Related papers: An Indicator Function Limit Theorem in Dynamical S…

We prove that the electron density function of a real physical system can be uniquely determined by its values on any finite subsystem. This establishes the existence of a rigorous density-functional theory for any open electronic system.…

Let $(Z_i)_{i\geq 1}$ be an independent, identically distributed sequence of random variables on $\RRR^d$. Under mild conditions on the density of $Z_1$, we provide a nonstandard uniform functional limit law for the following processes on…

In Nevzorov's $F^\alpha$-scheme, one deals with a sequence of independent random variables whose distribution functions are all powers of a common continuous distribution function. A key property of the $F^\alpha$-scheme is that the record…

Consider a class of probability distributions which is dense in the space of all probability distributions on $\mathbb{R}^{d}$ with respect to weak convergence, for every $d\in\mathbb{N}$. Then, we construct various explicit classes of…

Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of events on a probability space $(\Omega,\mathcal{F},\mathbf{P})$. We show that if $\lim_{m\to\infty}\sum_{n=1}^{m}w_n\mathbf{P}(A_n)=\infty$ where each $w_n\in\mathbb{R}$, then…

Let $s_1,s_2,\ldots s_n$ be states of a general probability theory, and $\mathcal A$ be the set of all subsets of indices $H \subset [n]\equiv\{1,2,\ldots n\}$ such that the states $(s_j)_{j\in H}$ are jointly perfectly distinguishable. All…

This paper introduces indefinite proximities inherent in the collection of physical objects found in a dynamical system. Axiomatically, these indefinite proximities lead to a new form of Hausdorff topology, which is indefinite…

In this paper we study representations of real numbers in a numeral system with the base $a>1$ and alphabet (digits set) $A\equiv\{0,1,...,r\}$, $a-1<r\in N$ given by \[x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{a^n}\equiv…

In the paper we describe some properties of function $$ y=r(x)=\lim_{n\to\infty}\frac{1}{n}\sum^{\infty}_{k=1}\alpha_k(x), \text{ where } x=\sum^{\infty}_{k=1}\alpha_k(x)4^{-k} $$ of $4$-adic digits asymptotic mean of fractional part of…

We argue that simple dynamical systems are factors of finite automata, regarded as dynamical systems on discontinuum. We show that any homeomorphism of the real interval is of this class. An orientation preserving homeomorphism of the…

A convenient framework for dealing with asymptotic limit problems of probabilistic nature is provided. These problems include questions such as finding the asymptotic proportion of terms of a sequence falling inside a given interval, or the…

The existence of unimodular forms with small norms on sequence spaces is crucial in a variety of problems in modern analysis. We prove that the infimum of $\left\Vert A\right\Vert $ over all unimodular $d$-linear (complex or real) forms $A$…

Fix a constant $C\geq 1$ and let $d=d(n)$ satisfy $d\leq \ln^{C} n$ for every large integer $n$. Denote by $A_n$ the adjacency matrix of a uniform random directed $d$-regular graph on $n$ vertices. We show that, as long as $d\to\infty$ with…

The Bohr-Jessen limit theorem is a probabilistic limit theorem on the value-distribution of the Riemann zeta-function in the critical strip. Moreover their limit measure can be written as an integral involving a certaindensity function. The…

Let $\Psi_n$ be a product of $n$ independent, identically distributed random matrices $M$, with the properties that $\Psi_n$ is bounded in $n$, and that $M$ has a deterministic (constant) invariant vector. Assuming that the probability of…

For a wide class of sequences of integer domains $\mathcal{D}_n\subset\mathbb{N}^d$, $n\in\mathbb{N}$, we prove distributional limit theorems for $F(X_1^{(n)},\ldots,X_d^{(n)})$, where $F$ is a multivariate multiplicative function and…

For each vector $x\in \ell^{\infty}$, we can define the non-empty compact set $L_x$ of accumulation points of $x$. Given an infinite subset $A$ of $\mathbb{N}\backslash\{1\}$, we can therefore investigate under which conditions on $A$, the…

The random map model is a deterministic dynamical system in a finite phase space with n points. The map that establishes the dynamics of the system is constructed by randomly choosing, for every point, another one as being its image. We…

The purpose of this article is to construct a toolbox, in Dynamical Systems, to support the idea that ``whenever we can prove a limit theorem in the classical sense for a dynamical system, we can prove a suitable almost-sure version based…

Let $A$ be a finite set with $|A|\geqslant 2$, let $n$ be a positive integer, and let $A^n$ denote the discrete $n$-dimensional hypercube (that is, $A^n$ is the Cartesian product of $n$ many copies of $A$). Given a family $\langle D_t:t\in…