Geometric Discrepancy: An Illustrated Guide / Edition 1

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What is the "most uniform" way of distributing n points in the unit sq uare? How big is the "irregularity" necessarily present in any such di stribution? 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 p arts, it also provides a comprehensive guide to recent research. Inclu ding 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 m athematics and computer science. Besides professional mathematicians, it will be of interest to specialists in fields where a large collecti on of objects should be "uniformly" represented by a smaller sample (s uch as high-dimensional numerical integration in computational physics or financial mathematics, efficient divide-and-conquer algorithms in computer science, etc.).

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Editorial Reviews

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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)
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Product Details

  • ISBN-13: 9783540655282
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 6/22/1999
  • Series: Algorithms and Combinatorics Series, #18
  • Edition description: 1999
  • Edition number: 1
  • Pages: 289
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.75 (d)

Table of Contents

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

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