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Journal articles on the topic 'Equiangular lines'

Author: Grafiati

Published: 18 May 2024

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1

Et-Taoui, Boumediene. "Quaternionic equiangular lines." Advances in Geometry 20, no. 2 (April 28, 2020): 273–84. http://dx.doi.org/10.1515/advgeom-2019-0021.

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AbstractLet 𝔽 = ℝ, ℂ or ℍ. A p-set of equi-isoclinic n-planes with parameter λ in 𝔽r is a set of pn-planes spanning 𝔽r each pair of which has the same non-zero angle arccos $\begin{array}{} \sqrt{\lambda} \end{array}$. It is known that via a complex matrix representation, a pair of isoclinic n-planes in ℍr with angle arccos $\begin{array}{} \sqrt{\lambda} \end{array}$ yields a pair of isoclinic 2n-planes in ℂ2r with angle arccos $\begin{array}{} \sqrt{\lambda} \end{array}$. In this article we characterize all the p-tuples of equi-isoclinic planes in ℂ2r which come via our complex representation from p-tuples of equiangular lines in ℍr. We then construct all the p-tuples of equi-isoclinic planes in ℂ4 and derive all the p-tuples of equiangular lines in ℍ2. Among other things it turns out that the quadruples of equiangular lines in ℍ2 are all regular, i.e. their symmetry groups are isomorphic to the symmetric group S4.

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2

Et-Taoui, B. "Equiangular lines in Cr." Indagationes Mathematicae 11, no. 2 (June 2000): 201–7. http://dx.doi.org/10.1016/s0019-3577(00)89078-3.

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3

Deza, M., and V. P. Grishukhin. "L-polytopes and equiangular lines." Discrete Applied Mathematics 56, no. 2-3 (January 1995): 181–214. http://dx.doi.org/10.1016/0166-218x(94)00086-s.

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4

Greaves, Gary, Jacobus H. Koolen, Akihiro Munemasa, and Ferenc Szöllősi. "Equiangular lines in Euclidean spaces." Journal of Combinatorial Theory, Series A 138 (February 2016): 208–35. http://dx.doi.org/10.1016/j.jcta.2015.09.008.

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5

Deza, M., and V. P. Grishukhin. "Cut Lattices and Equiangular Lines." European Journal of Combinatorics 17, no. 2-3 (February 1996): 143–56. http://dx.doi.org/10.1006/eujc.1996.0013.

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6

Et-Taoui, B. "Equiangular lines in Cr (part II)." Indagationes Mathematicae 13, no. 4 (2002): 483–86. http://dx.doi.org/10.1016/s0019-3577(02)80027-1.

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7

Mondal, Bishwarup, Roopsha Samanta, and Robert W. Heath. "Congruent Voronoi tessellations from equiangular lines." Applied and Computational Harmonic Analysis 23, no. 2 (September 2007): 254–58. http://dx.doi.org/10.1016/j.acha.2007.03.005.

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8

Lin, Yen-Chi Roger, and Wei-Hsuan Yu. "Equiangular lines and the Lemmens–Seidel conjecture." Discrete Mathematics 343, no. 2 (February 2020): 111667. http://dx.doi.org/10.1016/j.disc.2019.111667.

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9

Balla, Igor, Felix Dräxler, Peter Keevash, and Benny Sudakov. "Equiangular lines and subspaces in Euclidean spaces." Electronic Notes in Discrete Mathematics 61 (August 2017): 85–91. http://dx.doi.org/10.1016/j.endm.2017.06.024.

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10

Guiduli, B., and M. Rosenfeld. "Ubiquitous Angles in Equiangular Sets of Lines." Discrete & Computational Geometry 24, no. 2 (September 2000): 313–24. http://dx.doi.org/10.1007/s004540010038.

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11

Khatirinejad, Mahdad. "On Weyl-Heisenberg orbits of equiangular lines." Journal of Algebraic Combinatorics 28, no. 3 (November 6, 2007): 333–49. http://dx.doi.org/10.1007/s10801-007-0104-1.

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12

Coutinho, G., C. Godsil, H. Shirazi, and H. Zhan. "Equiangular lines and covers of the complete graph." Linear Algebra and its Applications 488 (January 2016): 264–83. http://dx.doi.org/10.1016/j.laa.2015.09.029.

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13

Balla, Igor, Felix Dräxler, Peter Keevash, and Benny Sudakov. "Equiangular lines and spherical codes in Euclidean space." Inventiones mathematicae 211, no. 1 (July 12, 2017): 179–212. http://dx.doi.org/10.1007/s00222-017-0746-0.

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14

Neumaier, A. "Graph representations, two-distance sets, and equiangular lines." Linear Algebra and its Applications 114-115 (March 1989): 141–56. http://dx.doi.org/10.1016/0024-3795(89)90456-4.

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15

Jedwab, Jonathan, and Amy Wiebe. "Large sets of complex and real equiangular lines." Journal of Combinatorial Theory, Series A 134 (August 2015): 98–102. http://dx.doi.org/10.1016/j.jcta.2015.03.007.

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16

Godsil, Chris, and Aidan Roy. "Equiangular lines, mutually unbiased bases, and spin models." European Journal of Combinatorics 30, no. 1 (January 2009): 246–62. http://dx.doi.org/10.1016/j.ejc.2008.01.002.

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17

Greaves, Gary R. W., Joseph W. Iverson, John Jasper, and Dustin G. Mixon. "Frames over finite fields: Equiangular lines in orthogonal geometry." Linear Algebra and its Applications 639 (April 2022): 50–80. http://dx.doi.org/10.1016/j.laa.2021.11.024.

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18

Bukh, Boris. "Bounds on Equiangular Lines and on Related Spherical Codes." SIAM Journal on Discrete Mathematics 30, no. 1 (January 2016): 549–54. http://dx.doi.org/10.1137/15m1036920.

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19

Lin, Yen-chi Roger, and Wei-Hsuan Yu. "Saturated configuration and new large construction of equiangular lines." Linear Algebra and its Applications 588 (March 2020): 272–81. http://dx.doi.org/10.1016/j.laa.2019.12.002.

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20

Glazyrin, Alexey, and Wei-Hsuan Yu. "Upper bounds for s-distance sets and equiangular lines." Advances in Mathematics 330 (May 2018): 810–33. http://dx.doi.org/10.1016/j.aim.2018.03.024.

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21

Jedwab, Jonathan, and Amy Wiebe. "Constructions of complex equiangular lines from mutually unbiased bases." Designs, Codes and Cryptography 80, no. 1 (March 22, 2015): 73–89. http://dx.doi.org/10.1007/s10623-015-0064-8.

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22

King, Emily J., and Xiaoxian Tang. "New Upper Bounds for Equiangular Lines by Pillar Decomposition." SIAM Journal on Discrete Mathematics 33, no. 4 (January 2019): 2479–508. http://dx.doi.org/10.1137/19m1248881.

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23

Yu, Wei-Hsuan. "New Bounds for Equiangular Lines and Spherical Two-Distance Sets." SIAM Journal on Discrete Mathematics 31, no. 2 (January 2017): 908–17. http://dx.doi.org/10.1137/16m109377x.

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24

Kolar-Begović, Zdenka, Ružica Kolar-Šuper, and Vladimir Volenec. "EQUICEVIAN POINTS AND EQUIANGULAR LINES OF A TRIANGLE IN AN ISOTROPIC PLANE." Sarajevo Journal of Mathematics 11, no. 1 (2015): 101–7. http://dx.doi.org/10.5644/sjm.11.1.08.

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25

Greaves, Gary R. W., Joseph W. Iverson, John Jasper, and Dustin G. Mixon. "Frames over finite fields: Basic theory and equiangular lines in unitary geometry." Finite Fields and Their Applications 77 (January 2022): 101954. http://dx.doi.org/10.1016/j.ffa.2021.101954.

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26

Appleby, Marcus, Steven Flammia, Gary McConnell, and Jon Yard. "Generating ray class fields of real quadratic fields via complex equiangular lines." Acta Arithmetica 192, no. 3 (2020): 211–33. http://dx.doi.org/10.4064/aa180508-21-6.

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27

Jiang, Zilin, and Alexandr Polyanskii. "Forbidden Subgraphs for Graphs of Bounded Spectral Radius, with Applications to Equiangular Lines." Israel Journal of Mathematics 236, no. 1 (March 2020): 393–421. http://dx.doi.org/10.1007/s11856-020-1983-2.

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28

Greaves, Gary R. W., and Jeven Syatriadi. "Real equiangular lines in dimension 18 and the Jacobi identity for complementary subgraphs." Journal of Combinatorial Theory, Series A 201 (January 2024): 105812. http://dx.doi.org/10.1016/j.jcta.2023.105812.

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29

Chien, Tuan-Yow, and Shayne Waldron. "Projective Symmetry Group of a Finite Frame." New Zealand Journal of Mathematics 48 (December 31, 2018): 55–81. http://dx.doi.org/10.53733/35.

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We define the projective symmetry group of a finite sequence of vectors (a frame) in a natural way as a group of permutations on the vectors (or their indices). This definition ensures that the projective symmetry group is the same for a frame and its complement. We give an algorithm for computing the projective symmetry group from a small set of projective invariants when the underlying field is a subfield of which is closed under conjugation. This algorithm is applied in a number of examples including equiangular lines (in particular SICs), MUBs, and harmonic frames.

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30

Greaves, Gary R. W., and Pavlo Yatsyna. "On equiangular lines in $17$ dimensions and the characteristic polynomial of a Seidel matrix." Mathematics of Computation 88, no. 320 (April 9, 2019): 3041–61. http://dx.doi.org/10.1090/mcom/3433.

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31

Bengtsson, Ingemar. "Algebraic units, anti-unitary symmetries, and a small catalogue of SICs." Quantum Information and Computation 20, no. 5&6 (May 2020): 400–417. http://dx.doi.org/10.26421/qic20.5-6-3.

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In complex vector spaces maximal sets of equiangular lines, known as SICs, are related to real quadratic number fields in a dimension dependent way. If the dimension is of the form n^2+3, the base field has a fundamental unit of negative norm, and there exists a SIC with anti-unitary symmetry. We give eight examples of exact solutions of this kind, for which we have endeavoured to make them as simple as we can---as a belated reply to the referee of an earlier publication, who claimed that our exact solution in dimension 28 was too complicated to be fit to print. An interesting feature of the simplified solutions is that the components of the fiducial vectors largely consist of algebraic units.

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32

Okuda, Takayuki, and Wei-Hsuan Yu. "A new relative bound for equiangular lines and nonexistence of tight spherical designs of harmonic index 4." European Journal of Combinatorics 53 (April 2016): 96–103. http://dx.doi.org/10.1016/j.ejc.2015.11.003.

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33

Dewhurst, Peter, and Sriruk Srithongchai. "An Investigaton of Minimum-Weight Dual-Material Symmetrically Loaded Wheels and Torsion Arms." Journal of Applied Mechanics 72, no. 2 (March 1, 2005): 196–202. http://dx.doi.org/10.1115/1.1831295.

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A cylindrically symmetric layout of two opposite families of logarithmic spirals is shown to define the layout of minimum-weight, symmetrically loaded wheel structures, where different materials are used for the tension and compression members, respectively; referred to here as dual-material structures. Analytical solutions are obtained for both structure weight and deflection. The symmetric solutions are shown to form the basis for torsion arm structures, which when designed to accept the same total load, have identical weight and are subjected to identical deflections. The theoretical predictions of structure weight, deflection, and support reactions are shown to be in close agreement to the values obtained with truss designs, whose nodes are spaced along the theoretical spiral layout lines. The original Michell solution based on 45 deg equiangular spirals is shown to be in very close agreement with layout solutions designed to be kinematically compatible with the strain field required for an optimal dual-material design.

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34

Titumir, Rashed Al Mahmud, and Md Zahidur Rahman. "Strategic Implications of China’s Belt and Road Initiative (BRI): The Case of Bangladesh." China and the World 02, no. 03 (September 2019): 1950020. http://dx.doi.org/10.1142/s2591729319500202.

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The paper presents a new framework in explaining the geostrategic compulsions arising out of China’s Belt and Road Initiative (BRI) with particular emphasis on finding the implications on Bangladesh amidst a translation of erstwhile pacifist Indo-Pacific region to a point of strategic importance. The framework, on the contrary to the exuberances of voluminous literature by liberals and realists, analyzes the internal compulsions stemming from a particular political settlement of the countries involved. While most available accounts typically urge to strike a “delicate balance”, but hardly any exercise has been carried out on how to achieve such balance. The paper makes an attempt to work out the balancing mechanism. The paper also identifies the conditions for mutual stability and growth by outlining equiangular development diplomacy — the optimal outcome that can be reached if there is an alignment of necessary, sufficient and sustainability conditions amongst the collaborating and/or contending partners. The sustenance of such partnership is dependent upon normative legitimacy arising from broad-based social approval along the lines of particular political settlement.

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35

ZAUNER, GERHARD. "QUANTUM DESIGNS: FOUNDATIONS OF A NONCOMMUTATIVE DESIGN THEORY." International Journal of Quantum Information 09, no. 01 (February 2011): 445–507. http://dx.doi.org/10.1142/s0219749911006776.

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This is a one-to-one translation of a German-written Ph.D. thesis from 1999. Quantum designs are sets of orthogonal projection matrices in finite(b)-dimensional Hilbert spaces. A fundamental differentiation is whether all projections have the same rank r, and furthermore the special case r = 1, which contains two important subclasses: Mutually unbiased bases (MUBs) were introduced prior to this thesis and solutions of b + 1 MUBs whenever b is a power of a prime were already given. Unaware of those papers, this concept was generalized here under the notation of regular affine quantum designs. Maximal solutions are given for the general case r ≥ 1, consisting of r(b2 - 1)/(b - r) so-called complete orthogonal classes whenever b is a power of a prime. For b = 6, an infinite family of MUB triples was constructed and it was — as already done in the author's master's thesis (1991) — conjectured that four MUBs do not exist in this dimension. Symmetric informationally complete positive operator-valued measures (SIC POVMs) in this paper are called regular quantum 2-designs with degree 1. The assigned vectors span b2 equiangular lines. These objects had been investigated since the 1960s, but only a few solutions were known in complex vector spaces. In this thesis further maximal analytic and numerical solutions were given (today a lot more solutions are known) and it was (probably for the first time) conjectured that solutions exist in any dimension b (generated by the Weyl–Heisenberg group and with a certain additional symmetry of order 3).

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36

De Caen, D. "Large Equiangular Sets of Lines in Euclidean Space." Electronic Journal of Combinatorics 7, no. 1 (November 9, 2000). http://dx.doi.org/10.37236/1533.

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A construction is given of ${{2}\over {9}} (d+1)^2$ equiangular lines in Euclidean $d$-space, when $d = 3 \cdot 2^{2t-1}-1$ with $t$ any positive integer. This compares with the well known "absolute" upper bound of ${{1}\over {2}} d(d+1)$ lines in any equiangular set; it is the first known constructive lower bound of order $d^2$ .

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37

Jiang, Zilin, Jonathan Tidor, Yuan Yao, Shengtong Zhang, and Yufei Zhao. "Equiangular lines with a fixed angle." Annals of Mathematics 194, no. 3 (November 1, 2021). http://dx.doi.org/10.4007/annals.2021.194.3.3.

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38

DEMPWOLFF, ULRICH, and WILLIAM M. KANTOR. "ON 2-TRANSITIVE SETS OF EQUIANGULAR LINES." Bulletin of the Australian Mathematical Society, August 22, 2022, 1–12. http://dx.doi.org/10.1017/s0004972722000661.

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Abstract We determine all finite sets of equiangular lines spanning finite-dimensional complex unitary spaces for which the action on the lines of the set-stabiliser in the unitary group is 2-transitive with a regular normal subgroup.

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39

Greaves, Gary R. W., Jeven Syatriadi, and Pavlo Yatsyna. "Equiangular Lines in Low Dimensional Euclidean Spaces." Combinatorica, August 31, 2021. http://dx.doi.org/10.1007/s00493-020-4523-0.

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40

de Laat, David, Fabrício Caluza Machado, Fernando Mário de Oliveira Filho, and Frank Vallentin. "k-Point semidefinite programming bounds for equiangular lines." Mathematical Programming, April 21, 2021. http://dx.doi.org/10.1007/s10107-021-01638-x.

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AbstractWe propose a hierarchy of k-point bounds extending the Delsarte–Goethals–Seidel linear programming 2-point bound and the Bachoc–Vallentin semidefinite programming 3-point bound for spherical codes. An optimized implementation of this hierarchy allows us to compute 4, 5, and 6-point bounds for the maximum number of equiangular lines in Euclidean space with a fixed common angle.

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41

Sane, Sharad. "Equiangular lines in the real space $${\mathbb {R}}^d$$." Journal of Analysis, February 1, 2020. http://dx.doi.org/10.1007/s41478-020-00227-z.

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42

Jiang, Zilin, Jonathan Tidor, Yuan Yao, Shengtong Zhang, and Yufei Zhao. "Spherical Two-Distance Sets and Eigenvalues of Signed Graphs." Combinatorica, July 21, 2023. http://dx.doi.org/10.1007/s00493-023-00002-1.

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AbstractWe study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let $$N_{\alpha ,\beta }(d)$$ N α , β ( d ) denote the maximum number of unit vectors in $${\mathbb {R}}^d$$ R d where all pairwise inner products lie in $$\{\alpha ,\beta \}$$ { α , β } . For fixed $$-1\le \beta<0\le \alpha <1$$ - 1 ≤ β < 0 ≤ α < 1 , we propose a conjecture for the limit of $$N_{\alpha ,\beta }(d)/d$$ N α , β ( d ) / d as $$d \rightarrow \infty $$ d → ∞ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when $$\alpha +2\beta <0$$ α + 2 β < 0 or $$(1-\alpha )/(\alpha -\beta ) \in \{1, \sqrt{2}, \sqrt{3}\}$$ ( 1 - α ) / ( α - β ) ∈ { 1 , 2 , 3 } .Our work builds on our recent resolution of the problem in the case of $$\alpha = -\beta $$ α = - β (corresponding to equiangular lines). It is the first determination of $$\lim _{d \rightarrow \infty } N_{\alpha ,\beta }(d)/d$$ lim d → ∞ N α , β ( d ) / d for any nontrivial fixed values of $$\alpha $$ α and $$\beta $$ β outside of the equiangular lines setting.

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43

IVERSON, JOSEPH W., JOHN JASPER, and DUSTIN G. MIXON. "OPTIMAL LINE PACKINGS FROM FINITE GROUP ACTIONS." Forum of Mathematics, Sigma 8 (2020). http://dx.doi.org/10.1017/fms.2019.48.

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We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum distance. We introduce our program in terms of general Schurian association schemes before focusing on the special case of Gelfand pairs. Notably, our program unifies a variety of existing packings with heretofore disparate constructions. In addition, we leverage our program to construct the first known infinite family of equiangular lines with Heisenberg symmetry.

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44

Ishida, Sachiko, Taketoshi Nojima, and Ichiro Hagiwara. "Design of Deployable Membranes Using Conformal Mapping." Journal of Mechanical Design 137, no. 6 (June 1, 2015). http://dx.doi.org/10.1115/1.4030296.

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This paper proposes a new method for designing the crease patterns of deployable membranes that can be wrapped up compactly. The method utilizes conformal mapping and the origami folding technique. The mapping of the flow with circulation can be used to control the angles between the fold lines, produce elements of the same shape, and maintain regularity of the fold lines. The proposed method thus enables the systematic and efficient design of complex patterns based on simple ones. The proposed method was successfully used to produce the patterns of Nojima and other extended new patterns of deployable membranes consisting of discrete equiangular spirals. The patterns were wrapped and used to form pillars such as regular polygonal, rectangular, and diamond pillars. Toward the industrial application of the proposed method, this paper also discusses pattern design for space-saving storage and to reduce the effect of thickness when using versatile materials.

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45

Kopp, Gene S. "SIC-POVMs and the Stark Conjectures." International Mathematics Research Notices, October 31, 2019. http://dx.doi.org/10.1093/imrn/rnz153.

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Abstract The existence of $d^2$ pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in $d$-dimensional Hilbert space is known only for finitely many dimensions $d$. We prove that, if there exists a set of real units in a certain ray class field (depending on $d$) satisfying certain algebraic properties, a SIC-POVM exists, when $d$ is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at $s=0$ and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact SIC-POVM in dimension 23.

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