We establish properties of a new type of fractal which has partial self similarity at all scales. For any collection of iterated functions systems with an associated probability distribution and any positive integer V there is a corresponding class of V -variable fractal sets or measures with a natural probability distribution. These V -variable fractals can be obtained from the points on the a...

It is of great significance to understand the particle distribution characteristics at different heights effectively control pollution. Based on fractal theory, dimension outdoor particles in a high-rise building Xi’an and its relationship with concentration sizes are discussed analyzed this paper. The results indicate that atmosphere mainly composed fine average proportion ranging from 0 1.0 µ...

We have carried out a multi-fractal analysis of the distribution of galaxies in the three Northern slices of the Las Campanas Redshift Survey (LCRS). Our method takes into account the selection effects and the complicated geometry of the survey. In this analysis we have studied the scaling properties of the distribution of galaxies on length scales from 20h−1Mpc to 200h−1Mpc. Our main results a...

For fractal-like agglomerates with volume fractal dimensions Df ≤ 2 multiple scattering of incident light may be neglected to a good approximation. Therefore the Rayleigh-Debye-Gans (RDG) theory provides a reliable yet convenient route towards computing the differential and total scattering cross sections of fractal particles (Sorensen, 2001). So far, however, besides assuming spherical isotrop...

Wildfire cumulative frequency – area distributions of Mediterranean landscapes are examined for agreement with self-similar (fractal) behavior. Our results support landscape-specific restricted scaling regions of 1.5 – 3.5 orders of magnitude in size, which are delimited by breakpoints or ‘cut-offs’. By identifying the extent of such regions in the fractal frequency – area distribution of wildf...

In this work, the anisotropic linear elastic properties of two-phase composite materials, made up of square inclusions embedded in a matrix, are investigated. The inclusions present a fractal hierarchical distribution and are supposed to have the same Poisson's ratio as the matrix but a different Young's modulus. The effective elastic moduli of the medium are computed at each fractal iteration ...

We study the betweenness centrality of fractal and nonfractal scale-free network models as well as real networks. We show that the correlation between degree and betweenness centrality C of nodes is much weaker in fractal network models compared to nonfractal models. We also show that nodes of both fractal and nonfractal scale-free networks have power-law betweenness centrality distribution P(C...

Using fractal self-similarity and functional-expectation relations, the classical theory of box integrals—being expectations on unit hypercubes—is extended to a class of fractal “string-generated Cantor sets” (SCSs) embedded in unit hypercubes of arbitrary dimension. Motivated by laboratory studies on the distribution of brain synapses, these SCSs were designed for dimensional freedom—a suitabl...

We present an Expectation-Maximization algorithm for the fractal inverse problem: the problem of fitting a fractal model to data. In our setting the fractals are Iterated Function Systems (IFS), with similitudes as the family of transformations. The data is a point cloud in R with arbitrary dimension H. Each IFS defines a probability distribution on R , so that the fractal inverse problem can b...

This paper addresses the problem of dielectric breakdown in composite materials. The dielectric breakdown model was generalized to describe dielectric breakdown patterns in conductor-loaded composites. Conducting particles are distributed at random in the insulating matrix, and the dielectric breakdown propagates according to new rules to take into account electrical properties and particle siz...