Dear Colleagues,

The material sciences, with fractal nature analysis, open a new frontier within total matter.

Chaos defines a phenomenon that is in apparent disorder. Its complex behavior can be difficult to understand. It may appear everywhere and with different mathematical structure laws. The high sensitivity to initial conditions brings complexity to an analytical perspective. Fractals are a form of chaos, where order is ruled by self-similarity, i.e., where parts of the whole are repeated in a smaller scale in some portions of the total structure. A pure fractal is characterized by a chain of an infinite number of levels of self-similarity. In applied studies, when an object or a phenomenon present a complex structure, one way to analyze it is to work with fractals, by doing a fractal reconstruction (through fractal regression or interpolation), or by estimating its Hausdorff dimension, through geometric or numerical methods. Fractal coefficients (or contraction factors, either vertical factors) and the Hausdorff dimension are key indicators of the fractal characterization of real data. Bigger coefficients mean higher fractal structure oscillations. Fractalization may be understood as a process of approximating a given data set to a fractal function. Fractal regression is a method that includes a finite chain of fractal levels, and the estimates of fractal coefficients are obtained numerically.

One special case of fractals is so-called random fractals. Several approaches to the study of Brownian random functions are possible, such as fractional calculus. In this context, since the behavior of Brownian motion is intrinsically irregular and does not suit a traditional differential perspective, it is conceivable to establish relations between fractional calculus and fractals in this topic. This is one fact that may foresee some relationship between fractional calculus and fractals.

Both fractals and fractional mathematics integrate and open up new insights that can help toward a more complete understanding of the total nature of matter, including biophysical and technical sciences systems in the frame of overall reality.

Prof. Dr. Vojislav V. Mitic
Prof. Dr. Cristina Serpa
Prof. Dr. Hans-Jörg Fecht
Guest Editors

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• Fractal
• Fractal reconstruction
• Chaos
• Hausdorff dimension
• Fractalization
• Fractional