Bijection

Abstract: This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the ¢minimal structures. The definition of this class and the corresponding class of theories, the strongly dLminlmal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizations of 41-minimal ordered groups and rings. Several other simple results are collected in §3. The primary tool in the analysis of ¢;minimal structures is a strong analogue of ii forking symmetry," given by Theorem 4.2. This result states that any (parametri- cally) definable unary function in an Yminimal structure is piecewise either constant or an ore er-preserving or reversing bijection of intervals. The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all btO-categorical Aminimal structures (Theorem 6.1). 1. Introduction. The class of linearly ordered structures has long been an im- portant subject of concern to model theorists. Impressive results have been obtained in the study of models of several particular theories that extend the theory of linear order. Among those theories that have been approached successfully are Peano arithmetic, the theory of ordered abelian groups, that of real-closed fields, and that of linear order itself. Yet, very little has been done in the way of developing a general model theory for ordered structures. In this paper, we develop the model theory for a class of linearly ordered structures that we isolate by demanding that a structure in this class satisfy a condition whose effect is that the linear ordering and the algebraic part of the structure behave quite well with respect to one another. Let L be a finitary first-order language, and w an L-structure. A set of n-tuples A c X" is said to be parametrically definable if there is some L-formula

Abstract: The notions of convexity and convex polytopes are introduced in the setting of tropical geometry Combinatorial types of tropical polytopes are shown to be in bijection with regular triangulations of products of two simplices Applications to phylogenetic trees are discussed Theorem 29 and Corollary 30 in the paper, relating tropical polytopes to injective hulls, are incorrect See the erratum at this http URL

Abstract: is bijective for i < n - 1 and surjective for i = n - 1. Several proofs of this theorem are to be found in the literature (see [5] for an account of the problem). Recently Thom has given a proof (unpublished) which, as far as we know, is the first to use Morse's theory of critical points. We present in ? 3, in a slightly more general setting, an alternate proof inspired by Thom's discovery. Our statement is given in the equivalent language of cohomology. The proof is derived from a theorem on Stein manifolds which is presented in ? 2. Some standard properties of the distance function which we require are assembled in ? 1 for the sake of completeness.

Abstract: We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of $[n]$, as well as over all matchings on $[2n]$. As a corollary, the number of $k$-noncrossing partitions is equal to the number of $k$-nonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no $k$-crossing (or with no $k$-nesting).