The fractal application formalism in mathematical description of the structure

V. I. Bolshakov, V. N. Volchuk, Yu. I. Dubrov

Abstract


Formulation of the problem. The practice of fractal formalism for identifying corresponding structures has shown that in this case, often there is the inadequacy of their perception as a fractal. This is probably due to the fact that any object recognition occurs on the basis of formation of the model (if recognition is made directly by man, the pattern formed in his mind). An inadequate recognizable model leads to the loss dramatically. This is often observed in the endeavors of researchers, to create mathematical models (MM) in the field of materials science, in which the structure, such as metal, intractable deterministic description, they tried to synthesize MM, using statistical analysis. Objective. In this regard, we suggest that if it is the dimension will be determined at the stage of determination of the type MM, its appearance will be apparent. Results and discussion. Unexpectedly for many researchers in accessing MM such a system, which, for example, represents the structure of a material, it was not possible to obtain an one-to-one correspondence between her and every structure of MM. Identifying the reasons for this discrepancy has led scientists to determine the factors influencing this disparity, the main ones were: factors not included in the MM, significantly affecting the function; MM does not display non-linearity states identify the object space (SIOS). Most likely, this is the root cause nonlinearity SIOS appearance of fractal dimensions. Conclusions. Thus, the pattern presented in the form of a function, be sure to include metric SIOS. Once again, we note that the reason for the inadequacy of the model often lies not in the shortcomings of the model, and in the nature of the phenomenon under study, which can be identified by the use of fractal formalism. After an extensive literature review, the authors of this article had the impression that the fractal dimension of the relation with the physical nature of the object displayed in the undeservedly little publications that reflect the practical applicability of this phenomenon. Probably the latter due to the fact that for each material, with its corresponding fractal characteristics, there is one and only one, only his inherent fractal dimension.


Keywords


mathematical model; structure; fractal dimension; linear space; function

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