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

Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/93548

Title:	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
Authors:	葉政叡;Yeh, Cheng-Jui
Contributors:	數學系
Keywords:	球面碼;距離集合;等角直線叢;半正定規劃;spherical codes;s-distance sets;equiangular lines;semidefinite programming
Date:	2023-07-20
Issue Date:	2024-09-19 17:12:21 (UTC+8)
Publisher:	國立中央大學
Abstract:	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。;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.
Appears in Collections:	[Graduate Institute of Mathematics] Electronic Thesis & Dissertation

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