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Abstract: Large random combinatorial structures tend to exhibit great statistical regularity. The objects considered (typically, words, trees, graphs, or permutations) are given by construction rules of the kind classically studied by combinatorial analysts via generating functions. A fundamental problem is then to extract asymptotic information on coefficients of a generating function either explicitly given by a formula or implicity determined by a functional equation. This talk reviews an approach largely developed by Flajolet-Odlyzko and called "Singularity Analysis" that bases itself on properties of generating functions in the complex domain, especially analytic continuation and singularities. As a consequence, numerous asymptotic counting results regarding classical structures can be obtained in a simple way; it becomes possible to discuss abstract combinatorial schemas, while probabilistic laws associated to many characteristics of large random combinatorial structures can be estimated within a unified conceptual framework. [Cf also the book "Analytic Combinatorics" by Flajolet-Sedgewick freely available on the WWW].
Colloquium - 16:00
Abstract: Performing local modifications to hexahedral mesh topology is difficult because these modifications tend to propagate throughout the mesh in order to satisfy conformity constraints between the hex elements. Previous efforts have described either ad-hoc groups of modification operations, or operation sets based on some theory but which are subsets of the operations described in the current work. In this presentation, a set of operations is described which is small (consisting of only three operations), local (each operation modifies connectivity of only a small set of contiguous hexahedra), and, most importantly, atomic (none of the operations can be reproduced using a combination of other atomic operations). The application of this operation set to improvement of hexahedral mesh quality will also be discussed. The presentation will conclude by relating this work to the work of Schwartz & Ziegler on dual manifolds of odd cubical 4-polytopes.