Related papers: Kissing number in hyperbolic space
Delsarte's method and its extensions allow to consider the upper bound problem for codes in 2-point-homogeneous spaces as a linear programming problem with perhaps infinitely many variables, which are the distance distribution. We show that…
We obtain lower bounds on the rank of the kappa ring of the Delign-Mumford compactification of the moduli space of curves in different degrees. For this purpose, we introduce a quotient of the kappa ring, the combinatorial kappa ring, and…
I. J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a nonnegative linear combination of Gegenbauer polynomials. This fact play a crucial role in Delsarte's method for finding…
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 boundedness of the kissing numbers of convex bodies has been known to Hadwiger for long. We present an application of it to the sum-product estimate…
We investigate the following question: how close can two disjoint lattice polytopes contained in a fixed hypercube be? This question stems from various contexts where the minimal distance between such polytopes appears in complexity bounds…
Let $K: \boldsymbol{\Omega}\times \boldsymbol{\Omega}$ be a continuous Mercer kernel defined on a compact subset of ${\mathbb R}^n$ and $\mathcal{H}_K$ be the reproducing kernel Hilbert space (RKHS) associated with $K$. Given a finite…
Let $E$ be the union of two real intervals not containing zero. Then $L_n^r(E)$ denotes the supremum norm of that polynomial $P_n$ of degree less than or equal to $n$, which is minimal with respect to the supremum norm provided that…
We briefly review known results on upper bounds for the minimal domination number $\gamma_n$ of a hypercube of dimension $n$, then present a new method for constructing dominating sets. Write $n =2^{\hat{n}}-1 +{\check{n}}$ with $0\leq…
We consider low energy configurations for the Heitmann-Radin sticky discs functional, in the limit of diverging number of discs. More precisely, we renormalize the Heitmann-Radin potential by subtracting the minimal energy per particle,…
We investigate the photonsphere in the background of regular asymptotically flat compact stars. The analysis includes the general hairy compact star considering the matter fields' backreaction on the metric in various gravity theories. We…
The width $w$ of a curve $\gamma$ in Euclidean space $R^n$ is the infimum of the distances between all pairs of parallel hyperplanes which bound $\gamma$, while its inradius $r$ is the supremum of the radii of all spheres which are…
A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be…
Let $ S $ be a hyperbolic surface. We investigate the topology of the space of all curves on $ S $ which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval $…
A binary code with covering radius $R$ is a subset $C$ of the hypercube $Q_n=\{0,1\}^n$ such that every $x\in Q_n$ is within Hamming distance $R$ of some codeword $c\in C$, where $R$ is as small as possible. For a fixed coordinate…
In [14] we found the large genus asymptotics of Hurwitz numbers for the Riemann sphere with a fixed number of general profiles and some (2,1^{d-2}) profiles. In this paper, motivated from [3], we generalize these results to Hurwitz numbers…
We discuss the high density behavior of a system of hard spheres of diameter d on the hypercubic lattice of dimension n, in the limit n -> oo, d -> oo, d/n=delta. The problem is relevant for coding theory. We find a solution to the…
We improve Larman's bound on the diameter of a polytope by showing that if $\Delta$ is a normal simplicial complex, all of whose missing faces have size at most $r$, then the diameter of the facet-ridge graph of $\Delta$ is not larger than…
In this paper, we obtain new asymptotic lower bounds for the chromatic numbers of spheres.
For a non-empty compact set $E$ in a proper subdomain $\Omega$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $\Omega$ by $d(E)$ and $d(E,\partial\Omega),$ respectively. The quantity…