It is proposed to create materials with a desired refraction coefficient in a bounded domain D ⊂ R by embedding many small balls with constant refraction coefficients into a given material. The number of small balls per unit volume around every point x ∈ D, i.e., their density distribution, is calculated, as well as the constant refraction coefficients in these balls. Embedding into D small bal...

The ball hull mapping β associates with each closed bounded convex setK in a Banach space its ball hull β(K), defined as the intersection of all closed balls containing K. We are concerned in this paper with continuity and Lipschitz continuity (with respect to the Hausdorff metric) of the ball hull mapping. It is proved that β is a Lipschitz map in finite dimensional polyhedral spaces. Both pro...

We analyze two classes of Kuramoto models on spheres that have been introduced in previous studies. Our analysis is restricted to ensembles identical oscillators with the global coupling. In such a setup, an additional assumption initial distribution uniform sphere, one can derive equations for order parameters closed form. The rate synchronization real model depends dimension sphere. Specifica...

Abstract: In the "tennis ball" problem we are given successive pairs of balls numbered (1,2), (3,4),... At each stage we throw one ball out of the window. After n stages some set of n balls is on the lawn. We find a generating function and a closed formula for the sequence 3,23,131,664,3166,14545,65187,287060,1247690,..., the n-th term of which gives the sum over all possible arrangements of th...

We study a variant of intersection representations with unit balls: unit disks in the plane and unit intervals on the line. Given a planar graph and a bipartition of the edges of the graph into near and far edges, the goal is to represent the vertices of the graph by unit-size balls so that the balls for two adjacent vertices intersect if and only if the corresponding edge is near. We consider ...

Consider two unit balls in a d-dimensional flat torus with edge length r, for d ≥ 2. The balls do not move by themselves but they are pushed by a Brownian motion. The balls never intersect—they reflect if they touch. It is proved that the joint distribution of the processes representing the centers of the balls converges to the distribution of two independent Brownian motions when r →∞, assumin...

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a “fat” one, if it contains the centers of its generating balls. The main result of this paper is a proof of the Illumination Conjecture for “fat” spindle convex b...

Consider sequential packing of unit volume balls in a large cube, in any dimension and with Poisson input. We show after suitable rescaling that the spatial distribution of packed balls tends to that of a Gaussian field in the thermodynamic limit. The results cover related applied models, including ballistic deposition and spatial birth-growth models.