Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications. 1.

One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fully-expansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof ...

Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call “tree entropy”, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasi-transitive amenable graphs, extending a result of Burton and Pemantle (1993). §1. Introduction. Methods of enumeration of spanning trees in a finite graph G and relations to various areas of mathematics and physics have been investigated for more than 150 years. The number of spanning trees is often called the complexity of the graph, denoted here by τ(G). The best known formula for the complexity, proved in every basic text on graph

Abstract. A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poincaré type. Special cases treated, with an explicit evaluation of limiting variances, are generalized Wigner and Wishart matrices. 1.

For fX i g i1 a sequence of i.i.d. random variables taking values in a Polish space \Sigma with distribution ¯, we obtain large and moderate deviation principles for the processes fn \Gamma1 P [nt] i=1 ffi X i ; t 0g n1 and fn \Gamma1=2 P [nt] i=1 (ffi X i \Gamma ¯); t 0g n1 , respectively. Given a class of bounded functions F on \Sigma, we then consider the above processes as taking values in the Banach space of bounded functionals over F and obtain the corresponding (uniform over F ), large and moderate deviation principles. Among the corollaries considered are functional laws of the iterated logarithm. 1 Introduction. We recall that, given a sequence fX i g i1 of random variables taking values in a measure space and a class of bounded measurable functions F , one may view the empirical measure L n = n \Gamma1 P n i=1 ffi X i as a bounded functional over F ; that is, as an element of l 1 (F ). Viewed Partially supported by NSF DMS92-09712 and DMS-9403553 grants, and by a...

Abstract. We show that there is a spectral gap for the transfer operator associated to a rational map f on the Riemann sphere. Using this and the method of pertubed operators we establish the (Local) Central Limit Theorem for the measure of maximal entropy of f, with an estimate on the speed of convergence. 1.

Based on Stein’s method, we derive upper bounds for Poisson process approximation in the L1-Wasserstein metric d (p) 2, which is based on a slightly adapted Lp-Wasserstein metric between point measures. For the case p = 1, this construction yields the metric d2 introduced in [Barbour, A. D. and Brown, T. C. (1992), Stochastic Process. Appl. 43(1), pp. 9–31], for which Poisson process approximation is well studied in the literature. We demonstrate the usefulness of the extension to general p by showing that d (p) 2-bounds control differences between expectations of certain p-th order average statistics of point processes. 1

Abstract. This material was used for a course on stochastic analysis at UW–Madison in fall 2003. The text covers the development of the stochastic integral of predictable processes with respect to cadlag semimartingale integrators, Itô’s formula in an open domain in R n, and an existence and uniqueness theorem for an equation of the type dX = dH + F (t, X) dY where Y is a cadlag semimartingale. The text is self-contained except for certain basics of integration theory and probability theory which are explained but not proved. In addition, the reader needs to accept without proof two basic martingale theorems: (i) the existence of quadratic variation for a cadlag local martingale; and (ii) the so-called fundamental theorem of local martingales that states the following: given a cadlag local martingale M and a positive constant c, M can be decomposed as N + A where N and A are cadlag local martingales, jumps of N are bounded by c, and A has paths of bounded variation.

Contents 1 The wellorder library 1

Abstract. Given a density function f on an compact subset of R d, we look at the problem of finding the best approximation of f by discrete measures ν = P ciδxi in the sense of the p-Wasserstein distance, subject to size constraints of the form P h(ci) ≤ α where h is a given weight function. This is an important problem with applications in economic planning of locations, in information theory and in shape optimization problems. The efficiency of the approximation can be measured by studying the rate at which the minimal distance tends to zero as α tends to infinity. In this paper, we introduce a rescaled distance which depends on a small parameter and establish a representation formula for its limit as a function of the local statistics for the distribution of the ci’s. The asymptotic problem for large α can be then treated in the case of quite general entropy functions h. Keywords: Monge-Kantorovich transportation, Wasserstein distance, entropy, optimal partitions, rate of approximation, asymptotics, Γ-convergence, functionals on measures, integral representation. 1.