1-\alpha}}f(\tau,x(\tau))\; d\tau
+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_a^{t_1}\frac{\tau^{\rho-1}}{(t^{\rho}-\tau^{\rho})^{1-\alpha}}f(\tau,x(\tau))\; d\tau.
\end{equation}
Because on the interval $[a,t_1]$, equation \eqref{eq:Volterra2} possesses a unique solution, we can rewrite \eqref{eq:Volterra2}
as follows
\begin{equation*}\label{eq:Volterra3}
x(t)=x_{01}(t)+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{t_1}^{t}\frac{\tau^{\rho-1}}{(t^{\rho}-\tau^{\rho})^{1-\alpha}}f(\tau,x(\tau))\; d\tau,
\end{equation*}
where
\begin{equation*}
x_{01}(t)=x_{0}(t)+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t_1}\frac{\tau^{\rho-1}}{(t^{\rho}-\tau^{\rho})^{1-\alpha}}f(\tau,x(\tau))\; d\tau
\end{equation*}
is the known function. By the same argument as before, we prove that there exists a unique solution $x^{*2}\in C([t_1,t_2])$ to equation \eqref{eq:Volterra}
on $[t_1,t_2]$. Repeating the previous reasoning, choosing $t_{k}=t_{k-1}+h_{k-1}$ such that $h_{k-1}>0$, $t_k<b$ and
\begin{equation}\label{eq:tk}
0<L\frac{\rho^{-\alpha}}{\Gamma(\alpha+1)}(t_{k}^{\rho}-t_{k-1}^{\rho})^{\alpha}<1,
\end{equation}
we see that equation \eqref{eq:Volterra} possesses a solution
$x^{*k}\in C([x_{k-1},x_k])$ on each interval $[x_{k-1},x_k]$ ($k=1,\dots,l$), where $a=x_0<x_1<\dots<x_l=b$ and we conclude that for problem \eqref{eq:CauchyEq}-\eqref{eq:BoundCond}, there exists a unique solution $x\in C([a,b])$.\\
It remains to prove that $x\in\mathcal{U}$, i.e., we need to show that ${^CD_{a+}^{\a,\r}} x\in C([a,b])$. Recall that our solution $x$ can be approximated
by the sequence $x_m(t)=\mathcal{S}^m x_0(t)$, i.e.,
\begin{equation*}
\lim\limits_{m\rightarrow\infty}\left\|x_m-x\right\|_{C([a,b])}=0,
\end{equation*}
with the choice of certain $x_m$ on each interval $[a,t_1],\dots,[t_{l-1},b]$.
Using \eqref{eq:CauchyEq} and the Lipschitz type condition \eqref{eq:LipType} we have
\begin{equation*}
\left\|{^CD_{a+}^{\a,\r}} x_m-{^CD_{a+}^{\a,\r}} x\right\|_{C([a,b])} =\left\|f(t,x_m)-f(t,x)\right\|_{C([a,b])}
\leq L\left\|x_m-x\right\|_{C([a,b])},
\end{equation*}
then, taking $m\rightarrow\infty$, one has
\begin{equation*}
\lim\limits_{m\rightarrow\infty}\left\|{^CD_{a+}^{\a,\r}} x_m-{^CD_{a+}^{\a,\r}} x\right\|_{C([a,b])}=0.
\end{equation*}
Since ${^CD_{a+}^{\a,\r}} x_m(t)=f(t,x_m(t))$ is continuous on $[a,b]$ we have that ${^CD_{a+}^{\a,\r}} x$ belongs to the space $C([a,b])$. The proof is complete.
\end{proof}

\section{Numerical approach to the Caputo--Katugampola fractional Cauchy problem}\label{sec:approx}

In this section we prove an approximation formula for the Caputo--Katugampola fractional derivative, that depends only on the first-order derivative of a function $x$. With this tool in hand, for a given Caputo--Katugampola fractional Cauchy problem we can consider a sequence of systems of $N+1$ ordinary differential equations with $N+1$ initial conditions, where $N\in \mathbb{N}$. A sequence $(x_N)$ of the solutions to those systems converges to the solution of the fractional problem.

\begin{theorem}\label{teo:approx} Let $x:[a,b]\rightarrow\mathbb{R}$ be a function of class $C^2$, and $N \geq 1$ an integer. Define the quantities
$$\begin{array}{ll}
A_N&=\displaystyle\frac{\rho^{\alpha-1}}{\Gamma(2-\alpha)}\sum_{k=0}^N\frac{\Gamma(k-1+\alpha)}{\Gamma(\alpha-1)k!},\\
B_{N,k}&=\displaystyle\frac{\rho^\alpha\Gamma(k-1+\alpha)}{\Gamma(2-\alpha)\Gamma(\alpha-1)(k-1)!},\quad  k=1,\ldots,N,\\
\end{array}$$
and functions $V_k:[a,b]\rightarrow\mathbb{R}$ by
$$V_k(t)=\int_a^t (\tau^\rho-a^\rho)^{k-1}x'(\tau)d\tau,\quad k=1,\ldots,N.$$
Then,
\begin{equation}\label{eq:mainformula}{^CD_{a+}^{\a,\r}} x(t)=A_N(t^\rho-a^\rho)^{1-\alpha} t^{1-\rho}x'(t)-\sum_{k=1}^N B_{N,k}(t^\rho-a^\rho)^{1-\alpha-k}V_k(t)+E_N(t),\end{equation}
with
$$\lim_{N\to\infty}E_N(t)=0, \quad \forall t\in[a,b].$$
\end{theorem}

\begin{proof} Set
$$u'(\tau)=(t^\rho-\tau^\rho)^{-\alpha}\tau^{\rho-1} \quad \mbox{and} \quad v(\tau)=\tau^{1-\rho}x'(\tau).$$
Then, integrating by parts
we obtain
$$\begin{array}{ll}
\displaystyle{^CD_{a+}^{\a,\r}} x(t) &=\displaystyle\frac{\rho^\alpha}{\Gamma(1-\alpha)}\int_a^t (t^\rho-\tau^\rho)^{-\alpha}x'(\tau) d\tau\\
&=\displaystyle\frac{\rho^{\alpha-1}}{\Gamma(2-\alpha)}(t^\rho-a^\rho)^{1-\alpha}a^{1-\rho}x'(a)+\frac{\rho^{\alpha-1}}{\Gamma(2-\alpha)}\int_a^t (t^\rho-\tau^\rho)^{1-\alpha}
\frac{d}{d\tau}\left(\tau^{1-\rho}x'(\tau)\right) d\tau.\\
\end{array}$$
Using the generalized binomial theorem, we get the decomposition
$$\begin{array}{ll}
(t^\rho-\tau^\rho)^{1-\alpha}& =\displaystyle (t^\rho-a^\rho)^{1-\alpha}\left(1-\frac{\tau^\rho-a^\rho}{t^\rho-a^\rho}\right)^{1-\alpha}\\
& =\displaystyle (t^\rho-a^\rho)^{1-\alpha}\sum_{k=0}^\infty \frac{\Gamma(k-1+\alpha)}{\Gamma(\alpha-1)k!}\left(\frac{\tau^\rho-a^\rho}{t^\rho-a^\rho}\right)^k.
\end{array}$$
Substituting this series into the expression of the fractional derivative, we obtain
$$\begin{array}{ll}
\displaystyle{^CD_{a+}^{\a,\r}} x(t) &=\displaystyle \frac{\rho^{\alpha-1}}{\Gamma(2-\alpha)}(t^\rho-a^\rho)^{1-\alpha}a^{1-\rho}x'(a)+\frac{\rho^{\alpha-1}}{\Gamma(2-\alpha)}\int_a^t (t^\rho-a^\rho)^{1-\alpha}\\
&\quad \displaystyle\times \sum_{k=0}^N \frac{\Gamma(k-1+\alpha)}{\Gamma(\alpha-1)k!}\left(\frac{\tau^\rho-a^\rho}{t^\rho-a^\rho}\right)^k \frac{d}{d\tau}\left(\tau^{1-\rho}x'(\tau)\right) d\tau+E_N(t),\\
\end{array}$$
where
$$E_{N}(t)=\frac{\rho^{\alpha-1}}{\Gamma(2-\alpha)}\int_a^t (t^\rho-a^\rho)^{1-\alpha}\sum_{k=N+1}^\infty \frac{\Gamma(k-1+\alpha)}{\Gamma(\alpha-1)k!}\left(\frac{\tau^\rho-a^\rho}{t^\rho-a^\rho}\right)^k \frac{d}{d\tau}\left(\tau^{1-\rho}x'(\tau)\right) d\tau.$$
Now, let us split the sum into $k=0$ and $k=1,\ldots, N$, that is,
$$\begin{array}{ll}
\displaystyle{^CD_{a+}^{\a,\r}} x(t) &=\displaystyle \frac{\rho^{\alpha-1}}{\Gamma(2-\alpha)}(t^\rho-a^\rho)^{1-\alpha}t^{1-\rho}x'(t)+\frac{\rho^{\alpha-1}}{\Gamma(2-\alpha)}(t^\rho-a^\rho)^{1-\alpha} \\
&\quad \displaystyle\times \sum_{k=1}^N \frac{\Gamma(k-1+\alpha)}{\Gamma(\alpha-1)k!(t^\rho-a^\rho)^k}\int_a^t(\tau^\rho-a^\rho)^k \frac{d}{d\tau}\left(\tau^{1-\rho}x'(\tau)\right) d\tau+E_N(t).\\
\end{array}$$
Setting
$$u(\tau)=(\tau^\rho-a^\rho)^k \quad \mbox{and} \quad v'(\tau)=\frac{d}{d\tau}\left(\tau^{1-\rho}x'(\tau)\right)$$
and then integrating by parts, we obtain
$$\begin{array}{ll}
\displaystyle{^CD_{a+}^{\a,\r}} x(t) &=\displaystyle \frac{\rho^{\alpha-1}}{\Gamma(2-\alpha)}(t^\rho-a^\rho)^{1-\alpha}t^{1-\rho}x'(t)\left[1+\sum_{k=1}^N\frac{\Gamma(k-1+\alpha)}{\Gamma(\alpha-1)k!}\right]\\
&\quad \displaystyle-\frac{\rho^{\alpha}}{\Gamma(2-\alpha)}\sum_{k=1}^N \frac{\Gamma(k-1+\alpha)}{\Gamma(\alpha-1)(k-1)!}(t^\rho-a^\rho)^{1-\alpha-k}
\int_a^t(\tau^\rho-a^\rho)^{k-1}x'(\tau)d\tau+E_N(t).\\
\end{array}$$
It remains to show that
$$\lim_{N\to\infty}E_N(t)=0, \quad \forall t\in[a,b],$$
and to do this we determine an upper bound for the error. Since $\tau\in[a,t]$, we have
$$\sum_{k=N+1}^\infty \left|\frac{\Gamma(k-1+\alpha)}{\Gamma(\alpha-1)k!}\left(\frac{\tau^\rho-a^\rho}{t^\rho-a^\rho}\right)^k \right|\leq \sum_{k=N+1}^\infty \frac{\exp((1-\alpha)^2+1-\alpha)}{k^{2-\alpha}}$$
$$\leq\int_N^\infty \frac{\exp((1-\alpha)^2+1-\alpha)}{k^{2-\alpha}}\, dk= \frac{\exp((1-\alpha)^2+1-\alpha)}{N^{1-\alpha}(1-\alpha)}.$$
Taking
$$M(t):=\max_{\tau\in[a,t]}\left|\frac{d}{d\tau}\left(\tau^{1-\rho}x'(\tau)\right) \right|,$$
we get the following upper bound:
$$\begin{array}{ll}
\displaystyle\left|E_N(t)\right|& \displaystyle \leq M(t)\frac{\exp((1-\alpha)^2+1-\alpha)}{N^{1-\alpha}(1-\alpha)}\frac{\rho^{\alpha-1}}{\Gamma(2-\alpha)}\int_a^t (t^\rho-a^\rho)^{1-\alpha}d\tau\\
&\displaystyle= M(t)\frac{\exp((1-\alpha)^2+1-\alpha)\rho^{\alpha-1}}{N^{1-\alpha}(1-\alpha)\Gamma(2-\alpha)}(t^\rho-a^\rho)^{1-\alpha}(t-a),\\
\end{array}$$
which converges to zero as $N$ goes to $\infty$, for al $t\in[a,b]$.
\end{proof}

\begin{remark} We stress that, when $\rho=1$, Theorem~\ref{teo:approx} reduces to the one proven in \cite{Atan1,Pooseh1}, and as $\rho\to0^+$, we obtain the main result of \cite{Pooseh0}.
\end{remark}

In an analogous way we obtain a decomposition formula for the right Caputo--Katugampola fractional derivative. The proof is omitted, since it is similar to the proof of Theorem~\ref{teo:approx}.

\begin{theorem} Let $x:[a,b]\rightarrow\mathbb{R}$ be a function of class $C^2$, and $N \geq 1$ an integer. Define the quantities
$$\begin{array}{ll}
A_N&=\displaystyle\frac{\rho^{\alpha-1}}{\Gamma(2-\alpha)}\sum_{k=0}^N\frac{\Gamma(k-1+\alpha)}{\Gamma(\alpha-1)k!},\\
B_{N,k}&=\displaystyle\frac{\rho^\alpha\Gamma(k-1+\alpha)}{\Gamma(2-\alpha)\Gamma(\alpha-1)(k-1)!},\quad k=1,\ldots,N,\\
\end{array}$$
and functions $W_k:[a,b]\rightarrow\mathbb{R}$ by
$$W_k(t)=\int_t^b (b^\rho-\tau^\rho)^{k-1}x'(\tau)d\tau,\quad  k=1,\ldots,N.$$
Then,
$${^CD_{b-}^{\a,\r}} x(t)=-A_N(b^\rho-t^\rho)^{1-\alpha} t^{1-\rho}x'(t)+\sum_{k=1}^N B_{N,k}(b^\rho-t^\rho)^{1-\alpha-k}W_k(t)+E_N(t),$$
such that
$$\lim_{N\to\infty}E_N(t)=0, \quad \forall t\in[a,b].$$
\end{theorem}

{ Due to } the complexity of solving analytically fractional differential equations, numerical tools are often applied to solve the given problems (see e.g. \cite{ford1,ford2,Gracia,Sousa,Yan,Yang}).
We now present a simple but efficient method to solve a fractional differential equation as presented in Section~\ref{sec:FDE}. The idea is to replace the fractional derivative ${^CD_{a+}^{\a,\r}} x$ in the differential equation by an approximation given in Theorem~\ref{teo:approx} up to order $N\in\mathbb N$. In this way we obtain a system of $N+1$ ODE's, with $N+1$ initial conditions.
{ The advantage of this procedure is that after replacing the fractional derivative by the expansion which involves ordinary derivatives only, we no longer deal with a fractional differential equation but with a system of ordinary differential equations. With this, we can apply any known technique from the literature (analytical or numerical) to solve the problem.}
The solution to this system $x_N$ obviously depends on $N$. However, the sequence $(x_N)$ converges to the solution of the fractional differential equation in a neighborhood of the initial point. We refer to \cite{Atan2}, where similar problems, with dependence of Caputo and Riemann-Liouville fractional derivatives, are addressed.

Recall that the Cauchy type problem \eqref{eq:CauchyEq}--\eqref{eq:BoundCond} is equivalent to the problem of finding solutions to the Volterra integral equation \eqref{eq:Volterra}. This last equation, using \eqref{Derivative_Integral}, can be rewritten in the following way:
$$x(t)=x(a)+{^CD_{a+}^{1-\alpha,\rho}}\left[\int_a^t f(\tau,x(\tau)) \tau^{\rho-1}\,d\tau\right].$$
 Applying the decomposition formula given in Theorem~\ref{teo:approx}, we obtain
\begin{equation}\label{solExact}
x(t)=x(a)+A_N(t^\rho-a^\rho)^{\alpha}f(t,x(t))-\sum_{k=1}^N B_{N,k}(t^\rho-a^\rho)^{\alpha-k}V_k(t)+E_N(t),
\end{equation}
where
$$\begin{array}{ll}
A_N&=\displaystyle\frac{\rho^{-\alpha}}{\Gamma(1+\alpha)}\sum_{k=0}^N\frac{\Gamma(k-\alpha)}{\Gamma(-\alpha)k!},\\
B_{N,k}&=\displaystyle\frac{\rho^{1-\alpha}\Gamma(k-\alpha)}{\Gamma(1+\alpha)\Gamma(-\alpha)(k-1)!},\quad \mbox{for} \, k \in \{1,\ldots,N\},\\
V_k(t)&=\displaystyle\int_a^t (\tau^\rho-a^\rho)^{k-1}f(\tau,x(\tau))\tau^{\rho-1}d\tau,\quad \mbox{for} \, k \in \{1,\ldots,N\},
\end{array}$$
and the error is bounded by the formula
$$\left|E_N(t)\right|\displaystyle \leq M(t)\frac{\exp(\alpha^2+\alpha)\rho^{-\alpha}}{\alpha N^{\alpha}\Gamma(1+\alpha)}(t^\rho-a^\rho)^{\alpha}(t-a),$$
with
$$M(t):=\max_{\tau\in[a,t]}\left|\frac{d}{d\tau}f(\tau,x(\tau))\right|.$$
In order to get an approximated solution, $x_N$, we truncate the formula up to order $N$, receiving
\begin{equation}\label{solApprox}
x_N(t)=x(a)+A_N(t^\rho-a^\rho)^{\alpha}f(t,x_N(t))-\sum_{k=1}^N B_{N,k}(t^\rho-a^\rho)^{\alpha-k}V_{k,N}(t)
\end{equation}
where
$$V_{k,N}(t)=\int_a^t (\tau^\rho-a^\rho)^{k-1}f(\tau,x_N(\tau))\tau^{\rho-1}d\tau,\quad \mbox{for} \, k \in \{1,\ldots,N\},$$
(we remark that $x_N(a)=x(a))$.

\begin{theorem} Let $f:[a,b]\times \mathbb R \to \mathbb R$ be a continuous function and Lipschitz with respect to the second variable (see \eqref{eq:LipType}).
For $N\in\mathbb N$, let $x$ and $x_N$ be given by conditions \eqref{solExact} and \eqref{solApprox}, respectively. Also, let $T\in\mathbb R$ be a real in the open interval
$$a<T<\left(a^\rho+\rho\left(\frac{\Gamma(1+\alpha)}{L}\right)^{\frac1\alpha}\right)^{\frac1\rho}.$$
Then, for all $t\in[a,T]$, $x_N(t)\to x(t)$ as $N\to\infty$.
\end{theorem}

\begin{proof} Starting with relations \eqref{solExact} and \eqref{solApprox} we have
$$|x_N(t)-x(t)|\leq |A_N|(t^\rho-a^\rho)^{\alpha}|f(t,x_N(t))-f(t,x(t))|+\sum_{k=1}^N |B_{N,k}|(t^\rho-a^\rho)^{\alpha-k}|V_{k,N}(t)-V_k(t)|+|E_N(t)|$$
for all $t\in[a,T]$.
Define
$$\delta x_N:=\max_{t\in[a,T]}|x_N(t)-x(t)|.$$
As a consequence of the following relations:
$$|f(t,x_N(t))-f(t,x(t))|\leq L|x_N(t)-x(t)|\leq L \delta x_N;$$
$$\begin{array}{ll}
|V_{k,N}(t)-V_k(t)|&=\left|\displaystyle \int_a^t (\tau^\rho-a^\rho)^{k-1}|f(\tau,x_N(\tau))-f(\tau,x(\tau))|\tau^{\rho-1}d\tau\right|\\
&\displaystyle\leq L \delta x_N\int_a^t (\tau^\rho-a^\rho)^{k-1}\tau^{\rho-1}d\tau=\frac{L \delta x_N}{k\rho}(t^\rho-a^\rho)^k;
\end{array}$$
$$\begin{array}{ll}
    |A_N|&=\displaystyle\frac{\rho^{-\alpha}}{\Gamma(1+\alpha)}\left|\sum_{k=0}^N\frac{\Gamma(k-\alpha)}{\Gamma(-\alpha)k!}\right|=\frac{\rho^{-\alpha}}{\Gamma(1+\alpha)|\Gamma(-\alpha)|}\cdot \frac{\Gamma(N+1-\alpha)}{\alpha \Gamma(N+1)}\\
&    \displaystyle =\frac{\rho^{-\alpha}}{\Gamma(1+\alpha)\Gamma(1-\alpha)}\cdot \frac{\Gamma(N+1-\alpha)}{\Gamma(N+1)}
  \leq\frac{1}{\rho^\alpha\a\pi}\cdot \frac{\Gamma(N+1-\alpha)}{\Gamma(N+1)}\end{array},$$
by the  Euler's reflection formula and equation (3) in \cite{Garrappa},
 $$\begin{array}{ll}
 \displaystyle\sum_{k=1}^N |B_{N,k}|(t^\rho-a^\rho)^{\alpha-k}|V_{k,N}(t)-V_k(t)|&\displaystyle\leq \frac{L \delta x_N(t^\rho-a^\rho)^\alpha }{\rho^\alpha\Gamma(1+\alpha)|\Gamma(-\alpha)|} \sum_{k=1}^N\frac{\Gamma(k-\alpha)}{k!}\\
&\displaystyle\leq  \frac{L \delta x_N(t^\rho-a^\rho)^\alpha}{\rho^\alpha\Gamma(1+\alpha)|\Gamma(-\alpha)|} \left[\frac{\Gamma(N+1-\alpha)}{\alpha\Gamma(N+1)}+|\Gamma(-\alpha)|\right]\\
&\displaystyle \leq  \frac{L \delta x_N}{\rho^\alpha}(t^\rho-a^\rho)^\alpha \left[\frac{1}{\alpha\pi}\cdot \frac{\Gamma(N+1-\alpha)}{\Gamma(N+1)}+\frac{1}{\Gamma(1+\alpha)}\right],
\end{array}$$
we conclude that
$$|x_N(t)-x(t)|\leq \frac{L \delta x_N}{\rho^\alpha}(t^\rho-a^\rho)^\alpha \left[\frac{2}{\alpha\pi}\cdot \frac{\Gamma(N+1-\alpha)}{\Gamma(N+1)}+\frac{1}{\Gamma(1+\alpha)}\right]+|E_N(t)|$$
for all $t\in[a,T]$.
Taking the maximum, over $t\in[a,T]$, on both sides of the inequality, we get
\begin{equation}\label{relation1}\delta x_N\leq \frac{L \delta x_N}{\rho^\alpha}(T^\rho-a^\rho)^\alpha \left[\frac{2}{\alpha\pi}\cdot \frac{\Gamma(N+1-\alpha)}{\Gamma(N+1)}+\frac{1}{\Gamma(1+\alpha)}\right]+\max_{t\in[a,T]}|E_N(t)|.\end{equation}
It is obvious that
$$\lim_{N\to\infty}|E_N(t)|=0.$$
Moreover, as a consequence of the Stirling's formula (see e.g. \cite{Tricomi}), we have
$$\lim_{N\to\infty}\frac{\Gamma(N+1-\alpha)}{\Gamma(N+1)}=0.$$
Therefore, setting  $N\to\infty$ in \eqref{relation1} we obtain
$$\lim_{N\to\infty}\delta x_N\left[1-\frac{L}{\rho^\alpha\Gamma(1+\alpha)}(T^\rho-a^\rho)^\alpha\right]\leq0,$$
and by the definition of $T$, we must have $\delta x_N\to0$.
\end{proof}

{ From Eq. \eqref{relation1} we obtain that
$$\delta x_N\left[1-\frac{L}{\rho^\alpha}(T^\rho-a^\rho)^\alpha \left[\frac{2}{\alpha\pi}\cdot \frac{\Gamma(N+1-\alpha)}{\Gamma(N+1)}+\frac{1}{\Gamma(1+\alpha)}\right]\right]\leq\max_{t\in[a,T]}|E_N(t)|.$$
Since
$$0<\frac{L}{\rho^\alpha}(T^\rho-a^\rho)^\alpha<\Gamma(1+\alpha),$$
we have
$$-\frac{2\Gamma(1+\alpha)\Gamma(N+1-\alpha)}{\alpha\pi\Gamma(N+1)}\leq1-\frac{L}{\rho^\alpha}(T^\rho-a^\rho)^\alpha \left[\frac{2}{\alpha\pi}\cdot \frac{\Gamma(N+1-\alpha)}{\Gamma(N+1)}+\frac{1}{\Gamma(1+\alpha)}\right]<1.$$
Therefore, for $N$ sufficiently large, by
$$\lim_{N\to\infty}\frac{\Gamma(N+1-\alpha)}{\Gamma(N+1)}=0,$$
we get that there exists a function $C$ (depending on $\alpha,\rho,t$, but independent of $N$) such that
$$\delta x_N \leq C N^{\alpha-1}.$$}

\section{Examples}\label{Sec:examples}

In this section we provide two examples of applications of the results presented in Section~\ref{sec:approx}.

\begin{example}

Let $x(t)=(t^\rho-1)^2$, with $t\in[1,2]$, be a test function. By Lemma~\ref{lemma:power}, we have that
$${^CD_{1+}^{\alpha,\rho}} x(t)=\frac{2\rho^\alpha}{\Gamma(3-\alpha)}(t^\rho-1)^{2-\alpha}.$$
In Figures \ref{fig:1}, \ref{fig:2} and \ref{fig:3} (on the left side) the exact fractional derivative of the function (continuous line) with different numerical approximations of the fractional derivative (dot lines), for different values of $\alpha\in(0,1)$, $\rho>0$ and $N\in\mathbb N$, are compared. The error between the exact and the numerical approximations is measured by the absolute value of the difference between these two expressions, and it is shown on the right side of figures.

\begin{figure}[h!]
\begin{center}
\subfigure{\includegraphics[scale=0.3]{plot-05-06.eps}}
\subfigure{\includegraphics[scale=0.3]{error-05-06.eps}}
\caption{For $\alpha=0.5$ and $\rho=0.6$.}\label{fig:1}
\end{center}
\end{figure}

\begin{figure}[h!]
\begin{center}
\subfigure{\includegraphics[scale=0.3]{plot-07-02.eps}}
\subfigure{\includegraphics[scale=0.3]{error-07-02.eps}}
\caption{For $\alpha=0.7$ and $\rho=0.2$.}\label{fig:2}
\end{center}
\end{figure}

\begin{figure}[h!]
\begin{center}
\subfigure{\includegraphics[scale=0.3]{plot-04-15.eps}}
\subfigure{\includegraphics[scale=0.3]{error-04-15.eps}}
\caption{For $\alpha=0.4$ and $\rho=1.5$.}\label{fig:3}
\end{center}
\end{figure}
\end{example}

\begin{example} Consider the following Cauchy type problem
$$\left\{
\begin{array}{l}
\displaystyle {^CD_{1+}^{\alpha,\rho}} x(t)=x(t)+\frac{2\rho^\alpha}{\Gamma(3-\alpha)}(t^\rho-1)^{2-\alpha}-(t^\rho-1)^2, \quad t\in[1,2],\\
x(1)=0,\\
\end{array} \right.$$
It is easy to check that the solution to this system is $x(t)=(t^\rho-1)^2$. We shall compare this exact solution with an approximation solutions that can be obtained using the results presented in Section~\ref{sec:approx}. Under the assumption that the set of admissible functions is $C^2([1,2])$ we can replace the fractional operator by the formula given in Theorem~\ref{teo:approx}:
$${^CD_{1+}^{\alpha,\rho}} x(t)\approx A_N(t^\rho-1)^{1-\alpha} t^{1-\rho}x'(t)-\sum_{k=1}^N B_{N,k}(t^\rho-1)^{1-\alpha-k}V_k(t),$$
where $A_N$ and $B_{N,k}$ are given as in Theorem~\ref{teo:approx}, and $V_k$ are defined by the solution to the system
$$\left\{
\begin{array}{l}
\displaystyle V'_k(t)=(t^\rho-1)^{k-1}x'(t), \quad t\in[1,2],\\
V_k(1)=0,\\
\end{array} \right.$$
for $k=1,\ldots,N$. Thus, we obtain a system of $N+1$ first-order differential equations with $N+1$ initial conditions:
$$\left\{
\begin{array}{l}
\displaystyle  A_N(t^\rho-1)^{1-\alpha} t^{1-\rho}x'(t)-x(t)-\sum_{k=1}^N B_{N,k}(t^\rho-1)^{1-\alpha-k}V_k(t) =\frac{2\rho^\alpha}{\Gamma(3-\alpha)}(t^\rho-1)^{2-\alpha}-(t^\rho-1)^2,\\
\displaystyle V'_k(t)=(t^\rho-1)^{k-1}x'(t), \quad k=1,\ldots,N,\\
x(1)=0,\\
V_k(1)=0, \quad  k=1,\ldots,N.\\
\end{array} \right.$$
Solutions, for different values of $\alpha,\rho$ and $N$, are presented in Figures \ref{fig:4}, \ref{fig:5} and \ref{fig:6} (on the left side). The error between the exact and the numerical approximations is shown on the right side of figures.

\begin{figure}[h!]
\begin{center}
\subfigure{\includegraphics[scale=0.3]{plotFDE-05-06.eps}}
\subfigure{\includegraphics[scale=0.3]{errorFDE-05-06.eps}}
\caption{For $\alpha=0.5$ and $\rho=0.6$.}\label{fig:4}
\end{center}
\end{figure}

\begin{figure}[h!]
\begin{center}
\subfigure{\includegraphics[scale=0.3]{plotFDE-07-02.eps}}
\subfigure{\includegraphics[scale=0.3]{errorFDE-07-02.eps}}
\caption{For $\alpha=0.7$ and $\rho=0.2$.}\label{fig:5}
\end{center}
\end{figure}

\begin{figure}[h!]
\begin{center}
\subfigure{\includegraphics[scale=0.3]{plotFDE-04-15.eps}}
\subfigure{\includegraphics[scale=0.3]{errorFDE-04-15.eps}}
\caption{For $\alpha=0.4$ and $\rho=1.5$.}\label{fig:6}
\end{center}
\end{figure}
\end{example}

\begin{example} { The solution to
$$\left\{
\begin{array}{l}
\displaystyle {^CD_{0+}^{\alpha,\rho}} x(t)=\rho^\alpha x(t), \quad t\in[0,1],\\
x(0)=1,\\
\end{array} \right.$$
is the Mittag--Leffler function $x(t)=E_\alpha(t^{\rho\alpha})$. Replacing the fractional derivative by our approximation formula presented in Theorem~\ref{teo:approx}, we obtain the following system of ordinary differential equations
$$\left\{
\begin{array}{l}
\displaystyle  A_N t^{1-\rho \alpha} x'(t)-\sum_{k=1}^N B_{N,k}t^{\rho(1-\alpha-k)}V_k(t) =\rho^\alpha x(t),\quad t\in[0,1],\\
\displaystyle V'_k(t)=t^{\rho(k-1)}x'(t), \quad k=1,\ldots,N,\quad t\in[0,1],\\
x(0)=1,\\
V_k(0)=0, \quad  k=1,\ldots,N.\\
\end{array} \right.$$
Solutions for $N=5,10,15$ and different values of $\alpha$ and $\rho$ are shown in Figures \ref{fig:7} and \ref{fig:8} (on the left side). The errors of the approximation are presented on the right side of the figures.

\begin{figure}[h!]
\begin{center}
\subfigure{\includegraphics[scale=0.3]{plotFDE2-09-15.eps}}
\subfigure{\includegraphics[scale=0.3]{errorFDE2-09-15.eps}}
\caption{For $\alpha=0.9$ and $\rho=1.5$.}\label{fig:7}
\end{center}
\end{figure}

\begin{figure}[h!]
\begin{center}
\subfigure{\includegraphics[scale=0.3]{plotFDE2-05-50.eps}}
\subfigure{\includegraphics[scale=0.3]{errorFDE2-05-50.eps}}
\caption{For $\alpha=0.5$ and $\rho=5$.}\label{fig:8}
\end{center}
\end{figure}}
\end{example}

\section{Conclusion}\label{con}

In this paper we present a new fractional operator, which generalizes the Caputo and the Caputo--Hadamard fractional derivative. Some fundamental properties of this operator are presented and proved. Then, an existence and uniqueness theorem for a fractional Cauchy type problem is given.
In order to apply a numerical procedure for solving the mentioned fractional differential problem, we provide a decomposition formula for the new operator in a way that we can rewrite it using only the first-order derivative. Doing so, we can approximate the fractional differential equation by an ordinary one, and thus to determine an approximate solution to the fractional problem. We hope to call the attention of other researchers to consider generalized fractional operators in order to avoid a proliferation of papers devoted to fractional operators.

\section*{Acknowledgments}

The first author is supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications,
and the Portuguese Foundation for Science and Technology (FCT-Funda\c{c}\~ao para a Ci\^encia e a Tecnologia), within project UID/MAT/04106/2013.
A. B. Malinowska is supported by the Bialystok University of Technology grant S/WI/1/2016. T. Odzijewicz by the Warsaw School of Economics grant KAE/S14/35/15.

redpajama_set_name: RedPajamaArXiv


text:
Irvine: Full marks to Ideye
Albion boss says striker's performance was just a glimpse of what he can do
ALAN Irvine insists Albion fans got just a glimpse of what is to come from Brown Ideye in today's goalless draw at Southampton.
The Baggies' club-record signing is still catching up on his fitness, having only arrived in the country just over a fortnight ago due to his wait for a visa.
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If you could pick any fruit to make you instantly healthier, it should be berries. I'm amazed at how much we are always learning about what they do to and for our bodies – and right now, they are being picked in abundance!
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One of my favorite things to do with raspberries is to give them a little mash, and then swirling in Greek yogurt. Recipe here.
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Happy end of the summer eating! Focus on getting in those antioxidants this week!


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An Interview with Joanna Waterfall, Founder of Yellow Co.
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