Related papers: Universal bounds for spherical codes: the Levensht…
We consider bounds on codes in spherical caps and related problems in geometry and coding theory. An extension of the Delsarte method is presented that relates upper bounds on the size of spherical codes to upper bounds on codes in caps.…
In this paper, we consider the Cauchy-Riemann equation $\bar\partial u= f$ in a new class of convex domains in $\C^n.$ We prove that under $L^p$ data, we can choose a solution in the Lipschitz space $\Lambda_{\alpha},$ where $\alpha$ is an…
We show bounds on five- and six-dimensional universal extra dimension (UED) models from the latest results of the Higgs searches at the LHC and from the electroweak precision data for the S and T parameters. We consider the minimal UED…
A spherical three-distance set is a finite collection $X$ of unit vectors in $\mathbb{R}^{n}$ such that for each pair of distinct vectors has three inner product values. We use the semidefinite programming method to improve the upper bounds…
We use linear programming techniques to find points of absolute minimum over the unit sphere $S^{d}$ in $\mathbb R^{d+1}$ of the total potential of a point configuration $\omega_N\subset S^{d}$ which is a spherical $(2m-1)$-design contained…
In the setting of a Gaussian channel without power constraints, proposed by Poltyrev, the codewords are points in an n-dimensional Euclidean space (an infinite constellation) and the tradeoff between their density and the error probability…
Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis. We introduce a new class of upper bounds on the log partition…
Certain simplicial complexes are used to construct a subset $D$ of $\mathbb{F}_{2^n}^m$ and $D$, in turn, defines the linear code $C_{D}$ over $\mathbb{F}_{2^n}$ that consists of $(v\cdot d)_{d\in D}$ for $v\in \mathbb{F}_{2^n}^m$. Here we…
We show that the spectral embeddings of all known triangle-free strongly regular graphs are optimal spherical codes (the new cases are $56$ points in $20$ dimensions, $50$ points in $21$ dimensions, and $77$ points in $21$ dimensions), as…
We investigate the most general phase space of configurations, consisting of the collection of all possible ways of assigning elementary attributes, "energies", to elementary positions, "cells". We discuss how this space defines a…
This paper concerns parameterized convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional Banach (resp. finite-dimensional) spaces and that are indexed by an arbitrary fixed set T…
For a large class of optimization problems, namely those that can be expressed as finite-valued constraint satisfaction problems (VCSPs), we establish a dichotomy on the number of levels of the Lasserre hierarchy of semi-definite programs…
Functional and linear-algebraic approaches to the Delsarte problem of upper bounds on codes are discussed. We show that Christoffel-Darboux kernels and Levenshtein polynomials related to them arise as stationary points of the moment…
We introduce in this article a new method to estimate the minimum distance of codes from algebraic surfaces. This lower bound is generic, i.e. can be applied to any surface, and turns out to be ``liftable'' under finite morphisms, paving…
We study the problem of constructing $k$-spectral minimal partitions of domains in $d$ dimensions, where the energy functional to be minimized is a $p$-norm ($1 \le p \le \infty$) of the infimum of the spectrum of a suitable Schr\"odinger…
We prove that the lacunary spherical maximal operator, defined on the $n$-dimensional real hyperbolic space, is bounded on $L^p(\mathbb{H}^n)$ for all $n\ge2$ and $1<p\le\infty$. In particular, the lacunary set is significantly larger than…
We develop lower bounds for the energy of configurations in $\mathbb{R}^d$ periodic with respect to a lattice. In certain cases, the construction of sharp bounds can be formulated as a finite dimensional, multivariate polynomial…
We introduce the notion of p-adic spherical codes (in particular, p-adic kissing number problem). We show that the one-line proof for a variant of the Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein upper bound for spherical codes,…
The celebrated Dvoretzky theorem asserts that every $N$-dimensional convex body admits central sections of dimension $d = \Omega(\log N)$, which is nearly spherical. For many instances of convex bodies, typically unit balls with respect to…
We give explicit time lower bounds in the Lebesgue spaces for all nontrivial solutions of nonlinear Schr\"odinger equations bounded in the energy space. The result applies for these equations set in any domain of $\R^N,$ including the whole…