Related papers: Functions that preserve p-randomness

Fix an odd prime $p$. If $r$ is a positive integer and $f$ a polynomial with coefficients in $\mathbb{F}_{p^r}$, let $P_{p,r}(f)$ be the proportion of $\mathbb{P}^1(\mathbb{F}_{p^r})$ that is periodic with respect to $f$. We show that as…

We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z in [0,1] is computably random if and only if each nondecreasing computable function…

Let f be a computable function from finite sequences of 0's and 1's to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y…

Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. It is based on a discrete mechanical framework that can be used to model computation over the real numbers. In this context the…

A function f is continuous iff the PRE-image f^{-1}[V] of any open set V is open again. Dual to this topological property, f is called OPEN iff the IMAGE f[U] of any open set U is open again. Several classical Open Mapping Theorems in…

Various problems on integers lead to the class of congruence preserving functions on rings, i.e. functions verifying $a-b$ divides $f(a)-f(b)$ for all $a,b$. We characterized these classes of functions in terms of sums of rational…

Convergence results for averages of independent replications of counting processes are established in a $p$-variation setting and under certain assumptions. Such convergence results can be combined with functional differentiability results…

Let X be a countably infinite set of real numbers and let Y_x, x \in X, be an independent family of stationary random subsets of the real numbers, e.g. homogeneous Poisson point processes. We give criteria for the a.s. existence of various…

In multicentric calculus one takes a polynomial $p$ with distinct roots as a new variable and represents complex valued functions by $\mathbb C^d$-valued functions, where $d$ is the degree of $p$. An application is e.g. the possibility to…

We characterize the variation functions of computable Lipschitz functions. We show that a real z is computably random if and only if every computable Lipschitz function is differentiable at z. Beyond these principal results, we show that a…

A rational function $f(x)$ is rationally summable if there exists a rational function $g(x)$ such that $f(x)=g(x+1)-g(x)$. Detecting whether a given rational function is summable is an important and basic computational subproblem that…

We prove a version of both Jacobi's and Montel's Theorems for the case of continuous functions defined over the field $\mathbb{Q}_p$ of $p$-adic numbers. In particular, we prove that, if \[ \Delta_{h_0}^{m+1}f(x)=0 \ \ \text{for all}…

We extend the notion of randomness (in the version introduced by Schnorr) to computable Probability Spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is typical for some dynamic, if it follows the…

Suppose $1 < p < \infty$. Carleson's Theorem states that the Fourier series of any function in $L^p[-\pi, \pi]$ converges almost everywhere. We show that the Schnorr random points are precisely those that satisfy this theorem for every $f…

Within the last fifteen years, a program of establishing relationships between algorithmic randomness and almost-everywhere theorems in analysis and ergodic theory has developed. In harmonic analysis, Franklin, McNicholl, and Rute…

Let $[a,b]\subset\mathbb{R}$ be a non empty and non singleton closed interval and $P=\{a=x_0<\cdots<x_n=b\}$ is a partition of it. Then $f:I\to\mathbb{R}$ is said to be a function of $r$-bounded variation, if the expression…

This expository paper advocates an approach to physics in which ``typicality" is identified with a suitable form of algorithmic randomness. To this end various theorems from mathematics and physics are reviewed. Their original versions…

We study the random \'etale algebra generated by a random polynomial with i.i.d. coefficients distributed according to Haar measure normalized on $\mathbb{Z}_p$. We determine the probability that this random algebra is an unramified field,…

The functional equation f(p(z))=g(q(z)) is studied, where p,q are polynomials and f,g are trancendental meromorphic functions in C. We find all the pairs p,q for which there exist nonconstant f,g satisfying our equation and there exist no…

An approach to build Probabilistic Arithmetic in which initial values of all correlated random variables are known, but with varying degrees of accuracy. As a result of the proposed Probabilistic Arithmetic operations, variable values,…