[en] We present a general introduction to the world of fractals. The attention is mainly devoted to stress how fractals do indeed appear in the real world and to find quantitative methods for characterizing their properties. The idea of multifractality is also introduced and it is presented in more details within the framework of the percolation problem

No abstract available

[en] It is shown that multivalued fractals have the same address structure as the associated hyperfractals. Hyperfractals may be used to model self-similar diffusion limited aggregations, structure of urban settlements, and clusters of nanoparticles. We establish that the Hausdorff dimensions of a particular class of hyperfractals can be calculated by means of the Moran–Hutchinson formula

[en] Complete text of publication follows. The previous experiments have detected that processes in the atmosphere at heights 10-20 km are extremely influenced by the galactic cosmic rays (GCR) with the protons' energies of 10¹¹/10¹⁵ eV. Strong variations of these rays (a few tens of percents) coincide with the solar activity cycles and atmospheric perturbation variations induced by the separate flares on the Sun. The spectrum of turbulent pulsations induced in the atmosphere by the galactic-cosmic rays is defined. A possible manifestation of genesis of fractal dimensions in the system of 'spectrum of turbulent pulsations of cosmic plasma - galactic-cosmic rays' spectrum - spectrum of atmospheric turbulent pulsations' is analyzed. It is considered possibility for the existence of spectrum of Kolmogorov-Obukhov turbulent kinetic energy dissipation induced by the GCR in the atmosphere and it establishes the attractive problem associated with the genesis of scaling invariance and scaling representation of turbulent spectrums.

[en] The use of fractals and fractal-like forms to describe or model the universe has had a long and varied history, which begins long before the word fractal was actually coined. Since the introduction of mathematical rigor to the subject of fractals, by Mandelbrot and others, there have been numerous cosmological theories and analyses of astronomical observations which suggest that the universe exhibits fractality or is by nature fractal. In recent years, the term fractal cosmology has come into usage, as a description for those theories and methods of analysis whereby a fractal nature of the cosmos is shown.

[en] The presented work belongs to the issue of searching for the effective kinetic properties of macroscopically disordered environments (MDE). These properties characterize MDE in general on the sizes which significantly exceed the sizes of macro inhomogeneity. The structure of MDE is considered as a complex of interpenetrating percolating and finite clusters consolidated from homonymous components, topological characteristics of which influence on the properties of the whole environment. The influence of percolating clusters’ fractal substructures (backbone, skeleton of backbone, red bonds) on the transfer processes during crossover (a structure transition from fractal to homogeneous condition) is investigated based on the offered mathematical approach for finding the effective conductivity of MDEs and on the percolating cluster model. The nature of the change of the critical conductivity index t during crossover from the characteristic value for the area close to percolation threshold to the value corresponded to homogeneous condition is demonstrated. The offered model describes the transfer processes in MDE with the finite conductivity relation of «conductive» and «low conductive» phases above and below percolation threshold and in smearing area (an analogue of a blur area of the second-order phase transfer). (paper)

[en] Highlights: • Fractal-fractional differentiation. • Fractal-fractional integration. • New numerical scheme for fractal-fractional operators. • New model of Darcy scale describing flow in a dual medium. - Abstract: New operators of differentiation have been introduced in this paper as convolution of power law, exponential decay law, and generalized Mittag-Leffler law with fractal derivative. The new operators will be referred as fractal-fractional differential and integral operators. The new operators aimed to attract more non-local natural problems that display at the same time fractal behaviors. Some new properties are presented, the numerical approximation of these new operators are also presented with some applications to real world problem.

[en] Relationship between fractional calculus and fractal functions has been explored. Based on prior investigations dealing with certain fractal functions, fractal dimensions including Hausdorff dimension, Box dimension, K-dimension and Packing dimension is shown to be a linear function of order of fractional calculus. Both Riemann-Liouville fractional calculus and Weyl-Marchaud fractional derivative of Besicovitch function have been discussed.

[en] In this paper, we introduce a separation property for self-similar sets which is necessary to reach the equality between the similarity dimension and the Hausdorff dimension of these spaces. The similarity boundary of a self-similar set is investigated from the viewpoint of that property. In this way, the strong open set condition (in the self-similar set setting) posed by Keesling and Krishnamurthi has been weakened leading to a Moran type theorem. Moreover, both a result based on a conjecture posed by Deng and Lau as well as an improved version of a theorem due to Bandt and Rao have been contributed regarding the size of the overlaps among the pieces of a self-similar set. Several (equivalent) conditions leading to the equality between the similarity dimension and a new Hausdorff type dimension for attractors described in terms of finite coverings are also provided. Finally, we list some open questions.

[en] We investigate the dynamics of the eigenstate of an infinite well under an abrupt shift of the well's wall. It is shown that when the shift is small compared to the initial well's dimensions, the short-time behavior changes from the well-known t^3/2 behavior to t^1/2. It is also shown that the complete dynamical picture converges to a universal function, which has fractal structure with dimensionality D=1.25.