Related papers: Invariant measures for continued fraction algorith…
In this paper we recall some results and some criteria on the convergence of matrix continued fractions. The aim of this paper is to give some properties and results of continued fractions with matrix arguments. Then we give continued…
The theory of uniform approximation of real numbers motivates the study of products of consecutive partial quotients in regular continued fractions. For any non-decreasing positive function $\varphi:\mathbb{N}\to [2,\infty)$, we determine…
We establish a notion of random entropy solution for degenerate fractional conservation laws incorporating randomness in the initial data, convective flux and diffusive flux. In order to quantify the solution uncertainty, we design a…
Let $\beta > 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $\beta$-expansion of $x$…
Let x be a quadratic irrational and let P be the set of prime numbers. We show the existence of an infinite subset S of P such that the statistics of the period of the continued fraction expansions along the sequence {px: p\in S} approach…
Let $\mathcal{F}$ be a countable collection of functions $f$ defined on the integers with integer values, such that for every $f\in \mathcal{F}$, $f(n)\to +\infty$ as $n\to +\infty$. This paper primarily investigates the Hausdorff dimension…
In this paper we obtain multifractal generalizations of classical results by L\'evy and Khintchin in metrical Diophantine approximations and measure theory of continued fractions. We give a complete multifractal analysis for Stern--Brocot…
As the conclusion of a line of investigation undertaken in two previous papers, we compute asymptotic frequencies for the values taken by numerators of differences of consecutive Farey fractions with denominators restricted to lie in…
For any real number $x \in [0,1)$, we denote by $q_n(x)$ the denominator of the $n$-th convergent of the continued fraction expansion of $x$ $(n \in \mathbb{N})$. It is well-known that the Lebesgue measure of the set of points $x \in [0,1)$…
We extend previously proposed measures of complexity, emergence, and self-organization to continuous distributions using differential entropy. This allows us to calculate the complexity of phenomena for which distributions are known. We…
We present a computationally efficient algorithm for stable numerical differentiation from noisy, uniformly-sampled data on a bounded interval. The method combines multi-interval Fourier extension approximations with an adaptive domain…
We establish a central limit theorem for counting large continued fraction digits $(a_n)$, i.e. we count occurrences $\{a_n>b_n\}$, where $(b_n)$ is a sequence of positive integers. Our result improves a similar result by Philipp which…
We use the complexity function of an invariant, not necessary closed, subset of a two-sided shift space to compute the polynomial entropy of the induced dynamics on the hyperspace of continua for certain one-dimensional dynamical systems.…
In this article we study the numerical approximation of a variable coefficient fractional diffusion equation. Using a change of variable, the variable coefficient fractional diffusion equation is transformed into a constant coefficient…
We investigate the dynamics of continued fractions and explore the ergodic behaviour of the products of mixed partial quotients in continued fractions of real numbers. For any function $\Phi:\mathbb N\to [2,+\infty)$ and any integer $d\geq…
For a closed-loop control system with a digital channel between the sensor and the controller, the notion of invariance entropy quantifies the smallest average rate of information transmission above which a given compact subset of the state…
The fractional order generalization of Shannon entropy proposed by Ubriaco has been studied for discrete distributions. In the current paper, we conduct a detailed study of the continuous analogue of this entropy termed as fractional…
This paper concerns extension of the classical Lagrange theorem, on the eventual periodicity of continued fraction expansions of quadratic surds, and the versions of it found in the literature in the case of complex numbers. In this…
We attempt to investigate a two-dimensional Gauss-Kuzmin theorem for R\'enyi-type continued fraction expansions. More precisely speaking, our focus is to obtain specific lower and upper bounds for the error term considered which imply the…
We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, %given a nonnegative reals $\hat{\nu},$ we calculate the Hausdorff dimension of the…