Related papers: Size dependence of the largest distance between ra…

This paper deals with maximization of classical $f$-divergence between the distributions of a measurement outputs of a given pair of quantum states. $f$-divergence $D_{f}$ between the probability density functions $p_{1}$ and $p_{2}$ over a…

In [3], algorithms to compute the density of the distance to a random variable uniformly distributed in (a) a ball, (b) a disk, (c) a line segment, or (d) a polygone were introduced. For case (d), the algorithm, based on Green's theorem,…

We investigate large deviations for the empirical measure of the position and momentum of a particle traveling in a box with hot walls. The particle travels with uniform speed from left to right, until it hits the right boundary. Then it is…

In the maximum independent set of convex polygons problem, we are given a set of $n$ convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the…

Diffusion invariably involves motion within a medium. An universal behavior observed is that self diffusivity exhibits a maximum as a function of the size of the diffusant when the diffusant is confined to a medium, as a result of what is…

The convergence of a sequence of point processes with dependent points, defined by a symmetric function of iid high-dimensional random vectors, to a Poisson random measure is proved. This also implies the convergence of the joint…

Gaussian random fields on Euclidean spaces whose variances reach their maximum values at unique points are considered. Exact asymptotic behaviors of probabilities of large absolute maximum of theirs trajectories have been evaluated using…

For a finite set of points $P$ in $R^d$, the function $d_P: R^d \to R^+$ measures Euclidean distance to the set $P$. We study the number of critical points of $d_P$ when $P$ is a Poisson process. In particular, we study the limit behavior…

Let $p$ be an unknown and arbitrary probability distribution over $[0,1)$. We consider the problem of {\em density estimation}, in which a learning algorithm is given i.i.d. draws from $p$ and must (with high probability) output a…

Circles of a single size can pack together densely in a hexagonal lattice, but adding in size variety disrupts the order of those packings. We conduct simulations which generate dense random packings of circles with specified size…

We consider the following variant of the Monge-Kantorovich transportation problem. Let S be a finite set of point sites in d dimensions. A bounded set C in d-dimensional space is to be distributed among the sites p in S such that (i) each p…

We study the sizes of delta-additive sets of unit vectors in a d-dimensional normed space: the sum of any two vectors has norm at most delta. One-additive sets originate in finding upper bounds of vertex degrees of Steiner Minimum Trees in…

We obtain large deviation results for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space…

Optimal transport and Wasserstein distances are flourishing in many scientific fields as a means for comparing and connecting random structures. Here we pioneer the use of an optimal transport distance between L\'{e}vy measures to solve a…

Density based spatial clustering of points in $\mathbb{R}^n$ has a myriad of applications in a variety of industries. We generalise this problem to the density based clustering of lines in high-dimensional spaces, keeping in mind there…

Consider a geometric range space $(X,\c{A})$ where each data point $x \in X$ has two or more values (say $r(x)$ and $b(x)$). Also consider a function $\Phi(A)$ defined on any subset $A \in (X,\c{A})$ on the sum of values in that range e.g.,…

In wireless networks, the knowledge of nodal distances is essential for several areas such as system configuration, performance analysis and protocol design. In order to evaluate distance distributions in random networks, the underlying…

Let $S$ be a set of $n$ points in $d$-dimensional Euclidean space. Assign to each $x\in S$ an arbitrary distance $r(x)>0$. Let $e_r(x,S)$ denote the number of points in $S$ at distance $r(x)$ from $x$. Avis, Erd\"os and Pach (1988)…

Given i.i.d samples from some unknown continuous density on hyper-rectangle $[0, 1]^d$, we attempt to learn a piecewise constant function that approximates this underlying density non-parametrically. Our density estimate is defined on a…

The aggregation of particles in the free molecular regime is determined approximately for situations with a high degree of translational energy equilibration. The mean particle sizes develop linearly in time. Scaling relations are used to…