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研究生:葉政叡
研究生(外文):Cheng-Jui Yeh
論文名稱:多項式方法於等角直線叢上的半正定規劃上界
論文名稱(外文):Polynomial Method in Semidefinite Programming Bounds for Equiangular Lines
指導教授:俞韋亘
指導教授(外文):Wei-Hsuan Yu
學位類別:碩士
校院名稱:國立中央大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2023
畢業學年度:111
語文別:英文
論文頁數:31
中文關鍵詞:球面碼距離集合等角直線叢半正定規劃
外文關鍵詞:spherical codess-distance setsequiangular linessemidefinite programming
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  • 被引用被引用:0
  • 點閱點閱:32
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  • 下載下載:2
  • 收藏至我的研究室書目清單書目收藏:0
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.
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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[2]  C. Bachoc and F. Vallentin. Optimality and uniqueness of the (4,10,1/6) spherical code. Journal of Combinatorial Theory, Series A, 116(1):195–204, 2009.
[3]  E. Bannai and N. J. Sloane. Uniqueness of certain spherical codes. Canadian Journal of Mathematics, 33(2):437–449, 1981.
[4] A. Barg and W.-H. Yu. New bounds for equiangular lines. Discrete geometry and algebraic combinatorics, 625:111–121, 2013.
[5] M.-Y. Cao, J. H. Koolen, Y.-C. R. Lin, and W.-H. Yu. The lemmens-seidel conjecture and forbidden subgraphs. Journal of Combinatorial Theory, Series A, 185:105538, 2022.
[6] H. Cohn, A. Kumar, S. Miller, D. Radchenko, and M. Viazovska. The sphere packing problem in dimension 24. Annals of Mathematics, 185(3):1017–1033, 2017. [7] H. Cohn and J. Woo. Three-point bounds for energy minimization. Journal of the American Mathematical Society, 25(4):929–958, 2012.
[8]  S. J. Einhorn and I. J. Schoenberg. On euclidean sets having only two distances between points ii. Indag. Math, 28:489–504, 1966.
[9]  A. Glazyrin and W.-H. Yu. Upper bounds for s-distance sets and equiangular lines. Advances in Mathematics, 330:810–833, 2018.
[10]  M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming, version 2.2. http://cvxr.com/cvx, mar 2020.
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[14]  O. R. Musin. Spherical two-distance sets. Journal of Combinatorial Theory, Series A, 116(4):988–995, 2009.
[15]  A. Numaier. Graph representations, two-distance sets, and equiangular lines. Linear Algebra and its Applications, 114:141–156, 1989.
[16]  A. M. Odlyzko and N. J. Sloane. New bounds on the number of unit spheres that can touch a unit sphere in n dimensions. Journal of Combinatorial Theory, Series A, 26(2):210–214, 1979.
[17]  J.-M. G. Philippe Delsarte and J. J. Seidel. Spherical codes and designs. Geometriae Dedicata, 6(3):363–388, 1977.
[18]  A. J. Scott and M. Grassl. Symmetric informationally complete positive-operator-valued measures: A new computer study. Journal of Mathematical Physics, 51(4), 2010.
[19] A. Schrijver. New code upper bounds from the terwilliger algebra and semidefinite program- ming. IEEE Transactions on Information Theory, 51(8):2859–2866, 2005.
[20] M. S. Viazovska. The sphere packing problem in dimension 8. Annals of mathematics, 185(3):991–1015, 2017.
[21]  N. I. Vilenkin. Special functions and the theory of group representations (Vol. 22). American Mathematical Soc., 1978.
[22]  S. F. Waldron. An introduction to finite tight frames. Basel: Birkhäuser, 2018.
[23]  W.-H. Yu. New bounds for equiangular lines and spherical two-distance sets. SIAM Journal on Discrete Mathematics, 31(2):908–917, 2017.
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