Finite Voronoi decompositions of infinite vertex transitive graphs

In this paper, we consider the Voronoi decompositions of an arbitrary infinite vertex-transitive graph G. In particular, we are interested in the following question: what is the largest number of Voronoi cells that must be infinite, given sufficiently (but finitely) many Voronoi sites which are sufficiently far from each other? We call this number the survival number s(G). The survival number of a graph has an alternative characterization in terms of covering, which we use to show that s(G) is always at least two. The survival number is not a quasi-isometry invariant, but it remains open whether finiteness of the s(G) is. We show that all vertex transitive graphs with polynomial growth have a finite s(G); vertex transitive graphs with infinitely many ends have an infinite s(G); the lamplighter graph LL(Z), which has exponential growth, has a finite s(G); and the lamplighter graph LL(Z^2), which is Liouville, has an infinite s(G)

Letter from Dr. Walter D. Finucane, Former Chapter President of Phi Delta Kappa, to Geraldine Ferraro

Letter and handwritten note from Dr. Walter D. Finucane, former president of the St. John\u27s Chapter of Phi Delta Kappa, to Geraldine Ferraro. References Ferraro visiting Dr. Finucane\u27s chapter in 1979 and giving a presentation titled, Violence and Discipline in the Public Schools. Contains copy of a pamphlet published by Dr. Finucane\u27s chapter for Ferraro\u27s presentation. Contains data entry sheet.https://ir.lawnet.fordham.edu/vice_presidential_campaign_correspondence_1984_new_york/1020/thumbnail.jp

Algebraically recurrent random walks on groups

Initial steps are presented towards understanding which finitely generated groups are almost surely generated as semigroups by the path of a random walk on the group.Comment: 9 page

A recursive construction of t-wise uniform permutations

We present a recursive construction of a (2t + 1)-wise uniform set of permutations on 2n objects using a (2t + 1) - (2n, n, \cdot) combinatorial design, a t-wise uniform set of permutations on n objects and a (2t+1)-wise uniform set of permutations on n objects. Using the complete design in this procedure gives a t-wise uniform set of permutations on n objects whose size is at most t^2n, the first non-trivial construction of an infinite family of t-wise uniform sets for t \geq 4. If a non-trivial design with suitable parameters is found, it will imply a corresponding improvement in the construction

U.S. Recognition of a Commander’s Duty to Punish War Crimes

This article explores the United States\u27 recognition of the doctrine of command responsibility. The doctrine has been invoked by those alleging that President Trump’s pardons of U.S. personnel convicted or accused of war crimes could amount to war crimes themselves. The article focuses on a commander’s duty to punish war crimes by his subordinates. It examines the United States’ past recognition of the duty to punish as an element of command responsibility under the law of war. The principle that a commander has an obligation to punish war crimes by his subordinates is not a progressive development of the law promoted by the advocacy community. Instead, the duty to punish stands out as an ancient legal norm interwoven into the domestic law of the United States and which the United States has incorporated into international legal instruments. The lesson from this history is clear, if not always appreciated: commanders who fail to punish their subordinates for war crimes may themselves be war criminals

Mistaken Identity and Its Effect on Contractual Validity: Some Cases Frm the English Courts

The doctrine of mistake has grown increasingly complicated with the passage of time. Nowhere is that complexity more obvious than in the area of unilateral mistake, of which the distinguishing feature is that the mistake made by one party is known to, and almost invariably induced by the fraud of, the other

Partners and Legal Pitfalls

Partnered military operations are an increasingly prominent feature of armed conflict and one which presents a distinct set of legal challenges to States assisting partners. This is particularly true of the war in Syria which is characterized both by States working with and through other States and non-State actors and by the widespread violation of the law of armed conflict (LOAC) by many of the parties. This article considers the legal implications of LOAC violations by a party to the conflict for the State or States providing it assistance and identifies risk mitigation measures that assisting States can adopt

Semiparametric Bayesian Density Estimation with Disparate Data Sources: A Meta-Analysis of Global Childhood Undernutrition

Undernutrition, resulting in restricted growth, and quantified here using height-for-age z-scores, is an important contributor to childhood morbidity and mortality. Since all levels of mild, moderate and severe undernutrition are of clinical and public health importance, it is of interest to estimate the shape of the z-scores' distributions. We present a finite normal mixture model that uses data on 4.3 million children to make annual country-specific estimates of these distributions for under-5-year-old children in the world's 141 low- and middle-income countries between 1985 and 2011. We incorporate both individual-level data when available, as well as aggregated summary statistics from studies whose individual-level data could not be obtained. We place a hierarchical Bayesian probit stick-breaking model on the mixture weights. The model allows for nonlinear changes in time, and it borrows strength in time, in covariates, and within and across regional country clusters to make estimates where data are uncertain, sparse, or missing. This work addresses three important problems that often arise in the fields of public health surveillance and global health monitoring. First, data are always incomplete. Second, different data sources commonly use different reporting metrics. Last, distributions, and especially their tails, are often of substantive interest.Comment: 41 total pages, 6 figures, 1 tabl