Related papers: Centering problems for probability measures on fin…

We prove that for any Borel probability measure $\mu$ on $\mathbb R^n$ there exists a set $X\subset \mathbb R^n$ of $n+1$ points such that any $n$-variate quadratic polynomial $P$ that is nonnegative on $X$ (i.e. $P(x)\geq 0$, for every $x…

The aim of this paper is to study the full $K-$moment problem for measures supported on some particular non-linear subsets $K$ of an infinite dimensional vector space. We focus on the case of random measures, that is $K$ is a subset of all…

The main result of this note, Theorem 2, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant under the action of the infinite unitary group and that admits well-defined projections onto the…

The space of unitary $C_{0}$-semigroups on separable infinite dimensional Hilbert space, when viewed under the topology of uniform weak convergence on compact subsets of $\mathbb{R}_{+}$, is known to admit various interesting residual…

Let $S$ be a Polish space and $(X_n:n\geq1)$ an exchangeable sequence of $S$-valued random variables. Let $\alpha_n(\cdot)=P(X_{n+1}\in \cdot\mid X_1,\...,X_n)$ be the predictive measure and $\alpha$ a random probability measure on $S$ such…

Let $(X,\mathcal{B},m,\tau)$ be a dynamical system with $\ds (X,\mathcal{B},m)$ a probability space and $\ds \tau$ an invertible, measure preserving transformation. The present paper deals with the almost everywhere convergence in…

Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are…

With a new proof approach we prove in a more general setting the classical convergence theorem that almost everywhere convergence of measurable functions on a finite measure space implies convergence in measure. Specifically, we generalize…

It is shown that for every measure $m$ on projections in a $W^*$-algebra of type $I_2$, there exists a Hilbert-valued orthogonal vector measure $\mu$ such that $\|\mu(p)\|^2= m(p)$ for every projection $p$. With regard to J. Hamhalter's…

Let $\mu$ be a Borel probability measure associated with an iterated function system consisting of a countably infinite number of contracting similarities and an infinite probability vector. In this paper, we study the quantization…

The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue…

We prove existence and uniqueness of solutions for an entropic version of the semi-geostrophic equations. We also establish convergence to a weak solution of the semi-geostrophic equations as the entropic parameter vanishes. Convergence is…

This paper deals with three major types of convergence of probability measures on metric spaces: weak convergence, setwise converges, and convergence in the total variation. First, it describes and compares necessary and sufficient…

Let $\mathscr{H}$ be a finite-dimensional complex Hilbert space and $\mathscr{D}$ the set of density matrices on $\mathscr{H}$, i.e., the positive operators with trace 1. Our goal in this note is to identify a probability measure $u$ on…

The geometric tangent cone to a probability measure $\mu$ is a set of measure-valued applications that are almost geodesics. This is a nonlocal condition, typically lost when conditioning the measure on a given set. We show that if one…

Given a compact metric space $X$, the collection of Borel probability measures on $X$ can be made into a compact metric space via the Kantorovich metric. We partially generalize this well known result to projection-valued measures. In…

Given a set of points, clustering consists of finding a partition of a point set into $k$ clusters such that the center to which a point is assigned is as close as possible. Most commonly, centers are points themselves, which leads to the…

The center of distances of a metric space $(X,d)$ is the set $C(X)$ of all $t\in \mathbb R^+$ for which the equation $d(x,p)=t$ has a solution for each $p\in X$. We prove the inequality $|C(X)| \le 1 + \lfloor \log_2 n \rfloor$ for all…

We consider the well-studied Robust $(k, z)$-Clustering problem, which generalizes the classic $k$-Median, $k$-Means, and $k$-Center problems. Given a constant $z\ge 1$, the input to Robust $(k, z)$-Clustering is a set $P$ of $n$ weighted…

We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an `almost smooth' structure, with…