Related papers: Exact formula on upper box dimension of generic H\…

We determine the Hausdorff and box dimension of the fractal graphs for a general class of Weierstrass-type functions of the form $f(x) = \sum_{n=1}^\infty a_n \, g(b_n x + \theta_n)$, where $g$ is a periodic Lipschitz real function and…

We improve existing lower bounds of the hyperbolic dimension for meromophic functions that have a logarithmic tract {\Omega} which is a H\"older domain. These bounds are given in terms of the fractal behavior, measured with integral means,…

For a given second order elliptic operation $\mathcal{L}$ in a domain $\Omega\subset{\mathbb{R}}^\mathbf{N}$, $\mathbf{N}\ $, and a compact set $\mathbf{K}\subset\Omega$, order $\mathbf{N}$-$2$-Ahlfors-David regular, we define the space…

We prove a packing exponent formula for the upper box-counting dimension of attractors of certain projective iterated function systems. This partially affirms a conjecture of De Leo, and gives that the box-counting dimension of the Rauzy…

We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with $\alpha$-H\"older derivatives (for some $0<\alpha\leq 1$). The smooth approximation is given by means of an…

For a given set of dilations $E\subset [1,2]$, Lebesgue space mapping properties of the spherical maximal operator with dilations restricted to $E$ are studied when acting on radial functions. In higher dimensions, the type set only depends…

In this paper, we investigate the dimension theory of the one parameter family of Okamoto's function. We compute the Hausdorff, box-counting and Assouad dimensions of the graph for a typical choice of parameter. Furthermore, we study the…

We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in C (the "dust"). In two dimensions, we…

We prove that if ${\mathcal E} \subset {\Bbb R}^{2d}$, $d \ge 2$, is an Ahlfors-David regular product set of sufficiently large Hausdorff dimension, denoted by $dim_{{\mathcal H}}({\mathcal E})$, and $\phi$ is a sufficiently regular…

A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order $\alpha, 0 < \alpha \leq 1$, called $F^\alpha$-integral, is defined, which is suitable to integrate functions with fractal…

Let $f, g^1, \dots, g^d : \mathbb{R}^d \longrightarrow \mathbb{R}$ be H\"older continuous functions. If the H\"older exponents of these functions are less than $1$ but sufficiently large, we use the integral introduced by Z\"ust to…

This paper provides a summary of the fractal calculus framework. It presents higher-order homogeneous and nonhomogeneous linear fractal differential equations with $\alpha$-order. Solutions for these equations with constant coefficients are…

We continue the study of model-independent constraints on the unitary Conformal Field Theories in 4-Dimensions, initiated in arXiv:0807.0004. Our main result is an improved upper bound on the dimension \Delta of the leading scalar operator…

We study fractal properties of unbounded domains with infinite Lebesgue measure via their complex fractal dimensions. These complex dimensions are defined as poles of a suitable defined Lapidus fractal zeta function at infinity and are a…

Fractal dimension constitutes the main tool to test for fractal patterns in Euclidean contexts. For this purpose, it is always used the box dimension, since it is easy to calculate, though the Hausdorff dimension, which is the oldest and…

We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several…

Hausdorff and box dimension are two familiar notions of fractal dimension. Box dimension can be larger than Hausdorff dimension, because in the definition of box dimension, all sets in the cover have the same diameter, but for Hausdorff…

We establish a general Kronecker limit formula of arbitrary rank over global function fields with Drinfeld period domains playing the role of upper-half plane. The Drinfeld-Siegel units come up as equal characteristic modular forms…

In this article, we provide a simple and systematic way to represent general (inhomogeneous) fractals that may look different at different scales and places. By using set-valued compression maps, we express these general fractals as…

This paper provides a new model to compute the fractal dimension of a subset on a generalized-fractal space. Recall that fractal structures are a perfect place where a new definition of fractal dimension can be given, so we perform a…