Multifractal analysis

For each case above, the reaction probabilities of different surface sites are recorded and analyzed using multifractal analysis. 

Grout, H, A.M. Tarquis, and M.R. Wiesner. 1998. Multifractal analysis of particle size distributions in soil. Environ. Sci. Technol. 32 1176-1182. 

Muller, J. 1996. Characterization of pore space in chalk by multifractal analysis. J. Hydrol. 187 215-222. 

Grout H., Tarquis A.M., Wiesner M.R. (1998), Multifractal analysis of particle size distribution in soil. 

Multifractal analysis provides a complete description of fault-network fractal properties. It could more veritably describe the complexity and essentiality of fault structure compared to single fractal analysis. The multifractal theory has been widely applied in various fields of earth science. More details about the properties of multifractal spectra in earth sciences applications can be found in References (Agterberg et al. 1996, Zhao et al. 2011, Kiyashchenko et al. 2004). Fault tectonic has multifractal structure characteristics (Panahi and Cheng 2004). But it was not found on the coalfield fault tectonics. 

However, Table 1 shows that the difference in the capacity dimensions of the three regions is very small, thus f j is not considered as a characteristic parameter. It is clear that multifractal analysis could more veritably describe the complexity and essentiality of fault structure compared to single fractal analysis. 

Vstovskii, G. V., Kohnakov, L. G., Terent ev, V. F. (1993). The Multifractal Analysis of Molybdenum Near-Surface Layers Fracture Features. Izvestiya AN, Metally 4,164-178. 

Kozlov, G. V. (2002). The Multifractal Analysis of Diffusion Processes in Semisrystal-hne Polyethylene and it s Melt. In New Perspectives in Chemistry and Biochemistry. Ed. Zaikov, G. E. New York, Nova Science Publishers Inc., 57-65. 

Kozlov, G. V., Aloev, V. Z., Yanovskii, Yu. G. (2004). The Multifractal Analysis of Polymeiization-Filled Compositions Sohd-Phase Extrusion Processes. Mekhanika Kom-pozitsionnykh Materialov i Konstruktsii 2, 267-275. 

Very recently a multifractal analysis of electroconvective turbulence has been performed [170]. 286... 

In the present monograph these problems were studied from the principally new point of view, when object structure, either macromolecular coil (microgel) or condensed state structure, was considered in terms of chemical or physical processes. Such an approach is possible with the availability of appropriate structural models, of which fractal (multifractal) analysis and the cluster model of the amorphous state structure of polymers were used. The first of the indicated approaches is a general mathematical calculus, whereas the second represents itself as a purely polymeric concept. This circumstance defines the excellent combination and addition to one another of crosslinked polymers. 

We have applied the multifractal scaling [55] which relates the analysis of the distribution of the reaction probabilities over the length of the rough surfaee. The steps in the multifractal scaling analysis are given below. 

There are three basic equations in the multifractal scaling analysis. 

Kantelhardt J.W. Zschiegner S.A. Koscielny-Bunde E. Havlin S. Bunde A. and Stanley H.E. (2002). Multifractal detrended fluctuation analysis of nonstationary time series. Physica, A316(14), 87-114. 

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 this chapter, the application of multifractal inverse distance weighted (MIDW) interpolation method and a fractal filtering technique, named spatial and spectral analysis or simply (S-A) method (Cheng, 2003), will be illustrated to evaluate geochemical background at the regional and local scale. For this purpose, two case studies will be discussed for two different Italian areas ... 

---