scholarly journals k-Point semidefinite programming bounds for equiangular lines

2021 ◽  
Author(s):  
David de Laat ◽  
Fabrício Caluza Machado ◽  
Fernando Mário de Oliveira Filho ◽  
Frank Vallentin
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Semidefinite Programming ◽  
Spherical Codes ◽  
Equiangular Lines

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.

scholarly journals Upper bounds for packings of spheres of several radii

2014 ◽  
Vol 2 ◽  
Author(s):  
DAVID DE LAAT ◽  
FERNANDO MÁRIO DE OLIVEIRA FILHO ◽  
FRANK VALLENTIN
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Semidefinite Programming ◽  
Unit Sphere ◽  
Upper Bound ◽  
Convex Bodies ◽  
Classical Problem ◽  
Upper Bounds ◽  
Sphere Packings ◽  
Spherical Caps

AbstractWe give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds for packings of spherical caps and of convex bodies through the use of semidefinite programming. We perform explicit computations, obtaining new bounds for packings of spherical caps of two different sizes and for binary sphere packings. We also slightly improve the bounds for the classical problem of packing identical spheres.

scholarly journals Equiangular lines and spherical codes in Euclidean space

2017 ◽  
Vol 211 (1) ◽  
pp. 179-212 ◽  
Author(s):  
Igor Balla ◽  
Felix Dräxler ◽  
Peter Keevash ◽  
Benny Sudakov
Keyword(s):  
Euclidean Space ◽  
Spherical Codes ◽  
Equiangular Lines

scholarly journals Large Equiangular Sets of Lines in Euclidean Space

10.37236/1533 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
D. De Caen
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Euclidean Space ◽  
Upper Bound ◽  
Equiangular Lines

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 $d^2$ .

Minimum norm interpolation in the ℓ1(ℕ) space

2020 ◽  
Vol 19 (01) ◽  
pp. 21-42
Author(s):  
Raymond Cheng ◽  
Yuesheng Xu
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Interpolation Problem ◽  
Dual Space ◽  
Original Problem ◽  
Dual Problem ◽  
Dimensional Euclidean Space ◽  
Minimum Norm ◽  
Finite Dimensional ◽  
Sparse Interpolation

We consider the minimum norm interpolation problem in the [Formula: see text] space, aiming at constructing a sparse interpolation solution. The original problem is reformulated in the pre-dual space, thereby inducing a norm in a related finite-dimensional Euclidean space. The dual problem is then transformed into a linear programming problem, which can be solved by existing methods. With that done, the original interpolation problem is reduced by solving an elementary finite-dimensional linear algebra equation. A specific example is presented to illustrate the proposed method, in which a sparse solution in the [Formula: see text] space is compared to the dense solution in the [Formula: see text] space. This example shows that a solution of the minimum norm interpolation problem in the [Formula: see text] space is indeed sparse, while that of the minimum norm interpolation problem in the [Formula: see text] space is not.

A GENERAL ALGORITHM FOR THE GENERATION OF QUASILATTICES BY MEANS OF THE CUT AND PROJECTION METHOD USING THE SIMPLEX METHOD OF LINEAR PROGRAMMING AND THE MOORE-PENROSE GENERALIZED INVERSE

1989 ◽  
Vol 03 (05) ◽  
pp. 773-786
Author(s):  
J. L. ARAGÓN ◽  
G. VÁZQUEZ POLO ◽  
A. GÓMEZ
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Projection Method ◽  
Simplex Method ◽  
Computational Algorithm ◽  
Generalized Inverse ◽  
Dimensional Space ◽  
General Algorithm ◽  
Higher Dimensional

A computational algorithm for the generation of quasiperiodic tiles based on the cut and projection method is presented. The algorithm is capable of projecting any type of lattice embedded in any euclidean space onto any subspace making it possible to generate quasiperiodic tiles with any desired symmetry. The simplex method of linear programming and the Moore-Penrose generalized inverse are used to construct the cut (strip) in the higher dimensional space which is to be projected.

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