Euclidean space, fractal sets

That Dfractai givcs the expected result for simple sets in Euclidean space is easy to see. If A consists of a single point, for example, we have N A, e) = 1, Ve, and thus that / fractal = 0. Similarly, if A is a line segment of length L, then N A,e) L/e so that Dfraciai = 1. In fact, for the usual n dinien.sional Euclidean sets, the fractal dimension equals the topological dimension. There are nrore complicated sets, however, for wliic h the two measures differ. 

Surfaces of most materials, including natural and synthetic, porous and non-porous, and amorphous and crystalline, are fractal on a molecular scale. Mandelbrot defines that a fractal object has a dimension D which is greater than the geometric or physical dimension (0 for a set of disconnected points, 1 for a curve, 2 for a surface, and 3 for a solid volume), but less than or equal to the embedding dimension in an enclosed space (embedding Euclidean space dimension is usually 3). Various methods, each with its own advantages and disadvantages, are available to obtain... 

For arbitrary dimension d of Euclidean space in which the fractal set Qf is embedded and arbitrary form of the initial element, the mass (the number of bonds) can be defined by singling out the factorial geometric coefficient F ... 

Fractals are self-similar objects invariant with respect to local dilatations, i.e., objects that reproduce the same shape during observation at various magnifications. The concept of fractals as self-similar ensembles was introduced by Mandelbrot [1], who defined a fractal as a set for which the Hausdorff-Besicovitch dimension always exceeds the topological dimension. The fractal dimension D of an object inserted in a d-dimensional Euclidean space varies from 1 to d. Fractal objects are the natural filling of the gap between known Euclidean objects with integer dimensions 0,1, 2, 3. .. The majority of naturally existing objects proved to be fractal, which is the main reason for the vigorous development of the methods of fractal analysis. 

As it has been shown in Ref [72], the fractional exponent v coincides with the fractal dimension of Cantor s set and indicates a fraction of the system states, being preserved during the entire evolution time t. Let us remind that Cantor s set is considered in one-dimensional Euclidean space (d=l) and therefore its fractal dimension d < by virtue of the fractal definition [86]. For fractal objects in Euclidean spaces with higher dimensions d> ) as V one should accept fractional part or [76, 77] ... 

Thus, from the said above it follows, that thermooxidative degradation process of PAr and PAASO melts proceeds in the fractal space with dimension A In such space degradation process can be presented schematically as devil s staircase [33]. Its horizontal sections correspond to temporal intervals, where the reaction does not proceed. In this case the degradation process is described with fractal time t using, which belongs to Cantor s setpoints [34]. If the reaction is considered in Euclidean space, then time belongs to real numbers sets. 

The authors of Ref. [53] have shown, that frictional properties of fractal clusters can be different essentially for the usual results for compact (Euclidean) structures. It is known through Ref. [54], that the polymer melt structure can be presented as a macromolecular coils sets, which are fractal objects. Therefore, the authors [55] proposed general structural treatment of polymer melt viscosity within the framework of fractal analysis, using the model [53]. Within the framework of the indicated model the derivations for translational friction coefficient f(N) of clusters from N particles in three-dimensional Euclidean space were received, calculated according to Kirkwood-Riseman theory in the presence of hydrodynami-cal interaction between the cluster particles. The fundamental relationship of this theory is the following equation [53] ... 

Devalues, obtained for PAr and PUAr poly condensation process, showed, that the indicated processes were realized by aggre tion cluster-cluster mechanism [49], i.e., by small macromolecular coils joining in larger ones [23], Thus, polycondensation process is a fractal object with dimension D. reaction. Such reaction can be presented schematically in a form of devil s staircase [80], Its horizontal parts correspond to temporal intervals, in which reaction is not realized. In this case polycondensation process is described with irsing fractal time t, which belongs to Kantor s set points [81], If polycondensation process is considered in Euclidean space, then time belongs to a real number set. 

If we again adopt that / =y=0.6, then the value =1.67 accurately coincides with the dimensions of the fractal of the chain df=(d + 2)/3=5/3 and the value ) =V, coincides with the Flory index. In this case the dimensions of fractal of traps and characteristic times d according to equations (8.62) and (8.59), is equal to characteristic times set z, of macroradical relaxation is completely determined by the fractality of traps, i.e., by dimension di. In the case that the set of traps is not a fractal (for example, at the uniform distribution of traps in the reactive zone when the dimension of the f ractal of traps formally coincides with the dimension of the Euclidean space di=d), then, in accordance with equation (8.59) the set of characteristic times r, would be characterized by a fractal dimension equal to infinity and t would not depend on chain length and time of chain propagation, and would be presented by a constant value T, Tp p ,. Under this variant we have 1= 1, 1 and stretched exponential law transforms into simple exponential law. 

Lately the mathematical apparatus of fractional integration and differentiation [58, 59] was used for fractal objects description, which is amorphous glassy polymers structure. It has been shown [60] that Kantor s set fractal dimension coincides with an integral fractional exponent, which indicates system states fraction, remaining during its entire evolution (in our case deformation). As it is known [61], Kantor s set ( dust ) is considered in onedimensional Euclidean space d = ) and therefore, its fractal dimension obey the condition d Euclidean spaces with d > 2 (d = 2, 3,. ..) the fractional part of fractal dimension should be taken as fractional exponent [62, 63] ... 

Now we will introduce briefly the concept of self-similar and self-affine fractals by considering the assumption that fractals are sets of points embedded in Euclidean E-dimensional space. 

The similarity transformation transforms a set of points S at position x = (xj,...,xE) in Euclidean E-dimensional space into a new set of points r(S) at position x = (rXj,...,rxE) with the same value of the scaling ratio 0self-similar with respect to a scaling ratio r if S is the union of N nonoverlapping subsets SU...,SN, each of which is congruent to the set r(S). Here congruent means that the set of points. S is identical to the set of points r(S) after possible translations and/or rotations. For the deterministic self-similar fractal, the selfsimilar fractal dimension dFss is clearly defined by the similarity... 

Figure 4.9. The A = 123 Sierpinski gasket, a two-dimensional uncountable set with zero measure and Hausdorff (fractal) dimension 3/ 2 = 1.584962... the companion Euclidean lattice referred to in the text is a space filling triangular lattice of (interior valency v = 6.

Hence, a solid-phase polymers deformation process is realized in fractal space with the dimension, which is equal to structure dimension d. In such space the deformation process can be presented schematically as the devil s staircase [39]. Its horizontal sections correspond to temporal intervals, where deformation is absent. In this case deformation process is described with using of fractal time t, which belongs to the points of Cantor s set [30]. If Euclidean object deformation is considered then time belongs to real numbers set. 

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