Theory of fractals

Because of the highly unstable nature of the acid attack in most of the carbonate reservoirs (propagation of wormholes), the development of a descriptive model of the skin evolution was not possible until the recent advent of the theory of fractals. In addition, the characteristics of the damaged zone greatly affect the behavior of the skin during acid injection in any type of reservoir, but particularly in carbonate ones. 

From the most general point of view, the theory of fractals (Mandelbrot [1977]), one-, two-, three-, m-dimensional figures are only borderline cases. Only a straight line is strictly one-dimensional, an even area strictly two-dimensional, and so on. Curves such as in Fig. 3.11 may have a fractal dimension of about 1.1 to 1.3 according to the principles of fractals areas such as in Fig. 3.12b may have a fractal dimension of about 2.2 to 2.4 and the figure given in Fig. 3.14 drawn by one line may have a dimension of about 1.9 (Mandelbrot [1977]). Fractal dimensions in analytical chemistry may be of importance in materials characterization and problems of sample homogeneity (Danzer and Kuchler [1977]). 

Experimentally accessible is D by means of scattering methods [144], The corresponding fractal analysis of scattering data is gaining special attractivity from its intriguing simplicity. In a double-logarithmic plot of I (s) v.v. s the fractal dimension is directly obtained from the slope of the linearized scattering curve. It follows from the theory of fractals that... 

See, for instance, M. Lax, in Multiple Scattering and Waves in Random Media, P. L. Chow, W. E. Kohler, and G. C. Papanicolaou, Eds., North Holland, 1981, and references therein J. Klafter and M. S. Schlesinger, Proc. Nat. Acad. Sci. U.S.A. 83, 848 (February 1986), and references therein R. Brown, Thesis, University of Bordeaux 1,1987 R. Brown et al. J. Phys. C20, L649 (1987) 21 (1988) in press. (This last work thoroughly discusses the applicability of fractal theory to isotopically mixed crystals as disordered system. Serious criticism is presented both of the analysis of the experimental data and of the fundementals of their description in the present theory of fractals. For this reason we omit all works treated there. 

M. Suzuki, /. Math. Phys., 32, 400 (1991). General Theory of Fractal Path Integrals with Applications to Many Body Theories and Statistical Physics. 

The regularities revealed in the theory of fractals and percolation have turned out to be generally true for heterogeneous stochastic media and, in particular, for composite materials. 

As shown below, an attempt is made to solve this problem using the ideas of the renormalization group transformation method and the theory of fractals, which is also called the geometry of chaos. 

The interfacial capacitance may also be measured at solid polarizable electrodes in an impedance experiment using phase-sensitive detection. Most experiments are carried out with single crystal electrodes at which the structure of the solid electrode remains constant from experiment to experiment. Nevertheless, capacity experiments with solid electrodes suffer from the problem of frequency dispersion. This means that the experimentally observed interfacial capacity depends to some extent on the frequency used in the a.c. impedance experiment. This observation is attributed to the fact that even a single crystal electrode is not smooth on the atomic scale but has on its surface atomic level steps and other imperfections. Using the theory of fractals, one can rationalize the frequency dependence of the interfacial properties [9]. The capacitance that one would observe at a perfect single crystal without imperfections is that obtained at infinite frequency. Details regarding the analysis of impedance data obtained at solid electrodes are given in [10]. 

Recently, visible progress has been achieved in the theoretical and experimental investigations of non-linear phenomena. Among vital achievements in this area, the theory of solitons, strange attractors, the theory of fractals, chemical reactions of complex dynamics should be mentioned. 

The theory of fractals and its application to physical and chemical processes has been developing vigorously in recent years [1-8]. To facilitate the understanding of the results presented in this chapter, we shall introduce some notions and definitions and consider briefly the grounds for applying the principles of synergetics and fractal analysis to the description of structures and properties of polymers. 

The theory of fractal dimension may be used in bioimpedance signal analysis, for example, for studying time series. Such analysis is often done by means of Hurst s rescaled range analysis (R/S analysis), which characterizes the time series by the so-called Hurst exponent H = 2 — D. Hurst found that the rescaled range often can be described by the empirical relation... 

Mandelbrot also pointed out that these unique mathematical concepts are now the fundamental tools defining natural phenomena in the world around us. He referenced particular mathematicians who made singular contributions to the theory of fractals their theories and the objects named after them are standards of the canon of fractal geometry. 

A fairly simple possibility to do this is to realize that contrary to a PSSS, Fig. 1. 20, the volume (V ) of an irregular porous sorbent being impenetrable for molecules of a sorptive gas, generally will depend on the size of the gas molecules, cp. Figs. 1.1, 1.21. This is well known in the mathematical theory of fractal surfaces [1.67] and thermodynamic phases of fractal dimension [1.68]. Also the range of surface forces V = V(A), cp. (1.14), of a sorbent material is limited in space but also depend on the sorptive gas molecules. Hence one may consider the volume of the joint sorbent/sorbate phase (V ) as... 

As an example, we may consider the approaches now used to model processes on the stock market (e.g. using elements from the theory of fractals). Many such models are very familiar to networking engineers, where e.g. the self-similar properties of data traffic have been identified a few years ago. 

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