Rhythmic precipitate patterns and fractal structure

Liesegang patterns of parallel precipitate bands are obtained when solutions containing co-precipitate ions interdiffuse in a 1D gel matrix.The sparingly soluble salt formed,displays a beautiful stratification of discs of precipitate perpendicular to the 1D tube axis.The Liesegang structures are analyzed from the viewpoint of their fractal nature.Geometric Liesegang patterns are constructed in conformity with the well-known empirical laws such as the time,band spacing and band width laws.The dependence of the band spacing on the initial concentrations of diffusing(outer)and immobile(inner)electrolytes(A0 and B0,respectively)is taken to follow the Matalon-Packter law.Both mathematical fractal dimensions and box-count dimensions are calculated.The fractal dimension is found to increase with increasing A0 and decreasing B0.We also analyze mosaic patterns with random distribution of crystallites,grown under different conditions than the classical Liesegang gel method,and report on their fractal properties.Finally,complex Liesegang patterns wherein the bands are grouped in multiplets are studied,and it is shown that the fractal nature increases with the multiplicity.     

A localized Jarník–Besicovitch theorem

Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form {x∈R:δx=δ}, where δ?1 and δx is the Diophantine approximation exponent of an irrational number x. We go beyond the classical results by computing the Hausdorff dimension of the sets {x∈R:δx=f(x)}, where f is a continuous function. Our theorem applies to the study of the approximation exponents by various approximation families. It also applies to functions f which are continuous outside a set of prescribed Hausdorff dimension.     

The Gompertz-Pareto income distribution

This work analyzes the Gompertz-Pareto distribution (GPD) of personal income, formed by the combination of the Gompertz curve, representing the overwhelming majority of the economically less favorable part of the population of a country, and the Pareto power law, which describes its tiny richest part. Equations for the Lorenz curve, Gini coefficient and the percentage share of the Gompertzian part relative to the total income are all written in this distribution. We show that only three parameters, determined by linear data fitting, are required for its complete characterization. Consistency checks are carried out using income data of Brazil from 1981 to 2007 and they lead to the conclusion that the GPD is consistent and provides a coherent and simple analytical tool to describe personal income distribution data.     

Semi-flexible compact polymers on fractal lattices

Semi-flexible compact polymers modeled by Hamiltonian walks with bending rigidity are studied on 3 and 4-simplex fractal lattices. Hamiltonian walks are weighted according to the number of bends in the walk, and total weights are obtained by an exact recursive treatment. Asymptotic form of the partition function, with temperature dependent scaling parameters, as well as the corresponding critical exponents, is determined. Various thermodynamic quantities are calculated numerically and presented graphically, and the possibility of phase transition between a compact molten globule and an ordered ‘crystal’ state is investigated. No phase transition is found on either of these two lattices, meaning that fractal geometry here prevents any kind of orientational order.     

COMPLEXITY,MULTIRESOLUTION, NON-STATIONARITY AND ENTROPIC SCALING: TEEN BIRTH THERMODYNAMICS

This paper presents a statistical methodology for analyzing a complex phenomenon in which deterministic and scaling components are superimposed. Our approach is based on the wavelet multiresolution analysis combined with the scaling analysis of the entropy of a time series. The wavelet multiresolution analysis decomposes the signal in a scale-by-scale manner. The scale-by-scale decomposition generates smooth and detail curves that are evaluated and studied. A wavelet-based smoothing filtering is used to estimate the daily birth rate and conception rate during the year. The scaling analysis is based on the Diffusion Entropy Analysis (DEA). The joint use of the DEA and the wavelet multiresolution analysis allows: 1) the separation of the deterministic and, therefore, non-scaling component from the scaling component of the signal; 2) the determination of the stochastic information characterizing the teen birth phenomenon at each time scale. The daily data cover the number of births phenomenon at each time scale. The daily data cover the number of births to teens in Texas during the period 1964-1999.     

Combined SANS and SAXS study of the action of ultrasound on the structure of amorphous zirconia gels

In the present work, we have studied for the first time the combined effect of both sonication and precipitation pH on the structure of amorphous zirconia gels synthesized from zirconium(IV) propoxide. The techniques of small-angle neutron and X-ray scattering (SANS and SAXS) and low temperature nitrogen adsorption provided the integral data on the changes in the microstructure and mesostructure of these materials caused by ultrasonic (US) treatment. Amorphous ZrO₂·xH₂O synthesized under ultrasonic treatment was found to possess a very structured surface, characterized by the surface fractal dimension 2.9–3.0, compared to 2.3–2.5 for the non US-assisted synthesis, and it was also found to possess a higher specific surface area, while the sizes of the primary particles remain unchanged.     

Optimisation of multifractal analysis at the 3D Anderson transition using box-size scaling

We study various box-size scaling techniques to obtain the multifractal properties, in terms of the singularity spectrum f(α), of the critical eigenstates at the metal-insulator transition within the 3-D Anderson model of localisation. The typical and ensemble averaged scaling laws of the generalised inverse participation ratios are considered. In pursuit of a numerical optimisation of the box-scaling technique we discuss different box-partitioning schemes including cubic and non-cubic boxes, use of periodic boundary conditions to enlarge the system and single and multiple origins for the partitioning grid are also implemented. We show that the numerically most reliable method is to divide a system of linear size L equally into cubic boxes of size l for which L/l is an integer. This method is the least numerically expensive while having a good reliability.     

Semilinear equations on fractal blowups

The paper concerns existence of a ground state for a nonlinear scalar field equation on a blowup fractal, where imbedding of the energy space into Lp is not compact. In absence of invariant transformations involved in conventional concentration-compactness argument, the paper develops convergence reasoning based on the fractal's self-similarity.     

Schmidt's game, badly approximable matrices and fractals

We prove that for every M,N∈N, if τ is a Borel, finite, absolutely friendly measure supported on a compact subset K of RMN, then K∩BA(M,N) is a winning set in Schmidt's game sense played on K, where BA(M,N) is the set of badly approximable M×N matrices. As an immediate consequence we have the following application. If K is the attractor of an irreducible finite family of contracting similarity maps of RMN satisfying the open set condition (the Cantor's ternary set, Koch's curve and Sierpinski's gasket to name a few known examples), then

dimK=dimK∩BA(M,N).     

Acceleration waves on random fields with fractal and Hurst effects

In this study, we determine the effect of spatial randomness on the probability of shock formulation and the distance to form shocks from acceleration waves as a function of the initial amplitude. The noise is applied to the dissipation and elastic nonlinearity of the system for two different cases: (i) two variables with the same noise of varying intensity and (ii) four variables with the same noise of varying intensity. The random fields used here are unique as they can capture and decouple the field’s fractal dimension and Hurst parameter. We focus on determining the driving parameter, either fractal or Hurst, which is significant in altering the response of the system.