歐式空間二距離集合之探討

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林軒宇(Xuan-Yu Lin) 查詢紙本館藏

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數學系

論文名稱

歐式空間二距離集合之探討
(A study of two-distance set in Euclidean space)

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摘要(中)

在此論文中，首先先介紹二距離集合的定義，以及相關的文獻探討。之後介紹二個計算最大二距離集合個數的方法，線性規劃和半正定規劃。以及列出在 3 和 4 維中，若固定兩個內積值，去找此集合上界為整數的構造，與計算這些構造的最小能量是否為現在找出來的最小的解。然後用線性規劃證明當內積值為 −1、0，最大二距離集合的上限為 2n。最後列出 3 維中，特殊角的構造和 3 維二距離集合個數為 5 個點和 6 個點的所有構造。

摘要(英)

In this thesis, we first introduce the definition of the two-distance set and the related literature discussion. Second, two methods for calculating the maximum two-distance set
are introduced, graph representation, linear programming and semidefinite programming method. Third, we try to find the structures if the upper bound of such two-distance set are integer, and check whether it is an energy minimization configuration. Fourth, when the inner product values are −1 and 0, using linear programming to prove that, the maximum two-distance set is 2n. Finally, the constructions of two-distance set of the special angles and the cardinality of two-distance set with 5 points and 6 points are listed in 3-dimensions.

關鍵字(中)

★ 二距離集合
★ 球面二距離集合

關鍵字(英)

★ Two-distance set
★ Spherical two-distance set

論文目次

摘要 iv
Abstract v
誌謝 vi
目錄 vii
圖目錄 ix
表目錄 x
使用符號與定義 xi
一、緒論 1
二、二距離集合與球面二距離集合 2
2.1 定義與定理 2
2.2 最小能量 4
三、實驗方法介紹 6
3.1 線性規劃方法 6
3.2 半正定規劃方法 10
3.3 圖表示法 11
3.4 問題敘述和實作流程 11
3.4.1 固定兩個內積值的最大球面二距離集合 11
3.4.2 在 R3 中是 5 和 6 個點的構造 14
四、結果與討論 17
4.1 固定兩個內積值的最大球面二距離集合 17
4.2 內積值為 −1 和 0 時，每維最大都是 2n 18
4.3 特殊角度上界構造 19
4.4 三維中 5 個點和 6 個點所有的圖 19
4.4.1 三維中 6 個點所有的圖 19
參考文獻 22
附錄 A 特殊角度的構造圖 24
附錄 B 三維中 5 個點的圖 29
附錄 C 線性規劃程式碼 32

參考文獻

參考文獻
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[9] A. Barg and W.-H. Yu, “New bounds for spherical two-distance sets,” Experimental Mathematics, vol. 22, no. 2, pp. 187–194, 2013.
[10] W.-H. Yu, “New bounds for equiangular lines and spherical two-distance sets,” SIAM Journal on Discrete Mathematics, vol. 31, no. 2, pp. 908–917, 2017.
[11] E. J. King and X. Tang, “New upper bounds for equiangular lines by pillar decomposition,” SIAM Journal on Discrete Mathematics, vol. 33, no. 4, pp. 2479–2508, 2019.
[12] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.1.” http://cvxr.com/cvx, Mar. 2014.
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[14] A. Glazyrin and W.-H. Yu, “Upper bounds for s-distance sets and equiangular lines,” Advances in Mathematics, vol. 330, pp. 810–833, 2018.
[15] E. Bannai and E. Bannai, “A survey on spherical designs and algebraic combinatorics on spheres,” European Journal of Combinatorics, vol. 30, no. 6, pp. 1392–1425, 2009.
[16] N. J. A. Sloane, “Neil j. a. sloane: Home page. http://NeilSloane.com, 6 2012.

指導教授

俞韋亘(Wei-Hsuan Yu)

審核日期

2020-10-20

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