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Abstract: The present survey describes the contribution of St. Petersburg mathematicians to the development of the theories of linear, classical, and algebraic groups. The first part is dedicated to the prehistory of the studies in the theory of linear groups in St. Petersburg, specifically, to the pedigree of the algebra schools created by Tartakovsky and Faddeev, and to an outline of the origin of the works by Borewicz and Suslin of the mid-1970s, which initiated systematic study in the field of classical groups and algebraic K-theory in St. Petersburg. PubDate: 2023-09-01

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Abstract: In this paper, we show how to select an adequate model of a stationary reversible moving-average process of finite order, given the appropriate number of sample correlations. We find the admissibility conditions, under which, for a reversible model of a moving-average process of no higher than the fifth order, a one-to-one correspondence between the coefficients and correlations of the process is established. If the admissibility conditions for sample correlations are met, it is possible to select a reversible stationary model. For higher-order moving-average processes, a mixed autoregression and moving-average model of no higher than the fifth order preliminarily approaches the initial data. This variant also has independent significance since even at small orders of the mixed model, good agreement between the correlations of the model and the sample correlations of the process is obtained. Particular attention is paid to the reversibility of the process since the prediction formulas assume fulfillment of this condition. PubDate: 2023-09-01

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Abstract: We consider a simplified model of a synchronous electric motor, which is described by a second-order differential equation, not containing electrical currents. F. Tricomi found that the phase portrait of this equation refers to one of three types, depending on whether the damping coefficient it contains is greater than, less than or equal to some critical value. Since an explicit expression for the critical value is not available, the efforts of many mathematicians were focused on deriving explicit upper and lower analytical estimates for this value. We use a computer to obtain phase portraits of this equation; the features of its phase trajectories are revealed, which are difficult to notice in known phase portraits obtained by analytical methods. Computer calculations are used to plot the critical value of the damping factor in this equation as a function of the main steady-state value of the angular variable. Linear and sinusoidal approximations of this curve are proposed, and the absolute and relative errors of such approximations are computed. PubDate: 2023-09-01

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Abstract: The term “mathematical diagnostics” was introduced by V. F. Demyanov in the early 2000s. The simplest problem of mathematical diagnostics is to determine the relative position of some point p and the convex hull C of a finite number of given points in n-dimensional Euclidean space. Of interest is the answer to the following questions: does the point p belong to the set C or not' If p does not belong to C, then what is the distance from p to C' In the general problem of mathematical diagnostics, two convex hulls are considered. The question is whether they have common points. If there are no common points, then it is required to find the distance between these hulls. From an algorithmic point of view, the problems of mathematical diagnostics reduce to special linear- or quadratic-programming problems, which can be solved by finite methods. However, the implementation of this approach in the case of large data arrays runs into serious computational difficulties. Such situations can be dealt with by infinite but easily implemented methods, which allow one to obtain an approximate solution with the required accuracy in a finite number of iterations. These methods include the MDM method. It was developed by Mitchell, Demyanov, and Malozemov in 1971 for other purposes, but later found application in machine learning. From a modern point of view, the original version of the MDM method can be used to solve only the simplest problems of mathematical diagnostics. This article gives a natural generalization of the MDM method, oriented towards solving general problems of mathematical diagnostics. In addition, it is shown how, using the generalized MDM method, a solution to the problem of the linear separation of two finite sets, in which the separating strip has the largest width, is found. PubDate: 2023-09-01

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Abstract: The motion of a heavy rigid thin rod on the surface of a right circular cylinder is considered. It is assumed that the angle between the generatrix of the cylinder and the direction of gravity is nonzero. The positions of equilibria of the rod on a cylinder form an equilibrium manifold (for all these equilibria the rod rests on the cylinder by its center of mass). The effect of transgression (nontrivial evolution along the equilibrium manifold) of the rod on the cylinder is studied using the normal form method. PubDate: 2023-09-01

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Abstract: As a generalization of a result obtained by Dubinin [7], Wali (preprint online) [14] recently proved the following: Let r ∈ \({{\mathcal{R}}_{n}}\) , where r has n poles at a1, a2, …, an and all its zeros lie in z ≤ 1, with s-fold zeros at the origin, then for z = 1 $$\left {r{\kern 1pt} '(z)} \right \geqslant \frac{1}{2}\left\{ {\left {\mathcal{B}{\kern 1pt} '(z)} \right + (s + m - n) + \frac{{\left {{{c}_{m}}} \right - \left {{{c}_{s}}} \right }}{{\left {{{c}_{m}}} \right + \left {{{c}_{s}}} \right }}} \right\}\left {r(z)} \right .$$ In this paper, instead of assuming that r(z) has a zero of order s at the origin as Wali did, we suppose that r(z) has a zero of multiplicity s at any point inside the unit circle and all other zeros are inside or outside a circle of radius k. Further, we prove some results which besides generalizing some inequalities for rational functions include refinements of some polynomial inequalities as special cases. PubDate: 2023-09-01

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Abstract: Non-isomorphic direct decompositions of torsion-free Abelian groups are reflected in their endomorphism ring decompositions which admit matrix representations. The set of possible direct decompositions of a special kind matrix rings into direct sums of one-sided indecomposable ideals is described. This leads to the combinatorial constructions of isomorphisms between non-commutative differently decomposable ring structures. PubDate: 2023-09-01

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Abstract: The review describes the creation and development of the scientific school of Sergei Vasilyevich Vallander at the Leningrad (now St. Petersburg) State University. We discuss the achievements of the scientific school in the development of methods of the kinetic theory of gases for the simulation of nonequilibrium flows, the construction of rigorous self-consistent mathematical models of varying complexity for strong and weak deviations from equilibrium, and the application of the developed models in solving modern problems of aerodynamics. Particular attention is paid to the study of nonequilibrium kinetics and transport processes in carbon dioxide, identifying the key relaxation mechanisms of polyatomic molecules, the development of physically reasonable reduced hybrid models, and the optimization of numerical simulation of flows using modern machine-learning methods. We discuss the problems of correctly accounting for electronic excitation in modeling the kinetics and transport processes, models of equilibrium gas flows with multiple ionization, and the peculiarities of simulating bulk viscosity in polyatomic gases. PubDate: 2023-09-01

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Abstract: This review, which consists of several works, is dedicated to the 300th anniversary of St. Petersburg State University (SPbSU) and is an attempt to analyze the scientific achievements of the St. Petersburg School of Mathematics and Mechanics in the field of rigid-body dynamics. This work, which is the first part of the review, covers the main achievements of the period from the founding of SPbSU to the mid-1970s. Due to the commemorative nature of this work, the scientific results obtained at SPbSU are considered in the context of events inextricably linked with the founding of the Academy of Sciences, the University and the gymnasium in 1724 and their further development over the subsequent 250 years. Due to the impossibility of covering even briefly all the publications that appeared during this period, attention is focused on the most important general areas of scientific thought and on those outstanding scientists of SPbSU, whose works enriched these areas. PubDate: 2023-09-01

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Abstract: New results on the asymptotic behavior of the probabilities of large deviations of combinatorial sums of independent random variables satisfying the Linnik condition are obtained. A zone, where these probabilities are equivalent to the tail of the standard normal law, is found. Such results were previously obtained by the author under Bernstein’s condition. The new results are proved by the truncation method. PubDate: 2023-09-01

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Abstract: A mathematical and numerical model of the joint dynamics of surface water and traction sediment is built, which takes into account the nonlinear dynamics of the fluid and bottom deformation. The dynamics of surface waters is described by the Saint-Venant equations, taking into account the spatially inhomogeneous distribution of the terrain. The transport of sediment loads is described by the nonlinear Exner equation, generalized to the case of a spatially inhomogeneous distribution of the parameters of the underlying surface. For numerical integration of the Saint-Venant and Exner equations, a stable and well-tested CSPH-TVD method of the second order of accuracy is used, the parallel CUDA algorithm of which is implemented as the software package “EcoGIS-Simulation” for high-performance computing on supercomputers with graphics processor units (GPU). Hydrodynamic modeling of the processes of erosion of the enclosing dam of a real hydraulic facility and flooding of adjacent territories is carried out. The parameters of penetration of the enclosing dam and flooding zones, formed as a result of the development of a hydraulic accident at the tailing dump, are determined. Based on the obtained results, it is concluded that the proposed method for numerical modeling of the joint dynamics of surface water and traction sediment can be more universal and efficient (has significantly better accuracy and performance) compared to available methods for calculating the parameters of the washout and flood zones, especially for hydrodynamic flows with complex geometry on an inhomogeneous bottom relief. PubDate: 2023-06-01 DOI: 10.1134/S1063454123020085

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Abstract: The study presents a brief overview of key results of research conducted at the Statistical Modeling Department of St. Petersburg State University. These results include mathematical substantiation of the computer simulation of randomness, stochastic methods for solving equations, stochastic optimization, and study of the stochastic stability and parallelism of Monte Carlo algorithms. In terms of experiment design, special attention is given to regression experiments with nonlinear parameterization. The references list includes mainly monographs authored by faculty members of the department, with the exception of some articles containing results not included in those. PubDate: 2023-06-01 DOI: 10.1134/S1063454123020048

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Abstract: In this paper, we will use the notion of \(\alpha \) -admissible mappings in Banach spaces, to introduce the concept of \({{T}_{\beta }}\) -contractive mappings and establish a fixed point theorem for this type of contractions. Our theorems generalize and improve many results in the literature. Moreover, we apply the main result to prove the existence of a solution for Volterra-integral equation, under more general assumptions than previously made. PubDate: 2023-06-01 DOI: 10.1134/S1063454123020164

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Abstract: The present work is the first part of a study of nonlinear dynamics of parametrically excited transverse oscillations of a clamped-clamped microbeam (the basic sensory element of a promising class of microsensors of various physical quantities) under exposure to laser-induced opto-thermal effects in the form of periodically generated pulses affecting a certain part of the beam-element surface. An analytical solution of the heat-transfer problem for steady harmonic temperature distribution in the resonator volume is found. The static and dynamic components of the temperature-induced axial force and transverse moment are determined. Discretization of the nonlinear coupled partial differential equations describing the longitudinal-transverse vibrations of the resonator is performed using the Galerkin method. Using the asymptotic method of multiple scales, an approximate analytical solution for the nonlinear dynamics problem under the conditions of primary parametric resonance is obtained. PubDate: 2023-06-01 DOI: 10.1134/S1063454123020127

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Abstract: In this work, we consider the problem of finite deformations of a dielectric tube under the action of an electric field. The tube consists of two layers, each of which is helically reinforced with fibers. The winding angles of the fibers in each layer are different. Flexible electrodes are deposited onto the inner and outer surfaces of the tube and between the layers. An electric field is induced by applying a voltage to the first or second layer, i.e., either to the electrodes of the inner surface and between the layers or to the electrodes between the layers and the outer surface. In the analysis, a simple model of an incompressible electroactive anisotropic material is considered. The potential-energy function is represented by the sum of the energy of an isotropic matrix in the Gent form, the simplest electrical component and the energy of the reinforcing fibers. Using a semi-inverse representation, the problem of the statics of a three-dimensional body is reduced to integral equations for the tube-deformation parameters: the radius of the outer layer, the multiplicity of the longitudinal elongation, and the angle of twist. The effect of the layer thickness on the deformation of the tube under a quasi-static increase in the electric voltage is studied. The aim of this study is to determine such ratios of the layer thicknesses, at which applying a voltage to different layers will lead to twisting of the tube in different directions. PubDate: 2023-06-01 DOI: 10.1134/S1063454123020097

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Abstract: In this paper, the lowest eigenfrequencies and vibration modes of a structure consisting of a closed circular cylindrical shell with an end cap attached to it in the shape of a shallow spherical segment are analyzed using numerical and analytical methods. Three types of free vibrations of the structure are described. Eigenfrequencies and modes of the first-type vibrations have been studied in previous papers, being close to the frequencies and vibration modes of a shallow spherical shell. In this study, modes and frequencies of second-type- (a cylindrical shell) and third-type vibrations (a cantilever beam with a load) are analyzed. An optimization problem is solved in order to determine the values of the structure parameters and the relative thickness of elements as well as the curvature of the end cap at which the minimum value of the eigenfrequency is maximal. A comparison between the asymptotic results and the numerical ones demonstrates their good agreement. PubDate: 2023-06-01 DOI: 10.1134/S1063454123020139

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Abstract: In this work, we consider the actual problem of separating the roots of nonlinear systems of equations in the case of many variables. The known method of reducing the problem of solving the system to an equivalent extremal problem, which is supposed to be solved by one of the stochastic optimization methods, is used. As the latter, the modeling method of annealing simulation and its modification, which are especially interesting because they allow effective implementation on quantum computers, are chosen. Since quantum computers based on simulated annealing demonstrate quantum superiority, the obtained results can be useful in solving systems of equations on these computers. PubDate: 2023-06-01 DOI: 10.1134/S1063454123020061

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Abstract: A closed self-consistent model of a nonequilibrium flow of a mixture of carbon dioxide and argon behind the front of a plane shock wave is developed. The generalized Chapman–Enskog method in the three-temperature approach, which takes into account different channels of vibrational relaxation in a carbon-dioxide molecule, is used. An extended system of Navier–Stokes–Fourier equations consisting of mass-, momentum-, and energy-conservation equations supplemented by diffusion equations for the mixture components and relaxation equations for vibrational modes of the CO2 molecule are written. Constitutive relations for the stress tensor, diffusion velocity, heat flux, and vibrational energy fluxes are obtained. An algorithm for calculating the coefficients of shear and bulk viscosity, the thermal conductivity of different degrees of freedom, diffusion and thermal diffusion are developed and implemented. The model is validated by comparing calculated transport coefficients with experimental data for the viscosity and thermal conductivity of carbon dioxide and argon and for the binary diffusion coefficient. Good agreement with the experiment is obtained. The dependence of transport coefficients on the gas temperature, vibrational-mode temperatures, and mixture composition is analyzed. The developed model is ready for use in the numerical simulation of shock waves in a CO2–Ar mixture. PubDate: 2023-06-01 DOI: 10.1134/S1063454123020024

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Abstract: In this paper, the class of Hölder functions in the sense of the Lp norm on certain compacts in \({{\mathbb{R}}^{m}}\) (m \( \geqslant \) 3) is analyzed, and theorems of the approximation by functions being harmonic in the neighborhoods of these compacts are proved. These compacts represent a generalization of the concept of a chord-arc curve in \({{\mathbb{R}}^{3}}\) to higher dimensions. The neighborhood size decreases along with an increase in the approximation accuracy. Estimates of the approximation rate as well as the gradient of the approximation functions are made in the same Lp norm. PubDate: 2023-06-01 DOI: 10.1134/S1063454123020140

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Abstract: In the context of tropical mathematics, the problem of solving a vector equation with two given matrices and unknown vectors, each part of which has the form of a product of one of the matrices and an unknown vector, is considered. Such an equation, which has an unknown vector on either side of the equal sign, is often called a two-sided equation. A new procedure for solving two-sided equations is proposed based on minimizing some distance function between the vectors of tropical vector spaces that are generated by the columns of each of the matrices. As a result of the procedure, a pair of vectors is obtained, which provides a minimum distance between the spaces and the value of the distance itself. If the equation has solutions, then the resulting vectors are the solution to the equation. Otherwise, these vectors define a pseudo-solution that minimizes the deviation of one side of the equation from the other. Execution of the procedure consists in constructing a sequence of vectors that are pseudo-solutions of the two-sided equation in which the left and right sides are alternately replaced by constant vectors. Unlike the well-known alternating algorithm, in which the corresponding inequalities are solved one by one instead of equations, the proposed procedure uses a different argument, looks simpler, and allows one to establish natural criteria for completing calculations. If the equation has no solutions, the procedure also finds a pseudo-solution and determines the value of the error associated with it, which can be useful in solving approximation problems. PubDate: 2023-06-01 DOI: 10.1134/S1063454123020103