A counterexample to the Hirsch conjecture

The Hirsch Conjecture (1957) stated that the graph of a $d$-dimensional polytope with $n$ facets cannot have (combinatorial) diameter greater than $n-d$. That is, that any two vertices of the polytope can be connected by a path of at most $n-d$ edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope with 48 facets which violates a certain generalization of the $d$-step conjecture of Klee and Walkup.Comment: 28 pages, 10 Figures: Changes from v2: Minor edits suggested by referees. This version has been accepted in the Annals of Mathematic

Normal Form for the Schr\"odinger equation with analytic non--linearities

In this paper we discuss a class of normal forms of the completely resonant non--linear Schr\"odinger equation on a torus. We stress the geometric and combinatorial constructions arising from this study. Further analytic considerations and applications to quasi--periodic solutions will appear in a forthcoming article. This paper replaces a previous version correcting some mistakes.Comment: 52 pages, 2 figure

Real equiangular lines and related codes

We consider real equiangular lines and related codes. The driving question is to find the maximum number of equiangular lines in a given dimension. In the real case, this is controlled by combinatorial phenomena, and until only very recently, the exact number has been unknown. The complex case appears to be driven by other phenomena, and configurations are conjectured always to meet the absolute bound of d^2 lines in dimension d. We consider a variety of the techniques that have been used to approach the problem, both for constructing large sets of equiangular lines, and for finding tighter upper bounds. Many of the best-known upper bounds for codes are instances of a general linear programming bound, which we discuss in detail. At various points throughout the thesis, we note applications in quantum information theory

View generated database

This document represents the final report for the View Generated Database (VGD) project, NAS7-1066. It documents the work done on the project up to the point at which all project work was terminated due to lack of project funds. The VGD was to provide the capability to accurately represent any real-world object or scene as a computer model. Such models include both an accurate spatial/geometric representation of surfaces of the object or scene, as well as any surface detail present on the object. Applications of such models are numerous, including acquisition and maintenance of work models for tele-autonomous systems, generation of accurate 3-D geometric/photometric models for various 3-D vision systems, and graphical models for realistic rendering of 3-D scenes via computer graphics

A survey of random processes with reinforcement

The models surveyed include generalized P\'{o}lya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.Comment: Published at http://dx.doi.org/10.1214/07-PS094 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

COMPUTATIONAL APPROACHES TO UNDERSTAND PHENOTYPIC STRUCTURE AND CONSTITUTIVE MECHANICS RELATIONSHIPS OF SINGLE CELLS

The goal of this work is to better understand the relationship between the structure and function of biological cells by simulating their nonlinear mechanical behavior under static and dynamic loading using image structure-based finite element modeling (FEM). Vascular smooth muscle cells (VSMCs) are chosen for this study due to the strong correlation of the geometric arrangement of their structural components on their mechanical behavior and the implications of that behavior on diseases such as atherosclerosis. VSMCs are modeled here using a linear elastic material model together with truss elements, which simulate the cytoskeletal fiber network that provides the cells with much of their internal structural support. Geometric characterization of single VSMCs of two physiologically relevant phenotypes in 2D cell culture is achieved using confocal microscopy in conjunction with novel image processing techniques. These computer vision techniques use image segmentation, 2D frequency analysis, and linear programming approaches to create representative 3D model structures consisting of the cell nucleus, cytoplasm, and actin stress fiber network of each cell. These structures are then imported into MSC Patran for structural analysis with Marc. Mechanical characterization is achieved using atomic force microscopy (AFM) indentation. Material properties for each VSMC model are input based on values individually obtained through experimentation, and the results of each model are compared against those experimental values. This study is believed to be a significant step towards the viability of finite element models in the field of cellular mechanics because the geometries of the cells in the model are based on confocal microscopy images of actual cells and thus, the results of the model can be compared against experimental data for those same cells