arXiv subjects

David de Laat

Publications and source records attributed to David de Laat.

21 records · Page 2Linked to original sources

arXiv2016

Optimal densities of packings consisting of highly unequal objects

Let $Δ$ be the optimal packing density of $\mathbb R^n$ by unit balls. We show the optimal packing density using two sizes of balls approaches $Δ+ (1 - Δ) Δ$ as the ratio of the radii tends to infinity. More generally, if $B$ is a body and $D$ is a finite set of bodies, then the optimal density $Δ_{\{rB\} \cup D}$ of packings consisting of congruent copies of the bodies from $\{rB\} \cup D$ converges to $Δ_D + (1 - Δ_D) Δ_{\{B\}}$ as $r$ tends to zero.

math.MG↗

arXiv2013

Fourier analysis on finite groups and the Lovász theta-number of Cayley graphs

We apply Fourier analysis on finite groups to obtain simplified formulations for the Lovász theta-number of a Cayley graph. We put these formulations to use by checking a few cases of a conjecture of Ellis, Friedgut, and Pilpel made in a recent article proving a version of the Erdős-Ko-Rado theorem for $k$-intersecting families of permutations. We also introduce a $q$-analog of the notion of $k$-intersecting families of permutations, and we verify a few cases of the corresponding Erdős-Ko-Rado assertion by computer.

math.OC↗

arXiv2012

Upper bounds for packings of spheres of several radii

We 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 bounds for the classical problem of packing identical spheres.

math.MG↗