A method based on the pattern roughness was introduced for determination of the fractal dimension and tested for fractals like the Sierpiński carpet, the Sierpiński triangle, standard Cantor set, the Menger sponge and the Sierpiński tetrahedron. It was tested for non-fractal pattern like two- and four-dimensional gray scale random dust as well. It was found that for all these patterns the Hausdorff dimension is reproduced with relatively high accuracy. Roughness method is based on simple, fast and easy to implement algorithm applicable in any topological dimension. It is particularly suited for patterns being composed of the hierarchy of structures having the same topological dimension as the space embedding them. It is applicable to "fuzzy" patterns with overlapping structures, where other methods are useless. It is designed for pixelized structures, the latter structures resulting as typical experimental data sets.
We define a specific class of fractals as "net fractals" and prove that in the logarithmic scale they are isomorphic with some bulk crystals. Furthermore, with the use of logarithmic coordinates, we prove that in the "net fractal" magnetic system the indirect exchange, by itinerant electrons can be presented in the form that is reminiscent of the Ruderman-Kittel-Kasuya-Yosida interaction characteristic of a system of fractional spectral dimension.
A localized spin system of fractal symmetry and Heisenberg exchange between nearest neighbors is considered. We define a specific class of fractals: "net fractals" and prove that in the logarithmic scale they are isomorphic with some bulk crystals. Further, with the use of logarithmic coordinates we show that the "net fractal" magnetic fractons can be presented as the conventional magnons.
Distribution of the topological point defects observed microscopically by Nagaya, Hotta, Orihira and Ishibashi in Schlieren texture of N-(4-n-methoxy benzylidene) 4'-n'-butylaniline has been analyzed. The same fractal dimensionality D_{f}=1.4 has been estimated for several sets of defects obtained in subsequent stages of evolution of the nematic sample after the transition from the isotropic liquid phase.
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