Related papers: Universal minima of potentials of certain spherica…

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…

This article is devoted to the study of discrete potentials on the sphere in $\mathbb{R}^n$ for sharp codes. We show that the potentials of most of the known sharp codes attain the universal lower bounds for polarization for spherical…

We use methods of approximation theory to find the absolute minima on the sphere of the potential of spherical $(2m-3)$-designs with a non-trivial index $2m$ that are contained in a union of $m$ parallel hyperplanes, $m\geq 2$, whose…

In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions…

Three-point semidefinite programming bounds are one of the most powerful known tools for bounding the size of spherical codes. In this paper, we use them to prove lower bounds for the potential energy of particles interacting via a pair…

We identify universal polar dual pairs of spherical codes $C$ and $D$ such that for a large class of potential functions $h$ the minima of the discrete $h$-potential of $C$ on the sphere occur at the points of $D$ and vice versa. Moreover,…

We give some new explicit examples of putatively optimal projective spherical designs. i.e., ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in…

We derive universal lower and upper bounds for max-min and min-max problems (also known as polarization) for the potential of spherical $(k,k)$-designs and provide certain examples, including unit-norm tight frames, that attain these…

We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m…

We consider the problem of finding an $N$-point configuration on the sphere $S^d\subset \RR^{d+1}$ with the smallest absolute maximum value over $S^d$ of its total potential. The potential induced by each point ${\bf y}$ in a given…

In this paper we present the results of computer searches using a variation of an energy minimization algorithm used by Kottwitz for finding good spherical codes. We prove that exact codes exist by representing the inner products between…

We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense -- they cannot be improved by employing polynomials of the same or lower degrees…

We derive general linear programming bounds for spherical $(k,k)$-designs. This includes lower bounds for the minimum cardinality and lower and upper bounds for minimum and maximum energy, respectively. As applications we obtain a universal…

The first 195 spherical codes for the global minima of 1 to 65 points on S2 have been obtained for 3 types of potentials: logarithmic, Coulomb, called the Thomson problem, and the inverse square law, with 77, 38, and 38 digits precision…

Given a set $P$ of points and a set $U$ of axis-parallel unit squares in the Euclidean plane, a minimum ply cover of $P$ with $U$ is a subset of $U$ that covers $P$ and minimizes the number of squares that share a common intersection,…

This paper investigates the existence and properties of spherical $5$-designs of minimal type. We focus on two cases: tight spherical $5$-designs and antipodal spherical $4$-distance $5$-designs. We prove that a tight spherical $5$-design…

In this article we investigate the $N$-point min-max and the max-min polarization problems on the sphere for a large class of potentials in $\mathbb{R}^n$. We derive universal lower and upper bounds on the polarization of spherical designs…

Let $K$ be a centrally symmetric spherical and simplicial polytope, whose vertices form a $\frac{1}{4n}-$net in the unit sphere in $\mathbb{R}^n$. We prove a uniform lower bound on the norms of all hyperplane projections $P: X \to X$, where…

We prove that there exists a>0 such that for any integer d>2 and any topological types S_1,...,S_n of plane curve singularities, satisfying $\mu(S_1)+...+\mu(S_n) \leq ad^2$, there exists a reduced irreducible plane curve of degree d with…

Alon and F\"uredi (European J. Combin. 1993) gave a tight bound for the following hyperplane covering problem: find the minimum number of hyperplanes required to cover all points of the n-dimensional hypercube {0,1}^n except the origin.…