Self-Similar Data

FIGURE 4.2.3 Fractals are those patterns repeated at different scales. (From Liebovitch, L.S., Fractals and Chaos Simplified for the Life Sciences, Oxford University Press, New York, 1998. With permission.) 

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For a terse introduction to the mathematical background and the associated literature on the solution of inverse problems, the reader is referred to the publication of Sathyagal et al, (1995) and to Wright and Ramkrishna (1992). In regard to the mathematical literature on inverse problems we refer to Tikhonov and Arsenin. (1977). The solution of inverse problems of the type O = Xa is best established for situations in which the input data represented by the vector O has noise and the operator X is noise-free. Since, in the context of our discussion, X inherits the noise of the self-similar data, the presmoothing process discussed in the text is an important issue. The quality of the inversion can of course be assessed by evaluating the ability of the forward problem to predict back the data used for inversion. 

Figure 6.2.3 shows the self-similar data on (/>(f/) when 10% noise is added to the exact self-similar distribution. In this situation, the worst results for the inverted aggregation frequency are obtained in the absence of regulariz-... 

The calculations of Wright and Ramkrishna clearly demonstrate that regularization is a significant step in solving for the aggregation frequency from self-similar data with noise. 

As a second example, we consider self-similar data for the sum frequency for which it is known that 0(f/) = The expansion (6.2.13) holds... 

The results of the inverse problem are recounted later (i) with no added error, and (ii) with 10% random error to the exact average particle size as it evolves through time as well as to the self-similar data. 

In the examples just presented, the quality of the inversion could be assessed by directly comparing the aggregation frequencies obtained with the actual frequencies used to generate the self-similar data. However, in dealing with experimental data from aggregating systems for which the aggregation frequency is unknown, the test of the inversion lies in being able to predict from forward simulations the transient size distribution data... 

Therefore, it follows that the mean volume increases linearly with granulation time. From Eq. (52) we see that the size distribution is uniquely defined by a dimensionless size V/V, and in that sense it is self-similar. In order to compare with the experimental data, it is necessary to transform the size attribute from volume to diameter. From Eq. (52) it can be shown that (K3)... 

Adolf and Martin [15] postulated, since the near critical gels are self-similar, that a change in the extent of cure results in a mere change in scale, but the functional form of the relaxation modulus remains the same. They accounted for this change in scale by redefinition of time and by a suitable redefinition of the equilibrium modulus. The data were rescaled as G /Ge(p) and G"/Ge(p) over (oimax(p). The result is a set of master curves, one for the sol (Fig. 23a) and one for the gel (Fig. 23 b). 

In view of the self-similar character of the solution, the loop does not change as A t —> 0 even though the strain and velocity fields converge to the constant initial data everywhere outside the point x = Xo. This means, that by selecting the point Xq we have supplemented constant initial data with a singularpartrepresentedby a parametric measure (in the state space) located at x = Xo. We conclude that, contrary to the behavior of, say, genuinely nonlinear systems (o w) 0) (see Di Perna, 1985), the choice of a short time... 

Since the energy ofthe nucleus is identically zero, the integral impact of this localized contribution to the initial data can be measuredby the corresponding energy density which is finite. For our self-similar solution (2.5) one can equivalently calculate the rate of dissipation R (Dafermos, 1973)... 

The mean velocity field is found to be self-similar for all the nozzle exit velocities, distances between nozzles and turbulence generators tested. This similarity allows the mean axial velocity traverses to be normalized so that all the measured data lie on the same curve. The shape of the curve is similar to that given in Fig. 1.8. 

Irrespective of the origin of fractals or fractal-like behavior in experimental studies, the investigator has to derive an estimate for df from the data. Since strict self-similarity principles cannot be applied to experimental data extracted from irregularly shaped objects, the estimation of df is accomplished with methods that unveil either the underlying replacement rule using self-similarity principles or the power-law scaling. Both approaches give identical results and they will be described briefly. 

Power-law expressions are found at all hierarchical levels of organization from the molecular level of elementary chemical reactions to the organismal level of growth and allometric morphogenesis. This recurrence of the power law at different levels of organization is reminiscent of fractal phenomena. In the case of fractal phenomena, it has been shown that this self-similar property is intimately associated with the power-law expression [28]. The reverse is also true if a power function of time describes the observed kinetic data or a reaction rate higher than 2 is revealed, the reaction takes place in fractal physical support. 

Conversely, the relationship (7.2) expresses a time-scale invariance (selfsimilarity or fractal scaling property) of the power-law function. Mathematically, it has the same structure as (1.7), defining the capacity dimension dc of a fractal object. Thus, a is the capacity dimension of the profiles following the power-law form that obeys the fundamental property of a fractal self-similarity. A fractal decay process is therefore one for which the rate of decay decreases by some exact proportion for some chosen proportional increase in time the self-similarity requirement is fulfilled whenever the exact proportion, a, remains unchanged, independent of the moment of the segment of the data set selected to measure the proportionality constant. 

We are drawn to the conclusion that log-log fractal plots are useful for the correlation of adsorption data - especially on well-defined porous or finely divided materials. A derived fractal dimension can also serve as a characteristic empirical parameter, provided that the system and operational conditions are clearly recorded. In some cases, the fractal self-similarity (or self-affine) interpretation appears to be straightforward, but this is not so with many adsorption systems which are probably too complex to be amenable to fractal analysis. 

ZOpT can be used to study both self-similar and self-affine fractal objects. The data at low frequencies (u and v <10) is not to be included in the calculation of D j. Figure 17.25 from Tang and Marangoni (2006) illustrates how Df, and ZOpT are calculated from the double logarithmic plot ofX vs. Y for polarized light microscopy images of the fat crystal networks. 

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