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研究生： 葉政叡
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
Polynomial Method in Semidefinite Programming Bounds for Equiangular Lines
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： 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 codes, s-distance sets, equiangular lines, semidefinite programming
相關次數： 點閱：12  下載：0 
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    • 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

    [1]  C. Bachoc and F. Vallentin. New upper bounds for kissing numbers from semidefinite program- ming. Journal of the American Mathematical Society, 21(3):909–924, 2008.
    [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.
    [11]  P. W. H. Lemmens and J. J. Seidel. Equiangular lines. Journal of Algebra, 24(3):494–512, 1973.
    [12]  P. Lisonk. New maximal two-distance sets. Journal of combinatorial theory, Series A, 77(2):318–338, 1997.
    [13]  O. R. Musin. The kissing number in four dimensions. Annals of Mathematics, pages 1–32, 2008.
    [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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