106 research outputs found
New upper bounds for kissing numbers from semidefinite programming
Recently A. Schrijver derived new upper bounds for binary codes using semidefinite programming. In this paper we adapt this approach to codes on the unit sphere and we compute new upper bounds for the kissing number in several dimensions. In particular our computations give the (known) values for the cases n = 3, 4, 8, 24
High accuracy semidefinite programming bounds for kissing numbers
The kissing number in n-dimensional Euclidean space is the maximal number of non-overlapping unit spheres which simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing number based on semidefinite programming. This paper is a report on high accuracy calculations of these upper bounds for n <= 24. The bound for n = 16 implies a conjecture of Conway and Sloane: There is no 16-dimensional periodic point set with average theta series 1 + 7680q^3 + 4320q^4 + 276480q^5 + 61440q^6 + ..
Template-based searches for gravitational waves: efficient lattice covering of flat parameter spaces
The construction of optimal template banks for matched-filtering searches is
an example of the sphere covering problem. For parameter spaces with
constant-coefficient metrics a (near-) optimal template bank is achieved by the
A_n* lattice, which is the best lattice-covering in dimensions n <= 5, and is
close to the best covering known for dimensions n <= 16. Generally this
provides a substantially more efficient covering than the simpler hyper-cubic
lattice. We present an algorithm for generating lattice template banks for
constant-coefficient metrics and we illustrate its implementation by generating
A_n* template banks in n=2,3,4 dimensions.Comment: 10 pages, submitted to CQG for proceedings of GWDAW1
The contact polytope of the Leech lattice
The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits
Fourier analysis, linear programming, and densities of distance avoiding sets in {}
In this paper we derive new upper bounds for the densities of measurable sets in R^n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions 2,..., 24. This gives new lower bounds for the measurable chromatic number in dimensions 3,..., 24. We apply it to get a new, short proof of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson, Weiss and Bourgain and Falconer about sets avoiding many distances
The isodiametric problem with lattice-point constraints
In this paper, the isodiametric problem for centrally symmetric convex bodies
in the Euclidean d-space R^d containing no interior non-zero point of a lattice
L is studied. It is shown that the intersection of a suitable ball with the
Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among
all bodies with the same volume. It is conjectured that these sets are the only
extremal bodies, which is proved for all three dimensional and several
prominent lattices.Comment: 12 pages, 4 figures, (v2) referee comments and suggestions
incorporated, accepted in Monatshefte fuer Mathemati
New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry
In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the -norm) and of Platonic and Archimedean solids having tetrahedral symmetry.
These bounds give strong indications that some of the lattice packings of superballs found in 2009 by Jiao, Stillinger, and Torquato are indeed optimal among all translative packings. We improve Zong's recent upper bound for the maximal density of translative packings of regular tetrahedra from to , getting closer to the best known lower bound of .
We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of packings of round spheres. The proofs of our new upper bounds are computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization
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