Mathematical fractals

Successive increments of mathematical fractal random processes are independent of the time step. Here D = 1.5 corresponds to a completely uncorrelated random process r = 0, such as Brownian motion, and D = 1.0 corresponds to a completely correlated process r= 1, such as a regular curve. Studies of various physiologic time series have shown the existence of strong long-time correlations in healthy subjects and demonstrated the breakdown of these correlations in disease see, for example, the review by West [56]. Complexity decreases with convergence of the Hurst exponent H to the value 0.5 or equivalently of the fractal dimension to the value 1.5. Conversely, system complexity increases as a single fractal dimension expands into a spectrum of dimensions. 

In mathematics, fractals appear as a result of the opposition and unity of two fields of mathematics One of these fields studies numbers (discrete objects), while the other studies shapes (continuous objects). 

In the case of regular mathematical fractals such as the Cantor set, the Koch curves and Sierpinski gaskets constructed by recurrent procedures, the Renyi dimension d does not depend on q but [16] ... 

The Fractal Factory is a DMS which reflects the mathematical fractal concept thus, it is based on the pattem-inside-pattem concept. The factory consists of small components, called fractal entities. The smallest component can decide on its own behavior and characteristics, without relying on higher entities. Its main properties are self-similarity, self-organization, and dynamics (Tharumarajah et al. 1996). 

Can fractals, defined as in Section 2.4.2, serve as appropriate representations of natural objects or processes The answer to this question is (surprisingly perhaps, yet uncompromisingly) no. Strictly speaking, there are no true (mathematical) fractals in... 

Processes in nature often result in fractal-like forms that differ from the mathematical fractals such as the Koch curve in two ways (a) the self-similarity is not exact but is a congruence in a statistical sense and (b) the number of repeated splittings is finite and random fractals have an upper and a lower cutoff length. A spatial example of a random fractal is the colloidal gold particle agglomerate shown earlier in Figure 7.4. 

Unlike mathematical fractals, real fractals (including polymers) have two natural length scales and (Figure 2.1) objects below and above are not fractal [23]. The lower limit is connected with the finite size of the structural elements and the upper one with uneven aspiration for the limit d As was noted above, for... 

It turns out that many surfaces (and many line patterns such as shown in Fig. XV-7) conform empirically to Eq. VII-20 (or Eq. VII-21) over a significant range of r (or a). Fractal surfaces thus constitute an extreme departure from ideal plane surfaces yet are amenable to mathematical analysis. There is a considerable literature on the subject, but Refs. 104-109 are representative. The fractal approach to adsorption phenomena is discussed in Section XVI-13. 

K. Falconer, Fractal Geometry Mathematical Foundations and Application, Wiley, New York, 1990. 

Here, D< )s is the fractional diffusivity defined as (4-2d,) (2d,-4) )(3-dP) j js a constant related to the fractal dimension and R0 is the side length of a square electrode), and dv dtv means the Riemann-Liouville mathematical operator of fractional derivative ... 

In their theoretical work,43 the various self-affine fractal interfaces were mathematically constructed employing the Weierstrass function /ws(x), 151>152... 

The aggregation of spherical particles such as small particles of silica have been modelled by a fractal approach.16,17 Fractal mathematics can... 

A mathematically definable structure which exhibits the property of always appearing to have the same morphology, even when the observer endlessly enlarges portions of it. In general, fractals have three features heterogeneity, setf-similarity, and the absence of a well-defined scale of length. Fractals have become important concepts in modern nonlinear dynamics. See Chaos Theory... 

Parameter that provides a mathematical description of the fractal structure of a polymer network, an aggregated particulate sol, or of the particles that comprise them. 

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