Multifractal formalism

A. Arneodo, B. Audit, N. Decoster, J.-F. Muzy, and C. Vaillant, Wavelet based multifractal formalism Application to DNA sequences, satellite images of the cloud structure and stock market data, in The Science of Disasters Climate Disruptions, Heart Attacks, and Market Crashes, Springer-Verlag, Berlin, 2002, pp. 26-102. 

The method proposed by Kolmogorov (53) and Hinze (54) was extended by Ba+dyga and Podgorska (56) and Ba+dyga and Bourne (57) to the case of more realistic intermittent turbulence, which was described by means of multifractal formalism. In this model drop size in the inertial subrange also depends on the integral scale of turbulence, which is related to the scale of the system. This new formalism predicts... 

Let us consider this question in detail. As Meakin [52] has shown, the relation exists between specific surface energy y value and parameter a, which within the frameworks of multifractal formalism is the scaling index, characterizing singularities concentration ... 

In Ref [51], the Halsey multifractal formalism [39], modified by Williford [58], application for particulate-filled pol5mier composites structure and properties description was considered. The Cantor set ( dust ) was used as mathematical model. It is assumed, that section of length /j and probabilistic... 

The data of Figs. 4.10 and 4.11 comparison shows, that the value obtained according to the dependences D- (q) and D (q), corresponds to the value X, at which fracture stress o -drop (Fig. 14.10) or interfacial boundaries polymer-filler fracture begins [2]. Thus, within the frameworks of multifractal formalism interfacial boundaries fracture of componors in solid-phase extrusion process is realized by polymer matrix structure regular fractal state achievement [56]. 

Hence, the cited above results shown correctness and expediency of multifractal formalism in it s the simplest variant for analysis of structure changes at polymerization-filled compositions on the basis of UHMPE during solid-phase extrusion. The observed experimentally during extrusion process effects were received the quantitative description within the frameworks of this formalism. Let us note purely geometrical character of main multifractal characteristics calculation, independent on polymer matrix and filler properties [56]. 

Semenov, B. L, Agibalov, S. N., Kohnakov, A. G. (1999). The Description of Casting Aluminium-Matrix Composite Shucture with Multifractal Formalism Method Using. Materialovedenie, 5, 25-33. 

Novikov, V. U., Kozlov, G. V, Bihbin, A. V. (1998). Polymer Composites Fracture Analysis within the Frameworks of Multifractal Formalism. Materialovedenie, 10, 14-19. 

Novikov, V. U., Kozitskii. D. V, Ivanova, D. V. (1999). The Materials Structure Analysis Computer Technique Development with Multifractal Formalism Using. Materialo-vedenie, 8,12-16. 

In Figure 1.15 the dependence of the density of the macromolecular entanglements clnster network on is shown. As might be expected, chaos intensification X increase) reduces the value, i.e., the local ordering degree in the polymer amorphous state structure [68]. More precise interpretation of the polymer structure chaotic character within the frameworks of multifractal formalism will be given below. 

At present it is established that the structures of both natural and many model objects cannot be described with the aid of only one value of fractal dimension. For more precise description of disordered structures, including polymers, it is necessary to calculate a spectrum of different dimensions, i.e., to use the multifractal formalism [23-25]. At present a number of papers exist that show correspondence of either... 

Hence, the results stated above have shown that analysis of structural properties for epoxy polymers, which are considered as natural nanocomposites, can be carried out within the frameworks of multifractal formalism in its most simple variant. The structure adaptability resource is reduced as the crosslinking density increases and is defined by the relative fraction of the loosely packed matrix. The properties of epoxy polymers are a function of their structure adaptability. 

J. Muzy, E. Bacry, Multifractal formalism for fractal signals The structure-function approach versus the wavelet-transform modulus-maxima method, Phys. Rev. E 47 (1993) 875-884. 

The multifractal behavior of time series such as SRV, HRV, and BRV can be modeled using a number of different formalisms. For example, a random walk in which a multiplicative coefficient in the random walk is itself made random becomes a multifractal process [59,60], This approach was developed long before the identification of fractals and multifractals and may be found in Feller s book [61] under the heading of subordination processes. The multifractal random walks have been used to model various physiological phenomena. A third method, one that involves an integral kernel with a random parameter, was used to model turbulent fluid flow [62], Here we adopt a version of the integral kernel, but one adapted to time rather than space series. The latter procedure is developed in Section IV after the introduction and discussion of fractional derivatives and integrals. 

Since the polymers structure is multifractal [53], then, following to Williford [42], the fracture surface can be considered as the first subfractal, having dimension d (information dimension, see the Eq. (4.49)) [48]. In this case within the fiiameworks of the indicated above formalism [42] a=f, where/is dimension of singularities a, equal to [47] ... 

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