Keizo Yamamoto, Sasuke Miyazima, Hiroshi Yamamoto, Toshiya Ohtsuki and Akihiro Fujihara

A model for power-law problem in high-income distribution is proposed. This model is so simple that we can examine the effect of some differences in rules that allocate different amounts of resources to winners and losers in economic competitions where we obtain different power-law exponents as well as exact analysis by the master equation.P. Allegrini, F. Barbi, P. Grigolini, and P. Paradisi

We describe a form of modulation, namely a dishomogeneous Poisson process whose event rate changes sporadically and randomly in time with a chosen prescription, so as to share many statistical properties with a corresponding non-Poisson renewal process. Using our prescription the correlation function and the waiting time distribution between events are the same. If we study a continuous-time random walk, where the walker has only two possible velocities, randomly established at the times of the events, we show that the two processes also share the same second moment. However, the modulated diffusion process undergoes a dynamical transition between superstatistics and a Lévy walk process, sharing the scaling properties of the renewal process only asymptotically. The aging experiment - based on the evaluation of the waiting time for the next event, given a certain time distance between another previous event and the beginning of the observation - seems to be the key experiment to discriminate between the two processes.Petra Jerabkova, Oldrich Zmeskal, Jan Haderka

This article describes a new method of determining fractal dimension and fractal measure using integral (e.g. wavelet – Haar) transformations. The main advantage of the new method is the ability to analyse both, black & white and grey scale (or colour) images. The fractal dimension (box counting fractal dimension) evaluated using the Haar transformation offers in difference to the classic box counting method much wider range of usability. It can be also used to determine the fractal parameters of surfaces or volume of three-dimensional structures e.g. distribution of the mass or electrical charge in the surface or space. The theory of the fractal structures in E-dimensional Euclidean space is described more closely in our previous work.Seiji Tokunaga and Hidetsugu Sakaguchi

A doublon is one of the fascinating patterns found in crystal growth. It forms a pair of symmetry broken fingers. In this paper, we investigate the doublon pattern growing with the effect of other doublons. We have found that as the result of some competition, some doublons break their own structure and the number of doublon keeping its structure depends on diffusion length.Pavla Stefkova, Oldrich Zmeskal, Radek Capousek

A new method of composite materials thermal parameters determination is described in this contribution. The transient pulse method is used for specific heat, thermal diffusivity and thermal conductivity determination. The evaluation is performed with the help of mathematical apparatus used for study of fractal structures properties. The method presented is more general then the classical transient method commonly used for homogenous materials description.Mario Markus, Malte Schmick, Eric Goles (Germany, Chile)

One-dimensional binary automata are allowed to interact with each other using a 2x2 binary payoff matrix. In this way, chaotic evolution can be controlled or chaos can be induced (anticontrol). For each matrix, a hierarchy of the automata can be established as follows: an automaton with a higher rank controls (or anticontrols) an automaton with a lower rank. For some matrices and some rule number intervals, the ranking depends monotonically on the rule number.B. Kaulakys, M. Alaburda, V. Gontis and T. Meskauskas

Multiplicative processes and multifractals have earned increased popularity in applications ranging from hydrodynamic turbulence to computer network traffic, from image processing to economics. We analyse the multifractality of the recently proposed point process models generating the signals exhibiting 1/f noise. The models may be used for modeling and analysis of stochastic processes in different systems. We show that the multiplicative point process models generate multifractal signals, in contrast to the formally constructed signals with 1/f noise and signals consisting of sum of the uncorrelated components with a wide-range distribution of the relaxation times.V.V. Titov, S.V. Titov and L.A. Reznitchenko

Analysis of multifractal spectra parameters of grains' patterns on the ferroelectric ceramics surfaces was performed. The samples were prepared from Nb2O5 with di erent phase compositions. It was found out that grains' patterns parameters under study are sensitive indicators of inhomogeneities in synthesized materials. These inhomogeneities also reflected by the mesoscopic parameters of crystallites structure such as microdeformation and coherent scattering regions. Conditions of Nb2O5 preliminary heat treatment necessary for processing materials with considerably improved desired characteristics were corrected.Nikitas Papasimakis and Fotini Pallikari (Greece)

Random multiplicative cascades, i.e. cascades, which redistribute a finite quantity from coarser to finer scales, have successfully modelled records of physical phenomena. In this work the redistribution of a quantity at each scale is regarded not to be performed at independent steps, but to depend on its distribution at previous coarser scales. This work considers the effect of memory within the random multiplicative process and its consequences on the multifractal behavior of the measure. The multipliers involved in the cascade assume two distinct values that occur with probabilities defined by a first-order, two-state Markov process. It is shown that in the non-Markov memory-less case the original quantity can be conserved on the average only if the two values of the multiplier satisfy a linear relation, while this is not true for the cascade presented here. It is also shown that when the two self-transition Markov probabilities are equal yet different from 0.5, the average occurrence of the multipliers converges to 50% as in the memoryless case. Nevertheless, the Markov memory has an effect on the scatter of multipliers about the average. Typically, multifractals are described by the spectrum function f(h), where h is the Holder exponent characterizing the singularity strength and f(h) is related to the probability of observing a singularity of strength h on any given scale. It is shown that the shape of the f(h) curve depends on the transition probabilities of the Markov chain and more specifically any asymmetry of f(h) can be considered as a result of a non-symmetrical transition matrix. Moreover, negative values of f(h) occur as a natural consequence of the anti-persistent character of the underlying Markov process. The conservation of measure now relaxes to convergence towards a nontrivial and finite value and the shape of singularity spectrum depends to a great extent on the Markov probabilities. Application of the model to turbulence data indicates an underlying anti-persistent Markov process. The proposed cascade is applied as a model to experimental data of the energy dissipation in turbulent flow, where negative values of f(h) have been observed. The best fit to the estimated f(h) curve is found in the symmetric case where both self-transition Markov probabilities are equal to 30% with multiplier values 0.91 and 0.19.Dane R. Camp

This exposition draws a connection between the Tower of Hanoi puzzle and four well-known fractals. These connections are illustrated by examining a graph theoretical picture of the solution to the puzzle and identifying patterns within the structure of the graph. The numbering scheme introduced by Ian Stewart, in an article he wrote for Scientific American, connects a graphical representation of the solution to the Tower of Hanoi to the structure of Sierpinski's Gasket. Further investigation of the solution by the author shows how connections can also be discovered between the Tower of Hanoi and three other fractals: the Dragon Curve, Cantor Dust, and the Von Koch Curve.Nicoletta Sala

Industrial Design is the field of developing physical solutions which might include products, vehicles, machinery, and even environments. For many years the industrial designers found inspiration by the symmetry and by the Euclidean shapes (e.g., polygons and polyhedra). The evolution of the technologies (from the hand-made to the Computerized Numeric Control) and the evolution of the materials (from glass to the plastic and the polymer composites) permitted to the designers to conceive shapes which pass the limits imposed by the Euclidean geometry. Modern design studies apply complex shapes based on the fractal geometry to create new kind of objects, realized in innovative materials. These objects sometimes mimic the complex shapes present in the nature. The aim of this paper is to describe an approach which introduces the connections between the Nature and some industrial design objects. Their complex shapes will be analysed using a fractal point of view.Xi-Zhe Zhang, zheng-Xuan Wang, Tian-Yang Lu

The iteration of complex function can generate beautiful and complex fractal images. This paper extends the existing iteration methods and tries to find a new way to generate fractal image. We present a novel method based on the iteration of the distance ratio with two points, which classifies the points according to their distance ratio iteration convergence speed. By using this method, we get a 4-dimension fractal set. Firstly, this paper proves the condition of distance ratio convergence and gives an inverse iteration layered method to render the distance ratio fractal. Secondly, take the quadratic complex function for example, we render the distance ratio fractal image by projecting the distance ratio fractal set into 2-dimension plane. Finally, we analysis some properties of distance ratio fractal image and prove its boundary is the Julia set. Comparing with traditional escape time algorithm, the new method uses iteration of two points instead of one point to generate more complex structure, which is different from existing fractal image.Tomek Martyn (Poland)

An approach to volume rendering 3D fractal measures is presented. Visualization is done with the adaptive level of detail at pixel-size accuracy. Using the IFS representation we show a generic construction of the measures, which can be thought as a variant of Caratheodory’s method when applied to the fractal context. Our construction holds a wide range of fractal measures, including the IFS and RIFS invariant measures. Then, we introduce the measure instancing technique, which allows one to approximate the measure of a set at any accuracy required. The technique enables fractal measures to be ray-traced with low memory and without storing a volume of cells in memory resources. Instead, we utilize a dynamic hierarchy of balls on the object instancing principles. Rather than with semi-lines, the measure is ray-traced with cones so as to speed up computation and to prevent aliasing. To illuminate measures, we use a generalization of the volume rendering integral which we adapt to the case of measures.P.K. Galenko and D.M. Herlach

Results of computational modeling are synthesized for the forms of crystals growing in undercooled liquid. Solidification patterns are analyzed as a result of the first-order phase transformation controlled by the heat diffusion, atomic kinetics, and interfacial anisotropy. It is shown that fractal patterns are observed at a vanishing anisotropy of surface energy and atomic kinetics of the solid-liquid interface. Simulated patterns are summarized into morphological spectrum which is considered as a sequence of growth shapes that forms versus undercooling (deviation from thermodynamic equilibrium) in solidifying system. A diagram ``complexity of growth forms'' as a function of ``microscopic disorder and deviation from equilibrium'' is presented.Antonio Turiel and Conrad Perez-Vicente

The application of the multifractal formalism to the study of some time series with scale invariant evolution has given rise to a rich framework of models and processing tools for the analysis of these signals. The formalism has been successfully exploited in different ways and with different goals: to obtain the effective variables governing the evolution of the series, to predict its future evolution, to estimate in which regime the series are, etc. In this paper, we discuss on the capabilities of a new, powerful processing tool, namely the computation of dynamical sources. With the aid of the source field, we will separate the fast, chaotic dynamics defined by the multifractal structure from a new, so-far unknown slow dynamics which concerns long cycles in the series. We discuss the results on the perspective of detection of sharp dynamic changes and forecasting.A. Bari, G. Ayad, S. Padulosi, T. Hodgkin, A. Martin, J. L. Gonzalez-Andujar and A. Brown (Italy, Spain, Australia)

Geographical distribution patterns in plants have been observed since primeval times and have been used by plant explorers to trace the origin of plants species. These patterns have been found to embody the presence of fundamental law-like processes. Diversity in plants has also been found to be proportionate with the area, and this scaling behavior is also known as fractal behavior. In the present study, we used fractal geometry to analyze the distribution patterns of wild taxa of cowpea with the objective to locate where their diversity would be the highest to aid in the planning of targeted explorations and conservation measures.Victor Korotkikh and Galina Korotkikh

It is shown that complex systems can be described in terms of self-organization processes of prime integer relations. The processes build up the prime integer relations from one level to the next by using the integers as the basic constituents. Prime integer relations specify nonlocal correlations between parts of a complex system and can be represented as two-dimensional geometric patterns with fractal properties. Controlled by arithmetic only, the self-organization processes of prime integer relations describe complex systems by absolutely true information and do not require deeper principles for explanations. This gives the possibility to develop an irreducible theory of complex systems.E.B. Postnikov, A. Loskutov and A.B. Ryabov

In the framework of the mean--field approximation we propose a new approach to the description of the growth of fractal structures which are formed as a result of the process of diffusion limited aggregation. It is based on the roughening of the time scale which takes into account the property of discreteness of such structures. The obtained system of partial differential equations allows to evaluate numerically the fractal dimension and the cluster density depending on the distance from its center. The results are in a quite good agreement with values found by the direct numerical simulations. The proposed approach is generalized for the case of the cluster description with different immiscible particles.L. S. Liebovitch, V. K. Jirsa, and L. A. Shehadeh (USA)

The expression of some genes increases or decreases the expression of other genes forming a complex network of interactions. Typically, correlations between the expression of different genes under different conditions have been used to identify specific regulatory links between specific genes. Instead of that "bottom up" approach, here we try to identify the global types of networks present from the global statistical properties of the mRNA expression. We do this by comparing the statistics of mRNA computed from different network models, including random and fractal scale free networks, to that found experimentally. The novel features of our approach are that: 1) we derive an explicit form of the connection matrix between genes to represent scale free models with arbitrary scaling exponents, 2) we extend Boolean networks to quantitative regulatory connections and quantitative mRNA expression concentrations, 3) we identify possible types of global genetic regulatory networks and their parameters from the experimental data.Bernd Sing

We investigate graph-directed iterated function systems in mixed Euclidean and p-adic spaces. Hausdorff measure and Hausdorff dimension in such spaces are defined, and an upper bound for the Hausdorff dimension is obtained. The relation between the Haar measure and the Hausdorff measure is clarified. Finally, we discus an example, for which we calculate upper and lower bounds for its Hausdorff dimension.Katsuya Honda, Soshu Kirihara, Yoshinari Miyamoto, Yoshiki Otobe and Mitsuo Wada-Takeda

We consider one-dimensional systems which are formed from a segment by employing the Cantor-set construction rule until a required stage. The rigorous expression of the transmission coefficient of electromagnetic waves through them is presented. The method to obtain the expression is explained in a straightforward way. Numerical demonstration displays the unpredictable results. As the stage is increased the behavior of the transmission spectra changes drastically. In the rather higher stages the periodicity hidden in the self-similar symmetry occurs.Przemyslaw Waliszewski and Jerzy Konarski

This paper describes a relationship between the stochastic processes at the microscopic scale and the emergence of universal, Gompertzian dynamics of growth and self-organization at the macroscopic scale. That fractal dynamics results from the complex coupling of probabilities. There is a relationship between fractal Gompertzian dynamics, non-Gaussian distribution of probability, and Kolmogorov-Sinai entropy. Moreover, Gompertzian dynamics appears in the case of the simpliest couplings, such as those occuring in the stationary differential Markov chain.

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September 2005