Fractals

Competitive ability of helical and two-dimensional models of generation of large-scale atmospheric hazardous events is discussed, and the conclusion is made that the helical model deserves more attention than it currently has in the world literature. A number of mechanisms of nonlinear stabilization of helical vortex instability are considered which are possible under different conditions. Intermittent nature of large-scale velocity field generated by such instability, and possibility of development of helical wave-turbulent instability are analyzed. The example of application of helical vortex instability for explanation of some phenomena observed after collision of Shoemaker-Levy 9 comet fragments with Jupiter in July 1994 is presented.

We discuss the behavior of the dynamic dimension exponents for families of fractals based on the Sierpinski gasket and carpet. As the length scale factor for the family tends to infinity, the lattice approximations to the fractals look more like the tetrahedral or cubic lattice in Euclidean space and the fractal dimension converges to that of the embedding space. However, in the Sierpinski gasket case, the spectral dimension converges to two for all dimensions. In two dimensions, we prove a conjecture made in the physics literature concerning the rate of convergence. On the other hand, for natural families of Sierpinski carpets, the spectral dimension converges to the dimension of the embedding Euclidean space. In general, we demonstrate that for both cases of finitely and infinitely ramified fractals, a variety of asymptotic values for the spectral dimension can be achieved.

Hospital emergency arrivals are often modeled as Poisson processes because of the similarity of the problem to a telephone exchange model, i.e. applications that involve counting the number of times a random event occurs in a given time, where the interval between individual counts follows the exponential distribution. In this paper, we propose a statistical modeling of hospital emergency ward arrivals, using self-similar processes. This modeling takes into account the fact that it is known empirically that the emergency time series consists of periods marked by "bursts" of high level demand peaks, followed by periods of lower demand. This is explained neither by a Poisson during fixed-lengths intervals nor by a lognormal distribution. We show that a self-similar distribution can model this phenomenon. We also show that the commonly used Poisson models seriously underestimate the "burstiness" of emergency arrivals over a wide range of time scales, and that the emergency time series cannot be modeled by a lognormal distribution. The self-similar distribution was tested by the Hurst parameter, which we have calculated using five different methods, all of which agree on the value of the parameter.

We measure the fractal dimension of an African plant that is widely cultivated as an ornamental – the Asparagus plumosus. This plant presents self-similarity, remarkable in at least two different scalings. In the following, we present the results obtained by analyzing this plant via the box counting method for three different scalings. We show in a quantitative way that this species is a fractal.

Quartz grain boundaries from metamorphic and igneous rocks may emphasize a complex geometry, characterized by self-similarity over one to two orders of magnitude. Their fractal analysis highlights scaling sub-domains, i.e. scale intervals with a particularly good correlation. Given the importance of these aspects for the deciphering of geological microstructures, the paper is dedicated to the detection and the objective depiction of the features of heterogeneous scaling intervals. A fractal analysis based on the divider method was followed by processing methods that (i) offer a global evaluation of the curve geometry from the point of view of the correlation sub-domains, and (ii) allow a local characterization of the curves in terms of scale, with special concern for the scaling intervals heterogeneity. The application of the proposed approach was exemplified both on natural and synthetic curves. On one hand, the grain boundary analysis highlighted scaling sub-domains most obviously in the case of microstructures that were subject to overprinting, due to successive processes. On the other hand, a pattern superposition in the case of the synthetic curves strongly emphasized scaling sub-domains, as compared to the unperturbed (recursively generated) curve geometry. These aspects were expressed quantitatively and highlighted in more detail on isocorrelation maps. The importance of a rigorous characterization of these sub-domains and, eventually, the detection of pattern overprinting phenomena in geological microstructures emphasize the relevance of such an approach.

In this paper, we analyze the fractal structure of long human language records by mapping large samples of texts onto time series. The particular mapping set up in this work is inspired on linguistic basis in the sense that is retains the word as the fundamental unit of communication. The results confirm that beyond the short-range correlations resulting from syntactic rules acting at sentence level, long-range structures emerge in large written language samples that give rise to long-range correlations in the use of words.

Scaling properties of self-potential and carbon dioxide flow rate data, concomitantly recorded in a CO₂ degassing area at Soos, Bohemia (Czech Republic), have been investigated by means of several fractal tools. The power spectra, behaving as power law function of the frequency, show the presence of colored noise-type dynamics. The Higuchi analysis reveals the self-potential and flow signals to have the property of fractal curves. The variograms, analyzing time-scales stretching from one hour to almost 50 hours, reveal the presence of approximately three time-scaling ranges, characterized by different features in terms of persistence of the processes underlying the signals.

An interesting analog circuit for simulating a signal with fluctuations having a probability density function with a power tail has recently been proposed and constructed. The exponent of the power law can be fixed by tuning an appropriate circuit element. The proposal is to use the circuit as a simulator-generator of financial market fluctuations and as a tool for risk estimations and forecasts. We present a discussion of the stability conditions for multiplicative noise and an exhaustive analysis of the power law fluctuation generator in connection with the electronic components and their parameters. From our studies, one can conclude that the proposal is not adequate to provide a confident experimental scheme to follow the fluctuations of the financial market due to both electronic implementation and to difficulties in parameter tuning.

Combat data collected from World War II and a cellular automaton combat model called MANA are shown to display fractal properties. This strongly supports our earlier hypotheses as to the nature of combat attrition data. It also provides a method by which we can judge a combat model's ability to produce realistic synthetic combat data. The data appear to display properties extremely similar to those of the fractal cascade models used to describe turbulent dynamics. Interestingly, the fractal parameters appear to depend on how the model is set up, implying that they are determined by the boundary and initial conditions. Examination of the dynamical rules used in the MANA model simulation suggests that the model entities need to respond to changes in the level of order on the battlefield grid for fractal behavior to occur. Such data imply that the entropy of the battlefield is dependent on the scale at which it is examined. We speculate that such formations in a military case effectively act to isolate the highest level of command from disorder in the lowest. If disorder within a force grows to the point where that force can no longer maintain a fractal-like distribution, the force distribution may tend to become uniformly random, effectively destroying its viability as a combat unit.