等角直線叢的研究

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謝欣妤(Hsin-Yu Heish) 查詢紙本館藏

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論文名稱

等角直線叢的研究
(A Study on Equiangular Lines)

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摘要(中)

本論文將會整理離散幾何中一個有趣的領域: 等角直線組（equiangular lines)的歷史演進與發展。以1973年 Lemmens-Seidel 的文章為主體，並加入後續的進展，例如 Barg-Yu 證明了24維度之後的半正定規劃的上界，Lin-Yu 對Neumann定理的推廣, Greaves et al 對於14和16維度決定最大條數的結果。我們整理這些相關文獻，把等角直線組的故事與發展說明得更完整，並且詳細寫下相關的例子或構造。本文以Lemmens-Seidel第四節和第五節為重，第四節說明柱(pillar)是甚麼和相關定理證明，第五節討論當角度固定在arccos(1/5)時，會說明上下界會如何變化。

摘要(英)

This paper dives into an intriguing realm of discrete geometry: the historical evolution and development of equiangular lines. It primarily builds upon the 1973 Lemmens-Seidel paper, incorporating subsequent advancements. For instance, Barg-Yu proved upper bounds for semidefinite programming beyond 24 dimensions, Lin-Yu extended Neumann′s theorem, and Greaves et al revealed results on determining the maximum number of lines in 14 and 16 dimensions. We′ll organize these relevant works, providing a more comprehensive narrative of the equiangular lines′ story and development, while delving into specific examples or constructions. The focus of this paper lies in Lemmens-Seidel′s fourth and fifth sections, the fourth section what a "pillar" is and proves associated theorems, while the fifth section how upper and lower bounds shift when the angle is fixed at arccos(1/5).

關鍵字(中)

★ 等角直線

關鍵字(英)

★ Equiangular Lines

論文目次

摘要第ix頁
Abstract 第xi頁
目錄第xiii頁
一、緒論第1頁
二、定義跟例子第5頁
三、最大等角直線數的上下界第11頁
四、柱第23頁
五、 v_{1/5}(r)的測定 (determination) 第33頁
六、結論第45頁
參考文獻第51頁

參考文獻

1] P. W. Lemmens and J. J.Seidel, “Equiangular lines,” Journal of Algebra, vol. 24,
no. 3, pp. 494–512, 1973.
[2] A. Barg and W.-H. Yu, “New bounds for equiangular lines.,” Discrete geometry and
algebraic combinatorics, vol. 625, pp. 111–121, 2013.
[3] Y.-C. R. Lin and W.-H. Yu, “Equiangular lines and the lemmens–seidel conjecture,”
Discrete Mathematics, vol. 343, no. 2, p. 111667, 2020.
[4] G. Greaves, J. H. Koolen, A. Munemasa, and F. Szöllősi, “Equiangular lines in
euclidean spaces,” Journal of Combinatorial Theory, Series A, vol. 138, pp. 208–
235, 2016.
[5] J. Haantjes, “Equilateral point-sets in elliptic two-and three-dimensional spaces,”
Nieuw Arch. Wiskunde (2), vol. 22, pp. 355–362, 1948.
[6] J. H. van Lint and J. J. Seidel, “Equilateral point sets in elliptic geometry,” Pro-
ceedings of the Koninklijke Nederlandse Akademie van Wetenschappen: Series A:
Mathematical Sciences, vol. 69, no. 3, pp. 335–348, 1966.
[7] D. de Caen, “Large equiangular sets of lines in euclidean space,” the electronic journal
of combinatorics, vol. 7, pp. R55–R55, 2000.
[8] Y.-c. R. Lin and W.-H. Yu, “Saturated configuration and new large construction of
equiangular lines,” Linear Algebra and its Applications, vol. 588, pp. 272–281, 2020.
[9] G. Greaves, J. Syatriadi, and P. Yatsyna, “Equiangular lines in euclidean spaces:
dimensions 17 and 18,” Mathematics of Computation, vol. 92, no. 342, pp. 1867–
1903, 2023.
[10] J. C. Tremain, “Concrete constructions of real equiangular line sets,” arXiv preprint
arXiv:0811.2779, 2008.
[11] P. Delsarte, J. Goethals, and J. J. Seidel, “Orthogonal matrices with zero diagonal.
ii,” Canadian Journal of Mathematics, vol. 23, no. 5, pp. 816–832, 1971.
[12] M.-Y. Cao, J. H. Koolen, Y.-C. R. Lin, and W.-H. Yu, “The lemmens-seidel conjec-
ture and forbidden subgraphs,” Journal of Combinatorial Theory, Series A, vol. 185,
p. 105538, 2022.
[13] G. Greaves and P. Yatsyna, “On equiangular lines in 17 dimensions and the charac-
teristic polynomial of a seidel matrix,” Mathematics of Computation, vol. 88, no. 320,
pp. 3041–3061, 2019.
[14] G. R. Greaves, J. Syatriadi, and P. Yatsyna, “Equiangular lines in low dimensional
euclidean spaces,” Combinatorica, vol. 41, no. 6, pp. 839–872, 2021.
[15] G. R. Greaves and J. Syatriadi, “Real equiangular lines in dimension 18
and the de caen-jacobi identity for complementary subgraphs,” arXiv preprint
arXiv:2206.04267, 2022.
[16] I. Balla, “Equiangular lines via matrix projection,” arXiv preprint arXiv:2110.15842,
vol. 3, no. 4, p. 6, 2021.

指導教授

俞韋亘(Wei-Hsuan Yu)

審核日期

2023-12-29

推文