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.
[13] M. Grant and S. Boyd, “Graph implementations for nonsmooth convex programs,” in Recent Advances in Learning and Control (V. Blondel, S. Boyd, and H. Kimura, eds.), Lecture Notes in Control and Information Sciences, pp. 95–110, Springer-Verlag Limited, 2008. http://stanford.edu/~boyd/graph_dcp.html.
[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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