Related papers: Renyi Dimension and Gaussian Filtering

Let $(M,g)$ be a compact, connected Riemannian manifold of dimension $n\ge 2$, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions $-\Delta_g e_j=\lambda_j^2 e_j$. Given a finite Borel measure $\mu$ on $M$,…

Let $\Lambda_s$ denote the Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$, which consists of all $f\in\mathfrak{C}\cap L^\infty$ such that, for some constant $L\in(0,\infty)$ and some integer $r\in(s,\infty)$, \begin{equation*}…

We discuss some properties of the generalized entropies, called Renyi entropies and their application to the case of continuous distributions. In particular it is shown that these measures of complexity can be divergent, however, their…

The $q$-Whittaker function $W_\lambda(\mathbf{x};q)$ associated to a partition $\lambda$ is a $q$-analogue of the Schur function $s_\lambda(\mathbf{x})$, and is defined as the $t=0$ specialization of the Macdonald polynomial…

We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several…

This paper deals with the quantitative normal approximation of non-linear functionals of Poisson random measures, where the quality is measured by the Kolmogorov distance. Combining Stein's method with the Malliavin calculus of variations…

In this paper we study algorithms to find a Gaussian approximation to a target measure defined on a Hilbert space of functions; the target measure itself is defined via its density with respect to a reference Gaussian measure. We employ the…

Given positive measures $\nu,\mu$ on an arbitrary measurable space $(\Omega, \mathcal F)$, we construct a sequence of finite partitions $(\pi_n)_n$ of $(\Omega, \mathcal F)$ s.t. $$ \sum_{A\in \pi_n: \mu(A)>0} 1_{A} \frac{\nu(A)}{\mu(A)}…

We introduce a universality theorem for functionals of measures on partitions which "behave like" the Ewens measure. Various limit theorems for the Ewens measure, most notably the Poisson-Dirichlet limit for the longest parts, the…

The concept of a hyperuniformity disorder length $h$ was recently introduced for analyzing volume fraction fluctuations for a set of measuring windows. This length permits a direct connection to the nature of disorder in the spatial…

Assume that we observe i.i.d.~points lying close to some unknown $d$-dimensional $\mathcal{C}^k$ submanifold $M$ in a possibly high-dimensional space. We study the problem of reconstructing the probability distribution generating the…

In this article we prove dimension free $L^p$-boundedness of Riesz transforms associated with a Bessel diferential operator. We obtain explicit estimates of the $L^p$-norms for the Bessel-Riesz transforms in terms of p, establishing a…

In a previous paper the boundedness properties of Riesz Potentials, Bessel potentials and Fractional Derivatives were studied in detail on Gaussian Besov-Lipschitz spaces $B_{p,q}^{\alpha}(\gamma_d)$. In this paper we will continue our…

This paper contributes to foundations of the geometric measure theory in the infinite dimensional setting of the configuration space over the Euclidean space $\mathbb R^n$ equipped with the Poisson measure $\pi$. We first provide a rigorous…

Probability functions figure prominently in optimization problems of engineering. They may be nonsmooth even if all input data are smooth.This fact motivates the consideration of subdifferentials for such typically just continuous…

Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every $d\geq 1$, if $P$ is a poset and the dimension of a subposet $B$ of $P$…

For a polynomial f: {-1, 1}^n --> C, we define the partition function as the average of e^{lambda f(x)} over all points x in {-1, 1}^n, where lambda in C is a parameter. We present a quasi-polynomial algorithm, which, given such f, lambda…

The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds…

Let $D\in\mathbb{N}$, $q\in[2,\infty)$ and $(\mathbb{R}^D,|\cdot|,dx)$ be the Euclidean space equipped with the $D$-dimensional Lebesgue measure. In this article, via an auxiliary function space $\mathrm{WE}^{1,\,q}(\mathbb R^D)$ defined…

We consider the value distribution of logarithms of symmetric power L-functions associated with newforms of even weight and prime power level. In the symmetric square case, under certain plausible analytical conditions, we prove that…