Related papers: On Divergence-based Distance Functions for Multipl…

The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the…

It has been recently established by the first and third author that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the…

Differential privacy is a widely studied notion of privacy for various models of computation. Technically, it is based on measuring differences between probability distributions. We study $\epsilon,\delta$-differential privacy in the…

We consider the (unoriented) long-range percolation on Z^d in dimensions d\ge1, where distinct sites x,y\in Z^d get connected with probability p_{xy}\in[0,1]. Assuming p_{xy}=|x-y|^{-s+o(1)} as |x-y|\to\infty, where s>0 and |\cdot| is a…

We prove that an open set $\Omega \subset \mathbb{R}^n$ can be approximated by smooth sets of uniformly bounded perimeter from the interior if and only if the open set $\Omega$ satisfies \begin{align*} &\qquad…

Divergence functions are interesting discrepancy measures. Even though they are not true distances, we can use them to measure how separated two points are. Curiously enough, when they are applied to random variables, they lead to a notion…

Distance Geometry is based on the inverse problem that asks to find the positions of points, in a Euclidean space of given dimension, that are compatible with a given set of distances. We briefly introduce the field, and discuss some open…

Letting $\mathcal{M}$ denote the space of finite measures on $\mathbb{N}$, and $\mu_\lambda\in\mathcal{M}$ denote the Poisson distribution with parameter $\lambda$, the function $W:[0,1]^2\to\mathcal{M}$ given by \[ W(x,y)=\mu_{c\log x\log…

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…

We prove that, if $\Omega\subset \mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial \Omega$ of the function $$\varphi(y) = \int_0^{\lambda(y)} \prod_{j=1}^{n-1}[1-t \kappa_j(y)]\, dt$$ implies…

Computing the reachability probability in infinite state probabilistic models has been the topic of numerous works. Here we introduce a new property called \emph{divergence} that when satisfied allows to compute reachability probabilities…

Consider an unlimited homogeneous medium disturbed by points generated via Poisson process. The neighborhood of a point plays an important role in spatial statistics problems. Here, we obtain analytically the distance statistics to $k$th…

We prove sharp near-diagonal pointwise bounds for the Green function $G_\Omega(x,y)$ for nonlocal operators of fractional order $\alpha \in (0,2)$. The novelty of our results is two-fold: the estimates are robust as $\alpha \to 2-$ and we…

Let $(N,g)$ be a Riemannian manifold with the distance function $d(x,y)$ and an open subset $M\subset N$. For $x\in M$ we denote by $D_x$ the distance difference function $D_x:F\times F\to \mathbb R$, given by…

This paper develops systematic approaches to obtain $f$-divergence inequalities, dealing with pairs of probability measures defined on arbitrary alphabets. Functional domination is one such approach, where special emphasis is placed on…

We extend the upper estimates obtained by M. Carlehed and B.-Y. Chen about the ratio of the classical and pluricomplex Green functions to the case of $\mathcal C^2$-smooth locally $\mathbb C$-convexifiable domains of finite type. We also…

We prove a number of double-sided estimates relating discrete counterparts of several classical conformal invariants of a quadrilateral: cross-ratios, extremal lengths and random walk partition functions. The results hold true for any…

A k-dissimilarity map on a finite set X is a function D : X \choose k \rightarrow R assigning a real value to each subset of X with cardinality k, k \geq 2. Such functions, also sometimes known as k-way dissimilarities, k-way distances, or…

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 when the local Lipschitz property of the real-valued function $g(z) = d_Y (f(z),A)$ implies the global Lipschitz property of the mapping $f:X\to Y$ between the metric spaces $(X,d_X)$ and $(Y,d_Y)$. Here, $d_Y(y,A)$ denotes…