Integral fractional exponent

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] ... 

For the evolutionary processes with fractal time description the mathematical calculus of fractional differentiation and integration is used [34]. As it has been shown in Ref [35], in this case the fractional exponent... 

The mathematical calculus of fractional differentiation and integration is used for the description of evolutionary processes with fractal time [81 ]. As it has been shown in paper [82], in this case the fractional exponent v coincides with fractal dimension of Kantor s set and indicates fraction of system states, maintaining during all evolution time t. Let us remind, that Kantor s set is considered in onedimensional Euchdean space (d = 1) and therefore its fractal dimension djfractal definition [52]. For fractal objects in Euclidean spaces with Mgher dimensions (d>l) dj,fractional part should be accepted as v or [83] ... 

The order of the reaction, n, can be defined as n = a + b. Extended to the general case, the order of a reaction is the numerical sum of the exponents of the concentration terms in the rate expression. Thus if a = b = 1, the reaction just described is said to be second-order overall, first-order relative to A, and first-order relative to B. In principle, the numerical value of a or b can be integral or fractional. 

The integral on the right-hand side of the above equation is the weight fraction of all molecules larger than N. This integral is dominated by the lower limit for critical exponent r > 2 and is proportional to N ... 

Figure 8-5. Integral mass distribution for a Wesslau distribution. The viscosity-average degree of polymerization (X,), and the mass fraction were measured. The viscosity-average degree of polymerization of the original material with the already known exponent the median value Xjvf, and equation (8-23) and (8-24) were used to calculate the number- and weight-average molecular weights.

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