Oriented Matroids / Edition 2

Oriented Matroids / Edition 2

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by Anders Bjorner, Michel Las Vergnas, Bernd Sturmfels, Neil White
     
 

Oriented matroids are a very natural mathematical concept which presents itself in many different guises, and which has connections and applications to many different areas. These include discrete and computational geometry, combinatorics, convexity, topology, algebraic geometry, operations research, computer science and theoretical chemistry. This is the first… See more details below

Overview

Oriented matroids are a very natural mathematical concept which presents itself in many different guises, and which has connections and applications to many different areas. These include discrete and computational geometry, combinatorics, convexity, topology, algebraic geometry, operations research, computer science and theoretical chemistry. This is the first comprehensive or accessible account of the subject. This book is intended for a diverse audience: graduate students who wish to learn the subject from scratch, researchers in the various fields of application who want to concentrate on certain aspects of the theory, specialists who need a thorough reference work, and others at points in between. A list of exercises and open problems ends each chapter, and the work is rounded off by an up-to-date and exhaustive reference list.

Product Details

ISBN-13:
9780521777506
Publisher:
Cambridge University Press
Publication date:
11/28/1999
Series:
Encyclopedia of Mathematics and its Applications Series, #46
Edition description:
Revised Edition
Pages:
564
Product dimensions:
6.14(w) x 9.21(h) x 1.14(d)

Table of Contents

Preface
Notation
1A First Orientation Session1
1.1Oriented matroids from directed graphs1
1.2Point configurations and hyperplane arrangements5
1.3Pseudoline arrangements14
1.4Topological Representation Theorem17
1.5Realizability20
1.6Combinatorial convexity24
1.7Linear programming26
1.8Computational geometry29
1.9Chirality in molecular chemistry33
1.10Allowable sequences35
1.11Slope problems39
2A Second Orientation Session46
2.1Real hyperplane arrangements46
2.2Zonotopes50
2.3Reflection arrangements65
2.4Stratification of the Grassmann variety77
2.5Complexified arrangements92
3Axiomatics100
3.2Circuits103
3.3Minors110
3.4Duality115
3.5Basis orientations and chirotopes123
3.6Modular elimination and local realizability135
3.7Vectors and covectors141
3.8Maximal vectors and topes146
3.9Historical sketch150
4From Face Lattices to Topology157
4.1The big face lattice158
4.2Topes I169
4.3Shellability and sphericity175
4.4Topes II181
4.5The affine face lattice186
4.6Enumeration of cells193
4.7Appendix: Regular cell complexes, posets and shellability200
5Topological Models for Oriented Matroids225
5.1Arrangements of pseudospheres225
5.2The Topological Representation Theorem232
5.3Pseudoconfigurations of points236
6Arrangements of Pseudolines247
6.1Arrangements of pseudospheres in low dimensions247
6.2Arrangements of pseudolines250
6.3How far can things be stretched?259
6.4Allowable sequences, wiring diagrams and homotopy264
6.5Three enumerative questions269
6.6Orientable matroids of rank 3272
7Constructions281
7.1Single element extensions281
7.2Lexicographic extensions and the extension lattice291
7.3Local perturbations and mutations296
7.4Many oriented matroids305
7.5Intersection properties and adjoints308
7.6Direct sum and union312
7.7Strong maps and weak maps318
7.8Inseparability graphs324
7.9Orientability329
8Realizability338
8.1The realization space of an oriented matroid338
8.2Constructions and realizability results342
8.3The impossibility of a finite excluded minor characterization348
8.4Algorithms and complexity results353
8.5Final polynomials and the real Nullstellensatz358
8.6The isotopy problem and Mnev's universality theorem363
8.7Oriented matroids and robust computational geometry369
9Convex Polytopes376
9.1Introduction to matroid polytopes376
9.2Convexity results and constructions381
9.3The Lawrence construction and its applications386
9.4Cyclic and neighborly matroid polytopes395
9.5The Steinitz problem and its relatives403
9.6Polyhedral subdivisions and triangulations408
10Linear Programming417
10.1Affine oriented matroids and linear programs419
10.2Pivot steps and tableaux433
10.3Pivot rules451
10.4Examples461
10.5Euclidean matroids472
Bibliography480
Index510

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