Related papers: Boundary density and Voronoi set estimation for ir…

We establish a density variant of the Frankl-R\"{o}dl theorem on the sphere $\mathbb{S}^{n-1}$, which concerns avoiding pairs of vectors with a specific distance, or equivalently, a prescribed inner product. In particular, we establish…

As a fundamental piece of multi-object Bayesian inference, multi-object density has the ability to describe the uncertainty of the number and values of objects, as well as the statistical correlation between objects, thus perfectly matches…

Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a 100-year-old lower bound on the…

In this paper, we introduce the boundary $\mathcal{U}X$ of a coarse proximity space $(X,\mathcal{B},{\bf b}).$ This boundary is a subset of the boundary of a certain Smirnov compactification. We show that $\mathcal{U}X$ is compact and…

The Voronoi Density Estimator (VDE) is an established density estimation technique that adapts to the local geometry of data. However, its applicability has been so far limited to problems in two and three dimensions. This is because…

We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss…

In this thesis we study sets of points in the plane and their Voronoi diagrams, in particular when the points coincide. We bring together two ways of studying point sets that have received a lot of attention in recent years: Voronoi…

We study the partial regularity problem of the three-dimensional incompressible Navier--Stokes equations. We present a new boundary regularity criterion for boundary suitable weak solutions. As an application, a bound for the parabolic…

In [Math. Meth. Appl. Sci. 19 (1996) 53-62], C. Marchioro examined the problem of vorticity confinement in the exterior of a smooth bounded domain. The main result in Marchioro's paper is that solutions of the incompressible 2D Euler…

We show that in any $d$-dimensional real normed space, unit balls can be packed with density at least \[\frac{(1-o(1))d\log d}{2^{d+1}},\] improving a result of Schmidt from 1958 by a logarithmic factor and generalizing the recent result of…

In the course of classifying generic sparse polynomial systems which are solvable in radicals, Esterov recently showed that the volume of the Minkowski sum $P_1+\dots+P_d$ of $d$-dimensional lattice polytopes is bounded from above by a…

It is well known that almost every dilation of a sequence of real numbers, that diverges to $\infty$, is dense modulo~1. This paper studies the exceptional set of points -- those for which the dilation is not dense. Specifically, we…

We consider the asymmetric simple exclusion process in $d\ge 3$ with open boundaries. The particle reservoirs of constant densities are modeled by birth and death processes at the boundary. We prove that, if the initial density and the…

Bernoulli convolutions form a one-parameter family of self-similar measures on the unit interval. We suggest to study their two-dimensional density which has an intricate combinatorial structure. Visualizing this structure we discuss…

Alexandrov's inequalities imply that for any convex body $A$, the sequence of intrinsic volumes $V_1(A),\ldots,V_n(A)$ is non-increasing (when suitably normalized). Milman's random version of Dvoretzky's theorem shows that a large initial…

We extend a well-known theorem by Jones and Makarov [JM] on the singularity of boundary distortion of planar conformal mappings. We use a different technique to recover the previous result and, moreover, generalize the result for…

We study the regularity of the distance function to the boundary of a domain in $\mathbb{R}^n$, with respect to the Minkowski functional of a convex polytope. We obtain the regularity of the distance function in certain cases. We also…

A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in \R^3. In this paper we show that the Minkowski…

We consider the imitative monomer-dimer model on the complete graph introduced in [1]. It was understood that this model is described by the monomer density and has a phase transition along certain critical line. By reverting the model to a…

We study in details the bias and variance of the entropy estimator proposed by Kozachenko and Leonenko for a large class of densities on $\mathbb{R}^d$. We then use the work of Bickel and Breiman to prove a central limit theorem in…