Radon measures belonging to the negative Sobolev space $H^{-1}(\mathbb{R}^2)$ are important from the point of view of fluid mechanics as they model vorticity of vortex-sheet solutions of incompressible Euler equations. In this note we discuss regular...
It is known that the heuristic principle, referred to as the multifractal formalism, need not hold for self-similar measures with overlap, such as the $3$-fold convolution of the Cantor measure and certain Bernoulli convolutions. In this paper we stu...
The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In particular, this...
This is a discussion on "Sparse graphs using exchangeable random measures" by Francois Caron and Emily B. Fox, published in Journal of the Royal Statistical Society, Series B, 2017....
Importance measures provide a systematic approach to scrutinize critical system components, which are extremely beneficial in making important decisions, such as prioritizing reliability improvement activities, identifying weak-links and effective us...
Associative measures are "mathematical formulas determining the strength of association between two or more words based on their occurrences and cooccurrences in a text corpus" (Pecina, 2010, p. 138). The purpose of this paper is to test the 12 assoc...
Deep Neural Networks can generalize despite being significantly overparametrized. Recent research has tried to examine this phenomenon from various view points and to provide bounds on the generalization error or measures predictive of the generaliza...
We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from zero, we...
Topic models extract representative word sets - called topics - from word counts in documents without requiring any semantic annotations. Topics are not guaranteed to be well interpretable, therefore, coherence measures have been proposed to distingu...
We study the computational complexity of two Boolean nonlinearity measures: the nonlinearity and the multiplicative complexity. We show that if one-way functions exist, no algorithm can compute the multiplicative complexity in time $2^{O(n)}$ given t...
Since the pioneering works of Jakobson and Benedicks & Carleson and others, it has been known that a positive measure set of quadratic maps admit invariant probability measures absolutely continuous with respect to Lebesgue. These measures allow one...
In this paper I study properties of the generators $\triangle_γ$ of non-local Dirichlet forms $\mathcal{E}^μ_γ$ on ultrametric spaces which are the path space of simple stationary Bratteli diagrams. The measures used to define the Dirichlet forms...
We investigate the estimation of an optimal transport map between probability measures on an infinite-dimensional space and reveal its minimax optimal rate. Optimal transport theory defines distances within a space of probability measures, utilizing...
The aim of this work is to provide a geometric characterization of the positive Radon measures $μ$ with compact support on the plane such that the associated Cauchy transform defines a compact operator from $L^2(μ)$ to $L^2(μ).$ It turns out that...
We derive new explicit bounds for the total variation distance between two convolution products of $n$ probability distributions, one of which having identical convolution factors. Approximations by finite signed measures of arbitrary order are con...
We consider the approximation of a convolution of possibly different probability measures by (compound) Poisson distributions and also by related signed measures of higher order. We present new total variation bounds having a better structure than th...
Peer-evaluation based measures of group research quality such as the UK's Research Assessment Exercise (RAE), which do not employ bibliometric analyses, cannot directly avail of such methods to normalize research impact across disciplines. This is se...
For many IT-security measures exact costs and benefits are not known. This makes it difficult to allocate resources optimally to different security measures. We present a model for costs and benefits of so called Honeynets. This can foster informed...
What causes house prices to rise and fall? Economists identify household access to credit as a crucial factor. "Loan-to-Value" and "Debt-to-GDP" ratios are the standard measures for credit access. However, these measures fail to explain the depth of...
We introduce an extension of the Pólya tree approach for constructing distributions on the space of probability measures. By using optional stopping and optional choice of splitting variables, the construction gives rise to random measures that are...