Oleg R. Musin

Spherical two-distance sets

A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b so that the inner products of distinct vectors of S are either a or b. It is known that the largest cardinality g(n) of spherical two-distance sets does not exceed n(n+3)/2. This upper bound is known to be tight for n=2,6,22. The set of mid-points of the edges of a regular simplex gives the lower bound L(n)=n(n+1)/2 for g(n). In this paper using the so-called polynomial method it is proved that for nonnegative a+b the largest cardinality of S is not greater than L(n). For the case a+b<0 we propose upper bounds on |S| which are based on Delsartes method. Using this we show that g(n)=L(n) for 6

The Strong Thirteen Spheres Problem

The thirteen spheres problem asks if 13 equal-size non-overlapping spheres in three dimensions can simultaneously touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schütte and van der Waerden only in 1953.A natural extension of this problem is the strong thirteen-sphere problem (or the Tammes problem for 13 points), which calls for finding the maximum radius of and an arrangement for 13 equal-size non-overlapping spheres touching the unit sphere. In this paper, we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on an enumeration of irreducible graphs.

Borsuk-Ulam type theorems for manifolds

This paper establishes a Borsuk-Ulam type theorem for PL-manifolds with a finite group action, depending on the free equivariant cobordism class of a manifold. In particular, necessary and sufficient conditions are considered for a manifold with a free involution to be a Borsuk-Ulam type.

Extensions of Sperner and Tucker's lemma for manifolds

The Sperner and Tucker lemmas are combinatorial analogs of the Brouwer and Borsuk-Ulam theorems with many useful applications. These classic lemmas are concerning labellings of triangulated discs and spheres. In this paper we show that discs and spheres can be substituted by large classes of manifolds with or without boundary.

Codes in spherical caps

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. Several new upper bounds on codes in caps are derived. Applications of these bounds to estimates of the kissing numbers and one-sided kissing numbers are considered. It is proved that the maximum size of codes in spherical caps for large dimensions is determined by the maximum size of spherical codes, so these problems are asymptotically equivalent.

The Tammes Problem for N = 14

The Tammes problem is to find the arrangement of N points on a unit sphere which maximizes the minimum distance between any two points. This problem is presently solved for several values of N, namely for N = 3, 4, 6, 12 by L. Fejes Tóth (1943); for N = 5, 7, 8, 9 by Schütte and van der Waerden (1951); for N = 10, 11 by Danzer (1963); and for N = 24 by Robinson (1961). Recently, we solved the Tammes problem for N = 13. The optimal configuration of 14 points was conjectured more than 60 years ago. In this article, we give a solution for this long-standing open problem in geometry. Our computer-assisted proof relies on an enumeration of the irreducible contact graphs.

Optimal Packings of Congruent Circles on a Square Flat Torus

We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason—the problem of “super resolution of images.” We have found optimal arrangements for

Positive definite functions in distance geometry

N=6