10 research outputs found
Spherical sets avoiding a prescribed set of angles
Let be any subset of the interval . A subset of the unit
sphere in will be called \emph{-avoiding} if for any
. The problem of determining the maximum surface measure of a -avoiding set was first stated in a 1974 note by Witsenhausen; there the
upper bound of times the surface measure of the sphere is derived from a
simple averaging argument. A consequence of the Frankl-Wilson theorem is that
this fraction decreases exponentially, but until now the upper bound for
the case has not moved. We improve this bound to using an
approach inspired by Delsarte's linear programming bounds for codes, combined
with some combinatorial reasoning. In the second part of the paper, we use
harmonic analysis to show that for there always exists an
-avoiding set of maximum measure. We also show with an example that a
maximiser need not exist when .Comment: 21 pages, 3 figure
Fourier analysis on finite groups and the Lov\'asz theta-number of Cayley graphs
We apply Fourier analysis on finite groups to obtain simplified formulations
for the Lov\'asz 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 Erd\H{o}s-Ko-Rado theorem for
-intersecting families of permutations. We also introduce a -analog of
the notion of -intersecting families of permutations, and we verify a few
cases of the corresponding Erd\H{o}s-Ko-Rado assertion by computer.Comment: 9 pages, 0 figure
Complete positivity and distance-avoiding sets
We introduce the cone of completely-positive functions, a subset of the cone
of positive-type functions, and use it to fully characterize maximum-density
distance-avoiding sets as the optimal solutions of a convex optimization
problem. As a consequence of this characterization, it is possible to reprove
and improve many results concerning distance-avoiding sets on the sphere and in
Euclidean space.Comment: 57 pages; minor corrections in comparison to the previous versio
Complete positivity and distance-avoiding sets
We introduce the cone of completely positive functions, a subset of the cone of positive-type functions, and use it to fully characterize maximum-density distance-avoiding sets as the optimal solutions of a convex optimization problem. As a consequence of this characterization, it is possible to reprove and improve many results concerning distance-avoiding sets on the sphere and in Euclidean space
Spectral bounds for the independence ratio and the chromatic number of an operator
We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L^2-space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numerical range of the operator. This provides a theoretical framework in which many packing and coloring problems for finite and infinite graphs can be conveniently studied with the help of harmonic analysis and convex optimization. The theory is applied to infinite geometric graphs on Euclidean space and on the unit sphere
SPECTRAL BOUNDS FOR THE INDEPENDENCE RATIO AND THE CHROMATIC NUMBER OF AN OPERATOR
We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L2-space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numerical range of the operator. This provides a theoretical framework in which many packing and coloring problems for finite and infinite graphs can be conveniently studied with the help of harmonic analysis and convex optimization. The theory is applied to infinite geometric graphs on Euclidean space and on the unit sphere
Complete positivity and distance-avoiding sets
We introduce the cone of completely positive functions, a subset of the cone of positive-type functions, and use it to fully characterize maximum-density distance-avoiding sets as the optimal solutions of a convex optimization problem. As a consequence of this characterization, it is possible to reprove and improve many results concerning distance-avoiding sets on the sphere and in Euclidean space.Optimizatio