This article continues the cycle of works by authors on the development of mathematical aspects methods of artificial intelligence for the processing of observations conducted under the guidance of academician A.D. Gvishiani, which was began in 2000. It is devoted to a new universal method of smoothing, originally intended for the analysis of geophysical time series. Gravitational smoothing formed the basis for studying the acceleration of the secular course of the Earth’s main magnetic field with using of the observational data of the INTERMAGNET network. But the properties of the smoothing operator have not been studied so far. This aticle is first step to this goal.

Methods of local and spline approximations for digital processing of geomagnetic observations are proposed for consideration. Algorithms for calculating piecewise-linear, sinusoidal and pol-ynomial local approximation models have been developed. An algorithm for calculating the spline approximation model is developed. The generated mathematical apparatus is focused on solving problems of parameter estimation, filtering and spectral analysis for geomagnetic observations.

The paper considers an approach to solving plastic forming problems using a variational approach, which allows to determine the stress-strain state and the technological parameters associated with it, taking into account the combination of rheological properties of the materials being processed. On the basis of the first principle of energy in mechanics, which is built on the theorems on extremal properties, when the absolute minimum of the total power of the formchanging process corresponds to a real velocity field, energy functional has been composed. The energy functional is a power balance of internal and external forces. As the power of internal forces, we understand the cost of power of plastic deformation; power associated with the presence of velocities discontinuity surfaces in the volume of a deformable medium; power of friction forces, on the contact boundary with the tool; inertial component of power expended on the change in kinetic energy. Such a formulation makes it possible to investigate the processes of high-speed strain in the same way. This functional characterizes the state of the material under these processing conditions. To solve this functional, the method of local variations is applied, which refers to the direct numerical methods of the calculus of variations. An algorithm for calculating the power of plastic strains for the process of reverse extrusion of a glass from an isotropic, rigid-plastic material is given as an example.

The paper is devoted to some aspects of application of pattern recognition algorithms in solving problems of strong earthquake-prone area determination that can be used for seismic hazard assessment. The main principles of the having a long-term history approach to recognition of strong earthquake-prone areas (EPA) in a region under consideration on the basis of its morphostructural zoning scheme applying the algorithms “CORA-3” and “HAMMING” are described. A review of the results obtained in this direction and work on the development of new algorithms based, in particular, on discrete mathematical analysis is given. The use of pattern recognition approaches to develop algorithms for medium-term earthquake prediction that can help to obtain an operative seismic hazard assessment is shown. The application of the Unified Scaling Law for Earthquakes for the earthquake hazard and risk assessment taking into account the EPA results is considered. A review of the EPA and earthquake hazard and risk assessment results for the Caucasus region is presented.

An overview of the three numerical models for calculating the electrodynamic parameters of the Earth’s high-latitude ionosphere is presented. The model of the global distribution of the ionospheric electric potential, constructed on the basis of the solution of the boundary value problem on the spreading of ionospheric currents, makes it possible to calculate the trajectories of convection of the ionospheric plasma in the northern and southern hemispheres. The model of the high-latitude ionosphere allows calculating the three-dimensional structure of the electron density in the altitude range of 120-500 km under various helio-geophysical conditions. Importance of the electric fields of magnetospheric origin is stressed. Concentration of the main ionospheric ions is determined by the solution of the photochemical balance equation and the convective-diffusion equation along the trajectory of the plasma tube convection, taking into account the parameters of the thermosphere. A methodology and algorithms for calculating the distribution of magnetic field over the ionosphere, which is created by electric currents of magnetospheric origin, are developed. The model is based on the solution of the equation for the vector magnetic potential and makes it possible to calculate a two-dimensional picture of the magnetic variations.

An example of the evolution of a random graph is used to discuss the approach to stochastic dynamics of complex systems based on evolutionary equations. For the case of a graph, these equations describe temporal changes in the structure of the graph associated with the process of randomly adding new bonds to it. Such a process is closely related to the coalescence of individual irreducible components of the graph and leads to the appearance of singularities in the spectra and their moments during finite time intervals. These singularities arise due to the appearance of a giant connected component whose order is comparable with the total order of the entire graph. The paper demonstrates a method for analyzing the dynamics of the process of evolution of a random graph based on the exact solution of an evolutionary equation that describes the time dependence of the generating functional for the probability of finding in the system a given distribution of connected components of the graph. A derivation of the nonlinear integral equation for the generating function distribution on the number of connected components is given and outlined the methods of its analysis. In the concluding part, the possibilities of applying this approach to solving a number of evolutionary problems of statistical geodynamics are discussed.

The paper considers the hyperbolic Zeta function of nets with weights and the distribution of error values of approximate integration with modifications of nets. Considered: parallelepipedal nets M(~a,p), consisting of points

Mk =({a1k/p },...,{ask/p }) (k = 1,2,...,p);

non-uniform nets M(P), the coordinates of which are expressed via power functions modulo P:

Mk =({k/P},{k2/P}...,{ks/P}) (k = 1,2,...,P),

where P = p or P = p² and p — odd prime number;

generalized uniform nets M(~n) of N = n1 ·...·ns points of the form

M k =({k1/n1},{k2/n2}...,{ks/ns}) (kj = 1,2,...,nj (j = 1,...,s));

algebraic nets introduced by K. K. Frolov in 1976 and generalized parallelepipedal nets, the study of which began in 1984.

In addition, the review of p-nets is considered: Hammersley, Halton, Faure, Sobol, and Smolyak nets.

In conclusion, the current problems of applying the number-theoretic method in geophysics are considered, which require further study.

The problem for diffraction a plane harmonic sound wave on an elastic sphere T with cavity near ideal plane Π is considered. The outer ball layer is nonhomogeneous. Решение проводится путем расширения области задачи до полного пространства The solution is carried out by expanding the scope of the problem to the full space. При этом вводится дополнительное препятствие, которое является копией T, расположенной зеркально по отношению к плоскости Π. In this case, an additional obstacle is introduced, which is a copy of T, which is located mirror with respect to the Π. A second incident plane wave is also added. This wave ensures the fulfillment of that condition at the points of the plane Π, which corresponds to the type of the half-space boundary in the initial formulation of the problem. Thus, the problem is transformed into the problem of scattering of two plane sound waves on two inhomogeneous spheres in unbounded space. The solution is based on the linear theory of elasticity and the model of propagation of small vibrations in an ideal fluid. In the outer part of the liquid, the solution is sought analytically in the form of an expansion in spherical harmonics and Bessel functions. In the spherical region, which includes two elastic balls and an adjacent layer of liquid, the finite element method (FEM) is used. The results of the calculation of the directivity patterns of the scattered sound field in the far zone are presented. These dependencies show the influence of the geometric and material parameters of the obstacle on the diffraction of sound.

Herein we give an overview of the basic mathematical concepts and constructions that underlie the methods of geoinformatics developed by the scientific school under the leadership of RAS academician A. Gvishiani. It is important to note that we understand geoinformatics more widely than the study and application of geographic information systems. Geoinformatics includes research on the creation of methods and algorithms that make it possible to automate the problem solution in the field of geosciences using observational data. The solution is understood as an adequate modelling of expert’s logic, who performs data analysis and decision making manually. It is the observation systems and recorded data on the processes in the Earth’s interior and near-Earth space that form the basis for fundamental studies in the field of geoinformatics and other geosciences. Particularly, under the leadership of RAS academician A. Gvishiani the system of geomagnetic field observations has been significantly developed. This article is mostly devoted to the mathematical apparatus, used to analyze observational data in order to subsequently identify new regularities in the processes of the Earth and near-Earth space.

By means of an continuous-non-uniform elastic coatings it is possible to change effectively scattering performances of bodies in determinate directions if to pick up corresponding the inhomogeneity laws for mechanical parametres of a coating. In paper the problem of diffraction of a spherical sound wave by a homogeneous isotropic elastic cylinder with radially non-uniform elastic coating is considered. It is believed that an infinite circular cylinder with a coating is placed in an ideal unlimited fluid, heterogeneity laws of a coating material are described by differentiable functions, on the body falls а harmonic spherical sound wave emitted by a point source. In the case of steady state oscillations the propagation of small perturbations in ideal fluid is described by the scalar Helmholtz’s equation, and in elastic homogeneous isotropic cylinder — scalar and vector Helmholtz’s equations. The oscillations of an inhomogeneous isotropic elastic cylindrical layer described by general motion equations of the continuous medium. The analytical solution of the viewed problem was obtained on the basis of the known solution for a similar problem of the diffraction of a plane wave. The velocity potential of a spherical wave is represented in integral form as a decomposition on wave cylindrical functions. The integrand turns out to be similar in form to the expression of the velocity potential of a plane wave. The velocity potential of the scattered wave in the case of a falling of a spherical wave on a cylinder with a coating is written as an integral, the integrand of which is similar in form to the expression of the potential of the scattered wave when a plane wave falls on the body. It is necessary to determine the displacement field in a non-uniform coating to calculate the integrand. For this the built boundary-value problem for the system of ordinary differential equations of the second order must be solved. The computational aspects of integral evaluation are considered.

Analysis and forecast of aftershocks of large earthquakes in the world practice is currently based exclusively on stochastic models of aftershock process. This makes it possible to use statistical methods of analysis, and also to apply the ”scenario” approach in forecasts by repeatedly generating random sequences of aftershocks and counting the frequency of repetition of the events of interest. Studies on the Russian Science Foundation project ”Development of information system for automatic seismic hazard assessment after large earthquakes based on geophysical monitoring” in 2016-2018 showed however that the effectiveness of such approaches has significant limitations. In this paper I give a critical review of statistical methods for the analysis and forecast of aftershocks, an interpretation of the effectiveness limits of forecasts using standard approaches, provide the rationale for the need to change the paradigm. As one of the search directions, the application of Discrete Mathematical Analysis (DMA) methods developed by Academician A.D. Gvishiani and his scientific school. An obvious advantage of this approach is demonstrated by the example of a simple algorithm for identification of aftershocks using fuzzy comparisons.

The well-known theorem, proved by Doffin and Scheffer, states that the boundedness of a power series with finite-valued coefficients in a certain sector of the unit circle is equivalent to the periodicity of its coefficients, starting from a certain number. The paper indicates the class of Dirichlet series with finite-valued coefficients bounded in any strip of the right half-plane of the complex complex plane by a constant depending only on the height of the strip, for which an analogue of Dauffin – Scheffer theorem is proved. Earlier, an analogue of the Dauffin – Scheffer theorem was obtained by the authors for Dirichlet series with multiplicative coefficients. The method of proving this result allowed, in particular, to solve the well-known problem of generalized characters posed in 1950 by Yu.V. Linnik and N.G. Eccentric In this paper, this technique is used to prove an analogue of the Duffin – Scheffer theorem for the indicated class of Dirichlet series with multiplicative coefficients.

For 2k > 0.5n(n+1)+1 0 ≤ l ≤ 0,5k−w−1,w = [lnn/lnp,] the asymptotic formulas was proved for the number of solutions of the system of congruences

{x1 +···+ xk ≡ y1 +···+ yk (mod p^m)

xⁿ/₁ +···+ xⁿ/_k ≡ yⁿ/₁ +···+ yⁿ/_k (mod p^m)},

where unknowns x1,...,xk,y1,...,yk run values up 1 to pm−l from the complete system residues by modulo pm. The finding formula for 2k ≤ 0.5n(n + 1) + 1 has no the place.

Let be 1 ≤ s < r < ··· < n,s + r +···+ n < 0.5n(n + 1),0 ≤ l ≤ 0,5k −w−1. Then as2 k > s + r +···+ n for the number of the system of congruencies

{x^s/₁ +•••+ xs/k ≡ y^s/₁ +•••+ y^s/_k (mod pm)

x^r/₁ +•••+ x^r/_k ≡ y^r/₁ +•••+ y^r/_k (mod pm)

xⁿ/₁ +•••+ xⁿ/_k ≡ yⁿ/₁ +•••+ yⁿ/_k (mod pm)},

, where unknowns x₁,...,x_k,y₁,...,y_k run values up 1 to p^m−l from the complete system residues by modulo pm, was found the asymptotic formula. This formula has no place as 2k ≤ s + r +···+ n.