Related papers: Supports of Measures in a free additive convolutio…

Given a parabolic cylinder $Q =(0,T)\times\Omega$, where $\Omega\subset \mathbb{R}^{N}$ is a bounded domain, we prove new properties of solutions of \[ u_t-\Delta_p u = \mu \quad \text{in $Q$} \] with Dirichlet boundary conditions, where…

We study the quantization for a class of in-homogeneous self-similar measures $\mu$ supported on self-similar sets. Assuming the open set condition for the corresponding iterated function system, we prove the existence of the quantization…

Let $ T\colon[0,1]^d\to [0,1]^d $ be a piecewise expanding map with an absolutely continuous invariant measure $ \mu $. Let $ \{H_n\} $ be a sequence of hyperrectangles or hyperboloids centered at the origin. Denote by $ \mathcal R(\{H_n\})…

The half-open real unit interval (0,1] is closed under the ordinary multiplication and its residuum. The corresponding infinite-valued propositional logic has as its equivalent algebraic semantics the equational class of cancellative hoops.…

Let \(\mu\) be a finite Borel measure on \((-\pi,\pi)\). Consider the one-dimensional Poisson equation \(-u''=\mu\), where equality holds in the sense of distributions, with Dirichlet boundary conditions \(u(\pm\pi)=0\). In this paper, we…

A metric measure space is a complete separable metric space equipped with probability measure that has full support. Two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure on…

We consider an ergodic invariant measure $\mu$ for a smooth action of $Z^k$, $k \ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $R^k$, $k \ge 2$ on a $(2k+1)$-dimensional manifold. We prove that if $\mu$ is…

In this note, we generalise a Bourgain's construction of finitely-supported symmetric measures whose Furstenberg measure has a smooth density from the case of $\mathrm{SL}_2(\mathbb{R})$ to that of a general simple Lie group. The proof is…

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces $L^{2}(\mu)$, with $\mu$ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix to the…

Let $G$ be an LCA group, $H$ a closed subgroup, $\varGamma$ the dual group of $G$ and $\mu$ be a regular finite non-negative Borel measure on $\varGamma$. We give some necessary and sufficient conditions for the density of the set of…

We study freely infinitely divisible $R$-diagonal elements in the unbounded setting and Brown measures for free additive perturbations by such elements. This class includes circular elements, circular Cauchy elements, and other previously…

We consider a measure of similarity for infinite words that generalizes the notion of asymptotic or natural density of subsets of natural numbers from number theory. We show that every overlap-free infinite binary word, other than the…

Multifractal formalism is designed to describe the distribution at small scales of the elements of $\mathcal M^+_c(\R^d)$, the set of positive, finite and compactly supported Borel measures on $\R^d$. It is valid for such a measure $\mu$…

We analyze the relationship between Borel measures and continuous linear functionals on the space $\mathrm{Lip}_0(M)$ of Lipschitz functions on a complete metric space $M$. In particular, we describe continuous functionals arising from…

Let $\mu$ be a positive measure on the real line with locally finite support $\Lambda$ and integer masses such that its Fourier transform in the sense of distributions is a purely point measure. An explicit form is found for an entire…

We prove an analogue of the portmanteau theorem on weak convergence of probability measures allowing measures which are unbounded on an underlying metric space but finite on the complement of any Borel neighbourhood of a fixed element.

We obtain a complete classification of a large class of non almost periodic free Araki-Woods factors $\Gamma(\mu,m)"$ up to isomorphism. We do this by showing that free Araki-Woods factors $\Gamma(\mu, m)"$ arising from finite symmetric…

Let $\{X_i\}_{i=1}^{\infty}$ be a sequence of independent copies of a random vector $X$ in $\mathbb{R}^n$. We revisit the question to determine the asymptotic shape of the random polytope $K_N={\rm conv}\{X_1,\ldots ,X_N\}$ where $N>n$. We…

Let $T:[0,1]^d \rightarrow[0,1]^d$ be a piecewise expanding map with an absolutely continuous (with respect to the $d$-dimensional Lebesgue measure $m_d$) $T$-invariant probability measure $\mu$. Let $\left\{\mathbf{r}_n\right\}$ be a…

We give necessary conditions for when a subset of $\mathbb{T}^n$ can contain the support of some non-zero RP-measure. Among other things we show that the support of a positive RP-measure cannot be contained in reflections of inverse images…