A Study On Kissing Number Problem

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题名:	A Study On Kissing Number Problem
作者:	陳佑銘;Chen, You-Ming
贡献者:	數學系
关键词:	吻球數
日期:	2020-07-24
上传时间:	2020-09-02 18:47:03 (UTC+8)
出版者:	國立中央大學
摘要:	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。;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.
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