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Geometric Discrepancy : An Illustrated Guide :  An Illustrated Guide - Jiri Matousek

Geometric Discrepancy : An Illustrated Guide

An Illustrated Guide

Hardcover Published: June 1999
ISBN: 9783540655282
Number Of Pages: 289

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What is the "most uniform" way of distributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? Such questions are treated in geometric discrepancy theory. The book is an accessible and lively introduction to this area, with numerous exercises and illustrations. In separate, more specialized parts, it also provides a comprehensive guide to recent research. Including a wide variety of mathematical techniques (from harmonic analysis, combinatorics, algebra etc.) in action on non-trivial examples, the book is suitable for a "special topic" course for early graduates in mathematics and computer science. Besides professional mathematicians, it will be of interest to specialists in fields where a large collection of objects should be "uniformly" represented by a smaller sample (such as high-dimensional numerical integration in computational physics or financial mathematics, efficient divide-and-conquer algorithms in computer science, etc.).

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)

Prefacep. v
Notationp. xi
Introductionp. 1
Discrepancy for Rectangles and Uniform Distributionp. 1
Geometric Discrepancy in a More General Settingp. 9
Combinatorial Discrepancyp. 16
On Applications and Connectionsp. 22
Low-Discrepancy Sets for Axis-Parallel Boxesp. 37
Sets with Good Worst-Case Discrepancyp. 38
Sets with Good Average Discrepancyp. 44
More Constructions: b-ary Netsp. 51
Scrambled Nets and Their Average Discrepancyp. 61
More Constructions: Lattice Setsp. 72
Upper Bounds in the Lebesgue-Measure Settingp. 83
Circular Discs: a Probabilistic Constructionp. 84
A Surprise for the L1-Discrepancy for Halfplanesp. 93
Combinatorial Discrepancyp. 101
Basic Upper Bounds for General Set Systemsp. 101
Matrices, Lower Bounds, and Eigenvaluesp. 105
Linear Discrepancy and More Lower Boundsp. 109
On Set Systems with Very Small Discrepancyp. 117
The Partial Coloring Methodp. 120
The Entropy Methodp. 128
VC-Dimension and Discrepancyp. 137
Discrepancy and Shatter Functionsp. 137
Set Systems of Bounded VC-Dimensionp. 145
Packing Lemmap. 155
Matchings with Low Crossing Numberp. 159
Primal Shatter Function and Partial Coloringsp. 164
Lower Boundsp. 171
Axis-Parallel Rectangles: L2-Discrepancyp. 172
Axis-Parallel Rectangles: the Tight Boundp. 176
A Reduction: Squares from Rectanglesp. 180
Halfplanes: Combinatorial Discrepancyp. 182
Combinatorial Discrepancy for Halfplanes Revisitedp. 193
Halfplanes: the Lebesgue-Measure Discrepancyp. 197
A Glimpse of Positive Definite Functionsp. 203
More Lower Bounds and the Fourier Transformp. 213
Arbitrarily Rotated Squaresp. 213
Axis-Parallel Cubesp. 230
An Excursion to Euclidean Ramsey Theoryp. 234
Tables of Selected Discrepancy Boundsp. 241
Bibliographyp. 245
Indexp. 265
Hintsp. 275
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540655282
ISBN-10: 354065528X
Series: Algorithms and Combinatorics
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 289
Published: June 1999
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 0.6

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