Titel: Geometric Discrepancy
Autor/en: Jiri Matousek
An Illustrated Guide.
Herausgegeben von Jiri Matousek
15. Dezember 2009 - kartoniert - XII
Discrepancy theory is also called the theory of irregularities of distribution. Here are some typical questions: What is the "most uniform" way of dis tributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? For a precise formulation of these questions, we must quantify the irregularity of a given distribution, and discrepancy is a numerical parameter of a point set serving this purpose. Such questions were first tackled in the thirties, with a motivation com ing from number theory. A more or less satisfactory solution of the basic discrepancy problem in the plane was completed in the late sixties, and the analogous higher-dimensional problem is far from solved even today. In the meantime, discrepancy theory blossomed into a field of remarkable breadth and diversity. There are subfields closely connected to the original number theoretic roots of discrepancy theory, areas related to Ramsey theory and to hypergraphs, and also results supporting eminently practical methods and algorithms for numerical integration and similar tasks. The applications in clude financial calculations, computer graphics, and computational physics, just to name a few. This book is an introductory textbook on discrepancy theory. It should be accessible to early graduate students of mathematics or theoretical computer science. At the same time, about half of the book consists of material that up until now was only available in original research papers or in various surveys.
1. Introduction 1.1 Discrepancy for Rectangles and Uniform Distribution 1.2 Geometric Discrepancy in a More General Setting 1.3 Combinatorial Discrepancy 1.4 On Applications and Connections 2. Low-Discrepancy Sets for Axis-Parallel Boxes 2.1 Sets with Good Worst-Case Discrepancy 2.2 Sets with Good Average Discrepancy 2.3 More Constructions: b-ary Nets 2.4 Scrambled Nets and Their Average Discrepancy 2.5 More Constructions: Lattice Sets 3. Upper Bounds in the Lebesgue-Measure Setting 3.1 Circular Discs: a Probabilistic Construction 3.2 A Surprise for the L 1-Discrepancy for Halfplanes 4. Combinatorial Discrepancy 4.1 Basic Upper Bounds for General Set Systems 4.2 Matrices, Lower Bounds, and Eigenvalues 4.3 Linear Discrepancy and More Lower Bounds 4.4 On Set Systems with Very Small Discrepancy 4.5 The Partial Coloring Method 4.6 The Entropy Method 5. VC-Dimension and Discrepancy 5.1 Discrepancy and Shatter Functions 5.2 Set Systems of Bounded VC-Dimension 5.3 Packing Lemma 5.4 Matchings with Low Crossing Number 5.5 Primal Shatter Function and Partial Colorings 6. Lower Bounds 6.1 Axis-Parallel Rectangles: L 2-Discrepancy 6.2 Axis-Parallel Rectangles: the Tight Bound 6.3 A Reduction: Squares from Rectangles 6.4 Halfplanes: the Combinatorial Discrepancy 6.5 Combinatorial Discrepancy for Halfplanes Revisited 6.6 Halfplanes: the Lebesgue-Measure Discrepancy 6.7 A Glimpse of Positive Definite Functions 7. More Lower Bounds and the Fourier Transform 7.1 Arbitrarily Rotated Squares 7.2 Axis-Parallel Cubes 7.3 An Excursion to Euclidean Ramsey Theory A. Tables of Selected Discrepancy Bounds Bibliography Index Hints
From the reviews:
"The book gives a very useful introduction to geometric discrepancy theory. The style is quite informal and lively which makes the book easily readable." (Robert F. Tichy, Zentralblatt MATH, Vol. 1197, 2010)