Fractals

We show that the predictability of letters in written English texts depends strongly on their position in the word. The first letters are usually the least easy to predict. This agrees with the intuitive notion that words are well defined subunits in written languages, with much weaker correlations across these units than within them. It implies that the average entropy of a letter deep inside a word is roughly 4–5 times smaller than the entropy of the first letter.

We study the origin of the rugged border of thin-layer ramified copper electrodeposits influenced by chemical perturbations. Our main interest is to look for the smallest growth building blocks toward the understanding of the transition from a compact to a rugged structure. Using electron microscopy, we found two building blocks, well differentiated in shape and size, consisting of: (a) Spikes or rods of a diameter between 100 and 500 nm and a maximum observed length of up to 10 μm; and (b) crystalline grains with a diameter between 200 nm and 2 μm. Through X-rays Bragg diffraction, we tentatively assessed a Cu₂O and metallic Cu chemical composition to the crystals and rods, respectively. The relative concentration of metallic Cu to Cu₂O changes as we move from the base to the front of the aggregation and this result could be associated with the Hecker effect.

In this paper, we focus on invariant measures arising from Iterated Function System with Probabilities (IFSP). We show the equivalence between an IFSP and a linear dynamical system driven by a white noise. Then, we use a multifractal analysis to obtain scaling properties of the resulting invariant measures, working within the framework of dynamical systems. Finally, as an application to fractal image generation, we show how this analysis can be used to obtain the most efficient choice for the probabilities to render the attractor of an IFS by applying the probabilistic algorithm known as “chaos game”.

The volcanic seismicity occurring in Campi Flegrei during the 1983–1984 volcanic crisis is quantitatively analyzed by means of fractal (scale invariant) geometry in all its five dimensional set (magnitude, time and space). The significant power laws, relating the number of earthquakes to magnitude, time and space, within a limited range of scales, show that the Campi Flegrei region is in a critical state during the whole crisis.

In this paper, a general class of fractional random fields, ℬ_α, 0≤α<2, is defined. The members of ℬ_α can be used to model natural scenes and textures. It is shown that the fractal dimension of random fields in ℬ_α is a linear nonincreasing function of a for 0≤α<α₀ and a linear nondecreasing function of α for α₀<α<2. The number α₀ corresponds to the Hausdorff-Besicovitch dimension of the random field. These linear relationships are significant for texture comparison and classification.

Wavelet analysis is used to measure directly the scale exponents of atmospheric turbulence at inertial range under different stratifications. It is found that the average values of the scale exponents under different stratifications are clearly different and are all lower than the Kolmogorov 1941 value of 1/3. The difference shows that, according to the intermittent turbulence model, turbulence under stable stratification behaves more intermittently than under unstable stratification. The reason may be that the gravity wave, which could exist only under stable stratification, modulates the state of nocturnal surface layer and causes this kind of intermittence.

The set of generalized purity functions, appearing in the problem of electromagnetic fields copropagating through a nonlinear Kerr medium, is characterized as a family of multifractal measures. The singularity spectra of these multifractal measures are obtained, and it is shown that they can be grouped into three distinct sets according to a characteristic moment m. The measures with m<3/2 and m>3 have a universal singularity spectrum, once the singularity strengths are properly scaled. For the set of intermediate moments the singularity spectrum exhibits a non-universal crossover.

A stochastic model for stretched or compressed exponential survival statistics is suggested based on the following assumptions. (1) The state of an individual is characterized by a finite set of random variables z=(y₁, y₂,…); the time evolution of these variables is described by a non-Markovian stochastic process with long memory. (2) In the state space of an individual there are some critical regions for which the death of an individual can occur; for a state vector z belonging to a critical region, there is a probability 0<p(z)≤1 different from zero that the individual dies. (3) Two characteristic fractal time laws are introduced, for the random behavior of an individual at a given moment and for the long range time correlations of the fluctuations in the state space of the individual, respectively. A systematic procedure for the evaluation of the average survival function of an individual is suggested, based on a cumulant expansion in terms of the bit number b= −ln(1−p(z)) of the probability of survival corresponding to a state vector z belonging to a critical region. Explicit computations are performed both for discrete and continuous time in the case of a non-Markovian and Gaussian process with long memory describing the time evolution of the bit number b. It is shown that the main cause for the occurrence of the stretched or compressed exponential statistics is the fractal time behavior for one-time fluctuations. The second fractal time law, which describes the long range correlations of state fluctuations leads only to a correction which increases the value of the survival function, i.e., the long memory fluctuations of the state variables increase the chances of survival of the individuals. Some implications of the model for the interpretation of the survival data in population biology and health statistics are investigated.

The search for low-dimensional chaos in ocean surface waves is nowadays a very active field. The interpretation of the results, however, is not always straightforward. The issue addressed in this paper is how time series analysis tools from dynamical systems theory behave for a class of Gaussian processes often used in the study of ocean surface waves. The study includes the largest Lyapunov exponent, the Grassberger and Procaccia correlation dimension and the self-similarity properties. Surprisingly, for certain parameter ranges, the correlation dimension is found to be finite, the largest Lyapunov exponent is found to be positive and structure appears on all time scales. These results suggest that improved techniques and data analysis procedures may be required in order to study chaos properties of ocean surface waves or of other Gaussian processes with similar power spectra.

We investigate the box dimension of a time series having an inverse power-law spectra in a high dimensional Euclidean space. The time series can be random (colored noise) or deterministic. Both isotropic and anisotropic cases are included in our investigation. We study both the graph dimension and trail dimension of the time series. We show that with the same inverse power-law spectra, the deterministic series has a lower graph dimension than that of the colored noise, though they both can have fractal dimensions. We also derive a sharp upper bound on the trial dimension of the time series.

Fractal parameters of soils has become increasingly important in understanding and quantifying transport and adsorption phenomena in soils. It is not known how soil plant development may affect fractal characteristics of soil pores. We estimated pore surface area fractal parameters from mercury porosimetry data on gray forest soil before and during crop development, in samples both containing and not containing soil carbohydrates known to be important structure-forming agents. Two distinct intervals with different fractal dimensions were found in the range of pore radii from 4 nm to 1 μm. This could be attributed to differences in mineral composition of soil particles of different sizes. The interval of the smallest radii had the highest average fractal dimension close to 3. Smaller surface area fractal dimensions corresponding to low surface irregularity were found in the next interval of radii. The plant development affected neither fractal dimensions nor the cutoff values of soil samples. The carbohydrate oxidation caused a significant increase in the fractal dimension in the interval of larger radii, but did not affect fractal dimension in the interval of small radii. The cutoff values decreased after carbohydrate oxidation.

An Abelian sandpile model is considered on the Husimi lattice of triangles with an arbitrary coordination number q. Exact expressions for the distribution of height probabilities in the Self-Organized Critical state are derived.