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Events and Comments
Research and Discussion
Saturday, January 29, 2022, 14:15 (GMT+7)
Thaison Group lifts its brand to a new height
Friday, January 28, 2022, 08:08 (GMT+7)
Over the years, under the leadership and direction of the Central Military Commission and the Ministry of National Defence (MND), Thaison Group has unceasingly devoted effort to surmounting difficulties, improving its task performance, enhancing its international integration, and gradually asserting the brand of "Thaison" both at home and abroad
Yen Bai province focuses on fulfilling its "dual goal"
Thursday, December 16, 2021, 13:41 (GMT+7)
It should be noted that during the 4th wave of COVID-19, the PPC Standing Board has opportunely released Directive 12-CT/TU, dated May 5th, 20221 on implementing urgent measures for COVID-19 prevention and control in tandem with the political task of 2021. Based on that Directive, the Provincial Steering Board on COVID-19 prevention and control and the Provincial People's Committee have issued nearly 200 guiding and directing documents
Several measures for marine governance and coastal zone management
Friday, December 10, 2021, 09:57 (GMT+7)
In order to keep effectively executing the Strategy for sustainable development of Vietnam's marine economy towards 2030, with a vision towards 2045, on March 5th, 2020, our Government issued Resolution 26/NQ-CP on promulgating a Master Plan and a Five-Year Plan for the 12th Party Central Committee's Resolution 36-NQ/TW. To realise the Master Plan towards 2030, with a vision towards 2045, according to the Government's Resolution, we will regularly review, evaluate, amend, and supplement laws and policies relating to seas and islands
Promoting movement titled "veterans help one another with poverty reduction and economic development"
Monday, December 06, 2021, 19:36 (GMT+7)
Over the years, Vietnam War Veterans' Association (VWVA) and its members have always "heightened self-reliance, helped one another in economic development, strived for sustainable poverty reduction and lawful enrichment, and actively, effectively taken part in national and local economic development programmes" as this association's consistent goal; therefore, they have always surmounted difficulties to obtain significant achievements in poverty reduction and economic development
Government's plan for implementing Resolution No. 36-NQ/TW
Monday, September 27, 2021, 09:37 (GMT+7)
Promote synergy, maintain independence, sovereignty and territorial integrity; protect the country's airspace, seas and islands. Firmly consolidate the people's security posture in association with the all-people's national defence posture on seas and islands.
Building and developing the defence industry as the spearhead of the national industry
Defence industry's capacity to research, design, manufacture and repair weapons and technical equipment has markedly changed and made a breakthrough, gradually mastering the technology of production, repair, improvement and digitisation of many types of weapons and technical equipment of new generation, high technology and modernity, in line with Vietnam's national defence strategy and military art.
The building of a socialist rule-of-law State in the Documents of the 13th National Party Congress
Thursday, September 09, 2021, 11:15 (GMT+7)
The building of a socialist rule-of-law State of Vietnam in the Documents of the 13th National Party Congress has been presented in a comprehensive, diverse manner with a lot of new points. Therefore, it is necessary for us to be more fully aware of this task
Combining economy with defence at Factory Z121
Thursday, August 26, 2021, 10:35 (GMT+7)
Founded on September 7th, 1966, the Factory Z121, aka the 21 Chemical One Member Limited Liability Company under the General Department of Defence Industry, is tasked with manufacturing and providing explosive accessories for the Military's training and combat readiness. At the same time, it is assigned to manufacture several dual-purpose products and fireworks for the country and export
Building and developing a modern, dual-purpose defence industry of Vietnam to meet the requirements of national construction and protection
Tuesday, August 24, 2021, 08:01 (GMT+7)
Building and developing a modern, dual-purpose defence industry of Vietnam to serve the Homeland defence and socio-economic development is an important strategy stipulated in the Resolution of the 13th National Party Congress. To realise that strategy, it is necessary to promote the synergy of the entire Party, Military and people, with Defence Industry playing a core role
The state economic sector's role in the country's socialist-oriented market economy is undeniable
Thursday, June 24, 2021, 13:47 (GMT+7)
The 35-year national renewal has proved that the state economic sector plays a leading role in the country's economy. According to the Documents of the 13th National Party Congress, "the socialist-oriented market economy continues to develop, the macroeconomic stability has been maintained, the economy's major balances have been ensured, the economic growth has been sustained at a rather high level, the economy's scale and potential have been increased, and the quality of economic growth has been improved
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Editorial: Appeal for National Resistance - its historical and epochal significance
With our nation's humanistic tradition and burning desire for peace, our Party and President Ho Chi Minh expressed goodwill and appeasement as much as possible to avoid further bloodshed and spare more time for national construction. "But the more we make concessions, the more the French colonialists encroach on us because they want to take over our country again." At that crucial moment of the nation, on the evening of December 19th, 1946, President Ho Chi Minh issued the Appeal for National Resistance
Registered at Ministry of Information and Communications No. 478/GP-BTTTT 27-7-2021.
Editor-in-chief: Major General Ta Quang Chuyen, MA
Deputy Editors-in-chief: Senior Colonel Do Hai Au, MA, Senior Colonel Hoang Van Truong, MA
Copyright © National Defence Journal 2013. All rights reserved
Add: 38A Ly Nam De str – Hoan Kiem dist – Hanoi
Tel: (024)38.457.044 – Fax: (024)37.479.956 - Email: tkts@tapchiqptd.vn; thukytoasoan.qptd@gmail.com


Set in 1914 St. Petersburg, Gloria Whelan's "Angel on the Square" is a story that expertly explores two sides of the complicated unrest that brought about the Russian Revolution.
Katya Ivanova's mother is an aristocrat and lady-in-waiting to the Empress Alexandra. As a teenager in this regal setting, Katya takes lessons and goes on vacations with the imperial family. Tsar Nicolai II is like a father to her, and his luxurious and seemingly civilized lifestyle has the weight and legitimacy of many years' tradition.
At the same time, Katya's friend Misha wants to teach her about the suffering and unrest fomenting outside the palace walls. When Misha is sent to fight in World War I and the old Russia crumbles under the ensuing chaos, Katya cannot avoid the conflict between the old way of life and the future of the Russia that she loves.
One of the most stunning aspects of Whelan's story is her poetic descriptions. (Her "Homeless Bird" won the National Book Award last year for young adult fiction.) Katya's aristocratic life is made luminously evident in an array of subtle, but sumptuous details. Marble bathrooms, sapphires that remind Katya of the "clearest, deepest blue water," trains that have been made into plush, ornamented drawing rooms, thousand-acre country estates - these are the things that Katya at first takes for granted, and the means by which the author transports her readers to another time.
The book's historical insights add substantially to its power. Whelan includes the brief career of Alexander Kerensky, the man who called for the abdication of Nicolai II, but who could not realize his dream of a democratic Russia, and Rasputin, the strange holy man who exerted so much power over Russia's last royal family.
A glossary of Russian terms also helps young readers make sense of this thrilling, if tragic, period.
And yet, like the goose-down quilts and sumptuous silks that adorn young Katya Ivanova's pre-revolutionary Russia, "Angel on the Square," may be enjoyed without effort.
Courtney Williamson is a freelance writer in Marblehead, Mass.


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IDF Honors Veterans of Israel&...
IDF Honors Veterans of Israel's Independence
Stand for Israel | May 2, 2022
(Photo: IDF Spokesperson's Unit)
This week, Israel will observe two special holidays – Yom HaZikaron (Memorial Day) and Yom HaAtzmaut (Independence Day). Ahead of these two times to remember those who served Israel, as well as the freedoms that all Israelis now enjoy, The Jerusalem Post's Anna Ahronheim reports on the IDF remembering its veterans of the 1948 Israel War of Independence:
Ahead of Remembrance Day for the Fallen of Israel's Wars, the IDF's Southern Command is embarking on an extensive project honoring the fighters of the War of Independence with a series of heritage ceremonies and marches as part of Operation LaOrem (Following their Path).
Cpt. Adi Perry, education officer in the IDF's 80th Edom Division in the Southern Command, told The Jerusalem Post that the military has been working on the project for over two months and that more than 1,500 troops are taking part in the three-day-long operation.
"It fills you with pride to hear officers explain why we need to learn and discuss the War of Independence," she said, explaining that yesterday officers spoke to troops about the generation of fighters who fought in the Independence War and why we continue to fight today…"
Defense Minister Benny Gantz said that he only recently learned that his father had taken part in the Battle of Be'erot Yitzhak that took place on July 15, 1948 against Egyptian forces.
Despite the fighters' lack of experience, Israeli forces managed to repel the Egyptian attack until reinforcements arrived and drove the Egyptians out.
"The story of the battle of Be'erot Yitzhak is largely the story of the State of Israel and the generation of 1948. They were new immigrants who gathered from all over the world, in order to establish a Jewish state in the Land of Israel…"
Support Israel Today
Share your blessings and help strengthen Israel and Her people.

redpajama_set_name: RedPajamaCommonCrawl


The all new Fonolo website is now up! It reveals more of the details about our enterprise product than ever before.
The Fonolo consumer service (that lets anyone Deep Dial to any of the 500 companies in our database) continues to work as before and is still completely free. We've just moved it to http://consumer.fonolo.com so update your bookmarks.
Our iPhone app continues to work as before.
This entry was posted in Fonolo and tagged fonolo on 2010/11/19 by mike.


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California Children's Services (CCS) is a statewide program that provides authorizations for medical care, case management, financial assistance, and medically necessary physical and occupational therapy services to children who meet the CCS eligibility criteria. The program is managed by the California Department of Health Care Services (DHCS) and is administered by the local county health care departments.
Which Medical Conditions Are Eligible?
Eligible conditions include severe physical disabilities resulting from congenital defects or those acquired through disease, accident or abnormal development. Examples include cerebral palsy, cystic fibrosis, cancer, heart conditions and orthopedic disorders.
CCS Public Health Nurses provide medical case management for complex medical conditions. CCS approved hospitals and providers are authorized to provide medical care for the CCS eligible medical condition. CCS partners with CalOptima, Medi-Cal, and Healthy Families to case manage medical care for children with a CCS eligible medical condition.
CCS provides financial assistance for children who have a CCS eligible medical condition and whose family meets the CCS financial eligibility criteria.
CCS provides medically necessary physical and occupational therapy services to children with medical conditions that are eligible for the Medical Therapy Program. Examples of conditions that are eligible for these services include cerebral palsy, spina bifida, traumatic brain injury, muscular dystrophy, juvenile rheumatoid arthritis, and other chronic disabilities. These services are provided in Medical Therapy Units located in public school settings throughout the county.
How Are CCS Services Obtained?
Referrals can be submitted by any agency involved with the child's care including physicians, hospitals, nurses, school districts, Regional Center, or the client or family directly.
A referral must include appropriate medical reports to confirm the medical condition, a Service Authorization Request that specifies what services are being requested from CCS, and a signed program application.


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International Retinal Research Foundation
Age-related macular
degeneration (AMD) ...
is the leading cause of blindness in people over 65 years of age. In the United States alone, more than 70 million people are expected to have reached this age by the year 2050. Up to 20% of individuals older than 65 and 37% by age 75 are at risk for developing AMD. Read more
IRRF Scholars
Kelman Award
Kelman Requirements
Callahan Award
Callahan Requirements
Rich Award
Rich Requirements
Charles D. Kelman
Alston Callahan
Loris Rich
Lasker/IRRF
Lasker/IRRF Initiative
Lasker Awards 2015
Janelia Farm
ARVO News
IRRF Archive
CALL FOR FUNDING APPLICATIONS:
The next submission date for IRRF Postdoctoral Scholar Awards is March 15, 2020. More Info.
Questions or request for information should be forwarded to Sandra Blackwood either by phone or email.
sblackwood@irrf.org
IRRF 2017-18 BIENNIAL REPORT:
QUANTIFYING SPATIAL RELATIONSHIPS FROM WHOLE RETINAL IMAGES:
Bioinformatics, "Quantifying Spatial Relationships from Whole Retinal Images," Brian E. Ruttenberg, Gabriel Luna, Geoffrey P. Lewis, Steven K. Fisher, and Ambuj K. Singh, Neuroscience
FOR ADDITIONAL INFORMATION OR OTHER INQUIRIES, PLEASE WRITE TO:
The International Retinal Research Foundation, Inc.
1720 University Boulevard
Attn: Sandra Blackwood, MPA
Or by email: sblackwood@irrf.org
NA3 glycan: a potential therapy for retinal pigment epithelial deficiency (SCROLL DOWN )
In Sumana R. Chintalapudi, XiangDi Wang; XiaoFei Wang; Yunfeng Shi, Mehmet Kocak; Mallika Palamoor; Raven N. Davis; T.J. Hollingsworth; Monica M. Jablonski.
Atrophic age-related macular degeneration (AMD) is the most common type of AMD, yet there is no United States Food and Drug Administration (FDA)-approved therapy. This disease is characterized by retinal pigment epithelial (RPE) insufficiency, primarily in the macula, which affects the structure and physiology of photoreceptors and ultimately, visual function. In this study, we evaluated the protective effects of a naturally derived small molecule glycan therapeutic – asialo-, tri-antennary complex-type N-glycan (NA3) – in two distinct preclinical models of atrophic AMD. In RPE-deprived Xenopus laevis tadpole eyes, NA3 supported normal retinal ultrastructure. In RCS rats, NA3 supported fully functioning visual integrity.(Above: Monica Jablonski and team members in University of Tennessee – Memphis labs)
Furthermore, structural analyses revealed that NA3 prevented photoreceptor outer segment degeneration, pyknosis of the outer nuclear layer, and reactive gliosis of Müller cells (MCs). It also promoted maturation of adherens junctions between MC and photoreceptors. Our results demonstrate the neuroprotective effects of a naturally derived small molecular glycan therapeutic-NA3-in two unique preclinical models with RPE insufficiency. These data suggest that NA3 glycan therapy may provide a new therapeutic avenue in the prevention and/or treatment of retinal diseases such as atrophic AMD.
Dr. Jablonski is an IRRF-supported scientist at the University of Tennessee – Memphis.
You may access this article HERE.
Collaborations That Provide Sustained Research Funding
Today's vision scientists face many funding challenges making it imperative that all support options are available to them. Similarly, in order to ensure continued funding for young scientists who are developing their independent research projects, the IRRF must maximize every dollar. The formation of partnerships and collaborations with outstanding institutions has made it possible to accomplish this while producing a collective impact. Since 2013, New York-based Fight For Sight (FFS) and the IRRF have combined resources to provide an annual funding award: FFS – The International Retinal Research Foundation Grant-in-Aid Award that is offered and administered by FFS.
2018 — Jordan Greco, PhD, University of Connecticut, for his work in the characterization of an ion-mediated protein-based retinal implant.
Dr. Greco obtained his PhD in physical chemistry at the University of Connecticut in 2015. Under the mentorship of Dr. Robert Birge, Dr. Greco's graduate thesis work primarily involved the investigation of the structure and function of photoactive proteins, using both spectroscopic and quantum mechanical approaches. Much of his work has contributed towards the application of the protein bacteriorhodopsin into photonic and biomimetic devices, such as protein-based optical memories and processors, photovoltaic cells, and the retinal implant developed by LambdaVision, Inc. Concurrent with his work on bacteriorhodopsin, Dr. Greco has contributed to numerous computational analyses for the excited state of behavior of heterocyclic conjugated compounds, (e.g., porphyrin, chlorins, and corroles), carotenoids (e.g., peridinin), and other polyene-based chromophores rooted in biological systems. Dr. Greco has presented this work to international audiences and he continues to remain active in the field via several multidisciplinary collaborations. (Reprinted from Crunchbase: www.crunchbase.com/person/jordan-greco)
2017 — John T. Pena, MD, PhD, Weill Cornell Medical College for his work in diabetic retinopathy.
Grant Title: Human ocular fluid contains an intercellular communication system of endogenous exosomes.
Summary: The vitreous humor of the eye is a clear gel-like structure comprised of collagen and water and fills the back of the eye.
Traditional thinking has been that the vitreous is biologically inactive. Dr. Pena's study showed a dense organized network of extracellular vesicles (EVs) in the human vitreous. However, attempts to image vitreous EVs in whole mount or tissue sections resulted in no evidence of EVs. Yet, electron microscope (EM) studies and nano-particle tracking analysis proved that millions of EVs exist in the vitreous. To solve this discrepancy and visualize the native anatomy of vitreous EVs a hypothesis emerged that the nanometer sized EVs were lost during tissue processing secondary to reversible formalin-fixation. Therefore, this team developed an innovative fixation technique to enable visualization of vitreous EVs in situ. In addition to identifying the vitreous EVs, it was proven that vitreous EVs are a highly potent vector that can be loaded with synthetic siRNAs or proteins, and subsequently transfects retinal cells in vitro and in vivo. The team has shown that vitreous EVs can be used as a vector to efficaciously deliver therapeutic recombinant proteins to tissues like the retina and choroid.
Current and Future Academic Plans: Dr. Pena's academic plans are to continue to grow and become a productive physician-scientist. He is currently an Assistant Professor of Ophthalmology at Weill Cornell Medicine and Principal Investigator of the laboratory. Dr. Pena plans to use his training from the clinic and basic sciences to ask pertinent questions that remain a challenge in vision research. He hopes to provide straightforward solutions that can be translated to benefit his patients and will take the next few years as an opportunity to develop strong academic relationships with his mentors and students.
Related Story:
Since joining the research group of Dr. Robert Birge in 2009, University of Connecticut, Jordan Greco has been actively involved in the research and development that has led to the creation of the protein-based retinal implant and the commercialization of this technology through LambdaVision. (LambdaVision is led by University of Connecticut alumni and former students in Dr. Birge's research group, Nicole Wagner, PhD and Jordan Greco, PhD.)
Dr. Greco's graduate thesis work influenced the design and development of the retinal implant construct and the manufacturing techniques used to produce the prosthetic. As Chief Scientific Officer, Dr. Greco is responsible for manufacturing the retinal implants and establishing standard operating procedures and quality control measures. Moreover, his research efforts helped to direct critical proof-of-concept experiments that investigated the efficacy of the retinal implant architecture.
Recently, the company's robotic system to manufacture films that could cure blindness was brought to the International Space Station U.S. National Laboratory by the SpaceX Dragon spacecraft.
On Earth, it takes LambdaVision approximately five days for each of its three robotic stations to produce an implant. The process involves a series of alternating dipping steps, which are subject to the effects of gravity. Once complete, the process results in a membrane approximately 60 um (micrometers) thick. A micrometer is one-millionth of a meter.
In the weightless conditions of the International Space Station, LambdaVision anticipates producing a more homogeneous and stable film. If successful, Wagner and Greco anticipate they can generate a similar signal with fewer layers of protein. This would drastically decrease the time for manufacturing, and save on the cost of materials.
IRRF News
Dopamine D1 receptor activation contributes to light-adapted changes in retinal inhibition to rod bipolar cells
Michael D. Flood; Johnnie M. Moore-Dotson; and Erika D. Eggers, University of Arizona. This study was conducted with IRRF support – Erika D. Eggers. (15 AUG 2018, Vol 120, Issue 2)
ABSTRACT: Dopamine modulation of retinal signaling has been shown to be an important part of retinal adaptation to increased background light levels, but the role of dopamine modulation of retinal inhibition is not clear. This team previously showed that light adaptation causes a large reduction in inhibition to rod bipolar cells, potentially to match the decrease in excitation after rod saturation. This study determined how dopamine D1 receptors in the inner retina contribute to this modulation. It was found that D1 receptor activation significantly decreased the magnitude of inhibitory light responses from rod bipolar cells. Whereas D1 receptor blockade during light adaptation partially prevented this decline. To determine what mechanisms were involved in the modulation of inhibitory light responses, the effect of D1 receptor activation on spontaneous currents and currents evoked from electrically stimulating amacrine cell inputs to rod bipolar cells was measured. D1 receptor activation decreased the frequency of spontaneous inhibition with no change in event amplitudes, suggesting a presynaptic change in amacrine cell activity in agreement with previous reports that rod bipolar cells lack D1 receptors. Additionally, it was found that D1 receptor activation reduced the amplitude of electrically evoked responses, showing that D1 receptors can modulate amacrine cells directly. These results suggest that D1 receptor activation can replicate a large portion but not all of the effects of light adaptation, likely by modulating release from amacrine cells onto rod bipolar cells.
To access the full abstract, CLICK HERE
Erika Eggers, PhD,is an assistant professor at the University of Arizona and received IRRF funding for her study, Testing the role of dopamine as a potential treatment for diabetic retinal dysfunction.
The Eivor and Alston Callahan, MD, Endowed Chair in Ophthalmology University of Alabama at Birmingham School of Medicine
Birmingham, Al: Maria Grant, MD is the inaugural holder of the Eivor and Alston Callahan, MD Endowed Chair in Ophthalmology. Dr. Grant's laboratory is interested in understanding the functional process of hematopoietic stem cells, and focusing on understanding how to best utilize human embryonic stem cells for vascular repair. The ability to repair vascular damage could have a profound impact on a large number of retinal diseases. The best possible approach for repair is targeted gene expression to force selective differentiation, therefore offering novel therapeutic possibilities for tissue repair throughout the body.
Dr. Grant (above) earned her medical degree from the University of Florida, where she also completed her residency. She completed a research fellowship in the Division of Endocrinology & Metabolism at the University of Florida and a research fellowship at The Wilmer Eye Institute at Johns Hopkins University.
The International Retinal Research Foundation established The Eivor and Alston Callahan, MD, Endowed Chair in Ophthalmology to support the work and mission of the Department of Ophthalmology through the recruitment and /or retention of a distinguished ophthalmologist who specializes in retina care. It is most fitting that this endowment bear the names of Dr. Alston Callahan and Mrs. Eivor Callahan, as recognition of their extraordinary legacy within the Birmingham community and the field of vision health. Dr. Callahan began the facility now known as The UAB Callahan Eye Hospital. Additionally, he was integral in the establishment of the UAB Department of Ophthalmology and served as the first Chairman.
Maria Grant, MD
EXPERIMENTAL EYE RESEARCH
Smc1 and the Cohesin Complex in Retinal Development and Disease: A New Mouse Model of Photoreceptor Degeneration
Rodrigo Martins, PhD.
Dr. Martins is an IRRF-Funded scientist at the Federal University of Rio de Janeiro, where he is an Assistant Professor. His funded project, Smc1 and the Cohesin Complex in Retinal Development and Disease: A New Mouse Model of Photoreceptor Degeneration, has resulted in a published paper in Developmental Biology, "N-myc regulates growth and fiber cell differentiation in lens development."
Rodrigo Martins, PhD. ABSTRACT:
Myc proto-oncogenes regulate diverse cellular processes during development, but their roles during morphogenesis of specific tissues are not fully understood. This study found that c-myc regulates cell proliferation in mouse lens development and previous genome-wide studies suggested functional roles for N-myc in developing lens. The role of N-myc was examined in mouse lens development. Genetic inaction of N-myc in the surface ectoderm or lens vesicle impaired eye and lens growth, while "late" inactivation in lens fibers had no effect. Unexpectedly, defective growth of N-myc-deficient lenses was not associated with alterations in lens progenitor cell proliferation or survival. Notably, N-myc-deficient lens exhibited a delay in degradation of DNA in terminally differentiating lens fiber cells. RNA-sequencing analysis of N-myc-deficient lenses identified a cohort of down-regulated genes associated with fiber cell differentiation that included DNaseIIβ. Further, an integrated analysis of differentially expressed genes in N-myc-deficient lens using normal lens expression patterns of iSyTE, N-myc-binding motif analysis and molecular interaction data from the String database led to the derivation of an N-myc-based gene regulatory network in the lens. Finally, analysis of N-myc and c-myc double-deficient lens demonstrated that these Myc genes cooperate to drive lens growth prior to lens vesicle stage. Together, these findings provide evidence for exclusive and cooperative functions of Myc transcription factors in mouse lens development and identify novel mechanisms by which N-myc regulates cell differentiation during eye morphogenesis.
TERMS: MYC – an immediate early response gene downstream of many ligand-membrane receptor complexes (Armelin et al., 1984; Kelly et al., 1983) binding to 10%-15% of genomic loci in mammals.
Oncogene – defined as a gene that encodes a protein that is capable of transforming cells in cultures or inducing cancer in animals.
Morphogenesis – can be defined as the processes that are responsible for producing the complex shapes of adults from the simple ball of cells that derives from division of the fertilized egg. (On-line Medical Dictionary, © 1997-98 Academic Medical Publishing & CancerWEB).
To read this paper in its entirety, please CLICK HERE.
Proceedings of the National Academy of
Sciences (PNAS) of the United States of America
Human complement factor H Y402H polymorphism causes an age-related macular degeneration phenotype and lipoprotein dysregulation in mice. Michael Landowski, Una Kelly, Mikael Klingeborn, Marybeth Groelle, Jin-Dong Ding, Daniel Grigsby and Catherine Bowes Rickman.
PNAS February 26, 2019 116 (9) 3703-3711. To read the entire article CLICK HERE.
This study was conducted with IRRF support through an RPB/IRRF Catalyst Award for Innovative Research.
The complement factor H (CFH) Y402H polymorphism (rs1061170) imparts the strongest risk for age-related macular degeneration (AMD), the leading cause of blindness in the elderly. Popular thinking holds that the CFH H402 variant increases complement activation in the eye, predisposing susceptibility to disease. However, clinical trials of complement inhibitors in AMD patients have failed. This study provides an explanation showing CFH variant-specific differences in the presentation of AMD-like pathologies. The study also shows that aged mice expressing the human H402, but not Y402 variant, (i) develop AMD-like symptoms and (ii) display differences in their systemic and ocular lipoprotein levels, but not in their complement activation, after diet. These findings support targeting lipoproteins for the treatment of AMD.
A Professor of Ophthalmology and Associate Professor in Cell Biology with tenure at Duke University School of Medicine, Dr. Bowes Rickman is currently receiving funds as a recipient of a Research to Prevent Blindness (RPB)/International Retinal Research Foundation (IRRF) Catalyst Award for Innovative Research Approaches for AMD. The RPB/IRRF Catalyst Awards provide seed money for high-risk/high-gain vision science research, which is innovative, cutting-edge and demonstrates out-of-the box thinking. Research related to both dry and wet forms of AMD are supported by this award.
Dr. Bowes Rickman's current studies involve the molecular mechanisms underlying the development of age-related macular degeneration, with a focus on development and studies of animal models of AMD, AMD pathogenesis and pre-clinical studies of novel therapies for AMD, and she has a strong track record of productive research in this field.
Dr. Bowes Rickman
Optical Coherence Tomography Features Preceding the Onset of Advanced Age-Related Macular Degeneration
Investigative Ophthalmology & Visual Science (IOVS), Optical Coherence Tomography Features Preceding the Onset of Advanced Age-Related Macular Degeneration
Daniela Ferrara, Rachel E. Silver, Ricardo N. Louzada, Eduardo A. Novais, Gilian K. Collins, and Johanna M. Seddon.
Ophthalmic Epidemiology and Genetics Service, Tufts Medical Center, 800 Washington Street #450, Boston, MA
02111. USA. (July, 2017) Invest Ophthalmol Vis Sci. 2017/58:3519-3529. DOI:10.1167/iovs.17-21696
This study was conducted with IRRF support – Johanna Seddon.
Age-related macular degeneration (AMD) is a progressive disease with multifactorial etiology. There is a need to identify clinical features that are harbingers of advanced disease. Evaluations were conducted on morphologic features of the retina and choroid on optical coherence tomography (OCT) to determine if they predict progression to advanced disease.
Abnormalities in the photoreceptors, retinal thickness, RPE, and choroid were associated with higher risk of developing advanced AMD. These findings provide insights into disease progression, and may be helpful to identify earlier endpoints for clinical studies.
To access this study: www.iovs.arvojournals.org/article.aspx?articleid=2644733
Convening of Private Vision Research Funding Foundations
For the sixth year, Research to Prevent Blindness (RPB) organized a convening of Private Vision Research Funding Foundations with federal agencies that support vision research and vision loss prevention research and vision surveillance, as well as approve new ophthalmic drugs and devices. With a theme this year of The Eye as a Window to Overall Health, the event featured keynote presentations on the use of visual imaging to diagnose and monitor the progression of various diseases and the use of Artificial Intelligence (AI) in ophthalmology. Attendees also gave updates on collaborations between the participating foundations. International Retinal Research Foundation (IRRF) Executive Director, Sandra Blackwood attended and acted as facilitator for one of the sessions.
RPB/IRRF Catalyst Award for Innovative Research Approaches for Age-Related Macular Degeneration (AMD)
2019 marked the fifth year of collaboration between the International Retinal Research Foundation (IRRF) and Research to Prevent Blindness (RPB) in which our collective resources were used to support focused research into the causes and possible cures for age-related macular degeneration (AMD). The American Macular Degeneration Foundation (AMDF) joined RPB and IRRF in establishing four more catalyst awards in 2019. These Catalyst Awards for Innovative Research Approaches for AMD will provide up to $300,000 per award, payable over 3 years.
THE GRANTS ARE AS FOLLOWS:
RPB/IRRF Catalyst Award for Innovative Research Approaches for AMD
Monica M. Jablonski, PhD, FARVO, Professor, University of Tennessee Health Science Center, will develop polygenetic models of AMD in order to better study disease pathogenesis and test innovative therapies.
RPB Catalyst Award for Innovative Research Approaches for AMD
Kevin L. Schey, PhD, Professor, Vanderbilt University School of Medicine, will develop a novel method of identifying early-stage AMD by correlating molecular and clinical information via a machine learning approach.
RPB/AMDF Catalyst Award for Innovative Research Approaches for AMD
Sabine Fuhrmann, PhD, Associate Professor, Vanderbilt University Medical Center will examine the potential of retinal pigment epithelium (RPE) cells to regenerate in mature mammalian eyes via specific signaling pathways.
Aparna Lakkaraju, PhD, Associate Professor, University of California, San Francisco, School of Medicine, will study RPE cell damage (a known precursor to AMD), with the goal of lerning about the mechanisms that initiate RPE damage and, subsequently, AMD.
The Catalyst Awards are aimed at researchers who are working on novel approaches to AMD research that has translational relevance or potential. A wide range of applications were considered from promoting a new understanding of AMD and to developing new treatments, including research related to both dry and wet forms of AMD. Assistant professor through full professor from any U.S. academic medical center and any relevant department were eligible to apply. However, the proposed research could not be funded – previously or at the time of application – by others, including government agencies/institutions, non-profits and private funders.
RPB and IRRF launched the first version of the Catalyst Awards in 2014, which focused on supporting novel stem cell-based approaches to AMD. In 2017, RPB and IRRF again joined together to launch the current version of the Catalyst Awards. This collaboration has been a very positive experience and the combining of our respective resources has allowed us to extend a significant award, and we feel confident that worthy recipients have again been selected. The IRRF welcomes the added participation of the American Macular Degeneration Foundation as this will further extend the impact of these awards.
About the International Retinal Research Foundation: The International Retinal Research Foundation (IRRF) provides financial support for vision research to scientists in every corner of the globe, while focusing on discovery of causes, preventions and cures of macular degeneration and diabetic retinopathy. In addition to research funding, IRRF supports training fellowships, public awareness programs, and promotes the exchange of scientific findings. To do this, IRRF must maximize every dollar. Forming partnerships and collaborations with outstanding institutions has made it possible to effectively achieve a collective impact, which will positively affect the lives of many individuals and further scientific knowledge in the vision research community. Learn more at www.irrf.org.
About Research to Prevent Blindness: Research to Prevent Blindness (RPB) is the leading nonprofit organization supporting eye research directed at the prevention, treatment, or eradication of all diseases that damage and destroy sight. RPB also supports efforts to grow and sustain a robust and diverse vision research community. RPB has awarded more than $368 million in research grants to the most talented vision scientists at the nation's leading medical schools. As a result, RPB has been associated with nearly every major breakthrough in the understanding and treatment of vision loss in the past 59 years. Learn more at www.rpbusa.org.
About the American Macular Degeneration Foundation: The American Macular Degeneration Foundation is a patient-centered foundation that supports potentially game-changing AMD research, education and advocacy in order to improve quality of life and treatment outcomes for all those affected by AMD. Learn more at www.macular.org.
The Callahan Endowed Chair in Ophthalmology University of Alabama at Birmingham School of Medicine
Smc1 and the Cohesin Complex in Retinal Development and Disease
Proceedings of the National Academy of Sciences (PNAS) of the United States of America
Current Research News
LASKER/IRRF INVESTIGATOR EXCHANGE FUNDS AWARDED
EFFECT OF AGING ON PHOTORECEPTOR FUNCTION, ALEXANDER V. KOLESNIKOV, JIE FAN, ROSALIE K. CROUCH, AND VLADIMIR J. KEFALOV.
LINK IN THE CAUSE OF VISION LOSS IN MACULAR DEGENERATION, DIABETIC RETINOPATHY, AND OTHER RETINAL DISEASES FOUND
The International Retinal Research Foundation, Inc. has been approved by the Internal Revenue Service as tax exempt
from federal tax under section 509(A) of the Internal Revenue Code as an organization described in section 501(c)(3).
Site created and maintained by RTWStudio.com
Attn: Sandra Blackwood, MPA, Executive Director


text:
\section{Proofs of Combinatorial Lemmas}\label{app-combin-proofs}

We repeat the combinatorial facts of Section~\ref{sub-combinatorics},
  this time with proofs.

\begin{qlemma}{\ref{lem-sum-combin-simple}}
\stmtlemsumcombinsimple
\end{qlemma}

\begin{proof}
Let us fix $0\le i_0<i_1<i_2<K$ such that both $i_2-i_1$ and $i_1-i_0$ are odd.
We want to show that the term $x_{i_0}x_{i_1}x_{i_2}$
  occurs $(K-3)/2$ times in each side of the equality.
The middle and right sides are trivial; it remains to check the left side.
Let us now fix an arbitrary~$k$.
For a term on the left hand side to equal $x_{i_0}x_{i_1}x_{i_2}$,
  it must be that the sets $\{(i'_0+k)\bmod K, (i'_1+k)\bmod K, (i'_2+k)\bmod K\}$
  and $\{i_0,i_1,i_2\}$ are equal.
In other words, once $i_0,i_1,i_2,k$ are fixed,
  the set $\{i'_0,i'_1,i'_2\}$ is uniquely determined.
Since, $i'_0<i'_1<i'_2$, the potential values of $i'_0,i'_1,i'_2$ are also uniquely determined.
The remaining question is for how many $k\in\{0,\ldots,K-1\}$
  it is the case that the values $i'_0,i'_1,i'_2$ so determined obey
  the other constraints.

There are three disjoint cases:
The smallest value in the set $\{i'_0,i'_1,i'_2\}$, namely~$i'_0$, is
  $(i_0-k)\bmod K$ or $(i_1-k)\bmod K$ or $(i_2-k)\bmod K$.
The case $i'_0=(i_1-k)\bmod K$ occurs exactly when
  (a)~$i_0<k<i_1$, and (b)~$k$~has the same parity as~$i_1$.
Let $\delta_0$ denote the size of the gap between $i_0$~and~$i_1$.
Then, there are $(\delta_0-1)/2$ values of $k$ that obey both (a)~and~(b).
The other two cases are similar,
  and so we conclude that the term $x_{i_0}x_{i_1}x_{i_2}$
  occurs on the left hand side
\[
  \frac{\delta_0-1}{2}+\frac{\delta_1-1}{2}+\frac{\delta_2-1}{2} = \frac{K-3}{2}
\]
times.
\end{proof}

\begin{qlemma}{\ref{lem-drop-terms}}
\stmtlemdropterms
\end{qlemma}

\begin{proof}
The proof below is a case analysis of where $i_1$ can be inserted in-between
  $0<i_2<i_3<i_4<K$.
As noted before, the task is to show that retaining the terms that occur in $f_5$
  gives a lower bound.
In other words, we want to show that those terms not occurring in $f_5$ have a positive sum.
We calculate this sum:
\begin{equation*}
\def\hskip-1em{\hskip-1em}
\def\hskip-2em{\hskip-2em}
\begin{aligned}
  &\sum_{\substack{1<i_2<i_1\\\text{$i_2$ even}}} v_0 v_{i_2} v_{i_1}
  -\hskip-1em\sum_{\substack{1<i_2<i_3<i_4<K\\i_2<i_1}}\hskip-2em \alpha v_0 v_{i_2} v_{i_1} v_{i_3} v_{i_4} \\
=\;
  &\sum_{\substack{1<i_2<i_1\\\text{$i_2$ even}}} v_0 v_{i_2} v_{i_1}
  \Bigl(
  1-
  \hskip-1em\sum_{\substack{i_2<i_3<i_4<K\\\text{$i_3$ odd, $i_4$ even}}}\hskip-1em \alpha v_{i_3} v_{i_4}
  \Bigr) \\
\ge\;
  &\sum_{\substack{1<i_2<i_1\\\text{$i_2$ even}}} v_0 v_{i_2} v_{i_1}
  \Bigl(
  1-
  \hskip-1em\sum_{\substack{3\le i_3<i_4<K\\\text{$i_3$ odd, $i_4$ even}}}\hskip-1em \alpha v_{i_3} v_{i_4}
  \Bigr) \quad\mathop{\ge}^\text{by~\eqref{eq-d2-nonpos}}\quad 0
\end{aligned}
\end{equation*}
\end{proof}

\begin{qlemma}{\ref{lem-fancy-sum}}
\stmtlemfancysum
\end{qlemma}

The proof is similar to that of Lemma~\ref{lem-sum-combin-simple}.

\begin{proof}
Let us fix $0\le i_0 <\cdots<i_{l-1}<K$ with odd gaps in-between.
We want to show that the term $x_{i_0}\ldots x_{i_{l-1}}$
  occurs $(l-1)K/2-l$ times on each side of the equation.
For the right side, it is trivial.
The general form of a term on the left side is
$
  x_k x_{i'_1+k} x_{i'_2+k} \ldots x_{i'_{l-1}+k}
$.
It must be that $k$ is one of $i_0,\ldots,i_{l-1}$.
Let us consider the case $k=i_0$; the others are similar.
If $k=i_0$, then, in fact,
\[
  (k, i'_1+k, i'_2+k,\ldots,i'_{l-1}+k) \bmod K \quad=\quad (i_0,i_1,\ldots,i_{l-1}).
\]
In particular, $i'_1$ equals the size of the gap between $i_0$~and~$i_1$.
Let us denote this gap by~$\delta_0$.
On the left hand side, the term is multiplied by $(K-i'_1-2)/2$,
  which is $(K-\delta_0-2)/2$.
The cases $k=i_1$, $k=i_2$, \dots are similar.
Because $\delta_0+\cdots+\delta_{l-1}=K$,
  we conclude that the term $x_{i_0}\ldots x_{i_{l-1}}$ occurs
\[
  \frac{K-\delta_0-2}{2} + \cdots + \frac{K-\delta_{l-1}-2}{2}
  = \frac{l-1}{2}K-l
\]
times on the left side.
\end{proof}

\section{Proofs of Properties of an Interior Local Maximum} \label{app-derivatives}

\begin{qlemma}{\ref{lem-derivatives}}
Let $\vec{v}$ be a local maximum of~$f^{(K)}$ in the interior of~$D^{(K)}$ and define $c \in \mathbb{R}$ by
\begin{align}
  c \ \ =
    \sum_{\substack{1<i_2<K\\\text{$i_2$ even}}} v_{i_2}
   \  - \quad
   \alpha   \hskip-1.5em
    \sum_{\substack{1<i_2<i_3<i_4<K\\\text{$i_2$, $i_4$ even}\\\text{$i_3$ odd}}}
    \hskip-1.5em
       v_{i_2} v_{i_3} v_{i_4}\,.
  \label{eq-proof-c-def_p}
  \end{align}
This expression holds for the same value of~$c$ if the indices are rotated by an arbitrary~$k$:
for all~$j$ the index~$i_j$ becomes $(i_j + k) \bmod K$. Further, we have
\begin{align}
  \sum_{\substack{3\le i_3<i_4<K\\\text{$i_3$ odd}\\\text{$i_4$ even}}} \!\!\!\! v_{i_3} v_{i_4}
  \le
  \frac{1}{\alpha}
  \label{eq-d2-nonpos_p}
\end{align}
Again, this inequality also holds when indices are rotated.
\end{qlemma}

\begin{proof}
We consider the second-order Taylor expansion of~$f^{(K)}$ along the direction $\vec{d}=(-1,0,1,0,\ldots,0)$
 (which is tangent to~$D^{(K)}$):
\[
  f^{(K)}(\vec{x}+\epsilon\vec{d}) =
    f^{(K)}(\vec{x}) + \epsilon Q(\vec{x}) + \epsilon^2 R(\vec{x}) + O(\epsilon^3)
  \;.
\]
Since $\vec{v}$ is a local maximum, we have $Q(\vec{v})=0$ and $R(\vec{v})\le 0$.
Proving~\eqref{eq-proof-c-def_p} boils down to calculating~$Q(\vec{x})$;
  proving~\eqref{eq-d2-nonpos_p} boils down to calculating~$R(\vec{x})$.

\medskip
First, we prove~\eqref{eq-proof-c-def_p}.
Let
\[
  f^{(K)}(x_0+\epsilon,x_1,\ldots,x_{K-1}) =
    f^{(K)}(\vec{x}) + \epsilon P(\vec{x}) + O(\epsilon^2).
\]
By the chain rule and using the rotational symmetry of~$f^{(K)}$ (Lemma~\ref{lem-f-properties}(a)), we find that
\[
  Q(x_0,\ldots,x_{K-1})=P(x_{0+2},x_{1+2},\ldots,x_{K-1+2})-P(x_0,x_1,\ldots,x_{K-1}).
\]
Now recall the definition of~$f^{(K)}$:
\begin{equation}
\label{fmnew}
  f^{(K)}(\vec{x}) :=
    \hskip-1em
    \sum_{\substack{0\le i_0<i_1<i_2<K\\\text{$i_2-i_1$ and $i_1-i_0$ odd}}}
    \hskip-2em
      x_{i_0} x_{i_1} x_{i_2}
    -
    \hskip-2em
    \sum_{\substack{0\le i_0<\cdots<i_4<K\\\text{$i_4-i_3,\ldots,i_1-i_0$ all odd}}}
    \hskip-2em
      \alpha
      x_{i_0} x_{i_1} x_{i_2} x_{i_3} x_{i_4}
\end{equation}
We differentiate $f^{(K)}$ with respect to $x_0$ to obtain
\begin{align}
  P(\vec{x}) =
    \sum_{\substack{0<i_1<i_2<K\\\text{$i_1$ odd}\\\text{$i_2$ even}}}
    \hskip-1em
      x_{i_1} x_{i_2}
    -
    \hskip-1em
    \sum_{\substack{0<i_1<\cdots<i_4<K\\\text{$i_1,i_3$ odd}\\\text{$i_2,i_4$ even}}}
    \hskip-2em
      \alpha
      x_{i_1} x_{i_2} x_{i_3} x_{i_4}.
\end{align}
Since $\vec v$~is a local maximum, $Q(\vec{v})=0$ and we have
\[
    \sum_{\substack{0<i_1<i_2<K\\\text{$i_1$ odd}\\\text{$i_2$ even}}}
    \hskip-1em
      v_{i_1} v_{i_2}
    -
    \hskip-1em
    \sum_{\substack{0<i_1<\cdots<i_4<K\\\text{$i_1,i_3$ odd}\\\text{$i_2,i_4$ even}}}
    \hskip-2em
      \alpha
      v_{i_1} v_{i_2} v_{i_3} v_{i_4}
=
    \sum_{\substack{0<i_1<i_2<K\\\text{$i_1$ odd}\\\text{$i_2$ even}}}
    \hskip-1em
      v_{i_1+2} v_{i_2+2}
    -
    \hskip-1em
    \sum_{\substack{0<i_1<\cdots<i_4<K\\\text{$i_1,i_3$ odd}\\\text{$i_2,i_4$ even}}}
    \hskip-2em
      \alpha
      v_{i_1+2} v_{i_2+2} v_{i_3+2} v_{i_4+2}
\]
Observe that the monomials not containing $v_1$ cancel each other out.
Dividing by $v_1=v_{1+K}$ (since $v_1>0$), we have
\[
    \sum_{\substack{1<i_2<K\\\text{$i_2$ even}}}
    \hskip-1em
      v_{i_2}
    -
    \hskip-1em
    \sum_{\substack{1<i_2<i_3<i_4<K\\\text{$i_3$ odd}\\\text{$i_2,i_4$ even}}}
    \hskip-2em
      \alpha
      v_{i_2} v_{i_3} v_{i_4}
=
    \sum_{\substack{0<i_1<K-1\\\text{$i_1$ odd}}}
    \hskip-1em
      v_{i_1+2}
    -
    \hskip-1em
    \sum_{\substack{0<i_1<i_2<i_3<K-1\\\text{$i_1,i_3$ odd}\\\text{$i_2$ even}}}
    \hskip-2em
      \alpha
      v_{i_1+2} v_{i_2+2} v_{i_3+2}
\]
Now we observe that the right hand side can be obtained from the left hand side
  by changing each index~$i$ into~$i+1$.
Taking into account rotations of the above equality, we conclude~\eqref{eq-proof-c-def_p}.

\medskip
Next, we prove~\eqref{eq-d2-nonpos_p}. To do so, we first calculate the terms of order $\epsilon^2$
  of $f^{(K)}(\vec{x}+\epsilon\vec{d})$.
Such terms occur only when $i_0=0$,\ $i_1=1$, and $i_2=2$.

In this case,
  the first sum reduces to $(x_0-\epsilon)x_1(x_2+\epsilon)$,
  and the second sum reduces to
\[
  \alpha
  (x_0-\epsilon)x_1(x_2+\epsilon)
  \hskip-1em
  \sum_{\substack{2<i_3<i_4<K\\\text{$i_3$ odd, $i_4$ even}}}
  \hskip-1em
    x_{i_3} x_{i_4}.
\]
Thus,
\[
  R(\vec{x}) =
  -x_1
  +\alpha x_1
  \hskip-1em
  \sum_{\substack{2<i_3<i_4<K\\\text{$i_3$ odd, $i_4$ even}}}
  \hskip-1em
    x_{i_3} x_{i_4}.
\]
Since the assumed interior local maximum $\vec{v}$ is in the interior of~$D^{(K)}$,
  we have that $v_1>0$,
  and so the condition $R(\vec{v})\le0$ is equivalent to $R({\vec{v}})/v_1\le0$:
\[
  -1 +
  \hskip-1em
  \sum_{\substack{2<i_3<i_4<K\\\text{$i_3$ odd, $i_4$ even}}}
  \hskip-1.5em
    \alpha v_{i_3} v_{i_4}
  \le0
\]
Up to trivial rearrangement, we obtained~\eqref{eq-d2-nonpos_p}.
\end{proof}

\begin{qcorollary}{\ref{cor-d2-nonpos-summed}}
\stmtcordtwononpossummed
\end{qcorollary}

\begin{proof}
We sum \eqref{eq-d2-nonpos_p} over all $K$ rotations:
\begin{align*}
  \frac{K}{\alpha}
    &\ge \sum_{i=0}^{K-1} \sum_{\substack{3\le i_3<i_4< K\\\text{$i_3$ odd}\\\text{$i_4$ even}}}
      \!\!\!v_{i_3+i} v_{i_4+i}
     = \sum_{\substack{3\le i_3<i_4< K\\\text{$i_3$ odd}\\\text{$i_4$ even}}} \sum_{i=0}^{K-1}
      v_{i_3+i} v_{i_4+i}
     = \sum_{\substack{3\le i_3<i_4< K\\\text{$i_3$ odd}\\\text{$i_4$ even}}}
        \!\!\! S_{i_4-i_3}(\vec{v}) \\
    &= \sum_{\substack{3\le i_3<i_3+i<K\\\text{$i$ odd, $i_3$ odd}}} \!\!\!\!S_i(\vec{v})
     = \sum_{\substack{1\le i\\\text{$i$ odd}}} \sum_{\substack{3\le i_3<K-i\\\text{$i_3$ odd}}}
       \!\!\!\!S_i(\vec{v})
     = \sum_{\substack{1\le i<K-2\\\text{$i$ odd}}} \frac{K-i-2}{2} S_i(\vec{v})
\end{align*}
\end{proof}

\section{Proof of Lemma~\ref{lem-incr-f5}}
\label{sec:incr-f5-proof}

Let $z:\{1,\ldots,K\}\rightarrow\{1,\ldots,N\}$ be a $K$-token
configuration on a ring with $N$ processes.  Recall that the
associated gap vector $\vec{g}(z) = (g_0,\ldots,g_{K-1}) \in \mathbb{N}^K$ is
defined by $g_0:=N+z(1)-z(K)$ and $g_i:=z(i+1)-z(i)$ for
$i=1,\ldots,K-1$.

Given $z$, consider the \emph{gap-increment vector}
$\Delta:=\vec{g}(z')-\vec{g}(z)$, where $z'$ is the random successor
configuration of $z$.  This is a random variable taking values in
$\{-1,0,+1\}^K$ where, for each $i\in\{0,\ldots,K-1\}$, $\Delta_i=0$
with probability $1/2$ (the two tokens adjacent to the $i$-th gap both
stay or both move clockwise), and $\Delta_i = \pm 1$ with probability
$1/4$ (one token stays and the other moves clockwise).

We will need the following two properties (\ref{eq:property1}) and
(\ref{eq:property2}) concerning the expectation of the random
variable $\Delta$.  First, it is straightforward
to verify by direct calculation that for $0\leq k< K$,
\begin{gather}
  \mathbb{E} (\Delta_i\Delta_{i+1}\ldots \Delta_{i+k}) =
\left\{ \begin{array}{ll}
        0 & \mbox{if $k$ is even}\\
        \left(-\frac{1}{4}\right)^{(k+1)/2} & \mbox{if $k$ is odd}
\end{array} \right .
\label{eq:property1}
\end{gather}
Secondly, suppose that $0\leq i_1 \leq i_2 < i_3 \leq i_4 < K$, with
$i_3 \not\equiv i_2+1$ and $i_1 \not\equiv i_4+1$ modulo $K$, that is,
$\{i_1,\ldots,i_2\}$ and $\{i_3,\ldots,i_4\}$ form two non-adjacent
intervals (treating $K-1$ and $0$ as adjacent).  Then
\begin{gather}
 \mathbb{E}(\Delta_{i_1}\ldots \Delta_{i_2}
     \Delta_{i_3}\ldots \Delta_{i_4})
  =  \mathbb{E}(\Delta_{i_1}\ldots \Delta_{i_2})
     \mathbb{E}(\Delta_{i_3}\ldots \Delta_{i_4}) \, .
\label{eq:property2}
\end{gather}
because $\Delta_{i_1},\ldots,\Delta_{i_2}$ and
$\Delta_{i_3},\ldots,\Delta_{i_4}$ are determined by the movements of
disjoints sets of tokens, and hence are independent.

For a given configuration $z$ we want to compute $\mathbb{E}
[f_5(\vec{g}(z)+\Delta)]$.  From the definition of $f_5$ and the linearity
of expectation, this is a sum of expressions of the form
\begin{align}
  \mathbb{E}
    (g_{i_0}+\Delta_{i_0})
    (g_{i_1}+\Delta_{i_1})
    (g_{i_2}+\Delta_{i_2})
    (g_{i_3}+\Delta_{i_3})
    (g_{i_4}+\Delta_{i_4})
\label{eq:expand}
\end{align}
over the set of indices $0\leq i_0<i_1<i_2<i_3<i_4<K$ of alternating parity.

Expression (\ref{eq:expand}) evaluates to a degree-5 polynomial in the
variables $\vec{g}$.  Observe that this polynomial has no monomials of even degree.
For example,
  all degree-2 monomials have coefficients of the form $\mathbb{E}(\Delta_i\Delta_j\Delta_k)$
  with $i<j<k$.
These coefficients are zero by \eqref{eq:property1}~and~\eqref{eq:property2}.
Degrees 0 and 4 are proved similarly.

There is a single degree-5 monomial in (\ref{eq:expand})---namely
$g_{i_0}\ldots g_{i_4}$.  Summing all such terms over indices $0\leq
i_0<i_1<i_2<i_3<i_4<K$ of alternating parity yields $f_5(\vec{g}(z))$.

Expanding the expression (\ref{eq:expand}) yields degree-3 monomials
of the form \[g_{j_0}g_{j_1}g_{j_2}\mathbb{E}(\Delta_{j_3}\Delta_{j_4})\] for
distinct indices $j_0<j_1<j_2$.  The coefficient of such a term is
$-1/4$ if $j_4 \equiv j_3+1$ or $j_3\equiv j_4+1$ and $0$ otherwise.
Moreover, if $j_0,j_1,j_2$ have alternating parity there are $(K-3)/2$
choices of $j_3$ such that
$g_{j_0}g_{j_1}g_{j_2}\mathbb{E}(\Delta_{j_3}\Delta_{j_3+1})$ appears in
(\ref{eq:expand}).
If $j_0,j_1,j_2$ do not have alternating parity
then there are no such terms in (\ref{eq:expand}).  We conclude that
the sum of all degree-3 monomials in $\mathbb{E}(f_5(\vec{g}(z)+\Delta)$ is
\[ -\frac{(K-3)}{8} f_3(\vec{g}(z)) \, . \]

Finally, consider the degree-1 monomials.  These have the form
\[g_{j_0}\mathbb{E}(\Delta_{j_1}\Delta_{j_2}\Delta_{j_3}\Delta_{j_4})\] for
distinct indices $j_0$ and $j_1<j_2<j_3<j_4$.  By Property
(\ref{eq:property2}), such terms are only non-zero if
$\{j_1,j_2,j_3,j_4\}$ comprises either a single block of adjacent
indices or two non-adjacent blocks of length $2$ (considering $K-1$
and $0$ to be adjacent).  In this case
$\mathbb{E}(\Delta_{j_1}\Delta_{j_2}\Delta_{j_3}\Delta_{j_4})= 1/16$, and
there are $\binom{(K-1)/2}{2}=(K-1)(K-3)/8$ such choices of
$\{j_1,j_2,j_3,j_4\}$ for each choice of $j_0$.  Thus $g_{j_0}$ has
total coefficient $(K-1)(K-3)/128$ in $\mathbb{E}(f_5(\vec{g}(z)+\Delta))$.
Moreover, since $g_0+\ldots+g_{K-1}=N$, the degree-1 terms in
$\mathbb{E}(f_5(\vec{g}(z)+\Delta))$ sum to $N(K-1)(K-3)/128$.

In summary, we have proved:
\begin{proposition}
For each $K$-token configuration $z$,
\[ \mathbb{E}(f_5(\vec{g}(z)+\Delta)) = f_5(\vec{g}(z)) - \frac{K-3}{8}f_3(\vec{g}(z)) +
\frac{(K-1)(K-3)N}{128} \, .\]
\label{prop:lemma8}
\end{proposition}

Lemma~\ref{lem-incr-f5} follows immediately from
Proposition~\ref{prop:lemma8} by scaling, since $V_5(z) = 4N^2
f_5(\vec{g}(z)/N)=\frac{4}{N^3}f_5(\vec{g}(z))$ is a linear multiple of
$f_5(\vec{g}(z))$.

\section{Conclusions} \label{sec-conclusion}

In this paper we have proved the Herman-Protocol Conjecture formulated by
McIver and Morgan in~\cite{McIverMorgan} a decade ago, which says that the
worst-case initial configuration consists of three maximally-separated tokens,
for $N$~multiple of~$3$. This follows from our result that the worst-case
self-stabilization time is at most $\frac{4}{27}N^2$, for any number of
processes~$N$ and any odd number of tokens~$K$.

The proof uses a Lyapunov function approach. To do so, we first find a
suitable Lyapunov function and then show that its maximum
is~$\frac{4}{27} N^2$. Then we show that this function gives an upper
bound for the self-stabilization time for \emph{each} possible
configuration in Herman's algorithm.

\section{Proof of Lemma~\ref{lem-f-max-induction}} \label{sec-f-max-induction}

In this section we prove Lemma~\ref{lem-f-max-induction}.
In Section~\ref{sub-derivatives} we state several properties that an interior local maximum of~$f^{(K)}$ would have to satisfy.
In Section~\ref{sub-proof-5} we prove Lemma~\ref{lem-f-max-induction} for $K=5$ for a first taste of the general argument.
In Section~\ref{sub-proof-7} we prove Lemma~\ref{lem-f-max-induction} for $K=7$
to illustrate some fine points that occur only for larger values of~$K$.
In Section~\ref{sub-combinatorics} we state some combinatorial facts needed for the general case.
Finally, in Section~\ref{sub-proof-f-max-induction} we prove Lemma~\ref{lem-f-max-induction}.

\subsection{Properties of an Interior Local Maximum} \label{sub-derivatives}

The following lemma is obtained by considering first and second derivatives of~$f$
evaluated at an interior local maximum.
\begin{lemma} \label{lem-derivatives}
Let $\vec{v}$ be a local maximum of~$f^{(K)}$ in the interior of~$D^{(K)}$ and define $c \in \mathbb{R}$ by
\begin{align}
  c \ \ =
    \sum_{\substack{1<i_2<K\\\text{$i_2$ even}}} v_{i_2}
   \  - \quad
   \alpha   \hskip-1.5em
    \sum_{\substack{1<i_2<i_3<i_4<K\\\text{$i_2$, $i_4$ even}\\\text{$i_3$ odd}}}
    \hskip-1.5em
       v_{i_2} v_{i_3} v_{i_4}\,.
  \label{eq-proof-c-def}
  \end{align}
This expression holds for the same value of~$c$ if the indices are rotated by an arbitrary~$k$:
for all~$j$ the index~$i_j$ becomes $(i_j + k) \bmod K$. Further, we have
\begin{align}
  \sum_{\substack{3\le i_3<i_4<K\\\text{$i_3$ odd}\\\text{$i_4$ even}}} \!\!\!\! v_{i_3} v_{i_4}
  \le
  \frac{1}{\alpha}\,.
  \label{eq-d2-nonpos}
\end{align}
Again, this inequality also holds when indices are rotated.
\end{lemma}
For example, for $K=7$ we have
$c =  v_2 + v_4 + v_6 - \alpha ( v_2 v_3 v_4 + v_2 v_3 v_6 + v_2 v_5 v_6 + v_4 v_5 v_6 )
   =  v_1 + v_3 + v_5 - \alpha ( v_1 v_2 v_3 + v_1 v_2 v_5 + v_1 v_4 v_5 + v_3 v_4 v_5 )$.

\begin{proof}[Proof of Lemma~\ref{lem-derivatives}] The idea of the proof is as follows. We pick a particular direction in $D^{(K)}$,
namely $\vec{d}=(-1,0,1,0,0,\ldots,0)$,
and consider the function $f(\vec{v}+\epsilon \vec{d})$ as a univariate function of~$\epsilon$.
Since $\vec{v}$ is a local maximum,
the first derivative must be zero and the second derivative must be nonpositive.
Exploiting the fact that $v_i > 0$ for all~$i$ holds in the interior,
we obtain \eqref{eq-proof-c-def}~and~\eqref{eq-d2-nonpos}, respectively. See \ifthenelse{\equal{\techRep}{true}}{Appendix~\ref{app-derivatives}}{\cite{BGKOW15-techrep}} for the detailed proof.
\end{proof}

Let $S_j^{(K)}(\vec{x})$ denote
  the scalar product of $\vec{x}$ with a copy of itself rotated $j$ times:
\[
  S_j^{(K)}(\vec{x}) := \sum_{i=0}^{K-1} x_i x_{i+j}
\]
In all formulas it will be the case that the subscript of~$S$ is odd.
Also, the superscript will be omitted when unimportant or understood from context.

\newcommand{\stmtcordtwononpossummed}{
Let $\vec{v}$ be a local maximum of $f^{(K)}$ in the interior of~$D^{(K)}$.
Then the following inequality holds:
\[
  \sum_{\substack{1 \le i < K-2\\\text{$i$ odd}}} \frac{K-i-2}{2} S_i(\vec{v}) \quad \le \quad \frac{K}{\alpha}
\]
}
\begin{corollary} \label{cor-d2-nonpos-summed}
\stmtcordtwononpossummed
\end{corollary}
For example, for $K=11$ we have
  $4S_1(\vec{v}) + 3S_3(\vec{v}) + 2S_5(\vec{v}) +S_7(\vec{v}) \le 11/\alpha$.

\begin{lemma}[Bound for $f_5$] \label{lem-bound-on-f5}
Suppose that $\vec{v} \in D^{(K)}$ satisfies $f^{(K)}(\vec{v}) > \frac{1}{27}$.
Then $\alpha f_5(\vec{v}) < \frac{1}{216}$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem-f3-max} we have $f_3(\vec{v}) \le \frac{1}{24}$ and hence
$
 \alpha f_5(\vec{v}) = f_3(\vec{v}) - f(\vec{v}) < \frac{1}{24} - \frac{1}{27} = \frac{1}{216}
$.
\end{proof}

\subsection{Proof of Lemma~\ref{lem-f-max-induction} for $K=5$} \label{sub-proof-5}

Let $K=5$. Then
\begin{align*}
 f(\vec{x}) \ &= \ f_3(\vec{x}) - \alpha f_5(\vec{x})
            \ =  \
x_0 x_1 x_2 +
x_0 x_1 x_4 +
x_0 x_3 x_4 +
x_1 x_2 x_3 +
x_2 x_3 x_4 - \alpha x_0 x_1 x_2 x_3 x_4
\end{align*}
Towards a contradiction, suppose that there is a local maximum $\vec{v}$ with $f(\vec{v}) > \frac{1}{27}$ in the interior of~$D$.
By~\eqref{eq-proof-c-def}, the value
\begin{equation} \label{eq-proof5-c-def}
 c \ = \ v_2 + v_4 - \alpha v_2 v_3 v_4
\end{equation}
is invariant under rotations. Indeed, $ v_{2+k} + v_{4+k} - \alpha v_{2+k} v_{3+k} v_{4+k} \equiv c$ for all~$k$,
  but we shall avoid explicitly mentioning rotations, for notational simplicity.
Summing~\eqref{eq-proof5-c-def} over all $K$~rotations we obtain:
\begin{equation} \label{eq-proof5-five-c}
 5 c \ = \ 2 - \alpha f_3(\vec{v})
\end{equation}
By~\eqref{eq-proof5-c-def} we have
$v_0 v_1 c =
v_0 v_1 v_2 +
v_0 v_1 v_4 - \alpha f_5(\vec{v})$
and, summing over all $K$~rotations,
\begin{equation} \label{eq-proof5-c-sum}
 c S_1(\vec{v}) \ = \ 2 f(\vec{v}) - 3 \alpha f_5(\vec{v})
 \end{equation}
Moreover,
\[
 c S_1(\vec{v}) \ \mathop{\le}^\text{Cor.~\ref{cor-d2-nonpos-summed}} \ \frac{5 c}{\alpha} \ \mathop{=}^\text{\eqref{eq-proof5-five-c}} \
 \frac{2}{\alpha} - f_3(\vec{v})  \ = \
 \frac{2}{\alpha} - f(\vec{v}) - \alpha f_5(\vec{v}).
\]
Combining this with~\eqref{eq-proof5-c-sum} gives:
\[
 \frac{2}{\alpha} \quad \ge \quad 3 f(\vec{v}) - 2 \alpha f_5(\vec{v})
 \quad \mathop{\ge}^\text{Lemma~\ref{lem-bound-on-f5}} \quad \frac{3}{27} - 2 \cdot \frac{1}{216}
\]
This implies $\alpha \le 216/11 \approx 19.6$, which is a contradiction as required (since  $\alpha=24$).
\qed

\subsection{Proof of Lemma~\ref{lem-f-max-induction} for $K=7$} \label{sub-proof-7}

Let $K=7$. Towards a contradiction, we suppose again that there is a local maximum $\vec{v}$ with $f(\vec{v}) > \frac{1}{27}$ in the interior of~$D$.
By~\eqref{eq-proof-c-def}, all $K$~rotations of the following hold with the same~$c\in\mathbb{R}$:
\begin{equation} \label{eq-proof7-c-def}
 c \ = \ v_2 + v_4 + v_6 - \alpha ( v_2 v_3 v_4 + v_2 v_3 v_6 + v_2 v_5 v_6 + v_4 v_5 v_6 )
\end{equation}
Summing~\eqref{eq-proof7-c-def} over $K$~rotations we obtain:
\begin{equation} \label{eq-proof7-five-c}
 7 c \ = \ 3 - 2 \alpha f_3(\vec{v})
\end{equation}
By~\eqref{eq-proof7-c-def} we have
\begin{equation} \label{eq-proof7-mult-c1}
 v_0 v_1 c \;=\;
v_0 v_1 v_2 +
v_0 v_1 v_4 +
v_0 v_1 v_6
 - \alpha (
v_0 v_1 v_2 v_3 v_4 +
v_0 v_1 v_2 v_3 v_6 +
v_0 v_1 v_2 v_5 v_6 +
v_0 v_1 v_4 v_5 v_6
)
\end{equation}
and
\begin{equation} \label{eq-proof7-mult-c2}
\begin{aligned}
v_0 v_3 c
 \;&=\;
v_0 v_3 v_4 +
v_0 v_3 v_6
- \alpha v_0 v_3 v_4 v_5 v_6
+ v_0 v_2 v_3 ( 1 - \alpha ( v_3 v_4 + v_3 v_6 + v_5 v_6 ) ) \\
\;&\ge\;
v_0 v_3 v_4 +
v_0 v_3 v_6
- \alpha v_0 v_3 v_4 v_5 v_6
\end{aligned}
\end{equation}
where the last inequality is by~\eqref{eq-d2-nonpos}.
Summing \eqref{eq-proof7-mult-c1}~and~\eqref{eq-proof7-mult-c2} over $K$~rotations we obtain:
\begin{equation} \label{eq-proof7-c-sum}
 c \bigl(2 S_1(\vec{v}) + S_3(\vec{v})\bigr)
  \ \ge \
  4 f_3(\vec{v}) - 9 \alpha f_5(\vec{v}) = 4 f(\vec{v}) - 5 \alpha f_5(\vec{v})
\end{equation}
Further we have:
\[
 c \bigl(2 S_1(\vec{v}) + S_3(\vec{v})\bigr)
 \ \mathop{\le}^\text{Cor.~\ref{cor-d2-nonpos-summed}} \
 \frac{7 c}{\alpha} \ \mathop{=}^\text{\eqref{eq-proof7-five-c}} \
 \frac{3}{\alpha} - 2 f_3(\vec{v})  \ = \
 \frac{3}{\alpha} - 2 f(\vec{v}) - 2 \alpha f_5(\vec{v})
\]
Combining this with~\eqref{eq-proof7-c-sum} gives:
\[
 \frac{3}{\alpha} \quad \ge \quad 6 f(\vec{v}) - 3 \alpha f_5(\vec{v})
 \quad \mathop{\ge}^\text{Lemma~\ref{lem-bound-on-f5}} \quad \frac{6}{27} - 3 \cdot \frac{1}{216}
\]
This leads to $\alpha \le 14.4$, which is a contradiction as desired.
\qed

\subsection{Combinatorial Lemmas} \label{sub-combinatorics}

In order to generalize the proofs from Sections \ref{sub-proof-5}~and~\ref{sub-proof-7}
 to any odd~$K$, we state some combinatorial lemmas in this subsection.
They are proved in \ifthenelse{\equal{\techRep}{true}}{Appendix~\ref{app-combin-proofs}}{\cite{BGKOW15-techrep}}.

In order to generalize \eqref{eq-proof5-five-c}~and~\eqref{eq-proof7-five-c}
we show the following lemma:

\newcommand{\stmtlemsumcombinsimple}{
We have:
\[
  \sum_{k=0}^{K-1}
  \sum_{\substack{1<i'_0<i'_1<i'_2<K\\\text{$i'_0$, $i'_2$ even}\\\text{$i'_1$ odd}}}
    x_{i'_0+k} x_{i'_1+k} x_{i'_2+k}
  =
  \frac{K-3}{2}
  \hskip-1em
  \sum_{\substack{0\le i_0<i_1<i_2<K\\\text{$i_2-i_1$, $i_1-i_0$ odd}}}
  \hskip-2em
    x_{i_0} x_{i_1} x_{i_2}
  =
  \frac{K-3}{2} f_3^{(K)}(\vec{x})
\]
}
\begin{lemma}\label{lem-sum-combin-simple}
\stmtlemsumcombinsimple
\end{lemma}
For example, if $K=5$, then we obtain that summing the $5$~rotations of
  $x_2 x_3 x_4$
gives $f_3^{(5)}(\vec{x})$.
As another example, if $K=7$, then we obtain that summing the $7$~rotations of
$
  x_2 x_3 x_4
  + x_2 x_3 x_6
  + x_2 x_5 x_6
  + x_4 x_5 x_6
$
gives $2 f_3^{(7)}(\vec{x})$.
These two instances of Lemma~\ref{lem-sum-combin-simple}
  help establish \eqref{eq-proof5-five-c}~and~\eqref{eq-proof7-five-c}.

In order to generalize the inequality in~\eqref{eq-proof7-mult-c2}
we need the following lemma:
\newcommand{\stmtlemdropterms}{
Let $\vec{v}$ be a local maximum of $f^{(K)}$ in the interior of~$D^{(K)}$.
If $i_1$ is odd and $0<i_1<K$, then the following inequality holds:
\[
v_0 v_{i_1}
  \biggl(
    \sum_{\substack{1<i_2<K\\\text{$i_2$ even}}}
      v_{i_2}
    -
    \hskip-1em
    \sum_{\substack{1<i_2<i_3<i_4<K\\\text{$i_2,i_4$ even}\\\text{$i_3$ odd}}}
    \hskip-2em
      \alpha v_{i_2} v_{i_3} v_{i_4}
  \biggr)
  \ge
v_0 v_{i_1}
  \biggl(
    \sum_{\substack{i_1<i_2<K\\\text{$i_2$ even}}}
      v_{i_2}
    -
    \hskip-1em
    \sum_{\substack{i_1<i_2<i_3<i_4<K\\\text{$i_2,i_4$ even}\\\text{$i_3$ odd}}}
    \hskip-2em
      \alpha v_{i_2} v_{i_3} v_{i_4}
  \biggr)
\]
}
\begin{lemma}\label{lem-drop-terms}
\stmtlemdropterms
\end{lemma}
The inequality says that if we drop those terms that do not occur in $f_3^{(K)}$ or $f_5^{(K)}$,
  then we obtain a lower bound.
The proof groups those terms that are not in either of $f_3^{(K)}$ or $f_5^{(K)}$,
  and then invokes~\eqref{eq-d2-nonpos} to show that their sum is nonnegative.

In order to generalize \eqref{eq-proof5-c-sum}~and~\eqref{eq-proof7-c-sum}
we need Corollary~\ref{cor-fancy-sum} below,
which is a consequence of the following lemma:
\newcommand{\stmtlemfancysum}{
Let $l$ be an odd, positive integer.
Then:
\begin{multline*}
  \sum_{k=0}^{K-1}
  \sum_{\substack{1\le i'_1< K-2\\\text{$i'_1$ odd}}}
    \frac{K-i'_1-2}{2}
  \hskip-1em
  \sum_{\substack{%
    i'_1<i'_2<\cdots<i'_{l-1}<K\\
    \forall j,\; i'_j \equiv j \pmod 2
  }}
    x_k x_{i'_1+k}
    \prod_{\substack{1<j<l}}
      x_{i'_j+k}
  =
\\
  =
  \Bigl( \frac{l-1}{2} K - l \Bigr)
  \hskip-1em
  \sum_{\substack{
    0\le i_0<\cdots<i_{l-1}<K\\
    \text{$i_j-i_{j-1}$ odd for $0<j<l$}}}
  \prod_{j=0}^{l-1}
    x_{i_j}
\end{multline*}
}
\begin{lemma}\label{lem-fancy-sum}
\stmtlemfancysum
\end{lemma}
For example, if $K=5$ and $l=3$, then we have that summing $5$~rotations of
$
  x_0 x_1 x_2 + x_0 x_1 x_4
$
gives $2f_3^{(5)}(\vec{x})$.
As another example, if $K=9$ and $l=3$, then summing $9$~rotations of
$
  3x_0x_1(x_2+x_4+x_6+x_8) + 2x_0x_3(x_4+x_6+x_8) + x_0x_5(x_6+x_8)
$
gives $6 f_3^{(9)}(\vec{x})$.

\begin{corollary}\label{cor-fancy-sum}
We have:
\[
  \sum_{k=0}^{K-1}
  \sum_{\substack{1 \le i_1 < K-2\\\text{$i_1$ odd}}}
     \frac{K-i_1-2}{2}
  \sum_{\substack{i_1<i_2<K\\\text{$i_2$ even}}}
    x_{0+k} x_{i_1+k} x_{i_2+k}
  =
  (K-3) f_3^{(K)}(\vec{x})
\]
and also
\[
  \sum_{k=0}^{K-1}
  \sum_{\substack{1 \le i_1 < K-2\\\text{$i_1$ odd}}}
     \frac{K-i_1-2}{2}
  \sum_{\substack{i_1<i_2<i_3<i_4<K\\\text{$i_2,i_4$ even}\\\text{$i_3$ odd}}}
  x_{0+k} x_{i_1+k}
  x_{i_2+k} x_{i_3+k} x_{i_4+k}
  =
  (2K-5) f_5^{(K)}(\vec{x})
\]
\end{corollary}

\begin{proof}
Instantiate Lemma~\ref{lem-fancy-sum} with $l=3$ and, respectively, $l=5$.
\end{proof}

\subsection{Proof of Lemma~\ref{lem-f-max-induction}} \label{sub-proof-f-max-induction}

Towards a contradiction, suppose that there is a local maximum $\vec{v}$ with $f(\vec{v}) > \frac{1}{27}$ in the interior of~$D$, i.e., $v_i > 0$ for all~$i \in \{0, \ldots, K-1\}$.
Summing up the $K$ rotations of~\eqref{eq-proof-c-def}  and using Lemma~\ref{lem-sum-combin-simple}, we obtain:
\begin{equation} \label{eq-proof-Mc}
 K c \ = \ \frac{K-1}{2} - \frac{K-3}{2} \alpha f_3(\vec{v})
\end{equation}
Multiplying \eqref{eq-proof-c-def} on both sides by
$
    \sum_{\substack{1 \le i_1 < K-2\\\text{$i_1$ odd}}} \frac{K-i_1-2}{2} v_0 v_{i_1}
$
we obtain:
\begin{align*}
c  \sum_{\substack{1 \le i_1 < K-2\\\text{$i_1$ odd}}}  \frac{K-i_1-2}{2} v_0 v_{i_1}  &=
  \sum_{\substack{1 \le i_1 < K-2\\\text{$i_1$ odd}}} \frac{K-i_1-2}{2} v_0 v_{i_1}
  \Biggl(
    \sum_{\substack{1<i_2<K\\\text{$i_2$ even}}} v_{i_2}
    -
    \hskip-1em
    \sum_{\substack{1<i_2<i_3<i_4<K\\\text{$i_2$, $i_4$ even}\\\text{$i_3$ odd}}}
    \hskip-1.5em
      \alpha v_{i_2} v_{i_3} v_{i_4}
  \Biggr) \\
  & \ge
  \sum_{\substack{1 \le i_1 < K-2\\\text{$i_1$ odd}}} \frac{K-i_1-2}{2} v_0 v_{i_1}
  \Biggl(
    \sum_{\substack{i_1<i_2<K\\\text{$i_2$ even}}} v_{i_2}
    -
    \hskip-1em
    \sum_{\substack{i_1<i_2<i_3<i_4<K\\\text{$i_2$, $i_4$ even}\\\text{$i_3$ odd}}}
    \hskip-1.5em
      \alpha v_{i_2} v_{i_3} v_{i_4}
  \Biggr)
\end{align*}
using Lemma~\ref{lem-drop-terms}. Summing $K$~rotations of this inequality yields:
\begin{equation} \label{eq-proof-c-sum}
\begin{aligned}
  c \!\!  \sum_{\substack{1 \le i_1 < K-2\\\text{$i_1$ odd}}} \frac{K-i_1-2}{2} S_{i_1}(\vec{v})
  \ &\ge\
  (K-3) f_3(\vec{v}) - (2K-5) \alpha f_5(\vec{v}) = (K-3) f(\vec{v}) - (K-2) \alpha f_5(\vec{v}).
\end{aligned}
\end{equation}
using Corollary~\ref{cor-fancy-sum}. Further we have:
\[
  c  \sum_{\substack{1 \le i_1 < K-2\\\text{$i_1$ odd}}} \frac{K-i_1-2}{2} S_{i_1}(\vec{v})
  \ \mathop{\le}^\text{Cor.~\ref{cor-d2-nonpos-summed}}\
  \frac{K c}{\alpha}
  \ \mathop{=}^\text{\eqref{eq-proof-Mc}}\
  \frac{K-1}{2\alpha}-\frac{K-3}{2} f_3(\vec{v})
\]
Combining this with~\eqref{eq-proof-c-sum} gives:
\[
  \frac{K-1}{2\alpha}
    \ \ge\
  \frac{3K-9}{2} f(\vec{v}) - \frac{K-1}{2} \alpha f_5(\vec{v})
    \ \mathop{\ge}^\text{Lemma~\ref{lem-bound-on-f5}}\
  \frac{K-3}{2}\cdot \frac{1}{9} - \frac{K-1}{2}\cdot \frac{1}{216}
\]
This implies
\[
  \alpha \le \frac{216(K-1)}{23K-71} < 19.7
\]
Since $\alpha=24$, this leads to a contradiction as desired.
\qed

\section{Proof of the Herman Conjecture} \label{sec-overview}

The function~$V_3$ from Lemma~\ref{lem-drop-f3} leaves room for improvement since $\mathbb{E}(V_3(z') \mid z) = V_3(z) - \frac{K-1}{2}$,
which is strictly less than $V_3(z) - 1$ for $K > 3$.
The idea for obtaining an optimal bound is to decrease the gap between $\frac{K-1}{2}$ and~$1$,
by decreasing the Lyapunov function~$V$.
One could think that the scaled function $\frac{2}{K-1} V_3$ is also a Lyapunov function satisfying~\eqref{eq-prop-lyapunov-step}, but this is not true;
in particular, note that the number of tokens~$K$ might be different for a configuration~$z$ and its successor~$z'$.
Since scaling does not work, we decrease the Lyapunov function by subtracting a quintic polynomial, as follows.
Define a quintic polynomial $f_5^{(K)} : D^{(K)} \to [0, \infty)$,
similar to~$f_3^{(K)}$:
\begin{align*}
  f_5^{(K)}(\vec{x}) =
    \hskip-2em
    \sum_{\substack{0\le i_0<i_1<\cdots<i_4<K\\\text{$i_4-i_3,\ldots,i_1-i_0$ odd}}}
    \hskip-2em
      x_{i_0} x_{i_1} x_{i_2} x_{i_3} x_{i_4}
\end{align*}
For instance,
$f_5^{(3)}(\vec{x}) = 0$,
$f_5^{(5)}(\vec{x}) =
x_0 x_1 x_2 x_3 x_4$,
and
$f_5^{(7)}(\vec{x}) =
x_0 x_1 x_2 x_3 x_4 +
x_0 x_1 x_2 x_3 x_6 +
x_0 x_1 x_2 x_5 x_6 +
x_0 x_1 x_4 x_5 x_6 +
x_0 x_3 x_4 x_5 x_6 +
x_1 x_2 x_3 x_4 x_5 +
x_2 x_3 x_4 x_5 x_6$.
We also define a polynomial $f^{(K)} : D^{(K)} \to [0, \infty)$:
\begin{equation} \label{eq-f}
f^{(K)}(\vec{x}) := f_3^{(K)}(\vec{x}) - \alpha f_5^{(K)}(\vec{x}) \qquad
\text{with } \alpha := 24
\end{equation}
For example,
$f^{(5)}(\vec{x}) =
x_0 x_1 x_2 +
x_0 x_1 x_4 +
x_0 x_3 x_4 +
x_1 x_2 x_3 +
x_2 x_3 x_4 - \alpha x_0 x_1 x_2 x_3 x_4$.
Throughout the paper we use $\alpha$ in the expression of $f^{(K)}$ for notational convenience. From now onwards we may drop the superscript $K$  from the domain $D^{(K)}$ of the functions $f_3^{(K)}$, $f_5^{(K)}$ and $f^{(K)}$ to avoid notational clutter when $K$ is understood.

The following properties of~$f$ are fundamental:
\begin{lemma}[Symmetry and continuity properties] \label{lem-f-properties}
The function~$f$ has the following properties.
\begin{itemize}
\item[(a)] It is symmetric with respect to rotation:
\[
f(x_0, \ldots, x_{K-1}) = f(x_1, \ldots, x_{K-1}, x_0)
\]
\item[(b)] It is continuous:
For $K \ge 5$ we have
\[
f^{(K)}(x_0, 0, x_2, x_3, \ldots, x_{K-1}) = f^{(K-2)}(x_0+x_2, x_3, \ldots, x_{K-1}).
\]
\end{itemize}
\end{lemma}
Analogous properties were shown for~$f_3$ in~\cite{FengZhang14}.
Their proof carries over to~$f_5$ and hence to~$f$.
The following lemma uses~$f$ to define a tighter Lyapunov function.
\begin{lemma}[Lyapunov function $V$] \label{lem-drop-f}
Define $V : Z \to [0, \infty)$ by
$V(z) := 4 N^2 f(\vec{g}(z)/N)$.
Let $z \in Z$ and denote by $z'$ the random successor configuration of~$z$.
Then $\mathbb{E}(V(z') \mid z) \le V(z) - 1$.
Hence, by Proposition~\ref{prop-lyapunov}, $\mathbb{E} T_z \le 4 N^2 f(\vec{g}(z)/N)$.
\end{lemma}
We remark that
  a similar Lyapunov function has been investigated in~\cite[Equation~(15)]{FengZhang14},
  but did not lead to a proof of the Herman conjecture.
It seems that $V(z)$~needs to be chosen with great care,
  since even slight variations do not work.

Lemma~\ref{lem-drop-f} suggests analyzing~$f$:
\begin{lemma}[Maximum of $f$] \label{lem-f-max}
For all $K \ge 3$ odd we have
\[
 \max_{\vec{x} \in D} f^{(K)}(\vec{x}) = \frac{1}{27}.
 \]
\end{lemma}
With this in hand our main result follows:
\begin{proof}[Proof of Theorem~\ref{thm-main}]
Immediate by combining Lemmas \ref{lem-drop-f}~and~\ref{lem-f-max}.
\end{proof}
It remains to prove Lemmas~\ref{lem-drop-f}~and~\ref{lem-f-max}.

\subsection{Proof of Lemma~\ref{lem-drop-f}} \label{sub-drop-f}

Towards Lemma~\ref{lem-drop-f} we show:

\begin{lemma}[Lyapunov function $V_5$] \label{lem-incr-f5}
Define $V_5 : Z \to [0, \infty)$ by $V_5(z) := 4 N^2 f_5(\vec{g}(z)/N)$.
Let $K \ge 5$ and $z \in Z$ and denote by $z'$ the random successor configuration of~$z$.
Then
\[
\mathbb{E}(V_5(z') \mid z) = V_5(z) + \frac{1}{32} \frac{(K-1)(K-3)}{N^2} - \frac{1}{2} (K - 3) f_3\left(\frac{\vec{g}(z)}{N}\right)\,.
\]
\end{lemma}
The proof in \ifthenelse{\equal{\techRep}{true}}{Appendix~\ref{sec:incr-f5-proof}}{\cite{BGKOW15-techrep}} requires an analysis of correlations among the changes in gaps between tokens in each step of the protocol.
Using Lemma~\ref{lem-incr-f5} one can readily prove Lemma~\ref{lem-drop-f}:
\begin{proof}[Proof of Lemma~\ref{lem-drop-f}]
For $K=3$ the statement follows from Lemma~\ref{lem-drop-f3}.
For $K \ge 5$ we have:
\begin{align*}
\mathbb{E}(V(z') \mid z)
& = \mathbb{E}( (V_3(z') - 24 V_5(z') ) \mid z) && \text{by the definitions} \\
& = \mathbb{E}(V_3(z') \mid z) - 24 \mathbb{E}(V_5(z') \mid z) && \text{linearity of expectation} \\
& = V_3(z) - \frac{K-1}{2} - 24 V_5(z) - \frac{3}{4} \frac{(K-1)(K-3)}{N^2} \\
& \ \qquad + 12 (K - 3) f_3\left(\frac{\vec{g}(z)}{N}\right) && \text{Lemmas \ref{lem-drop-f3}~and~\ref{lem-incr-f5}} \\
& \le V(z) - \frac{K-1}{2}  + 12 (K - 3) f_3\left(\frac{\vec{g}(z)}{N}\right) && \text{since $K \ge 3$} \\
& \le V(z) - \frac{K-1}{2} + \frac{K-3}{2} && \text{Lemma~\ref{lem-f3-max}} \\
& = V(z) - 1
\end{align*}
\end{proof}

\subsection{Proof of Lemma~\ref{lem-f-max}}  \label{sub-f-max}

Towards Lemma~\ref{lem-f-max} we show:

\begin{lemma}[Local maxima of $f$] \label{lem-f-max-induction}
Let $K \ge 5$ and odd.
There is no $\vec{v} \in D^{(K)}$ in the interior of~$D^{(K)}$
 such that $\vec{v}$ is a local maximum and $f^{(K)}(\vec{v}) > \frac{1}{27}$.
\end{lemma}
The proof in Section~\ref{sec-f-max-induction}
  involves a combinatorial analysis of inequalities arising
  from conditions on the derivatives of~$f^{(K)}$.
Using Lemma~\ref{lem-f-max-induction} one can readily prove Lemma~\ref{lem-f-max}:
\begin{proof}[Proof of Lemma~\ref{lem-f-max}]
We proceed by induction on~$K$.
For the induction base we have $K=3$.
It is straightforward to check that the maximum of
$f^{(3)}(\vec{x}) = f_3^{(3)}(\vec{x}) = x_0 x_1 x_2$
is $f^{(3)}(\frac{1}{3},\frac{1}{3},\frac{1}{3}) = \frac{1}{27}$.

For the induction step we have $K \ge 5$.
Let $\vec{v} \in D^{(K)}$ with $f^{(K)}(\vec{v}) = \max_{\vec{x} \in D^{(K)}} f^{(K)}(\vec{x})$.
If $\vec{v}$ is in the interior of~$D^{(K)}$,
then by Lemma~\ref{lem-f-max-induction} we have $f^{(K)}(\vec{v}) \le \frac{1}{27}$.
If $\vec{v}$ is at the boundary of~$D^{(K)}$,
then $v_i = 0$ for some~$i$.
By Lemma~\ref{lem-f-properties}(a) we can assume that $v_1=0$.
Using Lemma~\ref{lem-f-properties}(b) the statement follows from the induction hypothesis.
\end{proof}

\section{Introduction} \label{sec-prelim}

The notion of \emph{self-stabilization} was introduced in a seminal
paper of Dijkstra~\cite{D74}, and rose to prominence a decade later,
following (among others) an invited talk of Lamport during which he
pointed out that ``self-stabilization [is] a very important concept in
fault tolerance''~\cite{Lam84}. Both self-stabilization and fault
tolerance have since become central themes in distributed computing
(see, e.g.,~\cite{Dolev}), as recently witnessed by the award of the
2015 Edsger W. Dijkstra Prize in Distributed Computing to Michael
Ben-Or and Michael Rabin for ``starting the field of fault-tolerant
randomized distributed algorithms'' in the early 1980s.

In this paper, we examine an early self-stabilization algorithm known
as Herman's Protocol~\cite{Herman90}, whose exact mathematical
analysis has proven remarkably challenging over the two-and-a-half
decades since its inception. This algorithm considers a ring of $N$
processes (or nodes), where each process either holds or doesn't hold
a token. Starting from any initial configuration of $K$ tokens, where
$K$ is required to be odd, Herman's algorithm proceeds as follows: at
each time step, every process that holds a token either keeps it or
passes it to its clockwise neighbor with probability~$1/2$. All
updates happen synchronously, and if a process finds itself with two
tokens (having simultaneously kept one and received one from its
counterclockwise neighbor) then both tokens are annihilated. It is
straightforward to see that, starting from an odd number of tokens and
following this procedure, almost surely only one token eventually
remains, at which point the ring is said to have \emph{stabilized}.

Herman's original paper~\cite{Herman90} presents the algorithm in
a form amenable to implementation. Each process possesses a bit, which the
process can read and write. Each process can also read the bit of its
counterclockwise neighbor.  In this representation, having the same
bit as one's counterclockwise neighbor is interpreted as having a
token. At each time step, each process compares its bit with the bit
of its counterclockwise neighbor; if the bits differ, the process
keeps its bit, whereas if the bits are the same, the process flips its
bit with probability~$1/2$ and keeps it with
probability~$1/2$.
It is straightforward to verify that the bit-flipping version is
an implementation of the token-passing version:
in particular, a process
flipping its bit corresponds to passing its token to its clockwise
neighbor.  If the number of processes is odd, by construction this
bit representation forces the number of tokens to be odd as well,
which justifies the assumption that $K$, the number of tokens, is
always odd. In this paper we make no assumption about the parity of the
number of \emph{processes}, as we abstract from the bit
implementation, and simply assume that the number of tokens is odd
throughout.

Herman's original paper~\cite{Herman90} showed that the expected time
(number of synchronous steps) to stabilization is $O(N^2 \log N)$.
The same paper also mentions an improved upper bound of $O(N^2)$ due
to Dolev, Israeli, and Moran, without giving a proof or a further
reference.  In 2004, Fribourg et al.~\cite{Fribourg05} established an
upper bound of $2N^2$, and the following year Nakata~\cite{Nakata05}
gave a tighter upper bound of $0.936N^2$ and exhibited an initial
configuration with expected stablization time $\Omega(N^2)$.  At the
same time and independently, McIver and Morgan showed
in~\cite{McIverMorgan} that the initial configuration consisting of
three equally-spaced tokens has an expected stabilization time of
exactly $\frac{4}{27}N^2$, and conjectured that this value is an upper
bound on the expected time to stabilization starting from \emph{any}
initial configuration with \emph{any} (odd) number of tokens. The
conjecture is intriguing since increasing the initial number of tokens
might be thought to lengthen the expected time to stabilization, due
to the larger number of collisions required to achieve stabilization.

Nevertheless, McIver and Morgan's Herman-Protocol Conjecture is
supported by considerable amount of experimental
evidence~\cite{PrismHerman}, and in the intervening years a series of
papers have gradually reduced the upper bound on the constant $h$ such
that stabilization from any initial configuration takes expected time
at most $hN^2$: upper bounds of approximately $0.64$, $0.521$,
$0.167$, and $0.156$ are given respectively in
\cite{KMO11,FZ13,FengZhang14,Haslegrave14}, the last one provided last
year by Haslegrave, and coming relatively close to McIver and Morgan's
lower bound of $4/27 \approx 0.148$.

In this paper, we prove McIver and Morgan's conjecture and establish
that $h = 4/27$ is indeed optimal. Writing $T_z$ for the stabilization
time starting from an initial configuration~$z$, we seek to prove that
$\mathbb{E} T_z \leq \frac{4}{27}N^2$. To this end, one of the key ideas is to
work with a Lyapunov function $V(z)$ in lieu of the (more complicated)
function $\mathbb{E} T_z$.
The domain of the function~$V$ is \emph{continuous}:
a domain element describes a configuration in terms of the distances between adjacent tokens.
Combinatorial arguments exploiting the highly
symmetrical structure of $V(z)$ enable us to establish that, for an
arbitrary configuration~$z$, we have $\mathbb{E} T_z \leq V(z)$, with equality
holding for all three-token configurations.
Finally, in what constitutes the most technically challenging part of this paper,
we combine induction on the number of tokens with analytical techniques to show that $V$ is bounded by $\frac{4}{27}N^2$.
Taken together,
we obtain $\mathbb{E} T_z \le \frac{4}{27}N^2$, entailing the Herman-Protocol Conjecture.

The case of there being an \emph{even} number~$K$ of tokens is equally
natural from a mathematical point of view, although it does not
correspond to a concrete bit-flipping protocol.  It was established
in~\cite{FengZhang14} that the worst-case configuration in this
variant is the equidistant \emph{two}-token configuration, with an
expected stabilization time of~$\frac12 N^2$; the analysis underlying
that result is considerably simpler than what is required in case the
number of tokens is odd, as in the present paper.

Herman's protocol is also related to the notion of \emph{coalescing random
  walks}~\cite{Arratia-Coal-and-Anni81,Cox-Coal-89,AF02}.
There, one considers multiple independent random walks on~$\mathbb{Z}^d$ (or on the vertices of a connected graph).
When two walks meet, they coalesce into a new random walk.
A protocol for self-stabilizing mutual exclusion based on such random walks was proposed in~\cite{IsraeliJalfon}.
The expected coalescence time was studied in~\cite{Coppersmith93,Oliveira-12,Cooper-Coalescing}.

It is interesting to note that Herman's ring is closely related to
widely-studied models of random walks and Brownian motion in
statistical physics. Observe that by a simple modification of the
formalism, one may equivalently view Herman's model as a ring
in which tokens randomly move in discrete step in \emph{any}
direction, with pairwise collisions leading to annihilation; this
precisely corresponds to Fisher's \emph{vicious drunks}
model~\cite{Fis84} (with periodic boundary conditions). Similar models
have been studied in chemical physics~\cite{Gen68,Balding88,War91} and
statistical mechanics~\cite{GP06,RS11,SMC13}, among others.

The rest of the paper is organized as follows.  In
Section~\ref{sec-previous} we review previous results in the
literature that are relevant to our proof.  In
Section~\ref{sec-overview} we outline the structure of our proof,
identifying two key lemmas, Lemma~\ref{lem-incr-f5} and
Lemma~\ref{lem-f-max-induction}.  Those are proved in
\ifthenelse{\equal{\techRep}{true}}{Appendix~\ref{sec:incr-f5-proof}}{\cite{BGKOW15-techrep}} and
Section~\ref{sec-f-max-induction}, respectively.

Another solution of the conjecture, using different techniques, is independently shown in~\cite{Endre}.

\section{Relevant Previous Results} \label{sec-previous}

For the rest of the paper we fix the number~$N$ of processes.  We
assume that the number $K$ of tokens is odd, and both $N$ and $K$ are
at least $3$.

Processes are numbered from $1$ to~$N$, clockwise, according to their
position in the ring.  A configuration with $K$ tokens is formalized
as a function $z : \{1, \ldots, K\} \to \{1, \ldots, N\}$ with
$z(1) < \cdots < z(K)$, where the $i$th token
($i \in \{1, \ldots, K\}$) is held by the processor with the
number~$z(i)$.  We write $Z_K$ for the set of configurations with $K$
tokens, and $Z$ for the set of all possible configurations, that is,
$Z = Z_1 \cup Z_3 \cup Z_5 \cup \ldots$

For a fixed initial configuration~$z = z_0$ we write $(z_t)_{t \ge 0}$
for the stochastic process of configurations emanating from~$z$.  The
\emph{stabilization time} $T_z$ is the smallest $t \ge 0$ such that
$z_t\in Z_1$, i.e., the time until only one token is left.
In this paper we focus on the expectation~$\mathbb{E} T_z$.
It is shown in~\cite{McIverMorgan} that if $N$ is odd and a multiple of~$3$,
then there is a configuration~$z \in Z_3$
(with the 3 tokens maximally separated in an equilateral triangle)
such that $\mathbb{E} T_z = \frac{4}{27} N^2$.

In this paper we show:
\begin{theorem} \label{thm-main}
We have $\mathbb{E} T_z \le \frac{4}{27} N^2$ for all $z \in Z$.
\end{theorem}
Equivalently, the Herman conjecture states that for all odd $K\ge3$ and all $z \in Z_K$ we have $\mathbb{E} T_z \le \frac{4}{27} N^2$.
Only the case $K=3$ was previously known \cite{McIverMorgan}.

The following proposition has been used in a similar form in various papers on Herman's protocol, for instance in~\cite[Lemma~5]{McIverMorgan}.
It bounds the stabilization time by a Lyapunov function~$V$.
\begin{proposition}[Bound by a Lyapunov function] \label{prop-lyapunov}
Given $z \in Z$, denote by $z' \in Z$ the random successor configuration of~$z$.
Let $V : Z \to \mathbb{R}$ be a function with
\begin{align}
  \mathbb{E}(V(z') \mid z) \;&\le\; V(z) - 1
  &&\text{for all $z \in Z \setminus Z_1$, and}
  \label{eq-prop-lyapunov-step}
\\
  0 \;&\le\; V(z)
  &&\text{for all $z \in Z_1$.}
  \label{eq-prop-lyapunov-base}
\end{align}
Then $\mathbb{E} T_z \le V(z)$ for all $z \in Z$.
In particular, $V(z) \ge 0$ for all $z \in Z$.
\end{proposition}
Although this result is not new, we give a short proof based on a martingale argument.
The proof is inspired by~\cite{Haslegrave14}, and may provide some intuition.
\begin{proof}
Let $z \in Z$.
Consider the stochastic process $(z_t)_{t \ge 0}$ of configurations emanating from~$z=z_0$. Define $W_t := V(z_t) + t$.
By~\eqref{eq-prop-lyapunov-step} the process $(W_t)_{t \ge 0}$ is a supermartingale.
The stabilization time $T_{z} = T_{z_0}$ is a stopping time with finite expectation,
and the differences $|W_{t+1} - W_t|$ are bounded as the Markov chain reachable from~$z$ has finitely many states.
Hence, the optional stopping theorem applies, yielding
$
 \mathbb{E} W_{T_z} \le \mathbb{E} W_0 = V(z)
$.
By definition of~$W_t$ we have $\mathbb{E} W_{T_z} = \mathbb{E} V(z_{T_z}) + \mathbb{E} T_z$.
Since $z_{T_z}\in Z_1$,
  we have $\mathbb{E} T_z \le \mathbb{E} W_{T_z}$ by~\eqref{eq-prop-lyapunov-base}.
By combining the previous two inequalities, we obtain $\mathbb{E} T_z \le V(z)$.
\end{proof}

Following~\cite{FengZhang14,Haslegrave14} we associate with a configuration $z \in Z_K$
the \emph{gap vector} $\vec{g}(z) = (g_0, \ldots, g_{K-1}) \in \mathbb{N}^K$ by setting
$g_0 := N+z(1)-z(K)$,
and $g_i := z(i+1) - z(i)$ for $i \in \{1, \ldots, K-1\}$.
Then $\vec{g}(z)/N$ lives in the so-called standard $(K-1)$-simplex $D^{(K)}$, defined by
\[
 D^{(K)} := \left \{\vec{x} = (x_0, \ldots, x_{K-1}) \in [0,1]^K \ \mid \ x_0 + \cdots + x_{K-1} = 1\right \}.
\]
Towards a suitable Lyapunov function~$V$ we define the cubic polynomial $f_3^{(K)} : D^{(K)} \to [0, \infty)$ by
\[
  f_3^{(K)}(\vec{x}) :=
    \hskip-2em
    \sum_{\substack{0\le i_0<i_1<i_2<K\\\text{$i_2-i_1$, $i_1-i_0$ odd}}}
    \hskip-2em
       x_{i_0} x_{i_1} x_{i_2}.
\]
For instance, we have $f_3^{(5)}(\vec{x}) =
x_0 x_1 x_2 +
x_0 x_1 x_4 +
x_0 x_3 x_4 +
x_1 x_2 x_3 +
x_2 x_3 x_4$.

The following lemma was implicitly proved in previous works:
\begin{lemma}[%
  Lyapunov function $V_3$
  {\cite[Page 240, Proof of Theorem~1]{FengZhang14} and \cite[Theorem 4]{Haslegrave14}}%
]\label{lem-drop-f3}
Let $V_3 : Z \to [0, \infty)$ be defined by $V_3(z) := 4 N^2 f_3^{(K)}(\vec{g}(z)/N)$ for $z \in Z_K$.
Denote by $z' \in Z_1 \cup Z_3 \cup \ldots \cup Z_K$
the random successor configuration of~$z \in Z_K$.
Then $\mathbb{E}(V_3(z') \mid z) = V_3(z) - \frac{K-1}{2}$ for all $z \in Z_K$.
Hence, by Proposition~\ref{prop-lyapunov}, $\mathbb{E} T_z \le 4 N^2 f_3^{(K)}(\vec{g}(z)/N)$.
\end{lemma}
For $K=3$ Lemma~\ref{lem-drop-f3} gives
$\mathbb{E} T_z \le 4 N^2 f_3^{(K)}(\vec{g}(z)/N) = \frac{4}{N} g_0 g_1 g_2$.
In fact, for $K=3$ it was shown before in~\cite{McIverMorgan} that $\mathbb{E} T_z$ is identically equal to $\frac{4}{N} g_0 g_1 g_2$,
providing an exact formula for the expected stabilization time of configurations with three tokens.
Lemma~\ref{lem-drop-f3} suggests analyzing~$f_3$:
\begin{lemma}[Maximum of $f_3$ {\cite[Proof of Theorem~2]{FengZhang14}, \cite[Theorem 3]{Haslegrave14}}] \label{lem-f3-max}
For all $K \ge 3$ odd we have
\[
 \max_{\vec{x} \in D} f_3^{(K)}(\vec{x})
 \ = \ f_3^{(K)}\left(\frac{1}{K}, \ldots, \frac{1}{K}\right)
 \ = \ \frac{1}{24} \left(1 - \frac{1}{K^2} \right)\,.
\]
\end{lemma}
By combining Lemmas \ref{lem-drop-f3}~and~\ref{lem-f3-max} one obtains
$\mathbb{E} T_z \le \frac{N^2}{6}(1 - \frac{1}{K^2})$,
which is the bound obtained in~\cite{FengZhang14}.
A slightly better bound is given in~\cite{Haslegrave14}.

redpajama_set_name: RedPajamaArXiv


text:
Q: How do i group a table on a client and retreive the first admit in current year without having a null in prior discharge dates? I have a table that contains client information with Episodes of treatment, admissions and discharges for each separate episode of treatment.
Client    Episode    Admission_date    Discharge_date
   x         1         05-06-2020           Null
   x         2         09-26-2020        12-30-2020
   x         3         01-03-2021        In Process
   y         1         03-02-2020        05-20-2020
   y         2         08-13-2020        11-23-2020
   y         3         02-01-2021        03-31-2021
   y         4         04-02-2021        In Process

I want to retrieve clients Y Episode 3(just retrieve the client no need episode) because its the first admission in year 2021 1st Qtr and their previous discharge date is less than(has to be less) Episode 3's admission date. But i dont want to retreive x even though their current admission date is less than previous discharge date because their first episode has a null in discharge which means they are still active. Need inactive clients only.
Would appreciate any help. Thank you
Desired result:
Client    Episode    Admission_date    Discharge_date
y         3         02-01-2021        03-31-2021

redpajama_set_name: RedPajamaStackExchange


I saw this post in a book club meeting, and thought this is a great way to format reviews.
Use our general fiction questions when you can't find specific discussion questions. They're basic but smart.
2. Describe the main characters—personality traits, motivations, and inner qualities.
3. Are the main characters dynamic—changing or maturing by the end of the book? Do they learn about themselves, how the world works and their role in it?
• Is it engaging—do you find the story interesting?
• Is this a plot-driven book—a fast-paced page-turner?
• Does the plot unfold slowly with a focus on character?
• Were you surprised by complications, twists & turns?
• Did you find the plot predictable, even formulaic?
5. Talk about the book's structure.
• Is it a continuous story…or interlocking short stories?
• Does the time-line move forward chronologically?
• Does time shift back & forth from past to present?
• Is there a single viewpoint or shifting viewpoints?
the way he or she did?
you read or understand the book?


text:
, un village et une paroisse civile du Somerset en Angleterre.
 Lovington, un village du comté de Moultrie dans l'Illinois, aux États-Unis.
 Lovington, une ville du comté de Lea au Nouveau-Mexique, aux États-Unis.

Voir également
 Lovington Township dans l'Illinois.

redpajama_set_name: RedPajamaWikipedia


The DC Half-Torso Erratic Kicker is a professionally built steel-framed armature for use in an upside down hanging body/torso animation. The legs include separate cylinders allowing this body frame to produce individual leg movements that are extremely life like and extremely erratic like no other kicker on the market.
Includes 2 cylinders, 1 valve, all necessary fittings. Requires an air compressor with a steady 60-120PSI rating. Also includes a heavy duty chain with removable linkage, and ceiling mounting plate. A realtime recording controller such as the Animation maestro 1 will allow you to get extremely life/deathlike movement as well as be triggered via a infrared sensor. Check out our Controllers section for more information.
This armature also includes an ankle bar, that allows for the same movement as our standard kicker.