Related papers: A Variant Of Chaitin's Omega function

we prove that if $X$ is a locally compact $\sigma$-compact space then on its quotient, $\gamma(X)$ say, determined by the algebra of all real valued bounded continuous functions on $X$, the quotient topology and the completely regular…

Let $[q] = \{0,1,\ldots,q-1\}$, let $\Delta[q]$ denote the simplex of probability measures on $[q]$, and let $\gamma$ denote the Lebesgue measure normalized on $\Delta[q]$. We prove that for any symmetric monotone function $f \colon[q]^n…

The shift map $\sigma$ on $\omega^*$ is the continuous self-map of $\omega^*$ induced by the function $n \mapsto n+1$ on $\omega$. Given a compact Hausdorff space $X$ and a continuous function $f: X \rightarrow X$, we say that $(X,f)$ is a…

In this paper we prove that every collection of measurable functions $f_\alpha$, $|\alpha|=m$ coincides a.e. with $m$th order derivatives of a function $g\in C^{m-1}$ whose derivatives of order $m-1$ may have any modulus of continuity…

For an irrational $\alpha\in(0,1)$, we investigate the Ostrowski sum-of-digits function $\sigma_\alpha$. For $\alpha$ having bounded partial quotients and $\vartheta\in\mathbb R\setminus\mathbb Z$, we prove that the function $g:n\mapsto…

We consider two operations on the Mittag-Leffler function which cancel the exponential term in the expansion at infinity, and generate a completely monotonic function. The first one is the action of a certain differential-difference…

We prove a Lusin approximation of functions of bounded variation. If $f$ is a function of bounded variation on an open set $\Omega\subset X$, where $X=(X,d,\mu)$ is a given complete doubling metric measure space supporting a $1$-Poincar\'e…

Stein's formula states that a random variable of the form $z^\top f(z) - \text{div} f(z)$ is mean-zero for functions $f$ with integrable gradient. Here, $\text{div} f$ is the divergence of the function $f$ and $z$ is a standard normal…

Let $\Omega \subset \mathbb{R}^d$, $d \geq 2$, be a bounded convex domain and $f\colon \Omega \to \mathbb{R}$ be a non-negative subharmonic function. In this paper we prove the inequality \[ \frac{1}{|\Omega|}\int_\Omega f(x)\,dx \leq…

Let $G$ be a real reductive connected Lie group and $\sigma$ an involution of $G$. Let $H$ denote the identity component of the group of fixed points of $\sigma$, $\mathfrak g$ the Lie algebra of $G$ and $\mathfrak q$ the -1 eigenspace of…

Let h be a real-analytic function in the neighborhood of some compact set K on the plane. We show that for any complex measure on the Euclidean space of a finite total variation without singular components with the Fourier--Stieltjes…

Let K be the field of fractions of a Henselian discrete valuation ring O_K. Let X_K/K be a smooth proper geometrically connected scheme admitting a regular model X/O_K. We show that the index \delta(X_K/K) of X_K/K can be explicitly…

We consider the spaces $H_{F}^{\infty}(\Omega)$ and $\mathcal{A}_{F}(\Omega)$ containing all holomorphic functions $f$ on an open set $\Omega \subseteq \mathbb{C}$, such that all derivatives $f^{(l)}$, $l\in F \subseteq \mathbb{N}_0=\{…

Let $X=\{x_i:i\in\mathbb{Z}\}$, $\dots<x_{i-1}<x_i<x_{i+1}<\dots$, be a sampling set which is separated by a constant $\gamma>0$. Under certain conditions on $\phi$, it is proved that if there exists a positive integer $\nu$ such that…

We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space…

We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin's Omega number, halting probability, and program-size complexity. A classification of Turing machines…

Let $k$ be a perfect complete valued field with a nontrivial non-archimedean norm $|\cdot|$ and $\omega\in k$ with $0<|\omega|<1.$ Let $X$ be a reduced and normal $k$-analytic space. Then $O^{\circ}\simeq…

In this paper, we investigate the regularity of weak solutions $u\colon\Omega\to\mathbb{R}$ to elliptic equations of the type \begin{equation*} \mathrm{div}\, \nabla \mathcal{F}(x,Du) = f\qquad\text{in $\Omega$}, \end{equation*} whose…

In this paper, our focus lies on the study of the second-order variational analysis of orthogonally invariant matrix functions. It is well-known that an orthogonally invariant matrix function is an extended-real-value function defined on…

Let $f:{\mathbb R}_+\mapsto {\mathbb R}$ be a smooth function with $f(0)=0.$ A problem of estimation of a functional $\tau_f(\Sigma):= {\rm tr}(f(\Sigma))$ of unknown covariance operator $\Sigma$ in a separable Hilbert space ${\mathbb H}$…