Fractal notation

A fractal notation has been proposed by Mendenhall17,81 to address the complexity of cascade macromolecules, since they possess a high degree of molecular symmetry. This highly condensed description of a symmetrical molecule possessing a fractal architecture can be generalized in the following form  

Although this notation can be applied to diverse structures, the following rules and examples have been confined to very simple dendritic systems. 

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Thus, a summary of the basic fractal notation is (peripheral groups)numberfjunctures (coordination umber)(connectors)core. 

Thus, the complete fractal notation for the structure would be ... 

It was noted181 that further simplification could be made by a simple number replacement for consecutive methylene groups therefore, this complete fractal notation can be simplified (or condensed) to ... 

For applications to more complex macromolecules and the use of the additional rules, not cited herein, associated with the fractal notation, one should consult the Mendenhall article.181... 

Figure 4. The dependence of heterogeneity exponent h of reactive medium on relative fractal volume J for PAA solid state imidization. The notation is the same, that in figure 2.

In order to establish the relationship between the static and dynamic fractal dimensions, the initial conditions of the classical static percolation model must be considered for the solution of differential equation (89) which can be written as 0 = Qs = 1 for D = Ds. Here the notation s corresponds to the static percolation model, and the condition s = 1 is fulfilled for an isotropic cubic hyperlattice. The solution of (89) with the above-mentioned initial conditions may be written as... 

Recently, Matsunaga and Shibayama (Matsunaga and Shibayama 2007) investigated also the sol-gel transition of gelatin with both methods and found that the fractal exponent Dp in their notation, (g2(0 1 oc f and G oc G" oc co" with Dp... 

Using these data, one can calculate [q] and [q]g at the same molecular weight for the given polyarylates in different solvents and then compute D from Equation (16.11). Table 16.3 presents the comparison between the fractal dimensions D calculated by Equations (16.4) and (16.11) for seven polyarylates (for these, we retained Askadskii s notation [7]). As Table 16.3 shows, the fractal dimensions D calculated by both methods agree satisfactorily with each other, and the discrepancy between them does not exceed 10%. 

This Chapter is organized as follows Section 2 revisits general definitions of quantities of interest, for both deterministic and random fractals, and introduces the notation used. Section 3 contains the numerical techniques such as Monte Carlo (MC) methods and the exact enumeration (EE) technique as well as data analysis schemes. Section 4 provides a discussion of SAWs on deterministic fractals, specifically the Sierpinski square and... 

Critical exponents and fractal dimensions of the incipient percolation cluster, for spatial dimensions d = 2, 3, and d > 6. Note that usually d is called df in the fractal literature, but for the sake of consistency with our notation we refer here to this quantity as d. ... 

Figure 3. The dependence of fracture work U on structure fractal dimension <3 for carbon plastics. The notation is than one, that in Fig. 1...

FIGURE 4 The dependence of reactionary medium effective spectral dimension on macromolecular coil fractal dimension in imidization process. The notation is the same, as in Figure. 3 [8],... 

FIGURE 13 The dependence of the coefficient p. in the equation (21) on relative fractal fiee volume forPAAsolid-phaseimidization. The notation is the same that in Figure. 11 [29]. 

We estimate Kd for a column packed with a porous fractal material. In our notation, Vi PL and from Eq. (23)... 

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