Related papers: Packings, sausages and catastrophes
The paper is intended to be a survey on some topics within the framework of automorphisms of a relatively free groups of infinite rank. We discuss such properties as tameness, primitivity, small index, Bergman property, and so on.
For a finite set of integers such that the first few gaps between its consecutive elements equal $a$, while the remaining gaps equal $b$, we study dense packings of its translates on the line. We obtain an explicit lower bound on the…
We review several problems of conventional Grand Unification and some new approaches. In particular, we discuss strongly coupled Grand Unified Theories. Standard Model may emerge as a low energy effective theory of composite particles in…
In this article we propose a general method of obtaining infinite sums of products with functions that count patterns in numbers.
Circle packings with specified patterns of tangencies form a discrete counterpart of analytic functions. In this paper we study univalent packings (with a combinatorial closed disk as tangent graph) which are embedded in (or fill) a…
The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted volume with minimum weighted perimeter. According to Chambers' recent proof of the log-convex density conjecture, for many densities on…
We prove results about uniform convergence of densities in the free central limit theorem without assumptions of boundedness on the support.
We show the existence of generalized clusters of a finite or even infinite number of sets, with minimal total perimeter and given total masses, in metric measure spaces homogeneous with respect to a group acting by measure preserving…
Model sets (also called cut and project sets) are generalizations of lattices. Here we show how the self-similarities of model sets are a natural replacement for the group of translations of a lattice. This leads us to the concept of…
We prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds. More generally, we obtain finiteness results for open Riemannian manifolds whose topology is concentrated on compact domains of…
We review a recent development at the interface between discrete mathematics on one hand and probability theory and statistics on the other, specifically the use of Markov chains and their boundary theory in connection with the asymptotics…
A tractable nonparametric prior over densities is introduced which is closed under sampling and exhibits proper posterior asymptotics.
Finite Unified Theories (FUTs) are N=1 supersymmetric Grand Unified Theories that can be made all-loop finite. The requirement of all-loop finiteness leads to a severe reduction of the free parameters of the theory and, in turn, to a large…
In finite probability theory, events are subsets of the outcome set. Subsets can be represented by 1-dimensional column vectors. By extending the representation of events to two dimensional matrices, we can introduce "superposition events."…
We bound several quantities related to the packing density of the patterns 1(L+1)L...2. These bounds sharpen results of B\'ona, Sagan, and Vatter and give a new proof of the packing density of these patterns, originally computed by…
The tight span, or injective envelope, is an elegant and useful construction that takes a metric space and returns the smallest hyperconvex space into which it can be embedded. The concept has stimulated a large body of theory and has…
A mixture of Gaussians fit to a single curved or heavy-tailed cluster will report that the data contains many clusters. To produce more appropriate clusterings, we introduce a model which warps a latent mixture of Gaussians to produce…
A mixture of Gaussians fit to a single curved or heavy-tailed cluster will report that the data contains many clusters. To produce more appropriate clusterings, we introduce a model which warps a latent mixture of Gaussians to produce…
This brief paper develops a probability density that models processes for which the physical mechanism is unknown. It has desirable properties which are not realized by densities derived from Gaussian process or other classic methods. In…
The notion of density of a finite set is discussed. We proof a general theorem of set theory which refines Bose-Einstein distribution.