Oriented Matroids / Edition 2

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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.
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Editorial Reviews

From the Publisher
"Numerous exercises are included at the end of each chapter; thus the book is quite good literature for the student...the book contains a long list of references covering all essential papers published in oriented matroid theory thus far, as well as a few unpublished notes. Therefore, anybody who wants to study oriented matroids will not only find a compact survey concerning the theory but also a standard list of references." Bulletin of the AMS

"...comprehensive and authoritative....Without a doubt this book will become an indispensable tool for anyone working in an important and growing area of mathematics." SIAM Review

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Product Details

Table of Contents

Preface
Notation
1 A First Orientation Session 1
1.1 Oriented matroids from directed graphs 1
1.2 Point configurations and hyperplane arrangements 5
1.3 Pseudoline arrangements 14
1.4 Topological Representation Theorem 17
1.5 Realizability 20
1.6 Combinatorial convexity 24
1.7 Linear programming 26
1.8 Computational geometry 29
1.9 Chirality in molecular chemistry 33
1.10 Allowable sequences 35
1.11 Slope problems 39
2 A Second Orientation Session 46
2.1 Real hyperplane arrangements 46
2.2 Zonotopes 50
2.3 Reflection arrangements 65
2.4 Stratification of the Grassmann variety 77
2.5 Complexified arrangements 92
3 Axiomatics 100
3.2 Circuits 103
3.3 Minors 110
3.4 Duality 115
3.5 Basis orientations and chirotopes 123
3.6 Modular elimination and local realizability 135
3.7 Vectors and covectors 141
3.8 Maximal vectors and topes 146
3.9 Historical sketch 150
4 From Face Lattices to Topology 157
4.1 The big face lattice 158
4.2 Topes I 169
4.3 Shellability and sphericity 175
4.4 Topes II 181
4.5 The affine face lattice 186
4.6 Enumeration of cells 193
4.7 Appendix: Regular cell complexes, posets and shellability 200
5 Topological Models for Oriented Matroids 225
5.1 Arrangements of pseudospheres 225
5.2 The Topological Representation Theorem 232
5.3 Pseudoconfigurations of points 236
6 Arrangements of Pseudolines 247
6.1 Arrangements of pseudospheres in low dimensions 247
6.2 Arrangements of pseudolines 250
6.3 How far can things be stretched? 259
6.4 Allowable sequences, wiring diagrams and homotopy 264
6.5 Three enumerative questions 269
6.6 Orientable matroids of rank 3 272
7 Constructions 281
7.1 Single element extensions 281
7.2 Lexicographic extensions and the extension lattice 291
7.3 Local perturbations and mutations 296
7.4 Many oriented matroids 305
7.5 Intersection properties and adjoints 308
7.6 Direct sum and union 312
7.7 Strong maps and weak maps 318
7.8 Inseparability graphs 324
7.9 Orientability 329
8 Realizability 338
8.1 The realization space of an oriented matroid 338
8.2 Constructions and realizability results 342
8.3 The impossibility of a finite excluded minor characterization 348
8.4 Algorithms and complexity results 353
8.5 Final polynomials and the real Nullstellensatz 358
8.6 The isotopy problem and Mnev's universality theorem 363
8.7 Oriented matroids and robust computational geometry 369
9 Convex Polytopes 376
9.1 Introduction to matroid polytopes 376
9.2 Convexity results and constructions 381
9.3 The Lawrence construction and its applications 386
9.4 Cyclic and neighborly matroid polytopes 395
9.5 The Steinitz problem and its relatives 403
9.6 Polyhedral subdivisions and triangulations 408
10 Linear Programming 417
10.1 Affine oriented matroids and linear programs 419
10.2 Pivot steps and tableaux 433
10.3 Pivot rules 451
10.4 Examples 461
10.5 Euclidean matroids 472
Bibliography 480
Index 510
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