where (x, y) ∈ S. The effects of an affine transform on a set are depicted in fig. 1. The union of N affine transformations is called the Hutchinson operator: W = ⋃N n=1 wn. For a specified metric the distance h(A, B) between two sets A, B can be defined. Under certain conditions [2] the Hutchinson operator is contractive, h(W (A),W (B)) ≤ sh(A, B), s < 1. Successive iterations with Hutchinson ...

Monte Carlo methods for computing various statistical aspects of turbulent diffusion with long range correlated and even fractal random velocity fields are described here. A simple explicit exactly solvable model with complex regimes of scaling behavior including trapping, subdiffusion, and superdiffusion is utilized to compare and contrast the capabilities of conventional Monte Carlo procedure...

We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasimomentum space, the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a...

The parabolic-profile model for the singularity spectrum f(α) of fully-developed turbulence (FDT) is pursued further. This model is shown to provide not only considerable insight into the qualitative aspects of the intermittency problem but also the universal aspects associated with it. The parabolic-profile model also affords, unlike the multi-fractal model, an analytical calculation of probab...

In this paper, we generalize the zeta function for a fractal string (as in [18]) in several directions. We first modify the zeta function to be associated with a sequence of covers instead of the usual definition involving gap lengths. This modified zeta function allows us to define both a multifractal zeta function and a zeta function for higher-dimensional fractal sets. In the multifractal ca...

The parabolic-profile model for the singularity spectrum f(α) of fully-developed turbulence (FDT) is pursued further. This model is shown to provide not only considerable insight into the qualitative aspects of the intermittency problem but also the universal aspects associated with it. The parabolic-profile model also affords, unlike the multi-fractal model, an analytical calculation of probab...

This article presents a derivation of a new relative fractal dimension spectrum, DRq, to measure the dis-similarity between two finite probability distributions originating from various signals. This measure is an extension of the Kullback-Leibler (KL) distance and the Rényi fractal dimension spectrum, Dq. Like the KL distance, DRq determines the dissimilarity between two probability distibutio...

We study the proximity effect of a superconductor to a normal system with a fractal spectrum. We find that there is no gap in the excitation spectrum, even in the case where the underlying classical dynamics of the normal system is chaotic. An analytical expression for the distribution of the smallest excitation eigenvalue E1 of the hybrid structure is obtained. On small scales it decays algebr...

The Lévy stability analysis is carried out for ee collisions at Z mass using Monte Carlo method. The Lévy index μ is found to be μ = 1.701± 0.043. The self-similar generalized dimensions D(q) and multi-fractal spectrum f(α) are presented. The Rényi dimension D(q) decreases with increasing q. The self-similar multi-fractal spectrum is a convex curve with a maximum at q = 0, α = 1.169 ± 0.011. Th...