Related papers: On the equality of two-variable general functional…

We prove that a Radon measure $\mu$ on $\mathbb{R}^n$ can be written as $\mu=\sum_{i=0}^n\mu_i$, where each of the $\mu_i$ is an $i$-dimensional rectifiable measure if and only if for every Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$…

A remarkable theorem of Besicovitch is that an integrable function $f$ on $\mathbb{R}^2$ is strongly differentiable if and only if its associated strong maximal function $M_S f$ is finite a.e. We provide an analogue of Besicovitch's result…

We examine measure preserving mappings $f$ acting from a probability space $(\Omega, F,\mu) $ into a probability space $% (\Omega ^{*},F^{*},\mu ^{*}) ,$ where $\mu ^{*}=\mu (f^{-1})$. Conditions on $f$, under which $f$ preserves the…

Our goal in this paper is to find a characterization of $n$-dimensional bilinear Hardy inequalities \begin{align*} \bigg\| \,\int_{B(0,\cdot)} f \cdot \int_{B(0,\cdot)} g \,\bigg\|_{q,u,(0,\infty)} & \leq C \, \|f\|_{p_1,v_1,{\mathbb R}^n}…

We present and study semi-parametric estimators for the mean of functional outcomes in situations where some of these outcomes are missing and covariate information is available on all units. Assuming that the missingness mechanism depends…

Assume that $\mathfrak A$ is a real Banach space of finite dimension $n\geq2$. Consider any Borel probability measure $\nu$ supported on the unit ball $K$ of $\mathfrak A$. We show that \[\Delta(\nu)=\int_{x \in K}\int_{ y\in…

We investigate the set of invariant idempotent probabilities for countable idempotent iterated function systems (IFS) defined in compact metric spaces. We demonstrate that, with constant weights, there exists a unique invariant idempotent…

We establish analogs of Cheeger's inequality for probability measures with heavy tails. As one of the principal applications, suppose $\lambda > 3$ and define the (Pareto) probability measure $\mu_{\lambda}$ on $[1,\infty)$ by…

Using the spectral subspaces obtained in [HS], Brown's results on the Brown measure of an operator in a type II_1 factor (M,tr) are generalized to finite sets of commuting operators in M. It is shown that whenever T_1,..., T_n in M are…

Let $f: \mathbb{N}^2 \mapsto \mathbb{C}$ be an arithmetic function of two variables. We study the existence of the limit: \[\displaystyle \lim_{x \to \infty} \frac{1}{x^2 (\log x)^{k-1}} \sum_{n_1 , n_2 \le x} f (n_1, n_2) \] where $k$ is a…

In the context of generalized measurement theory, the Gleason-Busch theorem assures the unique form of the associated probability function. Recently, in Flatt et al. Phys. Rev. A 96, 062125 (2017), the case of subsequent measurements has…

Let a sequence of iid. random variables $\xi_1,...,\xi_n$ be given on a space $(X,\cal X)$ with distribution $\mu$ together with a nice class $\cal F$ of functions $f(x_1,...,x_k)$ of $k$ variables on the product space $(X^k,{\cal X}^k)$.…

We study the existence of solutions to the equation $-\Gd_pu+g(x,u)=\mu$ when $g(x,.)$ is a nondecreasing function and $\gm$ a measure. We characterize the good measures, i.e. the ones for which the problem as a renormalized solution. We…

Various inequalities (Boole inequality, Chung-Erd\"os inequality, Frechet inequality) for Kolmogorov (classical) probabilities are considered. Quantum counterparts of these inequalities are introduced, which have an extra `quantum…

Assume that $(\Omega,\mathcal A,P)$ is a probability space, $f\colon[0,1] \times \Omega\to[0,1]$ is a function such that $f(0,\omega)=0$, $f(1,\omega)=1$ for every $\omega\in\Omega$, $g\colon[0,1]\to\mathbb R$ is a bounded function such…

Novikov-Furutsu (NF) theorem is a well-known mathematical tool, used in stochastic dynamics for correlation splitting, that is, for evaluating the mean value of the product of a random functional with a Gaussian argument multiplied by the…

We show that for any locally compact second countable group $G$ and any continuous positive definite function $\phi:G\rightarrow\mathbb{C}$, there exists an ergodic measure preserving system $(X,\mathscr{B},\mu,\{T_g\}_{g \in G})$ and a…

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. For $\alpha>0$, the generalized Hankel matrix $\mathcal{H}_{\mu, \alpha}=(\mu_{n, k, \alpha})_{n, k \geq 0}$ with entries $\mu_{n, k, \alpha}=\int_{[0,1)}…

Monotonicity criteria are established for the generalized Marcum Q-function, $\emph{Q}_{M}$, the standard Nuttall Q-function, $\emph{Q}_{M,N}$, and the normalized Nuttall Q-function, $\mathcal{Q}_{M,N}$, with respect to their real order…

Properties of the mappings \begin{align*} C&\mapsto\frac1{(2\pi i)^2}\int_{\Gamma_1}\int_{\Gamma_2}f(\lambda,\mu)\,R_{1,\,\lambda}\,C\, R_{2,\,\mu}\,d\mu\,d\lambda, C&\mapsto\frac1{2\pi i}\int_{\Gamma}g(\lambda)R_{1,\,\lambda}\,C\,…