The study of the fractal properties of the surface of Nd100–xFex alloys in a wide range of concentrations х (х = 20–90) was carried out in the framework of the fractal thermodynamics model. To this end, we performed an analysis of images obtained by (scanning electron?) microscopy of the surfaces of a series of Nd100–xFex alloys synthesized by induction melting. A high degree of proximity of the surface structure of all the studied samples, both before and after etching, to fractals is shown. The values of the parameter δ characterizing the relative deviation of the studied samples from the fractal are in the range of 0.017–0.029. Three-dimensional diagrams of the fractal parameters Sf , Tf , Ef , x and two-dimensional diagrams of the same parameters: Sf , Tf , Ef , x, reflecting the nature of the state of the surfaces of Nd100–xFex alloy samples before and after etching, are constructed. For all investigated samples of alloys, the values of the parameters of the fractal equations of state arecalculated. The correlation of the maximum value of the coercive force Hc = 4.8 kE with the values of fractal entropy Sf = 39.86, fractal temperature Tf = 529, and fractal dimension D = 2.6530 of the Nd100–xFex alloys at x = 20 has been established.

The study of the fractal properties of the surface of Nd100 – xFex alloys in a wide range of concentrations х (х = 20–90) was carried out in the framework of the fractal thermodynamics model. To this end, we performed an analysis of scanning electron microscopy images of the surfaces of a series of Nd100 – xFex alloys synthesized by induction melting. A high degree of proximity of the surface structure of all the studied samples, both before and after etching, to fractals is shown. The values of the parameter δ characterizing the relative deviation of the studied samples from the fractal are in the range of 0.017–0.029. Three-dimensional diagrams of the fractal parameters Sf, Tf, Ef, and x and two-dimensional diagrams of the same parameters Sf, Tf, Ef, and x reflecting the nature of the state of the surfaces of Nd100 – xFex alloy samples before and after etching are constructed. For all investigated alloy samples, the values of the parameters of the fractal equations of state are calculated. The correlation of the maximum value of the coercive force Hc = 4.8 kE with the values of fractal entropy Sf = 39.86, fractal temperature Tf = 529, and fractal dimension D = 2.6530 of the Nd100 – xFex alloys at x = 20 has been established.

The article provides an analysis of the tracks of emerging particles in the mathematical model of fractal thermodynamics after the computer processing of simulated data from the BM@N experiment.

This paper presents studies of the influence of growth conditions of paratellurite single crystals on the side surface of grown boules and the possibility of assessing the quality of crystals based on the values and dynamics of the roughness parameters and fractal parameters of juvenile surfaces. Two single crystals were grown under similar technological conditions and differing from each other in structural quality. Their lateral surfaces were studied using the optical interference profilometer NanoMap 1000WLI employing SPIP and Gwyddion softwares. As a result, roughness parameters of profiles, and fractal parameters of crystal surfaces along the growth direction were obtained,. It was concluded that under conditions corresponding to the formation of stable flows in the melt, the values of the surface roughness over the entire length of the crystal is less than 5 mu m, and the fractal energy parameter can be used as a marker of the quality and homogeneity of crystals.

To describe the COVID-19 pandemic, we propose to use a mathematical model of multifractal dynamics, which is alternative to other models and free of their shortcomings. It is based on the fractal properties of pandemics only and allows describing their time behavior using no hypotheses and assumptions about the structure of the disease process. The model is applied to describe the dynamics of the COVID-19 pandemic from day 1 to day 699 from the beginning of the pandemic. The calculated parameters of the model accurately determine the parameters of the trend and the large jump in daily diseases in this time interval. Within the framework of this model and finite-difference parametric nonlinear equations of the reduced SIR (Susceptible-Infected-Removed) model, the fractal dimensions of various segments of daily incidence in the world and variations in the main reproduction number of COVID-19 were calculated based on the data of COVID-19 world statistics.

The paper considers the concept of fractal thermodynamics for studying the states of instantaneous cardiac rhythm (ICR) according to Holter monitoring (HM) data. An effective method cardiac rhythms analysis using the quantum phase space of the ICR Sq is presented. The space Sq is a powerful tool for studying nondeterministic chaotic systems which allows to identify statistical regularities adequately covering the patients' cardiovascular system states in the context of nondeterministic chaos of ICR states [3, 4]. The results of our papers show with high accuracy Sq is a fractal. DOI 10.1134/S1061920821020096

В докладе предложен алгоритм квантования классического фазового пространства мгновенного сердечного ритма с постоянной или шагом квантования h.В результате квантования фазовое пространство мгновенного сердечного ритма разбивается на ячейки конечной величины h.Для построения конкретных квантовых фазовых пространств мгновенного сердечного ритма постоянную квантования h мы взяли равной единице

Based on the findings of our previous studies of fast-rotating Newtonian polytropes, we found the relation between the minimum pulsar rotation period, the value of pulsar central density, and the polytropic index. From this relation we come to the conclusion that the value of minimum central density of a pulsar with a peak period is 2.5088 • 1014 g/cm3.

In this paper we propose a new method for calculation of parameters of the fractal dimension D and the fractal phase-space volume Γ of the phase space of instantaneous heart rhythm. The parameters of interest for D and Γ were calculated for four patients of the Tver Regional Cardiology Health Center based on the data obtained from 24-hour Holter monitoring. The investigated structure of the phase spaces of instantaneous heart rhythm for these patients is indicative of nondeterministic chaotic behavior of the instantaneous heart rhythm. The degree of similarity of the phase spaces of instantaneous heart rhythm to fractals in patients under examination was evaluated to be not greater than 0.0453 in C-metrics, and thus, the instantaneous heart rhythm chaos was identified as a fractal to the same degree of accuracy. Finally, we propose the foundation for identifying the state parameters for the fractal dimension D and the D-dimensional volume Γ of the phase space of instantaneous heart rhythm.

In this paper we offer an algorithm for quantization of the classical phase space of instantaneous heart rhythm with a constant or in quantization increment of h. The process of quantization of the phase space of instantaneous heart rhythm divides the phase space into cells with a finite size of h. The primary objective of this approach is visualisation of the quantum phase space of instantaneous heart rhythm. The phase cells of this space are color-coded with different colors depending on the values of their occupation numbers. The estimates showed the sufficiency of use about 10 colors. At that, the informativity level of visualisation of the quantum phase space of instantaneous heart rhythm remains invariant. Color visualisation of the quantum phase space of instantaneous heart rhythm clearly demonstrates that it can be used as a cardiovascular system state marker. We offer a method for description of the quantum phase space of instantaneous heart rhythm which will allow to detect statistical regularities of instantaneous heart rhythm chaos.

The paper is devoted to the study of cardiac rhythm variability (CRV) using the phase and extended phase spaces of instantaneous cardiac rhythm (ICR) obtained from the Holter monitoring (HM) data. In order to construct these spaces, a software package is developed and implemented. With specific references, the fractality of the ICR phase space is demonstrated. The fractal phase space volume and fractal entropy definitions of ICR are given. The paper justifies their availability for CRV quantitative assessment.

A system of equations and inequalities that allows one to determine the constraints on central density ρ c and the chemical composition, which is governed by parameter μ e , of the white dwarf RX J0648.0- 4418 with a record short period of rotation T = 13.18s and mass m = (1.28 ± 0.05) m ⊙, has been derived. The analysis of numerical solutions of this system reveal a complex dependence of μ e on ρ c . The intervals of variation of μ e and ρ c are as follows: 1.09 ≤ μ e ≤ 1.21 and 9.04 ≤ μ e /ρ 0 ≤ 10 3 (ρ 0 = 0.98 × 10 6 g/cm 3 ). This range of μ e values suggests that the white dwarf RX J0648.0-4418 is not made of pure hydrogen and should contain 9–21% of heavy elements. Calculations have been performed with the equation of state of an ideal degenerate electron gas. Approximate analytic expressions (with an accuracy of 10 -3 ) for the minimum period T min and mass m of the white dwarf are obtained. It is demonstrated that the white-dwarf mass is almost doubled (compared to the case of no rotation at a fixed central density) as period T approaches T min .

In this work the model of variations of average annual temperature of the Northern Hemisphere of the Earth is constructed. On its basis, in this research, the main regularities of temporary variability of temperature over the period from 1000 to 2010 have been revealed with the help of application of methods of multi-fractal dynamics. Over the period of 1010 years there were founded 26 periods and for each of them a value of fractal dimension of D was settled. Also, the function of jumps of difference h = D - D-0 (where D-0 is an equilibrium value of fractal dimension) describing the dynamics of fractal dimension for the period under consideration was constructed. The question of correlation of a historical process and behavior of function h has been discussed.

Based on test data, the self-similar character of the instantaneous heart rate is shown. Within the multifractal dynamics model, equations describing the piecewise-linear trend of the instantaneous heart rate are obtained. Using these equations, a classification of the types of its dynamics is proposed as relating to the value of the fractal D dimensionality of the heart rate curve. In the field of the D values near the D bifurcation points, there are bifurcation effects, of which the instant heart rate speed jumps of the piecewise linear trend are typical.

The presence of critical points and the bifurcation points of rotating Newtonian polytropes with an index of 1 ⩽ n ⩽ 1.6 has been shown for the first time in this paper. The symbolic-numerical calculation error in metric L 2 has reached the size of the 10−4 order. The approximate analytical solution to the problem to the abovementioned accuracy has been set forth. A critical value of the polytropes index n = n k = 1.51025 has been calculated which is the highest among the critical points and bifurcation points. The value n k corresponds to the infinitely slow polytropes rotation. Furthermore, in this paper the presence of the period jump at the bifurcation point T b has been predicted and the relative value of this jump ΔT b /T b ∼ B 0in 4/3 has been estimated.

In this work the variations of global temperature that have occurred in the period from 1860 up to now are analyzed on the basis of the concept of multifractal dynamics. The multifractal curve describing dynamics of global temperature for this period of time has the following values of fractal dimensions over 5 periods lasting for 30–31 years each, accordingly: D1 = 1.140; D2 = 1.166; D3 = 1.141; D4 = 1.203; D5 = 1.183. Such relatively small values of fractal dimensions are indicative of essentially determined character of processes responsible for variations of global temperature. Our predictive estimates provide 0.5◦C increase in global temperature by 2072, thereby confirming maintenance of the tendency of global warming in the near future. This is the e-book version of the article, published in the Russian Journal of Earth Sciences (doi:10.2205/2012ES000510). It is generated from the original source file using LaTeX’s epub.cls class.

In the present paper, the concept of multi-fractal dynamics is developed. The problem concerning catastrophes in this dynamics is studied in detail. In the framework of the concept of fractal curve as a thick curve, it is proved that the cell approach to measuring the fractal dimension D is equivalent to measuring the dependence of the length L of the line on the scope δ . The introduction of a fractal scale of temperatures T f is suggested.

This paper considers the polynomial and polytropic approximations in the equation of equilibrium post-Newtonian rotating configurations of a degenerate neutron gas. The concept of a configuration-averaged polytropic index is introduced.

An analytical representation of the density distribution of rapidly rotating superdense Newtonian polytropes is obtained in terms of polynomials in powers of flatness parameters e and the polytropic index approximating it with an error of 10−3. A scheme of the determination of critical points in the configuration density distribution is constructed. If n = 1.4, the values of the parameters characterizing an analytical representation of density near the critical points are obtained. The dynamics of critical points depending on the parameter e is studied. The formation of A 2- and A 3-type catastrophes is demonstrated. It is proven that a bubble domain in the shape of an elliptical torus is formed in the density distribution near the point e = 0.3503.