Abstract. 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.
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.
Let X be any subset of the interval [−1, 1]. A subset I of the unit sphere in R n will be called X-avoiding if u, v / ∈ X for any u, v ∈ I . The problem of determining the maximum surface measure of a {0}-avoiding set was first stated in a 1974 note by Witsenhausen;there the upper bound of 1/n 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 1/3 upper bound for the case n= 3 has not moved. We improve this bound to 0.313 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 n≥ 3, there always exists an X-avoiding set of maximum measure. We also show with an example that a maximizer need not exist when n= 2.
Abstract. We apply Fourier analysis on finite groups to obtain simplified formulations for the Lovász ϑ-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.
Abstract-The problem of joint zero-forcing (ZF) beamforming (BF) together with optimal power allocation (PA) and antenna selection (AS) for throughput maximization is considered in this paper for multi-user multiple input single output (MU-MISO) systems. We introduce a new formulation for the joint ZF and PA problem by adapting the algebraic subspace approach which finds a proper set for the optimization variable that inherently satisfies the ZF constraints. Also, the squared group Lasso penalty on the BF matrix is used to linearize (relax) the non-convex, NP-hard problem of joint BF and AS. Extensive simulations show that for the throughput problem, the proposed algorithm performs very closely to the optimal (exhaustive search) joint approach.
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