Multifractals structural properties

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. 

In the general case polymers structure is multifractal, for behavior description of which in deformation process in principle its three dimensions knowledge is enough fractal (Hausdorff) dimension d informational one d and correlation one d [82]. Each from the indicated dimension describes multifractal definite properties change and these dimensions combined application allows to obtain more or less complete picture of yielding process [73]. 

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. 

Yet another important aspect is the change in the fractal dimension of polymers when they are simulated on fractal rather than Euclidean lattices. This fact is also important from the practical standpoint for multicomponent polymer systems. The introduction of a dispersed filler into a polymer matrix results in structure perturbation in terms of fractal analysis, this is expressed as an increase in the fractal dimension of this structure. As shown by Novikov and co-workers [25], the particles of a dispersed filler form in the polymer matrix a skeleton which possesses fractal (in the general case, multifractal) properties and has a fractal dimension. Thus, the formation of the structure of the polymer matrix in a filled polymer takes place in a fractal rather than Euclidean space this accounts for the structure modifications of the polymer matrix in composites. 

It was widely accepted that the characteristics of outburst s partitions and zones are controlled by the fault structure largely, further, the fault s mechanical property and distribution are determined by the evolution process of its tectonic (Zhang 2009). So it is necessary to apply the multifractal theory to study the fault of coal mining areas for the purpose of revealing the relationships of the fault s multifractal characteristics and tectonic evolution process to coal and gas outburst. In this study, the multifractal theory was applied to study the multifractal characteristics of different regional faults in Jiaozuo Coalfield, Henan Province, China, then based on the domination of tectonic evolution process over coal and gas outburst, the relationship of fault s multifractal characteristics and outburst... 

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... 

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]. 

Kohnakov, A. G. (1996). The Intercormnunication of Multifractal Characteristics of Molybdenum Surface Structures with its Mechanical Properties. Izvestiya AN, Metally 6, 37-43. 

At present it is known [92, 93, 159] that the polymer structure is fractal (generally multifractal) with fractal dimension d 2 fluctuation free volume is its mirror, that it also possesses fractal properties. If this assumption is correct, the general relation should be fulfilled [18] ... 

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