多項式方法於等角直線叢上的半正定規劃上界

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葉政叡(Cheng-Jui Yeh) 查詢紙本館藏

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論文名稱

多項式方法於等角直線叢上的半正定規劃上界
(Polynomial Method in Semidefinite Programming Bounds for Equiangular Lines)

相關論文

★ A Study On Kissing Number Problem	★ 歐式空間二距離集合之探討
★ An Observation on 7-distance Set in Euclidean Plane	★ 等角直線叢的研究

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摘要(中)

考慮空間中過原點的若干條直線，若任兩條直線間所形成的夾角只有一種角度，我們稱該集合為等角直線叢。這樣的構造可以由空間中單位球面上的有限點集合（即球面碼）來描述。球面上的離散幾何極值問題有著相當悠久的歷史，知名的問題有吻球數問題、球面上最密堆積問題及能量最小化問題等。在這篇文章中，我們複習目前用來解以上問題的主流方法，即Delsarte的線性規劃及Bachoc-Vallentin的半正定規劃等最佳化方法，並考慮後者的對偶問題來重現歐式空間中維度介於$23$與$60$間且角度為acos(1/5)的等角直線叢的上界。

摘要(英)

Equiangular lines is a set of lines through the origin in the space with a single angle between any two of them. It can be identified as a finite set of points on the sphere which is known as spherical code. The search for extreme structures of spherical codes satisfying certain conditions has a long history in discrete geometry, such as the kissing number problem, Tammes′ problem, and energy minimizing problem. In this paper, we review two effective methods for dealing with those long-standing questions, namely, Delsarte′s linear programming and Bachoc-Vallentin′s semidefinite programming, and use the dual form of the latter to reproduce the bound on equiangular lines of angle acos(1/5) in R^n where 22<n<61.

關鍵字(中)

★ 球面碼
★ 距離集合
★ 等角直線叢
★ 半正定規劃

關鍵字(英)

★ spherical codes
★ s-distance sets
★ equiangular lines
★ semidefinite programming

論文目次

Chinese Abstract i
Abstract ii
Acknowledgement iii
Contents iv
1 Introduction 1
2 Optimization Method on Spherical Codes 3
2.1 Positive Definite Kernels on Sphere 3
2.2 Linear Programming Bound on Spherical Codes 5
2.3 Semidefinite Programming Bound on Spherical Codes 9
2.4 Dual Form of Semidefinite Programming 12
3 Upper Bounds on Equiangular Lines 16
3.1 Bounds on Equiangular Lines with Fixed Angle 16
3.2 Semidefinite Programming Bounds on Equiangular Lines 17
4 Conclusion 21
References 22

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指導教授

俞韋亘(Wei-Hsuan Yu)

審核日期

2023-7-20

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