Renyi dimensions

The question of whether proteins originate from random sequences of amino acids was addressed in many works. It was demonstrated that protein sequences are not completely random sequences [48]. In particular, the statistical distribution of hydrophobic residues along chains of functional proteins is nonrandom [49]. Furthermore, protein sequences derived from corresponding complete genomes display a distinct multifractal behavior characterized by the so-called generalized Renyi dimensions (instead of a single fractal dimension as in the case of self-similar processes) [50]. It should be kept in mind that sequence correlations in real proteins is a delicate issue which requires a careful analysis. 

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] ... 

FIGURE 14.11 The dependences of critical Renyi dimensions (1,2) and (3, 4) on extrusion draw ratio X for componors UHMPE-Al (1, 3) and UHMPE-bauxite (2, 4) [56]. 

As it has been noted in Ref. [60], the generalized Renyi dimension (or Z) characterizes the most concentrated system set. In Ref. [55] it has been shown, that for amorphous polymer matrix the cluster are such set and the linear correlationwas obtained. It is obvious that for the considered componors with polymer semicrystalline matrix the crystalline regions will be the most concentrated set that assumes a definite correlation between and K. Actually, such correlation exists, as it follows from the data of Fig. 14.15 and it is described by the simple analytical relationship [56] ... 

FICURE14.15 The dependence of critical generalized Renyi dimension on crystallinity... 

A polymeric material s structure can be considered as multifractal, possessing a pseudo-spectrum, when Renyi dimensions decrease with growth in the index q [22, 44]. In this case the structure adaptability resource can be determined according to the equation [44] ... 

The limiting Renyi dimensions and were calculated according to Equations 9.11 and 9.12, respectively, and dimension was determined as follows [22] ... 

The three dimensions introduced above, Dp, Di and Dq, are actually three members of an (uncountably infinite) hierarchy of generalized fractal dimensions introduced by Heiitschel and Procaccia [hent83]. The hierarchy is defined by generalizing the information function /(e) (equation 4.88) used in defining Dj to the i/ -order Renyi information function, Io e) -... 

Generalized Renyi Entropies and Dimensions A hierarchy of generalized entropies and dimensions, SQ B,t), s[ B,t), S B,t),. .. - analogous to the hierarchy of fractal dimensions, Dq, Di,. .., introduced earlier in equation 4.94 for continuous systems may also be defined ... 

Linear a number of entropies, dimensions, Renyi measures, multi-fractal and fractal measures, and surrogate based quantities, iii. Prediction... 

The Eqs. (14.21) and (14.22) together with estimated by considered above method parameters A nd allow to calculate the dimensions and [55]. In Fig. 14.11 the dependences of Renyi characteristic dimensions and on extrusion draw ratio for componors UHMPE-Al and UHMPE-bauxite. As one can see, at the definite values X XJ the componors structure transition from multifiractal (canonical spectrum, grows at q increase [59]) to regular fractal D- = occurs and then at X>X - again to multifiractal (pseudospectrum, D decreases at q growth). 

As has been noted above, the parameter which is equal to the difference of the Renyi limiting dimensions and (which are used in practice instead of and D, respectively), characterises the degree of chaos in a system, i.e., in the structure of the considered epoxy polymers [31]. Renyi limiting dimensions and for polymeric materials can be determined with the aid of the following equations [22] ... 

This value of 0=oid, referred to as the jamming limit in one dimension, was obtained originally by Renyi and others [59, 60]. Interestingly, a very similar value of the jamming limit (i.e. 0.7506) was obtained in the diffusion RSA process solved analytically in Ref [61]. 

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