An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

Revisiting Synthesis for One-Counter Automata

We study the (parameter) synthesis problem for one-counter automata with parameters. One-counter automata are obtained by extending classical finite-state automata with a counter whose value can range over non-negative integers and be tested for zero. The updates and tests applicable to the counter can further be made parametric by introducing a set of integer-valued variables called parameters. The synthesis problem for such automata asks whether there exists a valuation of the parameters such that all infinite runs of the automaton satisfy some omega-regular property. Lechner showed that (the complement of) the problem can be encoded in a restricted one-alternation fragment of Presburger arithmetic with divisibility. In this work (i) we argue that said fragment, called AERPADPLUS, is unfortunately undecidable. Nevertheless, by a careful re-encoding of the problem into a decidable restriction of AERPADPLUS, (ii) we prove that the synthesis problem is decidable in general and in N2EXP for several fixed omega-regular properties. Finally, (iii) we give a polynomial-space algorithm for the special case of the problem where parameters can only be used in tests, and not updates, of the counter

What future for gender equality policy in the UK after Brexit?

The evolution of gender equality policies in the UK prior to, and since, entering the Common Market in the 1970s illustrates the distinctive characteristics of the UK employment system. This can be understood as a movement between historically embedded voluntarism and periods of statutory compliance. European influence on British policies has been filtered through the interests and interaction of four key sets of actors: governments, the legal system, employers and unions. An historical analysis of gender equality policies suggests that the potential future consequences of Brexit are likely to be patterns of continuity, change and unintended consequences. Continuity indicates that existing regulations will persist, until they are challenged. Change will see a removal of appeal to EU-level adjudication. Some unintended consequences may emerge related to the impact of migration patterns and the behaviour of large-scale companies working in the UK which could have unexpected positive outcomes. The analysis in this paper suggests that this will remain a contested terrain

Revisiting the Relationship between Income Distribution and Economic Growth

Most models exploring the exploring the relationship between income inequality and economic growth postulate the existence of a ngative correlation between the two operating through diverse channels. Parallel to the theoretetical models a number of empirical studies have attempted to appraise this relationship. A broad consensus had built validating the existence of such negative correlation until some panel data studies tended to show the opposite result. The review of the ensuing debate as well as the empirical estimation undertook in this paper show that estimates may not be as robust as believed. It is suggested that country case studies may be a way to more appropriately explore this issue.********************************************************************************************************La mayoría de los modelos que exploran la relación entre la desigualdad en la distribución del ingreso y el crecimiento económico, postulan la existencia de una correlación negativa entre las dos que es generada a través de diferentes mecanismos. Paralelamente a los modelos teóricos, un número importante de estudios empíricos han tratado de evaluar esta relación. De este esfuerzo ha surgido un consenso amplio que valida la existencia de dicha relación negativa. No obstante, estudios recientes basados en el uso de datos de panel han producido el resultado contrario, documentando la presencia de una relación positiva entre desigualdad y crecimiento. El examen del debate generado a partir de estos resultados, así como el trabajo empírico adelantado en este estudio, indican que las estimaciones obtenidas en diversos trabajos pueden no ser tan robustas como se creía. En consecuancia, se sugiere que la realización de estudios de caso por país puede ser una mejor vía para explorar este tema.Income Distribution, Economic Growth, Panel Data Econometrics, Multiregional

John Moore Jr.: \u3ci\u3eMoore v. City of East Cleveland\u3c/i\u3e and Children’s Constitutional Arguments

This Article is divided into three parts. First, I retell the story of Moore from John Jr.’s perspective and frame his potential claims. Second, I explore constitutional arguments under existing doctrine, using contemporary equal protection and substantive due process analyses. Finally, I suggest how a children’s rights perspective might be even more persuasive as a strategy for John Jr. as well as for achieving opportunity and equality on behalf of contemporary children living amid and affected by structural inequalities that impact their developmental capacity

Baker v. Nelson: Flotsam in the Tidal Wave of Windsor\u27s Wake

Part I of this Article sketches the virtually unbroken string of pro-marriage decisions in the lower federal and state courts since the U.S. Supreme Court’s 2013 ruling in United States v. Windsor to give a sense of the size and magnitude of this “tidal wave” of precedent. Next, Part II briefly explores some of the reasons that might help account for the flood of litigation and overwhelmingly positive outcomes. Part III tentatively suggests one way this flow of decisions in favor of marriage equality might influence the Supreme Court as it returns to the issue. Part II then at some length shows one particular aspect of Windsor’s wake: the way it has helped lower federal courts unanimously and properly conclude that doctrinal developments after the Supreme Court summarily rejected a same-sex couple’s constitutional claims to a right to marry in Baker v. Nelson in 1972 have rendered that decision no longer dispositive. Although Baker would in no event prevent the Supreme Court itself from revisiting the constitutional issues, the ability to declare Baker doctrinally undermined has positive repercussions for the social equality and lived reality of same-sex couples across the country in the meantime. Finally, Part IV of the Article addresses some of the ways in which United States v. Windsor itself developed constitutional doctrine in ways that advance the cause of constitutional justice and same-sex couples’ rights to equal protection and to marry

Baker v. Nelson: Flotsam in the Tidal Wave of Windsor\u27s Wake

Part I of this Article sketches the virtually unbroken string of pro-marriage decisions in the lower federal and state courts since the U.S. Supreme Court’s 2013 ruling in United States v. Windsor to give a sense of the size and magnitude of this “tidal wave” of precedent. Next, Part II briefly explores some of the reasons that might help account for the flood of litigation and overwhelmingly positive outcomes. Part III tentatively suggests one way this flow of decisions in favor of marriage equality might influence the Supreme Court as it returns to the issue. Part II then at some length shows one particular aspect of Windsor’s wake: the way it has helped lower federal courts unanimously and properly conclude that doctrinal developments after the Supreme Court summarily rejected a same-sex couple’s constitutional claims to a right to marry in Baker v. Nelson in 1972 have rendered that decision no longer dispositive. Although Baker would in no event prevent the Supreme Court itself from revisiting the constitutional issues, the ability to declare Baker doctrinally undermined has positive repercussions for the social equality and lived reality of same-sex couples across the country in the meantime. Finally, Part IV of the Article addresses some of the ways in which United States v. Windsor itself developed constitutional doctrine in ways that advance the cause of constitutional justice and same-sex couples’ rights to equal protection and to marry