### 博碩士論文 107221022 詳細資訊

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(A Study On Kissing Number Problem)

 ★ 歐式空間二距離集合之探討

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This question seems simple. However, for hundreds of years, there are still few known answers. In 1694, there was a famous dispute occurred between Newton and Gregory. Newton believed that 12 was the correct answer to the kissing number in dimension 3. However, Gregory thought it was 13. In the end, Schütte and van der Waerden gave us the first proof of the kissing number in dimension 3, N = 12, in 1953.

During these hundreds of years, mathematicians had developed many ways to approximate the answer. There are some famous methods that are well known. For example, Odlyzko and Sloane′s linear programming solved the kissing number problem in dimension 8 and 24. Moreover, Musin′s extension of the linear programming solved the kissing number problem in dimension 4.

This study presents the linear programming method, and semidefinite programming method. We combine these methods to obtain a way to reproduce the result of the classical kissing number problem by a different method. We also offer the formula of four points semidefinite programming method to be the reference for latecomers.

Abstract II
Contents III
1 Introduction 1
2 Linear programming bounds 4
2.1 Notation and definition 4
2.2 Linear programming bounds 4
3 Semide nite programming bounds (SDP bounds) 6
3.1 Notation and de nition 6
3.2 SDP bounds 6
3.3 Computation 8
3.4 Coding 8
3.4.1 Linear programming 8
3.4.2 SDP 9
4 Outlook 10
4.1 Notation and de nition 10
4.2 Four points semide nite programming 11
4.3 Prospect 11
Appendix 13
List of Figures
1 The perfect kissing arrangement for n = 1 [15] 1
2 The perfect kissing arrangement for n = 2 [15] 1
3 The perfect kissing arrangement for n = 3[15] 2
List of Tables
1 List of kissing number[15] and the stars mean the exact answers 3

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