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[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
[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] 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] 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 t3/2 behavior to t1/2. It is also shown that the complete dynamical picture converges to a universal function, which has fractal structure with dimensionality D=1.25.
[en] We present a method for estimating the dynamical noise level of a ‘short’ time series even if the dynamical system is unknown. The proposed method estimates the level of dynamical noise by calculating the fractal dimensions of the time series. Additionally, the method is applied to EEG data to demonstrate its possible effectiveness as an indicator of temporal changes in the level of dynamical noise. - Highlights: • A dynamical noise level estimator for time series is proposed. • The estimator does not need any information about the dynamics generating the time series. • The estimator is based on a novel definition of time series dimension (TSD). • It is demonstrated that there exists a monotonic relationship between the • TSD and the level of dynamical noise. • We apply the proposed method to human electroencephalographic data.
[en] The present work studies the behaviour of continuous time quantum walks on regular hyperbranched fractals, whose centre is a trap. We focus on the variations of the eigenvalue spectrum of the transfer operator by tuning the trap strength from zero to infinity. We show that the degenerate eigenvalues are independent from the trap strength and can be obtained analytically. Due to this the mean survival probability is just in the intermediate range affected by the trap strength; moreover, because of the presence of real eigenvalues, the asymptotical probability of being outside the trap is not zero