Vestnik KRAUNTS. Fiziko-matematicheskie nauki

12-37

Authors: R. F. Shamoyan, V.V. Loseva

We discuss some new problems in several new mixed norm Hardy type spaces in products of bounded pseudoconvex domains with smooth boundary in Cn and then prove some new sharp decomposition theorems for multifunctional Hardy type spaces in the unit ball and then we show also similar results in pseudoconvex and convex domains of finite type extending previously known assertions obtained by first author earlier in Bergman spaces under certain Poisson integral type condition which vanishes in one functional case. Some new (in particular sharp in the unit ball) embeddings for some new mixed norm Hardy spaces in bounded pseudoconvex domains will be also indicated. Some new extensions of Poisson integral in the unit ball and some new assertions concerning them will be indicated and discussed in product domains. Some related multifunctional results are also given.Some new embedding theorems are also provided in some new mixed norm Hardy spaces in unbounded tubular domains over symmetric cones.

47-54

Authors: Е. R. Novikova

A study is conducted on the asymptotic stability of the rest points of the fractional oscillator Van der Pol-Duffing. The fractional van der Pol-Duffing oscillator is an oscillatory system of two differential equations with fractional order derivatives in the sense of Gerasimov-Caputo. The orders of fractional derivatives characterize the properties of the medium (memory effects) in which the oscillatory process takes place and can be the same (commensurate) or different (incommensurable). Using theorems for commensurable and incommensurable systems, the asymptotic stability of the rest points of the fractional van der Pol-Duffing oscillator is investigated with concrete examples. The results of the studies were confirmed by constructing the appropriate waveforms and phase trajectories.

74-94

Authors: I. I. Chekrygin, A. A. Faraonov

This article describes the problem of using parallel computing for the correlation analysis of large amounts of data.

105-116

Authors: G. I. Druzhin, V.M Pukhov, D. V. Sannikov, E. I. Malkin, I. E. Stasiy

In order to study the natural noise electromagnetic radiation at the Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, a VLF-recorder was developed, created and installed in Kamchatka, in the expedition point «Karymshina». Continuous observations carried out with the help of VLF-registrator allowed to study various geophysical effects associated with the propagation of radio waves, thunderstorm activity, earthquakes, cyclones.

17-27

Authors: A. Ebadian, A. Jabbari

Let A and B be Banach algebras and T: B→A be a continuous homomorphism. We consider left multipliers from A×TB into its the first dual i.e., A*×B* and we show that A×TB is a hyper-Tauberian algebra if and only if A and B are hyper-Tauberian algebras.

46-53

Authors: E. A. Gafurova, M. L. Mikhaylov, Yu.V. Grushko, R. I. Parovik, I. A. Kashutina

The paper proposes a new mathematical model of the dynamics of small enterprises with the participation of foreign investment, which takes into account the memory effect and affects the rate of change in the value of production assets. This memory effect can be considered as a property of the economic environment, for example, the influence on the production of external factors, in which the cost of production assets will depend on its previous values. This non-local effect can be written in terms of a fractional order derivative. In this paper, we will assume that the order of the fractional derivative is a function of time. Therefore, we will solve the initial model equation using numerical methods of the theory of finite difference schemes. Further, in the work, visualization and interpretation of the calculation result was carried out.

63-70

Authors: O. D. Lipko

In this paper, using the qualitative analysis, we studied the stability of the point of rest of the fractional oscillator FitzHugh-Nagumo in commensurate and incommensurable cases. For the corresponding point of rest, using the numerical method of the theory of finite difference schemes, phase trajectories were constructed. It is shown that quiescent points can be both asymptotically stable, which correspond to stable focus, and are asymptotically unstable (unstable focus), and for them the phase trajectories usually go to the limit cycle.

78-93

Authors: M. E. Semenov, T.Yu. Zablotskaya

In the paper, the biological neural network models are analyzed with a purpose to solve the problems of segmentation and pattern recognition when applied to the bio-liquid facies obtained by the cuneiform dehydration method. The peculiarities of the facies’ patterns and the key steps of their digital processing are specified in the frame of the pattern recognition. Feasibility of neural network techniques for the different image data level digital processing is reviewed as well as for image segmentation. The real-life biological neural network architecture concept is described using the mechanisms of the electrical input-output membrane voltage and both induced and endogenic (spontaneous) activities of the neural clusters when spiking. The mechanism of spike initiation is described for metabotropic and ionotropic receptive clusters with the nature of environmental exciting impact specified. Also, the mathematical models of biological neural networks that comprise ot only functional nonlinearities but the hysteretic ones are analyzed and the reasons are given for preference of the mathematical model with delay differential equations is chosen providing its applicability for modeling a single neuron and neural network as well.

100-109

Authors: L. D. Ostrоverhaya, O. K. Zhdanova

In this paper, an effective spiral method is proposed for selecting the roots belonging to a given interval (longer than 2π) when solving trigonometric equations.

17-23

Authors: V. N. Marichev, D. A. Bochkovsky

Lidar complex at small station of high-altitude atmospheric sensing in Institute of Atmospheric Optics, Siberian Branch, Russian Academy of Sciences makes it possible to obtain promptly and regularly the profiles of scattering ratio up to the altitude of 70 km and atmospheric temperature and density in the altitude range of 10-70 km with the spatial resolution of 192 m. The measurements are performed under the conditions of clear sky or, at least, broken clouds. Weather forecast is used to plan the measurements, which are performed when weather forecast coincides with actual weather under nighttime clear-sky conditions. Remote or automatic control of lidar complex will make it possible to increase the efficiency and quality (through introduction of measurements in variable clouds) of accumulated data record and increase the number of sessions of atmospheric lidar sensing.

34-41

Authors: I. N. Panchishkina, G. G. Petrova, A. I. Petrov, E. V. Egorov, A. V. SHevchenko, A. I. Kupinskaya

Proceeding from the results of measurements near the earth’s surface, nonlinear effects associated with the dependence of electrical conductivity on the electric field strength have been observed. This is manifested in the loss of proportionality of the conductivity current density from the field intensity, and the nonlinearity manifests itself more noticeably at low values of electrical conductivity. Based on the data obtained, empirical regression series have been constructed and equations of the dependence of polar electrical conductivity on the electric field strength have been obtained. In the interval of a sharp decrease in the values of polar conductivities with multiple amplification of the electric field strength, their interrelation is approximated by a linear function with a high level of reliability.

55-61

Authors: V. I. Kozlov, D. G. Baishev

A comparative analysis of parameters of the geomagnetic field components and electrical potentials at station of the SHICRA SB RAS near the city of Yakutsk far from industrial noises from autumn 2016 to winter 2018 is executed. Variations of Earth’s magnetic field and electric potentials during the strong magnetic storm of September 7-9, 2017 are compared. Calculated correlation coefficient is 0.5-0.9.

74-82

Authors: O. V. Sheremetyeva

A model of five modes of deformation activity based on the compound Poisson process and its fractal generalizations is considered. Comparison of the analytical dependences with the experimental ones obtained by the authors Carbon and Buck, was carried out [1, 2].

99-114

Authors: E. O. Makarov, P. P. Firstov, D. V. Kostylev, E. S. Rylov, I. P. Dudchenko

Radon monitoring in the world literature is recognized as a promising method for predicting earthquakes. In the last decade on the Kamchatka Peninsula, the prospect of radon method to forecast subduction earthquakes has been quite convincingly demonstrated. The southern part of Sakhalin Island is a region of high seismic risk. The results obtained on Kamchatka give grounds to hope for the detection of precursor anomalies in the radon field (Rn) and for weaker, but also dangerous, small-focus earthquakes on Sakhalin Island. A network consisting of three points for monitoring volume activity of radon (VA Rn) in the air of the subsoil is currently operating in the test mode in the south of Sakhalin Island. In order to better understand the conditions of Rn migration to the surface, seismic survey was performed in the vicinity of the installed radiometers. Volume activity of radon is registered by α-radiation using the method of forced convection. The manifestation of weak seismic activity in the south of Kuril-Kamchatka and in the north of the Japanese seismic regions was reflected in the dynamics of the subsurface radon at two points. After testing the equipment and installing stationary points, it is planned to use radon monitoring data as an additional parameter to substantiate conclusions about the seismic hazard of the south of Sakhalin Island.

19-28

Authors: S. Kh. Gekkieva

The mathematical modeling of different process types, for example, particle diffusion in a turbulent plasma, the propagation of heat in a thin rod, moisture transfer in soil, problems in mathematical biology and control problems, entails solving nonlocal boundary value problems. The paper considers a nonlocal boundary-value problem for the Aller – Lykov moisture transfer equation with a Riemann – Liouville time fractional derivative. The equation under consideration is a generalization of the Aller – Lykov equation obtained by introducing the concept of the fractal rate of humidity change, which explains the presence of flows moving against the water potential. For the solution to the problem, an a priori estimate has been obtained by the method of energy inequalities in terms of the fractional Riemann – Liouville derivative, which implies the uniqueness of the solution.

40-49

Authors: S. M. Mamajonov

In this paper, a number of boundary tasks for the equation of the fourth order of a parabolo-hyperbolic type look (a1 ∂/∂x+b1∂/∂y)(a2 ∂/∂x+b2 ∂/∂y)(Lu)=0 is put in pentagonal area. Unequivocal resolvability of one of put tasks is proved by methods of creation of the decision, the integrated and differential equations.

54-60

Authors: E. M. Shogenova

In this paper, the method of energy inequalities obtained a priori estimates of the first and third boundary value problems for the convection-diffusion equation of fractional order, from which follows the uniqueness and continuous dependence of the solution of the problems posed on the input data.

66-89

Authors: V. V. Bogdanov

The paper is devoted to the construction of a model of volumetric electrization of high-resistance dielectrics used in space technology and interacting with aggressive factors of space. The assumption of the Maxwell distribution of the charged particles flux by energy makes it possible to obtain a simple model that makes it possible to calculate the distribution of the electric field and the potential in the dielectric volume taking into account the secondary emission currents and the surface potential value for a given external electron flow. The complexity arises because of the nonlinearity: the surface potential depends on the external current, and the current – on this potential. This nonlinearity determines the emergence of the transcendent equation with respect to the potential φ0, which is solved numerically by iterative method. From the physics of the interaction of a stream of charged particles, described in the paper, its explanation of the formation of the plane of the zero values of the electric field with the withdrawal of the expressions defining the numerical values of the depth location of this plane xoreceives quite naturally.

97-108

Authors: T. S. Kumykov

In this paper, we investigate the effect of the medium with fractal structure on the growth of small cloud droplets at the initial condensation stage of cloud formation using a fractional differential equation. An electrodynamic model of coagulation of droplets under the action of an electric field is constructed in the cloud medium with a fractal structure. Numerical experiments are performed for assessment of the effect of the medium with fractal structure on the growth of cloud particles involving various combinations of microphysical parameters. A general dependence of the growth of cloud particles on different parameters of fractal structure in medium is established.

117-126

Authors: D. B. Litvin, I. P. Shepet, A. Y. Simonovsky

The analysis of the existing methodological apparatus of the use of accelerometric systems in inertial navigation is Carried out, its limitations are shown. The technique of optimal error dispersion to minimize the use of functional redundancy of information available in the measurements of distributed accelerometric systems is developed.

133-147

Authors: E. O. Makarov, P. P. Firstov

At the Petropavlovsk-Kamchatsky geodynamic test site, for almost 20 years, a network of monitoring stations for subsoil radon has been operating. During the work, a number of anomalies were registered, which preceded earthquakes with a magnitude M> 5, which by morphological features are divided into two types (A, B). The paper analyzed in detail the mechanism of occurrence of type B anomalies on the basis of a mathematical model for the propagation of radon in a stream of groundwater with complete transverse mixing. An attempt was made to link some features of the appearance of predictive anomalies of type B with the areas of occurrence of earthquakes.

158-165

Authors: M. A. Bagov, L. V. Skorikova

In the paper, the construction method of the basic (redundant) flow network node-link diagram is presented. Formalized design approach combined with optimization methods are used.

178-185

Authors: V. Ch. Kudaev, M. B. Abazokov

In this paper we present an algorithm for constructing an optimal P-rank flow network in a specified redundant geometric graph with potential links of nodes.

197-207

Authors: O. O. Lukovenkova, Yu. V. Marapulets, A. B. Tristanov, A. A. Kim, I. А. Kashutina

The paper is devoted to the development and comparison of different numerical methods which increase the adaptive property and improve the accuracy of matching pursuit algorithm in connection to geoacoustic signals. At each step of adaptive matching pursuit, a function is chosen which has the highest correlation with an initial signal. Then parameters of a chosen function are refined. The refinement is performed by the help of different grid methods and methods based on gradient direction search.

213-225

Authors: I. A. Larionov, E. I. Malkin, Yu. V. Marapulets, M. A. Mishchenko, A. A. Solodchuk

Methods and technical tools for complex analysis of geoacoustic and electromagnetic emissions signals have been developed to estimate automatically the state of the system «Earth surface layer – atmosphere lower layers» during the periods preceding hazardous seismic events. The investigation is a topical and one of the most important scientific directions, that is development of the theory of natural disaster mechanism modeling based on the consideration of the effects of a specified set of different physical factors. Technological processes which provide the detection, processing and analysis of geophysical signals from the near-surface lithosphere-atmosphere system of the Earth are described for the periods of background and anomalous activity in Kamchatka.

19-26

Authors: Zh. A. Balkizov

In this paper, we study the boundary-value problem with displacement for a model inhomogeneous parabolic-hyperbolic equation of the third order. We prove the uniqueness and existence theorems for a regular solution of the problem under study. In the case when the coefficients of the problem are constant real numbers, the solution of the problem under study is written out in explicit form.

36-41

Authors: R. A. Kirzhinov

In this paper the A. A. Dezin problem analogue is considered for inhomogeneous parabolic–hyperbolic type equation of the second order. We proved the solution uniqueness of the solution to the problem under investigation. The solution representation is written out by the Green’s function method.

48-56

Authors: L. Kh. Gadzova

We consider an initial value problem for ordinary differential equation with discretely distributed fractional differentiation operator. We give the Lagrange formula, prove the existence and uniqueness theorem and construct an explicit form of solution.

67-75

Authors: R. Kh. Makaova

In this paper, we study the Tricomi problem for a third-order hyperbolic equation with degeneracy of order inside a mixed domain. The existence and uniqueness theorem for a regular solution is proved.

83-90

Authors: О. Kh. Masaeva

The existence and uniqueness of the solution of the Neumann problem is proved for the generalized Laplace equation with a fractional derivative in the upper half-plane.

98-105

Authors: V. A. Kim, R. I. Parovik

In the study of nonlinear systems, one of the important problems is the determination of the type of oscillations-periodic, quasi-periodic, random, chaotic. It is especially difficult to distinguish between quasi-periodic oscillations from chaotic and random oscillations, since quasi-periodic oscillations often have a very complex shape, visually weakly distinguishable from «random». A feature of chaotic oscillations is their high sensitivity to small changes in the initial conditions. Therefore, one of the most reliable ways of detecting chaos is to determine the rate of run-off of trajectories, which is estimated using the Lyapunov exponent spectrum. Using the construction of the spectrum of Max Lyapunov exponents, depending on the values of the control parameters, chaotic regimes of the Duffing fractal oscillator with variable power memory were found, and its phase trajectories.

116-123

Authors: O. D. Lipko

In this paper we study the conditions for the existence of chaotic and regular oscillatory regimes of the hereditary oscillator FitzHugh-Nagumo (EFN), a mathematical model for the propagation of a nerve impulse in a membrane. To achieve this goal, using the non-local explicit finite-difference scheme and Wolf’s algorithm with the Gram-Schmidt orthogonalization procedure, the Lyapunov maximum exponent spectra were constructed as a function of the values of the control parameters of the model of the EFN. The results of the calculations showed that there are spectra of maximum Lyapunov exponents both with positive values and with negative values. The results of the calculations were also confirmed with the help of oscillograms and phase trajectories, which indicates the possibility of the existence of both chaotic attractors and limit cycles.

131-139

Authors: I. Kh. Mashukov, V. A. Shapovalov, A. A. Adzhieva

With the help of the Vaisala EFM550 electric field meter, long-term studies (2010-2017) of the course of the electric field strength in the area of the city of Nalchik were carried out. The article considers the use of digital processing to improve the quality of data obtained with a network of sensors operating in the hardware and software complex in the monitoring center of the geophysical situation over the southern region of the Russian Federation at the FSBI «High-Mountain Geophysical Institute». The possibilities of applying Kalman filtering to refine the data of instrumental measurements of the electric field strength of the atmosphere are investigated. A comparative analysis of the obtained data on the course of the electric field strength before and after filtration is given.

148-157

Authors: D. A. Tvoyrdyj

The Cauchy problem for the Riccati equation with non-constant coefficients and taking into account variable power memory is proposed. Power memory is defined by the operator of a fractional derivative of a variable order generalizing the Gerasimov-Caputo derivative. In work with the help of numerical methods: the Newton method and the explicit finitedifference scheme, the solution of the proposed Cauchy problem is found, and also their calculation accuracy is determined using the Runge rule. It is shown that both methods can be used to solve the proposed Cauchy problem, but Newton’s method converges faster. Further in this work, the calculated curves and phase trajectories were constructed for a different choice of the fractional order function of the differentiation operator. It is assumed that the proposed model can be used in describing economic cyclical processes.

168-179

Authors: V. A. Shapovalov

A three-dimensional numerical model of a convective cloud is developed taking into account thermodynamic, microphysical and electrical processes. The model uses a detailed microphysics. The system of equations of the cloud model describing the time variation of the dynamic and microphysical characteristics of the cloud consists of 3 equations of motion, heat and moisture balance equations, 137 equations describing the spectrum of cloud droplets, crystals, and microbubble particles. In addition, in order for the solution to satisfy the continuity equation, it is necessary to solve the three-dimensional elliptic equation for the pressure perturbation at each time step. One of the methods widely used for solving such problems is the splitting method developed by G.I. Marchuk, an improved version of this method, the predictor scheme with a divergent corrector, was successfully used in the modeling of cumulus clouds by R. Pastushkov. The conducted studies showed that, despite the certain complexity in the implementation of this scheme, it provides the necessary stability of the count, an approximation of the second order of accuracy in space and time, and is conservative. Splitting methods for physical processes and componentwise splitting are used (locally — one-dimensional schemes). The equations of the cloud model in finite-difference form were approximated by central and directional differences for spatial variables, as well as directed time differences. The resulting algebraic system was solved by a sweep method.

184-189

Authors: M. A. Kazakov

The article suggests a constructive method for training neural networks in which neurons added just before the current epoch of training assume the main training load on the new class to ensure the stability of the network in relation to learning on new data classes. The results of computational experiments are presented