Related papers: One distribution function on the Moran sets
We examine Frostman-type characterisations and other extremal measure criteria for a range of fractal dimensions of sets. In particular we derive properties of the less familiar modified lower box dimension and upper correlation dimension.…
This paper presents a review of the fractal approach for describing the large scale distribution of galaxies. We start by presenting a brief, but general, introduction to fractals, which emphasizes their empirical side and applications…
In many low-dimensional dynamical systems transport coefficients are very irregular, perhaps even fractal functions of control parameters. To analyse this phenomenon we study a dynamical system defined by a piece-wise linear map and…
We investigate Benford's law in relation to fractal geometry. Basic fractals, such as the Cantor set and Sierpinski triangle are obtained as the limit of iterative sets, and the unique measures of their components follow a geometric…
Michael Barnsley introduced a family of fractals sets which are repellers of piecewise affine systems. The study of these fractals was motivated by certain problems that arose in fractal image compression but the results we obtained can be…
In this paper, we introduce the concept of the $\alpha$-fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers. Also, we study some properties of such fractal…
We discuss a variation of Takagi curves based, however, more on algebraic than geometric principles. Namely, we construct functions of loops in a special binary representation. The graph of these functions usually has chaotic and fractal…
Several studies have investigated the fractal and multifractal nature of magnetic features in the solar photosphere and its variation with the solar magnetic activity cycle. Here we extend those studies by examining the fractal geometry of…
The cosmic web is one of the most complex systems in nature, consisting of galaxies and clusters of galaxies joined by filaments and walls, leaving large empty regions called cosmic voids. The most common method of describing the web is a…
In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More…
An intermittent nonlinear map generating subdiffusion is investigated. Computer simulations show that the generalized diffusion coefficient of this map has a fractal, discontinuous dependence on control parameters. An amended continuous…
The fractal dimension of large-scale galaxy clustering has been demonstrated to be roughly $D_F \sim 2$ from a wide range of redshift surveys. If correct, this statistic is of interest for two main reasons: fractal scaling is an implicit…
Galaxies and clusters distributions show two major properties: (i) the positions of galaxies and clusters are characterized by a power law distribution indicating properties with respect to their positions. (ii) The distribution of masses…
Given an integer $n\geq 2$ and a digit set ${\mathcal D}\subsetneq {0,1,...,n-1}^2$, there is a self-similar set $F \subset {\Bbb R}^2$ satisfying the set equation: $F=(F+{\mathcal D})/n$. We call such $F$ a fractal square. By studying a…
There is a general agreement that galaxy structures exhibit fractal properties, at least up to some small scale. However the presence of an eventual crossover towards homogenization, as well as the exact value of the fractal dimension, are…
In this paper we study various aspects of porosities for conformal fractals. We first explore porosity in the general context of infinite graph directed Markov systems (GDMS), and we show that, under some natural assumptions, their limit…
The power spectrum S of linear transects of the earth's topography is often observed to be a power-law function of wave number k with exponent close to -2: S(k) is proportional to k^-2. In addition, river networks are fractal trees that…
The fractal properties of models of randomly placed $n$-dimensional spheres ($n$=1,2,3) are studied using standard techniques for calculating fractal dimensions in empirical data (the box counting and Minkowski-sausage techniques). Using…
A way to add an extra dimension is briefly discussed.
Fractal geometry is a fundamental approach for describing the complex irregularities of the spatial structure of point patterns. The present research characterizes the spatial structure of the Swiss population distribution in the three…