Vol 25, No 3 (2025)
Mathematics
Multiplicities of some graded сocharacters of the matrix superalgebra M(2,2)(F)
Abstract
Let $F$ be an arbitrary field of characteristic zero, and let $M^{(m,k)}(F)$ be a matrix superalgebra over $F$. It is known from the theory of algebras with polynomial identities that the superalgebra $M^{(m,k)}(F)$ has a finite basis of $Z_2$-graded identities. Therefore, the problem of describing such a basis arises naturally. At the present moment of time, there is no such description. First of all, this is due to the fact that there are no effective methods for finding the usual or $Z_2$-graded identities of a superalgebra $M^{(m,k)}(F)$. Nevertheless, for some values of $m$, $k$, such identities can still be found. For this purpose, one uses either computer computations or the well-developed apparatus of the representation theory of the symmetric group $S_n$ and the general linear group $GL_p$. More precisely, to find $Z_2$-graded identities of a superalgebra $M^{(m,k)}(F)$ for small values of $m,k$, one studies the sequence $\{\chi_n\}$ of characters of representations of either groups $S_r\times S_{n-r}$ or group $GL_p\times GL_p$. For each such group, one constructs a vector $F$-space in the free algebra $F\{Y\bigcup Z\}$. At the same time, with respect to the action of group $S_r\times S_{n-r}$ ($GL_p\times GL_p$) on its vector space, it has the structure of a left $S_r\times S_{n-r}$ ($GL_p\times GL_p$) module. However, it turns out that it is computationally preferable to work with the characters representation sequence of the group $GL_p\times GL_p$. In this paper, we study the sequence of $GL_p\times GL_p$-characters $\{\chi_n\}$ of matrix superalgebra $M^{(2,2)}(F)$. This uses the fact that between pairs of partitions $(\lambda,\mu)$, where $\lambda\vdash r,\, \mu\vdash n-r$ and irreducible $GL_p\times GL_p$-modules, there is a one-to-one correspondence. Moreover, we investigate only those multiplicities in the decomposition of the character $\chi_n$ that are associated with irreducible $GL_p\times GL_p$-modules corresponding to pairs of partitions $(\lambda,\mu)$ of the form $(0,\mu)$. It is shown that if the height $h(\mu)$ of the Young diagram $D_\mu$ for a pair $(0,\mu)$ is no more than five, then the multiplicity $m_{0,\mu}$ of the irreducible $GL_p\times GL_p$-character $\chi_n$ is different from zero.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(3):306-315
306-315
Asymptotics of optimal investment behavior under a risk process with two-sided jumps
Abstract
We study an optimal investment control problem for an insurance company having two business branches, life annuity insurance and non-life insurance. The company can invest its surplus into a risk-free asset and a risky asset with the price dynamics given by a geometric Brownian motion. The optimization objective is to maximize the survival probability of the total portfolio over the infinite time interval. In the absence of investments, the portfolio surplus is described by a stochastic process involving two-sided jumps and a continuous drift. Downward jumps correspond to the claim sizes, and upward jumps are interpreted as random gains that arise at the final moments of the life annuity contracts' realizations (i.e., at the moments of death of policyholders) as a result of the release of unspent funds. The drift is determined by the difference between premiums in the non-life insurance contracts and the annuity payments. The solving to the optimization problem that yields the maximal survival probability, as well as the optimal strategy, is related to the classical solution of the corresponding Hamilton – Jacobi – Bellman (HJB) equation, if this solution exists. In the considered risk model, HJB includes integral operators of two types: Volterra and non-Volterra ones. The presence of the latter makes the asymptotic analysis of the solution quite complicated. However, for the case of small jumps (when the jumps have exponential distributions), we obtained asymptotic representations of solutions for both small and large values of the initial surplus.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(3):316-324
316-324
Solution of the inverse spectral problem for differential operators on a finite interval with complex weights
Abstract
Non-self-adjoint second-order ordinary differential operators on a finite interval with complex weights are studied. Properties of spectral characteristics are established, and the inverse problem of recovering operators from their spectral characteristics is investigated. For this class of nonlinear inverse problems, an algorithm for constructing the global solution is obtained. To study this class of inverse problems, we develop ideas of the method of spectral mappings.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(3):325-331
325-331
Mechanics
Oscillations of finite-dimensional models of an extensible catenary
Abstract
This article is devoted to the study of natural oscillation frequencies of finite-dimensional models of a stretchable flexible catenary. An analytical solution for two- and three-dumbbell models is presented, as well as the results of computer modeling of a twenty-dumbbell model of a stretchable catenary. In the case of an analytical approach, a coordinate solution method is used, in which the coordinates of the concentrated masses of dumbbell models in a deflected position are calculated. In the case of the numerical approach, the MSC.ADAMS software package is used, which allows analyzing the statics, kinematics, and dynamics of multibody systems. The results obtained for the considered stretchable catenary models are in good qualitative agreement with each other. Besides, when considering the limit transitions from the stretchable variant to the non-stretchable one, there is also a good consistency of the expected effects with the found results. For a finite-dimensional twenty-dumbbell model of a non-stretchable catenary with concentrated parameters, the first three dimensionless frequencies are compared with the frequencies of a continuous model, the values of which were found earlier. There is an excellent similarity of the results, confirming the applicability of the twenty-dumbbell scheme for describing the dynamics of catenary at low oscillation frequencies. In addition to determining the frequencies familiar to the classical non-stretchable catenary, an analysis of new ''migrating'' frequencies is carried out, which appear as a result of the emergence of additional degrees of freedom due to the consideration of stretchability. Frequency dependencies on the parameter characterizing the compliance of the catenary are constructed, which allows for estimating how quickly the ''migrating'' frequencies move from the high-frequency range to the low-frequency zone as the stiffness of the chain weakens. The formulas obtained and the models considered have both theoretical value and good applicability for applied tasks.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(3):332-344
332-344
Simulation models and research algorithms of thin shell structures deformation. Part II. Algorithms for studying shell structures
Abstract
Mathematical models of a thin shell deformation, which are considered in the first part of the article, constitute either a variational problem of energy functional minimum in terms of shell deformation or a boundary problem for differential equations of shell equilibrium. In both cases, the boundary conditions are also introduced according to the type of shell fixation. To solve the specified tasks, the different methods are considered. Using either the Ritz method for the variational problem of energy functional minimum for shell deformation or the Bubnov – Galerkin method for the boundary problem for differential equations of shell equilibrium, we will get systems of linear or nonlinear equations. The finite element method (FEM) in application to shell theory problems also leads to systems of linear equations, and the order of the equations can be very large. It is possible to use the Gauss method to solve the linear systems of algebraic equations in case the system order is less than 10$^3$. In another case, it is necessary to use iterative methods. For nonlinear tasks of thin shell theory, the parameter marching method is used. If the load is taken as a parameter, it is the V. V. Petrov's sequential loading method. It allows transforming the nonlinear tasks into a consistent linear solution with coefficients varying at each stage of loading. For solving nonlinear problems of shell theory, we consider the iteration method, when the nonlinear terms are transferred to the right side and successively changed at each iteration stage. In the article, it is also considered the method of quickest descent. A. L. Goldenweiser developed the special method: The asymptotic-integration method of thin shell theory, which is described in the article. If the equilibrium equation of the shell contains the discontinuous function (unit functions, delta-functions), then for this case there is a special G. N. Bialystochny's method, which is also specified in the article. Examples of the application of the described methods for solving specific problems of shell theory are also given.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(3):345-365
345-365
Statics and dynamics of an electrically driven mesh nanoplate
Abstract
The study object is a flexible plate of a mesh structure with an electrical drive with clamped edges. A source of electromotive force is connected to the gate and the plate. The gate is located at some distance below the plate. The volumetric ponderomotive forces of the electric field acting on the plate are modeled by the Coulomb force. The motion equations of a geometrically nonlinear plate, boundary, and initial conditions are obtained from the Ostrogradsky – Hamilton variational principle based on Kirchhoff's hypotheses. An isotropic, homogeneous material is considered. Scale effects are taken into account using modified couple stress theory. It is assumed that the fields of displacement and rotation are not independent. Geometric nonlinearity is taken into account according to the theory of T. von Karman. The mesh structure of the plate was modeled using the continuum theory of G. I. Pshenichny, which made it possible to replace the regular system of ribs with a continuous layer. The system of partial differential equations describing the nonlinear vibrations of the mesh plate under consideration was reduced to a system of ordinary differential equations using the finite difference method of second-order accuracy. The Cauchy problem was solved by the Runge – Kutta method of fourth-order accuracy. The mathematical model, solution algorithm, and software package were verified by comparing the calculation results with a full-scale experiment. An analysis of the static instability depending on the mesh geometry was carried out, as well as an analysis of the appearance of instability zones depending on the amplitude and frequency of the electrical voltage dynamic part.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(3):366-379
366-379
On the question of the physical interpretation of material constants of hyperelastic models
Abstract
There is a known demand for hyperelastic deformation models in the design of technical products using elastomeric materials (rubber and rubber-like polyurethanes, silicones, and TPE thermoplastic elastomers), which realize high (up to 500%) reversible deformations and damping capacity under cyclic and impact loading. Such products include car tires, shock absorbers, flexible gears, ''compliance mechanisms'' in robotics, etc. No less relevant and at the same time socially significant is the application of the theory of hyperelasticity for the purpose of developing implantable materials and devices for general, cardiac, and plastic surgery, including the replacement of soft biological tissues (skin, muscles, ligaments, etc.) with their functional analogues in the form of biocompatible synthetic materials. One of the unsolved problems in the mechanics of hyperelastic models remains the physical interpretation of their material constants. In this report, the material constants of the models are compared with the elastic moduli of the materials ($E_{0}$ and $G_{0}$) in the unstrained state. It is verified that for the neo-Hookean model, the relation $\mu=E_{0}/6$ holds, for the 2-parameter Mooney – Rivlin model $C_{01}+C_{10}=E_{0}/6$. It has been established that the same formula is valid for the 3-, 5-, and 9-parameter Mooney – Rivlin models and the nth order polynomial model. For the Ogden model $3\mu\alpha=2E_{0}$, Yeoh $C_{1}=E_{0}/6$, Veronda – Westmann $6(C_{1}C_{2}+C_{3})=E_{0}$. Material constants are indicators of the mechanical stability of hyperelastic models due to the Hill – Drucker condition. Using the example of a biomaterial, the results obtained using the derived formulas are compared with each other and with the indicators of elastic models: linear, bilinear, and exponential. A number of models characterize cases of small deformations unsatisfactorily.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(3):380-390
380-390
The effect of the geometric shape of an incision on the relaxation of residual stresses in a surface-hardened cylinder during thermal exposure
Abstract
A computational method is proposed for predicting residual stress relaxation during high-temperature creep following prior surface plastic deformation of solid cylinders with square and V-shaped notches. A series of parametric simulations was performed for cylindrical specimens made of EI698 alloy (20 mm length, 3.76 mm radius) with various notch geometries: depths of $\{0.1; 0.3\}$ mm for square notches, and depths of $\{0.1; 0.3\}$ mm with opening angles of $\{1^\circ, 5^\circ, 15^\circ\}$ for V-notches. The study demonstrates that residual stress field calculations after notching a strengthened cylindrical specimen require an elastoplastic formulation. The steady-state creep law was employed to simulate residual stress relaxation at 700$^\circ$C over 100 hours. A parametric analysis of notch geometry effects on stress relaxation was conducted. Results indicate that after the complete loading cycle ''hardening treatment at 20$^\circ$C — thermal loading (heating) to 700$^\circ$C — 100-hour creep at 700$^\circ$C — thermal unloading (cooling) to 20$^\circ$C'', despite relaxation, significant compressive residual stresses remain. This confirms the effectiveness of surface plastic strengthening for components with the investigated notch types under high-temperature creep conditions.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(3):391-405
391-405
Thermal force resonant loading of a three-layer plate
Abstract
The effect of thermal shock on forced vibrations of a circular three-layer plate excited by a resonant load is investigated. The plate is asymmetrical in thickness, thermally insulated on the lower surface and contour. The distribution of the non-stationary temperature over the thickness of the plate is calculated using an approximate formula obtained as a result of solving the problem of thermal conductivity by averaging the thermophysical properties of materials of a three-layer package. In accordance with the Neumann hypothesis, forced oscillations from a resonant load are superimposed on free oscillations caused by heat stroke (instantaneous drop in heat flow). The hypothesis of a broken line is used as a kinematic one: for high-strength thin bearing layers, the Kirchhoff hypothesis; for an incompressible thicker filler, the Timoshenko hypothesis about the straightness and incompressibility of a deformed normal that rotates by some additional angle (relative shift). The formulation of the initial boundary value problem includes differential equations of transverse vibrations of the plate in partial derivatives obtained by the variational method, homogeneous initial conditions and boundary conditions of the spherical support of the contour. The desired functions are plate deflection, relative shear, and radial displacement of the median plane of the filler. The analytical solution of the initial boundary value problem is constructed by decomposing the desired displacements into a series according to a system of proper orthonormal functions. The corresponding calculation formulas and the results of numerical parametric analysis of the dependence of the solution on the intensity and time of exposure to the heat flux, the magnitude of the power load are presented.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(3):406-418
406-418
Numerical study of coagulation of dispersed inclusions during injection of droplet fractions into a flow of dusty medium
Abstract
The paper presents a numerical solution to the problem of coagulation of solid particles and droplets during the injection of a gas-droplet flow into a gas suspension flow. It was assumed that a dusty medium moves in a flat channel, and a gas-droplet mixture is blown through the side surface of the channel. As a result of the coagulation of solid particles and droplets, the average density of the solid particle fraction decreases and the fractional composition of the droplet mixture changes. The calculations are based on a mathematical model of the dynamics of a polydisperse multi-velocity and multi-temperature gas suspension with a Lagrangian model of particle coagulation with relative velocity sliding. The mathematical model implemented a continuum technique for modeling the dynamics of multiphase media, which makes it possible to take into account the interphase interaction. The dynamics of the carrier medium is described by the Navier – Stokes equations for a compressible heat-conducting gas with interphase heat and momentum exchange. The aerodynamic drag force, the added mass force, and the dynamic Archimedes force were taken into account. The dispersed phase consisted of a number of fractions differing in the size of dispersed inclusions and the density of the particle material. The hydro- and thermodynamics of each dispersed fraction were described by a system of hydrodynamic equations, including the continuity equation, the equations for the conservation of momentum components, and the equation for the conservation of thermal energy, written taking into account the interphase thermal and force interaction. The system of equations for the dynamics of a multi-velocity multi-temperature polydisperse system was integrated by the explicit finite-difference McCormack method. The monotonicity of the solution was ensured by a nonlinear correction scheme. As a result of the calculations, time and space dependencies were obtained that characterize the evolution of the composition of a multi-fraction system when mixing flows of different dispersion.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(3):419-433
419-433
Computer Sciences
Dendrograms of electroencephalograms and their characterization based on metrics
Abstract
Dendrograms obtained from electroencephalograms are studied as maximal prefix codes. A dendrogram defines a distribution on the space of 2-adic integers and represents a partition, up to the set of zero Haar measure, into balls of nonzero radii. Non-Archimedean and Archimedean metrics are proposed for the characterization of dendrograms associated with the electroencephalograms of given mental classes. To more reliably distinguish one mental class from another, it is proposed to use the Gromov – Hausdorff distance between disconnected compact spaces: non-Archimedean in the form of a union of 2-adic balls represented by branches of a dedrogram, on the one hand, and Archimedean in the form of a (fat) Cantor set, on the other hand.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(3):434-441
434-441
Юбилеи
Legacy of Viktor Wagner. On the 90th anniversary of the Department of Geometry of Saratov State University
Abstract
The article is dedicated to the 90th anniversary of the Department of Geometry of Saratov State University, which was founded by the outstanding mathematician, honored scientist Viktor Vladimirovich Wagner. A brief description of the scientific work of V. V. Wagner and the main achievements in the field of geometry, algebra, and their applications is given. Prospects for further development of the scientific research areas founded by him are noted.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(3):442-453
442-453