Related papers: A class of random Cantor measures, with applicatio…
This paper introduced a way of fractal to solve the problem of taking count of the integer partitions, furthermore, using the method in this paper some recurrence equations concerning the integer partitions can be deduced, including the…
Singular vectors are those for which the quality of rational approximations provided by Dirichlet's Theorem can be improved by arbitrarily small multiplicative constants. We provide an upper bound on the Hausdorff dimension of singular…
We construct a class of homogeneous Cantor-Moran measures with all contraction ratios being reciprocal of integers, and prove that they are pointwise absolutely normal. Our approach relies on methods developed by Davenport, Erd{\H{o}}s, and…
The local magnetization in the one-dimensional random-field Ising model is essentially the sum of two effective fields with multifractal probability measure. The probability measure of the local magnetization is thus the convolution of two…
We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let $K$ be a self-similar subset of $\mathbb{R}^2$…
We study long-range power-law correlated disorder on square and cubic lattices. In particular, we present high-precision results for the percolation thresholds and the fractal dimension of the largest clusters as function of the correlation…
We construct a family of measures for random fields based on the iterated subdivision of simple geometric shapes (triangles, squares, tetrahedrons) into a finite number of similar shapes. The intent is to construct continuum limits of scale…
We report a numerical study of van der Waals adsoprtion and capillary condensation effects on self-similar fractal surfaces. An assembly of uncoupled spherical pores with a power-law distributin of radii is used to model fractal surfaces…
Recently, a new approach in the fine analysis of stochastic processes sample paths has been developed to predict the evolution of the local regularity under (pseudo-)differential operators. In this paper, we study the sample paths of…
A fractal is in essence a hierarchy with cascade structure, which can be described with a set of exponential functions. From these exponential functions, a set of power laws indicative of scaling can be derived. Hierarchy structure and…
Let $\Phi$ be a $C^{1+\gamma}$ smooth IFS on $\mathbb{R}$, where $\gamma>0$. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points $x$ that are absolutely normal. That is,…
We construct a wavelet and a generalised Fourier basis with respect to some fractal measures given by one-dimensional iterated function systems. In this paper we will not assume that these systems are given by linear contractions…
We study the critical behavior of various geometrical and transport properties of percolation in 6 dimensions. By employing field theory and renormalization group methods we analyze fluctuation induced logarithmic corrections to scaling up…
We consider the fragmentation process with mass loss and discuss self-similar properties of the arising structure both in time and space focusing on dimensional analysis. This exhibits a spectrum of mass exponents $\theta$, whose exact…
This work proposes algorithms for computing additive and multiplicative free convolutions of two given measures. We consider measures with compact support whose free convolution results in a measure with a density function that exhibits a…
We consider a Markov evolution of lozenge tilings of a quarter-plane and study its asymptotics at large times. One of the boundary rays serves as a reflecting wall. We observe frozen and liquid regions, prove convergence of the local…
In this article, we consider the concept of the decay of the Favard length of $\varepsilon$-neighborhoods of purely unrectifiable sets. We construct non-self-similar Cantor sets for which the Favard length decays arbitrarily with respect to…
Fractals define a new and interesting realm for a discussion of basic phenomena in quantum field theory and statistical mechanics. This interest results from specific properties of fractals, e.g., their dilatation symmetry and the…
We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of…
In this paper, we construct and then prove the up-to constants uniqueness of the natural measure on several random fractals, namely the SLE cut points, SLE boundary touching points, CLE pivotal points and the CLE carpet/gasket. As an…