中文名: 離散幾何中的研究問題
原名: Research Problems in Discrete Geometry
作者: (美)Peter Brass William Moser Janos Pach
資源格式: PDF
版本: 掃描版
出版社: Springer New York
書號: 1441920161
發行時間: 2010年
地區: 美國
語言: 英文
簡介:

內容簡介:
離散幾何有著150余年的豐富歷史，提出了甚至高中生都能理解的諸多公開問題。某些問題異常困難，並和數學其他領域的一些深層問題密切相關。然而，許多問題，甚至某些年代久遠的問題，都可能被聰明的大學本科生或者高中生運用精妙構思和數學奧林匹克競賽中的某些技巧所解決。
《離散幾何中的研究問題》是由Leo Moser牽頭，花費25年著成，書中包括500余個頗具吸引力的公開問題，理解其中許多問題並不需要太多的准備知識。書中的各章很大程度上內容自含，概述了離散幾何，介紹了各個問題的歷史細節及最重要的相關結果。..
本書可作為參考書，供致力數學研究，熱愛美妙數學問題並不遺余力地試圖加以解決的那些專業數學家和研究生查閱。
本書的顯著特色包括：
500多個公開問題，其中某些問題的歷史久遠，而某些問題為新近提出且從未出版；
　　
每章分為內容自含的各個部分，各部分均附有詳實的參考文獻；
　　
為尋找論文課題的研究生提供眾多研究問題；
　　
包含離散幾何的一個全面綜述，突出介紹離散幾何研究的前沿問題和發展前景；
　　
150多幅圖表；
　　
Paul Erdos生前為本書早期版本所寫的序言。
內容截圖:

目錄:
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