Related papers: Functions that preserve p-randomness

For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect…

This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for…

A word-to-word function is rational if it can be realized by a non-deterministic one-way transducer. Over finite words, it is a classical result that any rational function is regular, i.e. it can be computed by a deterministic two-way…

We establish new recurrence and multiple recurrence results for a rather large family $\mathcal{F}$ of non-polynomial functions which includes tempered functions defined in [11], as well as functions from a Hardy field with the property…

Let $R$ be a finite commutative ring with $1\ne 0$. The set $\mathcal{F}(R)$ of polynomial functions on $R$ is a finite commutative ring with pointwise operations. Its group of units $\mathcal{F}(R)^\times$ is just the set of all…

We prove that a (globally) subanalytic p-adic function which is locally Lipschitz continuous with some constant C is piecewise (globally on each piece) Lipschitz continuous with possibly some other constant, where the pieces can be taken…

$\DeclareMathOperator{\IntR}{Int{}^\text{R}}$Integer-valued rational functions are a natural generalization of integer-valued polynomials. Given a domain $D$, the collection of all integer-valued rational functions over $D$ forms a ring…

Intrinsic random functions (IRF) provide a versatile approach when the assumption of second-order stationarity is not met. Here, we develop the IRF theory on the circle with its universal kriging application. Unlike IRF in Euclidean spaces,…

When formalized, some diagonal arguments do not show the diagonal object to be impossible but rather reveal some other anomaly (e.g., that one of the relevant sets is ill-defined). This raises the possibility that some diagonal arguments…

In this note we propose a version of the classical Stone-Weierstrass theorem in the context of quantum operations, by introducing a particular class of quantum operations, dubbed polynomial quantum operations. This result permits to…

Let $P\in\mathbb{P}_1(\mathbb{Q})$ be a periodic point for a monic polynomial with coefficients in $\mathbb{Z}$. With elementary techniques one sees that the minimal periodicity of $P$ is at most $2$. Recently we proved a generalization of…

We propose and study a generalized continued fraction algorithm that can be executed in an arbitrary imaginary quadratic field, the novelty being a non-restriction to the five Euclidean cases. Many hallmark properties of classical continued…

This paper provides a new and more direct proof of the assertion that a Turing computable function of the natural numbers is primitive recursive if and only if the time complexity of the corresponding Turing machine is bounded by a…

A promising theory of quaternion-valued functions of one quaternionic variable, now called slice regular functions, has been introduced in 2006. The basic examples of slice regular functions are power series centered at 0 on their balls of…

Continued fractions in the field of $p$--adic numbers have been recently studied by several authors. It is known that the real continued fraction of a positive quadratic irrational is eventually periodic (Lagrange's Theorem). It is still…

We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define a notion of computable randomness associated with interval, rather than precise,…

One of the elegant achievements in the history of proof theory is the characterization of the provably total recursive functions of an arithmetical theory by its proof-theoretic ordinal as a way to measure the time complexity of the…

The purpose of this article is to formulate a number of probabilistic hidden-variable theorems, to provide proofs in some cases, and counterexamples to some conjectured relationships. The first theorem is the fundamental one. It asserts the…

Let $\mathcal{P}$ be a property of function $\mathbb{F}_p^n \to \{0,1\}$ for a fixed prime $p$. An algorithm is called a tester for $\mathcal{P}$ if, given a query access to the input function $f$, with high probability, it accepts when $f$…

For $q$ a prime power and $\phi$ a rational function with coefficients in $\mathbb{F}_q$, let $p(q,\phi)$ be the proportion of $\mathbb{P}^1(\mathbb{F}_q)$ that is periodic with respect to $\phi$. And if $d$ is a positive integer, let $Q_d$…