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| 研究生： |
林軒宇 Xuan-Yu Lin |
|---|---|
| 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 |
| 指導教授： |
俞韋亘
Wei-Hsuan Yu |
| 口試委員: | |
| 學位類別： |
碩士 Master |
| 系所名稱： |
理學院 - 數學系 Department of Mathematics |
| 論文出版年： | 2020 |
| 畢業學年度： | 109 |
| 語文別： | 中文 |
| 論文頁數： | 46 |
| 中文關鍵詞： | 二距離集合 、球面二距離集合 |
| 外文關鍵詞： | Two-distance set, Spherical two-distance set |
| 相關次數： | 點閱：15 下載：0 |
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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。
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.
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.
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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.
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