离散几何中的研究问题 Research Problems in Discrete Geometry 美 Peter Brass William Moser

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资源信息： 中文名: 离散几何中的研究问题 原名: Research Problems in Discrete Geometry 作者: (美)Peter Brass William Moser Janos Pach 资源格式: PDF 版本: 扫描版 出版社: Springer New York 书号: 1441920161 发行时间: 2010年 地区: 美国 语言: 英文 概述:

内容简介: 离散几何有着150余年的丰富历史，提出了甚至高中生都能理解的诸多公开问题。某些问题异常困难，并和数学其他领域的一些深层问题密切相关。然而，许多问题，甚至某些年代久远的问题，都可能被聪明的大学本科生或者高中生运用精妙构思和数学奥林匹克竞赛中的某些技巧所解决。 《离散几何中的研究问题》是由Leo Moser牵头，花费25年著成，书中包括500余个颇具吸引力的公开问题，理解其中许多问题并不需要太多的准备知识。书中的各章很大程度上内容自含，概述了离散几何，介绍了各个问题的历史细节及最重要的相关结果。.. 本书可作为参考书，供致力数学研究，热爱美妙数学问题并不遗余力地试图加以解决的那些专业数学家和研究生查阅。 本书的显著特色包括： 500多个公开问题，其中某些问题的历史久远，而某些问题为新近提出且从未出版； 　　 每章分为内容自含的各个部分，各部分均附有详实的参考文献； 　　 为寻找论文课题的研究生提供众多研究问题； 　　 包含离散几何的一个全面综述，突出介绍离散几何研究的前沿问题和发展前景； 　　 150多幅图表； 　　 Paul Erdos生前为本书早期版本所写的序言。 内容截图:

目录: 0. Definitions and Notations . 1. Density Problems for Packings and Coverings 1.1 Basic Questions and Definitions 1.2 The Least Economical Convex Sets for Packing 1.3 The Least Economical Convex Sets for Covering 1.4 How Economical Are the Lattice Arrangements? 1.5 Packing with Semidisks, and the Role of Symmetry 1.6 Packing Equal Circles into Squares, Circles, Spheres 1.7 Packing Equal Circles or Squares in a Strip 1.8 The Densest Packing of Spheres 1.9 The Densest Packings of Specific Convex Bodies 1.10 Linking Packing and Covering Densities 1.11 Sausage Problems and Catastrophes 2. Structural Packing and Covering Problems 2.1 Decomposition of Multiple Packings and Coverings 2.2 Solid and Saturated Packings and Reduced Coverings 2.3 Stable Packings and Coverings 2.4 Kissing and Neighborly Convex Bodies 2.5 Thin Packings with Many Neighbors 2.6 Permeability and Blocking Light Rays 3. Packing and Covering with Homothetic Copies 3.1 Potato Bag Problems 3.2 Covering a Convex Body with Its Homothetic Copies 3.3 Levi-Hadwiger Covering Problem and Illumination 3.4 Covering a Ball by Slabs 3.5 Point Trapping and Impassable Lattice Arrangements 4. Tiling Problems 4.1 Tiling the Plane with Congruent Regions 4.2 Aperiodic Tilings and Tilings with Fivefold Symmetry 4.3 Tiling Space with Polytopes 5. Distance Problems 5.1 The Maximum Number of Unit Distances in the Plane 5.2 The Number of Equal Distances in Other Spaces 5.3 The Minimum Number of Distinct Distances in the Plane 5.4 The Number of Distinct Distances in Other Spaces 5.5 Repeated Distances in Point Sets in General Position 5.6 Repeated Distances in Point Sets in Convex Position 5.7 Frequent Small Distances and Touching Pairs 5.8 Frequent Large Distances 5.9 Chromatic Number of Unit-Distance Graphs 5.10 Further Problems on Repeated Distances .. 5.11 Integral or Rational Distances 6. Problems on Repeated Subconfigurations 6.1 Repeated Simplices and Other Patterns 6.2 Repeated Directions, Angles, Areas 6.3 Euclidean Ramsey Problems 7. Incidence and Arrangement Problems 7.1 The Maximum Number of Incidences 7.2 Sylvester-Gallai-Type Problems 7.3 Line Arrangements Spanned by a Point Set 8. Problems on Points in General Position 8.1 Structure of the Space of Order Types 8.2 Convex Polygons and the Erdos-Szekeres Problem 8.3 Halving Lines and Related Problems 8.4 Extremal Number of Special Subconfigurations 8.5 Other Problems on Points in General Position 9. Graph Drawings and Geometric Graphs 9.1 Graph Drawings 9.2 Drawing Planar Graphs 9.3 The Crossing Number 9.4 Other Crossing Numbers 9.5 From Thrackles to Forbidden Geometric Subgraphs 9.6 Further Turan-Type Problems 9.7 Ramsey-Type Problems 9.8 Geometric Hypergraphs 10. Lattice Point Problems 10.1 Packing'La. ice Points in Suhspaces 10.2 Covering Lattice Points by Subspaces 10.3 Sets of Lattice Points Avoiding Other Regularities 10.4 Visibility Problems for Lattice Points 11. Geometric Inequalities 11.1 Isoperimetric Inequalities for Polygons and Polytopes 11.2 Heilbronn-Type Problems 11.3 Circumscribed and Inscribed Convex Sets 11.4 Universal Covers 11.5 Approximation Problems 12. Index 12.1 Author Index 12.2 Subject Index