Fractals

Bucket random permutations (shuffling) are used to modify the dependence structure of a time series, and this may destroy long-range dependence, when it is present. Three types of bucket permutations are considered here: external, internal and two-level permutations. It is commonly believed that (1) an external random permutation destroys the long-range dependence and keeps the short-range dependence, (2) an internal permutation destroys the short-range dependence and keeps the long-range dependence, and (3) a two-level permutation distorts the medium-range dependence while keeping both the long-range and short-range dependence. This paper provides a theoretical basis for investigating these claims. It extends the study started in Ref. 1 and analyze the effects that these random permutations have on a long-range dependent finite variance stationary sequence both in the time domain and in the frequency domain.

We have investigated wave scattering by chaotic fractured systems of fractal geometry with random spatial variation that causes energy loss of the directly propagated field. We have examined simple analytic solutions in fractal poroelastic media. These solutions may be characterized by their frequency-power-law (FPL) signature caused by wave dispersion and attenuation. It has been proved that medium memory effects cause smoothing of the wavefield in the vicinity of the wavefront and rapid amplitude decay far from the wavefront. It appears that finite-bandwidth signals are delayed with respect to the wavefront in comparable elastic media. To examine the FPL dependence of direct body waves propagating in a homogeneous medium containing fractal inhomogeneities, we compute acoustic finite-difference snapshots in the frequency range f = 20 - 200 Hz. Numerical results show that the fractal dimension can be estimated from the FPL dependence of the scattered wavefield. Applications to fracture characterization are considered. Results are important for multi-scale depth imaging, inverse Q filtering, fracture detection, and integrated geophysical reservoir monitoring.

An agent-based model is employed for the study of the group size distributions. A fixed number of homogeneous agents are distributed on a two-dimensional lattice system. The dynamics of the agents is described in terms of the inverse distance potential and the friction factor. From a random initial distribution, the agents move forming groups until all the agents come to a stationary position. For a squared system with L × L cells, the group size distribution showed a well defined power-law behavior up to the cut-off size. But when the system changed to an L × H non-squared one, a "geometric aging effect" emerged. Together with the phase transition, the geometric aging effect is considered to be a generic mechanism of the deviated power-law distributions, such as the "fall-off" and the three-bent-line distributions. The results are discussed in relation to the well-known physical or social phenomena such as the King Effect in the city size distributions, the fall-off distribution of the fish schools, the three-bent-line distributions of the Earth-crossing asteroids and 2D percolation problem.

The origin of fractal patterns is a fundamental problem in many areas of science. In ecological systems, fractal patterns show up in many subtle ways and have been interpreted as emergent phenomena related to some universal principles of complex systems. Recently, Lévy-type processes have been pointed out as relevant in large-scale animal movements. The existence of Lévy probability distributions in the behavior of relevant variables of movement, introduces new potential diffusive properties and optimization mechanisms in animal foraging processes. In particular, it has been shown that Lévy processes can optimize the success of random encounters in a wide range of search scenarios, representing robust solutions to the general search problem. These results set the scene for an evolutionary explanation for the widespread observed scale-invariant properties of animal movements. Here, it is suggested that scale-free reorientations of the movement could be the basis for a stochastic organization of the search whenever strongly reduced perceptual capacities come into play. Such a proposal represents two new evolutionary insights. First, adaptive mechanisms are explicitly proposed to work on the basis of stochastic laws. And second, though acting at the individual-level, these adaptive mechanisms could have straightforward effects at higher levels of ecosystem organization and dynamics (e.g. macroscopic diffusive properties of motion, population-level encounter rates). Thus, I suggest that for the case of animal movement, fractality may not be representing an emergent property but instead adaptive random search strategies. So far, in the context of animal movement, scale-invariance, intermittence, and chance have been studied in isolation but not synthesized into a coherent ecological and evolutionary framework. Further research is needed to track the possible evolutionary footprint of Lévy processes in animal movement.

Time series data of many biological processes are amenable to fractal analysis. In this study, fractal dimensions were calculated for continuous cultures of Saccharomyces cerevisiae with (1) no noise in the feed stream, (2) noise inflow but no filtering, and (3) with noise and an extended Kalman filter. Each situation was analyzed for dilution rates and the gas-liquid mass transfer coefficients for oxygen under which monotonic, oscillatory and chaotic outputs were generated. The fractal dimensions were compatible with the nature of the time-domain profiles and were sensitive to changes in the profiles and to the presence or absence of noise and a filter. Together with similar results for Escherichia coli, the fractal dimension thus appears to be a convenient and reliable index of fermentation performance.

In this paper, we study two kinds of fractional differential systems with multi-orders. One is a system of fractional differential equations with multi-order, , ; the other is a multi-order fractional differential equation, . By the derived technique, such two kinds of fractional differential equations can be changed into equations with the same fractional orders providing that the multi-orders are rational numbers, so the known theorems of existence, uniqueness and dependence upon initial conditions are easily applied. And asymptotic stability theorems for their associate linear systems, , , and , , , are also derived.

Two fractal image compression algorithms based on possibility theory are originally presented in this paper. Fuzzy sets are used to represent the edge character of each image block, and two kinds of membership function are designed. A fuzzy integrated judgement model is also proposed. The model generates an accurate value for each edge block, which would be a label during the search process. The edge possibility distribution function and the edge necessity level are designed to control the quantity of the blocks to be searched. Meanwhile the pre-restriction is proposed, the average intensity value at different locations is used to be a necessary condition before the MSE computations. It is shown by our experiments that the encoding times of our two algorithms, compared to that of Jacquin's approach, are reduced to 60%–70% and 10%–20%, respectively.

We give a characterization of the convex hull of self-similar sets in ℝ³ which extends the results of Panzone¹ in ℝ². As an application, we show a convex set which cannot correspond to the convex hull of a self-similar set.