A Study On Kissing Number Problem

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陳佑銘(You-Ming Chen) 查詢紙本館藏

畢業系所

數學系

論文名稱

(A Study On Kissing Number Problem)

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摘要(中)

Kissing number problem 吻球數問題(又稱為牛頓數)是問說，在n維歐氏空間中可以同時接觸中心單位球體的非重疊單位球體的最大數量N? 到目前為止，僅當n = 1; 2; 3; 4; 8和24時才知道其確切的值。
這個問題陳說起來很簡單，但是經過數百年，已知的答案仍然很少。1694年，Newton與Gregory之間對3維的吻球數發生了一場著名的爭論。Newton認為12是3維時吻球數的正確答案,而Gregory認為答案是13。最後，在關於3維吻球數的結尾是由Schutte和van der Waerden這兩位數學家在1953年給了我們第一個完整證明，即當維度n = 3時吻球數N = 12。
在這數百年中，數學家們開發了許多方法來近似吻球數的答案。其中有一些方法是眾所周知的。例如，Odlyzko和Sloane使用了線性規劃解決了8維和24的吻球數問題。此外，Musin則是對線性規劃做了一些拓展，從而解決了4維的吻球數問題。
本研究介紹了線性規劃，半正定規劃。我們將這些方法結合起來，以獲得通過另一種方法重現經典吻球數問題的結果。我們也嘗試用四點半正定規劃來計算，然而並未得到新結果，但我們仍提供數學的基本架構給後來的人參考。

摘要(英)

The kissing number problem asks for the maximum number N of pairwise non-overlapping unit spheres that can simultaneously touch a central unit sphere in n-dimensional Euclidean space. The value is only known when n = 1, 2, 3, 4, 8, and 24.

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.

關鍵字(中)

★ 吻球數

關鍵字(英)

論文目次

Chinese Abstract I
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 Semidenite programming bounds (SDP bounds) 6
3.1 Notation and denition 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 denition 10
4.2 Four points semidenite 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

參考文獻

[1] Christine Bachoc and Frank Vallentin. New upper bounds for kissing numbers from semidefinite programming. Journal of the American Mathematical Society,21(3):909–924, 2008.
[2] Christine Bachoc and Frank Vallentin. Optimality and uniqueness of the (4, 10,1/6) spherical code. Journal of Combinatorial Theory, Series A, 116(1):195–204,2009.
[3] Bill Casselman. The difficulties of kissing in three dimensions. Notices of the AMS,51(8):884–885, 2004.
[4] John Horton Conway and Neil James Alexander Sloane. Sphere packings, lattices and groups, volume 290. Springer Science & Business Media, 2013.
[5] Philippe Delsarte, Jean-Marie Goethals, and Johan Jacob Seidel. Spherical codes and designs. In Geometry and Combinatorics, pages 68–93. Elsevier, 1991.
[6] Vladimir Iosifovich Levenshtein. On bounds for packings in n-dimensional euclidean space. In Doklady Akademii Nauk, volume 245, pages 1299–1303. Russian Academy of Sciences, 1979.
[7] Hans D Mittelmann and Frank Vallentin. High-accuracy semidefinite programming bounds for kissing numbers. Experimental Mathematics, 19(2):175–179, 2010.
[8] Oleg R Musin. Multivariate positive definite functions on spheres.arXiv preprintmath/0701083, 2007.
[9] Oleg R Musin. The kissing number in four dimensions. Annals of Mathematics, pages 1–32, 2008.
[10] Andrew M Odlyzko and Neil JA Sloane. New bounds on the number of units pheres that can touch a unit sphere in n dimensions. Journal of Combinatorial Theory, Series A, 26(2):210–214, 1979.
[11] Pablo A Parrilo. Semidefinite programming relaxations for semialgebraic problems. Mathematical programming, 96(2):293–320, 2003.
[12] Florian Pfender and G ̈unter M Ziegler. Kissing numbers, sphere packings, and some unexpected proofs. 2004.
[13] Mihai Putinar. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, 42(3):969–984, 1993.
[14] Kurt Sch ̈utte and Bartel Leendert van der Waerden. Das problem der dreizehnkugeln.Mathematische Annalen, 125(1):325–334, 1952.
[15] Wikipedia.Kissingnumber—Wikipedia, the freee ncyclopedia.http://en.wikipedia.org/w/index.php?title=Kissing%20numberoldid=959780572,2020. [Online; accessed 16-June-2020].

指導教授

俞韋亘(Wei-Hsuan Yu)

審核日期

2020-7-24

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