Geometric Discrepancy : An Illustrated Guide

An Illustrated Guide

By: Jiri Matousek (Editor)

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Hardcover | 1 June 1999

At a Glance

Hardcover
306 Pages

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24.77 x 16.51 x 1.91

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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.).

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"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)

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You Can Find This Book In

Non-Fiction Mathematics Geometry Number Theory Medicine Medicine in General Education Combinatorics & Graph Theory Discrete Mathematics

Preface	p. v
Notation	p. xi
Introduction	p. 1
Discrepancy for Rectangles and Uniform Distribution	p. 1
Geometric Discrepancy in a More General Setting	p. 9
Combinatorial Discrepancy	p. 16
On Applications and Connections	p. 22
Low-Discrepancy Sets for Axis-Parallel Boxes	p. 37
Sets with Good Worst-Case Discrepancy	p. 38
Sets with Good Average Discrepancy	p. 44
More Constructions: b-ary Nets	p. 51
Scrambled Nets and Their Average Discrepancy	p. 61
More Constructions: Lattice Sets	p. 72
Upper Bounds in the Lebesgue-Measure Setting	p. 83
Circular Discs: a Probabilistic Construction	p. 84
A Surprise for the L₁-Discrepancy for Halfplanes	p. 93
Combinatorial Discrepancy	p. 101
Basic Upper Bounds for General Set Systems	p. 101
Matrices, Lower Bounds, and Eigenvalues	p. 105
Linear Discrepancy and More Lower Bounds	p. 109
On Set Systems with Very Small Discrepancy	p. 117
The Partial Coloring Method	p. 120
The Entropy Method	p. 128
VC-Dimension and Discrepancy	p. 137
Discrepancy and Shatter Functions	p. 137
Set Systems of Bounded VC-Dimension	p. 145
Packing Lemma	p. 155
Matchings with Low Crossing Number	p. 159
Primal Shatter Function and Partial Colorings	p. 164
Lower Bounds	p. 171
Axis-Parallel Rectangles: L₂-Discrepancy	p. 172
Axis-Parallel Rectangles: the Tight Bound	p. 176
A Reduction: Squares from Rectangles	p. 180
Halfplanes: Combinatorial Discrepancy	p. 182
Combinatorial Discrepancy for Halfplanes Revisited	p. 193
Halfplanes: the Lebesgue-Measure Discrepancy	p. 197
A Glimpse of Positive Definite Functions	p. 203
More Lower Bounds and the Fourier Transform	p. 213
Arbitrarily Rotated Squares	p. 213
Axis-Parallel Cubes	p. 230
An Excursion to Euclidean Ramsey Theory	p. 234
Tables of Selected Discrepancy Bounds	p. 241
Bibliography	p. 245
Index	p. 265
Hints	p. 275
Table of Contents provided by Publisher. All Rights Reserved.

Geometric Discrepancy : An Illustrated Guide

An Illustrated Guide

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

You Can Find This Book In

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