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 Vestnik St. Petersburg University: MathematicsJournal Prestige (SJR): 0.219 Number of Followers: 0      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1934-7855 - ISSN (Online) 1063-4541 Published by Springer-Verlag  [2469 journals]
• Liouvillian Solutions in the Problem of Rolling of a Heavy Homogeneous
Ball on a Surface of Revolution

Abstract: The problem of a heavy homogeneous ball rolling without slipping on a surface of revolution is a classical problem of the nonholonomic system dynamics. Usually, when considering this problem, following the E.J. Routh approach, it is convenient to define explicitly the equation of the surface on which the ball’s center is moving. This surface is equidistant from the surface on which the contact point is moving. It is known from the classic works by Routh and F. Noether that, if a ball rolls on a surface such that its center moves along a surface of revolution, then the problem is reduced to solving the second-order linear differential equation. Therefore, it is of interest to study for which surfaces of revolution the corresponding second-order linear differential equation admits a general solution expressed by Liouvillian functions. To solve this problem, it is possible to apply the Kovacic algorithm to the corresponding second-order linear differential equation. In this paper, we present our own method to derive the corresponding second-order linear differential equation. In the case in which the center of the ball moves along an ellipsoid of revolution, we prove that the general solution of the equation is expressed through Liouvillian functions.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040105

• Machine Learning and Artificial Intelligence in the Works of V.A.
Yakubovich

Abstract: This note precedes the republication of the article “Machine-Learning Pattern Recognition” by V.A. Yakubovich, which was first published by Leningrad University Press in the collection Methods of Computation in 1963.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040075

• On Mathematical Modeling of a Hypersonic Flow Past a Thin Wing with
Variable Shape

Abstract: This work is devoted to the further study of a spatial flow past a thin wing of variable shape by a hypersonic flow of a nonviscous gas. The head shock wave is considered to be attached to the leading edge of the wing. The use of the thin shock-layer method for solving the system of gas-dynamics equations allows constructing a mathematical model of the flow in question. Also note that the analysis of boundary conditions makes it possible to determine the structure of the expansion of sought values in a series and to construct approximate analytical solutions. In this case, in determining the first-approximation corrections, two equations are integrated independently of other equations. The application of the Euler–Ampere transform allows constructing a solution depending on two arbitrary functions and an unknown shape of the front of the head shock wave. To determine these functions, the integrodifferential system of equations was obtained previously. This paper proposes a variant of the semi-inverse method for constructing a solution (of this system) such that the formula for one arbitrary function is given. This approach allows additional assignment of the equation for the leading edge of the wing, as well as (in the case in which the head wave is attached along the entire leading edge) the inclination of the wing surface on it. The variant of the semi-inverse method presented in this paper for the nonstationary spatial problem of flow makes it possible to obtain a particular solution, which is a model solution for various regimes of a flow past a wing. We obtain formulas to determine the shape of the front of the shock wave, the shape of the surface of the streamlined body, the distance between the shock wave and the surface of the body, and the flow parameters on the wing surface.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040038

• On Regularization of the Solution of Integral Equations of the First Kind

Abstract: Ill-conditioned systems of linear algebraic equations (SLAEs) and integral equations of the first kind belonging to the class of ill-posed problems are considered. This class includes also the problem of inversion of the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to SLAEs with special matrices. To obtain a reliable solution, regularization methods are used. The general strategy is to use the Tikhonov stabilizer or its modifications or to represent the desired solution in the form of the orthogonal sum of two vectors, one of which is determined stably, while, to search for the second vector, it is necessary to use some kind of stabilization procedure. In this paper, the methods of numerical solving an SLAE with a symmetric positive definite matrix or with an oscillatory-type matrix with the use of regularization leading to an SLAE with a reduced condition number are considered. A method of reducing the problem of inversion of the integral Laplace transform to an SLAE with generalized Vandermonde oscillatory-type matrices, the regularization of which reduces the ill-conditioning of the system, is indicated.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040129

• Applying Adjustment Factors in the Rayleigh Method to Calculate the
Principal Frequency of the Vibrations of a Shell with a Rectangular Cross
Section

Abstract: The application of adjustment factors in the Rayleigh method to calculate the principal frequency of the vibrations of a shell with a rectangular cross section is considered in this paper. The behavior patterns of the adjustment factors are generalized. The relationship between the adjustment factors and properties of the approximate formulas is analyzed.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040063

• Contactless Capture and Removal of Space Debris Objects Using
Electromagnetic Induction

Abstract: The possibility of a contactless capture of the conductive space-debris objects (SDOs) using electromagnetic induction based on Faraday’s law of induction and Lenz’s law is analyzed. It is assumed that the spacecraft (SC) for capturing SDOs is equipped with a toroidal coil, which generates a sufficiently strong magnetic field and induces it on the SDO approaching the SC. The dynamics of the orbital motion of the SDO relative to the SC is modeled using the Clohessy–Wiltshire equations and studied numerically. The proposed method for the contactless electromagnetic capture of the conductive SDO can be used to move the SDO to a target orbit—in particular, to a graveyard orbit. The feasibility of the method is discussed based on modeling results. Directions for improvement and development of the method are identified.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040142

• Equilibria and Oscillations in a Reversible Mechanical System

Abstract: This paper investigates symmetric periodic motions (SPM) of reversible mechanical systems. A solution is given to the problem of bilateral continuation of a nondegenerate SPM to the global family of such SPMs. The result is applied to the general case of the Euler problem for a heavy rigid body, when the body parameters are not constrained by equality conditions. Two families of pendulum oscillations are found connecting the lower and upper equilibria.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040191

• Numerical Analysis of the Dynamics of Air Heating by an Interelectrode
Discharge

Abstract: The first 200 ns of an air pulsed interelectrode discharge are considered with gas dynamics taken into account. It is this initial stage that is of most importance for determination of the properties of the heating power released in the interelectrode gap. Data are presented on heating of the near-cathode and -anode layers and the gap by the moment when the transient stage in the development of the discharge ends. A spherical expanding shock wave is produced near the cathode.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040154

• Determination of a Preliminary Orbit by the
Cauchy–Kuryshev–Perov Geometric Method

Abstract: The determination of preliminary orbits of celestial bodies is of interest to observational astronomy in terms of discovering new bodies or identifying them with already known ones. The solution of this problem requires methods that are not limited both by the eccentricity of the orbit and by the time intervals between observations. This article considers the Cauchy–Kuryshev–Perov geometric method for determining a preliminary orbit. It is shown how to determine an orbit that does not lie in the observer’s plane of motion within the two-body problem, based only on geometric constructions, and using five angular observations. This method makes it possible to reduce the problem of determining a preliminary orbit to the algebraic system of equations relative to two dimensionless variables with a finite number of solutions. The method is suitable for determining both elliptical and hyperbolic orbits. It has no restrictions on the length of the orbital arc of the observed body and is not limited by the number of complete revolutions around the attractive center between observations. All possible combinations of positions of the body in the orbit are divided into 64 variants and described by the corresponding systems of equations. This article presents an algorithm for finding solutions to the problem without having prior information about the desired orbit. The solutions are sought in a bounded region in which triangulations are performed with triangles ranked according to the search conditions, thus eliminating the consideration of most of them at the initial stage. The solutions of the system are found by the Nelder–Mead method through the search for minima of the target function. The obtained orbits are compared by means of a representation of observations, and the best one is selected. An example of determining the orbit of the comet 2I/Borisov is given.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040117

• An Estimate of Average Case Approximation Complexity for Tensor Degrees of
Random Processes

Abstract: We consider random fields that are tensor degrees of a random process of second order with a continuous covariance function. The average case approximation complexity of a random field is defined as the minimal number of evaluations of linear functionals needed to approximate the field with a relative twofold average error not exceeding a given threshold. In the present paper, we estimate the growth of average case approximation complexity of random field for an arbitrarily high parametric dimension and for an arbitrarily small error threshold. Using rather weak assumptions concerning the spectrum of covariance operator of the generating random process, we obtain the necessary and sufficient condition that the average case approximation complexity has an upper estimate of special form. We show that this condition covers a wide class of cases and the order of the estimate of the average case approximation complexity coincides with the order of its asymptotic representation obtained by Lifshits and Tulyakova earlier.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040087

• Machines Learning Pattern Recognition

PubDate: 2021-10-01
DOI: 10.1134/S106345412104021X

• Nonclassical Vibrations of a Monoclinic Composite Strip

Abstract: A mathematical model of damped flexural–torsional vibrations of monoclinic composite strip of constant-length rectangular cross section is proposed. The model is based on the refined Timoshenko beam-bending theory, the theory of generalized Voigt–Lekhnitskii torsion, and the elastic–viscoelastic correspondence principle in the linear theory of viscoelasticity. A two-stage method for solving a coupled system of differential equations is developed. First, using the Laplace transform in spatial variable, real natural frequencies and natural forms are found. To determine the complex natural frequencies of the strip, the found real values are used as their initial values of natural frequencies, and then the complex frequencies are calculated by the method of third-order iterations. An assessment is given of the reliability of the mathematical model and method of numerical solution performed by comparing calculated and experimental values of natural frequencies and loss factors. The results of a numerical study of the effect of angles of orientation of reinforcing fibers and lengths by the values of natural frequencies and loss factors for free–free and cantilever monoclinic stripes are discussed. It is shown that, for the free–free strip, the region of mutual transformation eigenmodes of coupled vibration modes arise for quasi-bending and -twisting vibrations of either even or odd tones. In the cantilever strip of the region of mutual transformation of eigenforms of coupled modes, vibrations occur for both even and odd tones.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040166

• A Proof of Bel’tyukov–Lipshitz Theorem by Quasi-Quantifier
Elimination. II. The Main Reduction

Abstract: This paper is the second part of a new proof of the Bel’tyukov–Lipshitz theorem, which states that the existential theory of the structure $$\left\langle {\mathbb{Z};0,1, + , - , \leqslant , {\kern 1pt} } \right\rangle$$ is decidable. We construct a quasi-quantifier elimination algorithm (the notion was introduced in the first part of the proof) to reduce the decision problem for the existential theory of $$\left\langle {\mathbb{Z};0,1, + , - , \leqslant ,{\text{GCD}}} \right\rangle$$ to the decision problem for the positive existential theory of the structure $$\left\langle {{{\mathbb{Z}}_{{ > 0}}};1,{{{\{ a{\kern 1pt} \cdot {\kern 1pt} \} }}_{{a \in {{\mathbb{Z}}_{{ > 0}}}}}},{\text{GCD}}} \right\rangle$$ . Since the latter theory was proved decidable in the first part, this reduction completes the proof of the theorem. Analogues of two lemmas of Lipshitz’s proof are used in the step of variable isolation for quasi-elimination. In the quasi-elimination step we apply GCD-lemma, which was proved in the first part.
PubDate: 2021-10-01
DOI: 10.1134/S106345412104018X

• Discretization of the Parking Problem

Abstract: This paper considers a natural discretization of the Rényi problem known as the “parking problem.” Let l, n, and i be integers such that l ≥ 2, n ≥ 0, and 0 ≤ i ≤ n − l. We place an open interval (i, i + l) in the segment [0, n], where i is a random variable taking values 0, 1, 2, …, n − l with equal probability for all n ≥ l. If n < l, we say that the interval cannot be placed. After placing the first interval, two free intervals [0, i] and [i + l, n] are formed, which are filled independently of each other with intervals of length l according to the same rule, and so on. At the end of the process of filling the segment [0, n] with intervals of length l, the distance between any two neighboring intervals is at most l − 1. Let ξn, l denote the cumulative length of the placed intervals. The asymptotic behavior of the expectations of the aforementioned sequence of random variables has been studied before. This paper aims to continue the investigation of the behavior of the expectation E{ξn, l} as n → ∞ and to study the behavior of variances D{ξn, l} as n tends to infinity.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040099

• On the Constants in Inverse Theorems for the First-Order Derivative

Abstract: The known proofs of the inverse theorems in the theory of approximation by trigonometric polynomials and entire functions of exponential type are based on S.N. Bernstein’s idea to expand the function in a series with respect to the functions of its best approximation. In this paper, a new method to prove inverse theorems is proposed. Sufficiently simple identities are established that immediately lead to the aforementioned inverse theorems, with the constants being improved. This method can be applied to derivatives of any order—not necessarily integer—as well as (with certain modifications) to the estimates of some other functionals via their best approximations. In this paper, the case of the first-order derivative of the function itself and of its trigonometrically conjugate function is considered.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040208

• Approximation by Entire Functions on a Countable Set of Continua. The
Inverse Theorem

Abstract: In approximation theory, statements in which functions from certain classes are approximated by functions from other fixed classes (for example, by polynomials, rational functions, harmonic functions, etc.) and the accuracy of approximation is measured in a certain scale are called direct approximation theorems. Statements where the smoothness class of the approximated function is derived from the known accuracy of approximation of this function by polynomials, rational functions, and harmonic functions are called inverse approximation theorems. It is usually said that some class of generally smooth functions is constructively described in terms of the approximation by polynomials, rational functions, harmonic functions, etc., if functions from this class can be approximated in the chosen scale of the approximation accuracy and if the accuracy of the approximation in this scale yields the belonging of the approximated function to the class under consideration. Since the constructive description of classes of functions is a high-priority area of investigation in approximation theory, there exists a tendency to add inverse statements to the existing direct theorems for some classes of functions. The authors have previously proved the direct theorem concerning the approximation of a set of analytic functions defined on a countable set of continua by entire functions of exponential type. This paper presents the inverse statement. Section 1 assembles definitions and formulations, and Section 2 provides a proof of the main result.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040178

• On the Stability of the Zero Solution of a Differential Equation of the
Second Order in a Critical Case

Abstract: Differential equations of the form $$\ddot {x} + {{x}^{2}}\operatorname{sgn} x$$ = Y(t, x, $$\dot {x}$$ ) are considered, in which the right-hand side is a small periodic perturbation of t, a sufficiently differentiable function in the origin neighborhood with variables x, $$\dot {x}$$ . It is assumed that X perturbation is of an order of smallness not lower than the fifth if x is assigned the second order and $$\dot {x}$$ is assigned the third order. Periodic functions are introduced that are solutions of the equation above with a zero right-hand side. Since the differentiability of the quadratic part is bounded, the differentiability of the introduced functions is also bounded. These functions are used to switch from the initial equation to a system of equations in coordinates similar to polar. This system, with the help of polynomial replacement, is reduced to a system with Lyapunov constants. Replacement coefficients are found by partial fraction decomposition. A conclusion regarding the nature of the stability of the zero solution is drawn on the basis of the sign of the first nonzero constant. Due to the bounded differentiability of the introduced functions, the degree of the polynomial replacement must be limited. The system of differential equations for the replacement coefficients is solved recursively. The number of found Lyapunov constants is also bounded. This paper considers the in which when all found constants are zero. To study this problem, a method is used of isolating the main part of the introduced functions and their combinations as a result of the expansion of the latter in the Fourier series. The remainder of the series is assumed to be rather small, and it is shown that its presence can be neglected. The transition to the main parts instead of functions allows the lack of differentiability of the introduced functions to be compensated. In the case of such systems, polynomial replacement can be again used and the Lyapunov constant for each main part can be found. It is shown that the sign of the constant for any main part is preserved. Sufficient conditions for stability and instability are indicated.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040051

• Vibrations of a Plate with Periodically Changing Parameters

Abstract: Vibrations of a square plate with periodically changing parameters are considered. The averaged fourth-order partial differential equation for plate deflection w is presented. Solution of the problem is obtained with the approximate theory. The approximate results are presented by analytical formulas. Asymptotic averaging (implemented in Wolfram Mathematica) and the finite element method (ANSYS) are used to determine the values of eigenfrequencies. Numerical and asymptotic results are compared.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040130

• On the Compactness of Solutions to Certain Operator Inequalities Arising
from the Likhtarnikov–Yakubovich Frequency Theorem

Abstract: The compactness property of operator solutions to certain operator inequalities arising from the Likhtarnikov–Yakubovich frequency theorem for C0-semigroups is studied. It is shown that the operator solution can be described through solutions to an adjoint problem, as was previously known under a certain regularity condition. Thus, some regularity properties of the semigroup are connected with the compactness of the operator in the general case. Several results are proved that are useful for checking the noncompactness of operator solutions to Lyapunov inequalities and equations, into which the operator Riccati equation degenerates in certain cases that arise in applications. As an example, these theorems are applied for a scalar delay equation posed in a proper Hilbert space and it is shown that the operator solution cannot be compact. These results are related to the author’s recent work on the nonlocal reduction principle of cocycles (nonautonomous dynamical systems) in Hilbert spaces.
PubDate: 2021-10-01
DOI: 10.1134/S1063454121040026

• Conical Singular Points and Vector Fields

Abstract: We consider several examples of mechanical systems the configuration spaces of which have the form of smooth manifolds with a unique singular point: two intersecting (or tangent) curves on a two-dimensional torus, four curves on a four-dimensional torus with a common point, and a two-dimensional cone (cusp) in $${{\mathbb{R}}^{6}}$$ . The main problem presented in this paper is the calculation of the (co)tangent space above the singular point by means of various theoretical approaches. Outside singular points, the motion of the mechanisms in question is described in the context of classical mechanics. However, in the neighborhood of a singular point, terms like “tangent vector” and “cotangent vector” must have conceptually new definitions. In this paper, the approach of the theory of “differential spaces” is used. In the case of a conical singular point, to calculate (co)tangent space, we use two various differential structures: the algebra of functions locally constant near the cone vertex and the algebra of restrictions of smooth functions from enveloping space to a cone. In the first case, tangent and cotangent spaces at the cone vertex are zero. In the second case, the algebra of functions on the cotangent bundle consists of functions locally constant on the cotangent layer above the singular point.
PubDate: 2021-10-01
DOI: 10.1134/S106345412104004X

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