相关论文: How behave the typical $L^q$-dimensions of measure…

The problem of estimating, from a random sample of points, the dimension of a compact subset $S$ of the Euclidean space is considered. The emphasis is put on consistency results in the statistical sense. That is, statements of convergence…

A statistical measure is given expressing relative occurrences of quantities within a given data set. Application of this measure on several real life physical data sets and some abstract distributions are shown to yield consistent results.…

We imagine an experiment on an unknown quantum mechanical system in which the system is prepared in various ways and a range of measurements are performed. For each measurement M and preparation rho the experimenter can determine, given…

We say that $E$ is a microset of the compact set $K\subset \mathbb{R}^d$ if there exist sequences $\lambda_n\geq 1$ and $u_n\in \mathbb{R}^d$ such that $(\lambda_n K + u_n ) \cap [0,1]^d$ converges to $E$ in the Hausdorff metric, and…

The halfspace depth of a $d$-dimensional point $x$ with respect to a finite (or probability) Borel measure $\mu$ in $\mathbb{R}^d$ is defined as the infimum of the $\mu$-masses of all closed halfspaces containing $x$. A natural question is…

Contours may be viewed as the 2D outline of the image of an object. This type of data arises in medical imaging as well as in computer vision and can be modeled as data on a manifold and can be studied using statistical shape analysis.…

The Kullback-Leibler (KL) divergence is a foundational measure for comparing probability distributions. Yet in multivariate settings, its single value often obscures the underlying reasons for divergence, conflating mismatches in individual…

Using probabilistic ideas, we prove that the packing dimension of a mean porous measure is strictly smaller than the dimension of the ambient space. Moreover, we give an explicit bound for the packing dimension, which is asymptotically…

In this paper we propose numerical measures for evaluating the aesthetic interest of simple patterns. The patterns consist of elements (symbols, pixels, etc.) in regular square arrays. The measures depend on two characteristics of the…

In the framework of the generalized measure theory the decomposable probabilistic-valued set functions are introduced with triangle functions $\tau$ in an appropriate probabilistic metric space as natural candidates for the "addition",…

We study the dimensional properties of Moran sets and Moran measures in doubling metric spaces. In particular, we consider local dimensions and $L^q$-dimensions. We generalize and extend several existing results in this area.

Upper bounds are obtained for the $p$-capacity of compact sets in $\R^d$, with $d \ge 2$ and $1<p<d$. Upper and lower bounds are obtained for the product of $p$-capacity and powers of the $q$-torsional rigidity over the collection of all…

Relying on results due to Shmerkin and Solomyak, we show that outside a $0$-dimensional set of parameters, for every planar homogeneous self-similar measure $\nu$, with strong separation, dense rotations and dimension greater than $1$,…

Given a finite subset $\Sigma\subset\mathbb{R}$ and a positive real number $q<1$ we study topological and measure-theoretic properties of the self-similar set $K(\Sigma;q)=\big\{\sum_{n=0}^\infty…

For a finite partially ordered set we calculate the dimension of the variety of its subspace representations having fixed dimension vector. The dimension is given in terms of the Euler quadratic form associated with a partially ordered set,…

We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre in [SIAM J. Optim. 21(3) (2011), pp. 864--885], obtained by searching for an optimal probability…

Let $K$ be a finite dimensional compact metric space and $K^\mathbb{Z}$ the full shift on the alphabet $K$. We prove that its mean dimension is given by $\dim K$ or $\dim K-1$ depending on the "type" of $K$. We propose a problem which seems…

The Assouad and lower dimensions and dimension spectra quantify the regularity of a measure by considering the relative measure of concentric balls. On the other hand, one can quantify the smoothness of an absolutely continuous measure by…

We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set.…

In the present paper we study quantile risk measures and their domain. Our starting point is that, for a probability measure $ Q $ on the open unit interval and a wide class $ \mathcal{L}_Q $ of random variables, we define the quantile risk…