Vol 25, No 1 (2025)
Mathematics
On one consequence of the Chebyshev alternance
Abstract
The classical problem of the best approximation of a continuous function by a polynomial over a Chebyshev system of functions is considered. It is known that the solution of the problem is characterized by alternance. In addition, there is a linear growth function of the deviation of the target function of the coefficients of the polynomial from its minimum value with respect to the deviation of the vector of coefficients from the optimal one. In this article, the formula for the exact coefficient of this linear growth function is obtained by means of convex analysis. In contrast to those obtained earlier, it is expressed in a form constructive for realization through the values of the Chebyshev system functions at the points realizing alternance.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):4-14
4-14
Hopfian additive groups of rings
Abstract
A group is called Hopfian if it is not isomorphic to any of its proper factor groups, or, equivalently, any of its epimorphisms on itself is an isomorphism, i.e., an automorphism. This property was first proved by the Swiss mathematician H. Hopf for fundamental groups of Riemann surfaces. The results of the present paper concentrate around the problem of investigating general properties of Hopfian abelian groups and describing Hopfian groups in certain classes of abelian groups. Among the questions relating to Hopfian abelian groups, the study of the hopficity property in such a specific class of abelian groups as additive groups of rings occupies an important place. Additive groups of rings are one of the directions of research connecting the theory of abelian groups with the theory of rings. With regards to the methods of investigation and the nature of the results, this newly emerged direction, which appeared in the middle of the last century, is traditionally referred to the theory of abelian groups. When considering additive groups of particular classes of rings, some interesting examples of Hopfian abelian groups arise. The paper studies the hopficity in additive groups of $E$-rings (also called $E$-groups) and artinian rings. The work, in particular, proves that the additive group of an $E$-ring is Hopfian, and also gives a full description of how Hopfian additive groups of artinian rings are structured.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):15-23
15-23
A uniqueness theorem for mean periodic functions on the Bessel – Kingmann hypergroup
Abstract
One of the properties of a periodic function on the real axis is that it is completely determined by its values on the period. This fact admits the following nontrivial multidimensional generalization: if a function $f\in C^\infty (\mathbb R^n)$ $(n\ge 2)$ with zero integrals over all spheres (or balls) of fixed radius $r$ is zero in some ball of radius $r$, then $f$ is zero in $\mathbb R^n$. The condition of infinite smoothness of the function $f$ in this statement cannot be relaxed. In this paper, we study a similar phenomenon for solutions of convolution equations related to the generalized Bessel shift operator. First, we consider the case when the convolution factor in the equation is an indicator of a segment symmetric with respect to zero. It is shown that the solutions to such an equation are determined by their values on the specified segment. Further, a generalization of this property for the general Bessel convolution equation is given. The results obtained are analogues of the well-known uniqueness theorems for mean periodic functions belonging to F. John, Yu.I. Lyubich and A.F. Leontiev.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):24-33
24-33
On structure of isomorphisms of universal graphic automata
Abstract
Automata theory is one of the branches of mathematical cybernetics, that studies information transducers that arise in many applied problems. The major objective of automata theory is to develop methods by which one can describe and analyze the dynamic behavior of discrete systems. Depending on study tasks, automata are considered, for which the set of states and the set of output signals are equipped with additional mathematical structure preserved by transition and output functions of automata. We investigate automata over graphs and call them graphic automata. Universal graphic automaton $\mathrm{Atm}(G,H)$ is a universally attractive object in the category of such automata. The semigroup of input signals of the automaton is $S(G,H) = \mathrm{End}\ G \times \mathrm{Hom}(G,H)$. It can be considered as a derivative algebraic system of the mathematical object $\mathrm{Atm}(G,H)$, which contains useful information about the initial automaton. It is common knowledge that properties of the semigroup are interconnected with properties of the algebraic structure of the automaton. Hence, it is possible to study universal graphic automata by researching their input signal semigroups. Earlier the authors proved that a wide class of such kind of automata are determined up to isomorphism by their input signal semigroups. In this paper, we investigate a connection between isomorphisms of universal graphic automata and isomorphisms of their components — semigroups of input signals and graphs of states and output signals.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):34-45
34-45
A method for solving the Poincare boundary value problem for generalized harmonic functions in circular domains
Abstract
The paper considers a Poincare-type boundary value problem for a second-order elliptic differential equation that generates a class of generalized harmonic functions. It is established that in the case of circular domains the solution of the considered boundary value problem reduces to the solution of a Riemann-type differential boundary value problem in the classes of analytic functions of a complex variable. In addition, necessary and sufficient conditions for the solvability of the problem are obtained.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):46-52
46-52
Planarity ranks for varieties of equationally noetherian semigroups
Abstract
The problem of describing semigroup varieties with finite planarity rank is researched. In addition to the previously obtained results the author finds new countable infinite series of semigroup varieties with finite planarity rank.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):53-56
53-56
Mechanics
Dynamics of P. L. Chebyshev’s paradoxical mechanism
Abstract
The paradoxical mechanism of P. L. Chebyshev is the main issue of this article. The paradoxical mechanism could be considered as the union of the lambda-mechanism and a double pendulumthe free vertices of which are hingedly connected. One of the pendulum rods is usually replaced by a massive disk. Among all known mechanisms of P. L. Chebyshev, singular points of the configuration space or branching points arise only in the paradoxical mechanism. In particular configurations of the paradoxical mechanism the rods of the double pendulum become parallel. In this case, the mechanism could continue the motion from the singular points in two different ways. There are two types of smooth motion of the paradoxical mechanism, which differ by the direction of the disk rotation. In the first case, with a full circle of the drive link, the disk makes two full circles. In the second case, it makes four full circles. The trajectory of the free vertex of the lambda-mechanism in the paradoxical mechanism is located between two concentric circles and alternately touches each circle at three points. In addition, the curve that is located between two concentric circles and touches each circle in an arbitrary number of points is considered in the article. It is proved that it is possible to obtain any given number of disk revolutions in one revolution of the mechanism-driving link. The basic equations of dynamics are written down for the paradoxical mechanism. The expression for the moment of inertia forces of the disk is written down. The work of the external moment, which is applied to the leading link of the mechanism, and the work of the disk inertia forces are compared.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):57-69
57-69
Numerical study of the hydrodynamics of supercavitation flow around an underwater body
Abstract
The work is devoted to the study of the high-speed flow around an elongated body in water at various depths in the supercavitation regime. The aim of the research is to study the state of the environment around a submerged body and the possible influence of environmental disturbances on the movement of a group of bodies. The mathematical model of a compressible medium was used based on the Navier – Stokes equations. Two-phase, turbulence and phase transition were taken into account using the Mixture model, $k-\epsilon$ equations and Singhal full cavitation model, respectively. In the work, elongated conical bodies with different cavitator diameters and streamlined by a fluid flow at different speeds were considered. The numerical results were compared with the experimental results obtained by launching bodies on a hydroballistic track at the RIAMM TSU. Numerical simulation results showed that the proposed mathematical model can accurately predict the geometric shape and dimensions of the cavity. The numerical results are also in good agreement with the semi-empirical approximation for the cavity shape. Flow calculations show that a shock-wave flow pattern is formed near the body and flow disturbances propagate to a sufficient distance. On a cavitator at the front end of the body the flow is stalled and there is a sharp decrease of pressure to the values of saturated vapor pressure behind the shock wave. The dimensions of the cavity depend on the speed and ambient pressure — a greater flow rate leads to an increase in the size of the cavity. From calculations it follows that in the case of simulating deep-water launching under the same conditions of speed, following the medium pressure increase the volume of the cavity and the area of a disturbances propagation in the medium decreases, which can positively affect the accuracy of moving a group of bodies in a water.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):70-79
70-79
Asymptotic theory of the transient waves in shells of revolution at shock edge loading of the bending type
Abstract
The present work is devoted to completing the construction of the nonstationary stress-stain state asymptotic theory of shells of revolution at shock edge bending loading. There components with different values of the variability and dynamicity indices are used. This asymptotic model applies such components as the bending component of the Kirchoff – Love shell theory, the antisymmetric high-frequency short-wave component and the antisymmetric hyperbolic boundary layer in the vicinity of the dilatation wave front. The existence of the overlap regions is indicative of the exact statement of the boundary value problems for all the components and of the validity of the introduced separation scheme.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):80-90
80-90
Calculation of parameters of elastic and hyperelastic facial skin models
Abstract
The results of uniaxial mechanical tests of the facial (forehead) skin in vitro were compared with linear, bilinear and non-linear exponential, as well as five hyperelastic models. The results showed that the deformation properties of tissues are best described by the exponential function. Linear and bilinear elastic models are considered and the numerical values of the model parameters are determined. To study the hyperelastic properties of the skin, neohookean, Mooney – Rivlin, Ogden, polynomial and Veronda – Westmann phenomenological models were used. In order to find the most advanced algorithms for calculating the parameters of hyperelastic models, calculations were performed in the Mathcad 15 computer algebra system and the Ansys 2022 R2 multi-purpose software package. The parameters of the models and the closeness of the correlation between the exponential curve and the calculated data were determined, the correlation coefficient was used as a criterion for the correspondence of the models. The polynomial model and the Ogden model demonstrated the highest correlation with the experimental values, and the neohookean one demonstrated the lowest correlation. The values of Young's moduli and other elastic and hyperelastic characteristics of tissues were compared to study the factors affecting the mechanical behavior of human facial skin, and can be used in calculations in finite element analysis and in the development of replacement materials for plastic surgery.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):91-105
91-105
Computer Sciences
Multi-agent modeling of evacuation from premises with consideration of agent collisions
Abstract
The article proposes a software implementation of a developed multi-agent model for the evacuation of people of various age groups from premises of a complex shape. The model is distinguished by the possibility of taking into account the physical collisions of agents, their age dimensions in accordance with the methods from the order of the Ministry of Emergency Situations of Russia from 2022. The behavior of agents is based on a purposeful desire to exit. At the same time, in the event of congestion, they can wait until a place for movement is vacated. To simulate the collisions of agents with each other and with obstacles, we use the model of partially elastic impact. We carried out a series of experiments for different age groups: children, teenagers, adults and mixed groups. The total evacuation time from the premises was calculated and averaged for each age group. The program provides an opportunity to visualize the process of evacuation of people, set their initial arrangement randomly or manually. We compared the obtained results with the data of a similar experiment using the methods from the order of the Russian Emergencies Ministry of 2009. The comparison showed that in this experiment, children and teenagers are evacuated at about the same speed. However, a group of adults reaches the exit faster. The results of the comparison suggested that the features of the multi-agent approach used, taking into account the ability of agents to expect an exit, make it possible to increase the degree of organization of the modeled behavior of the group. This makes it possible to significantly reduce the evacuation time (more than twice for 100 people). The obtained and promising results of the project are intended for use in the development of digital twins of evacuation processes from commercial and public premises.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):106-115
106-115
Development and analysis of an algorithm for detecting multiple instances of an object in microscopic images using numerical methods
Abstract
This paper presents a method for object detection in microscopy images, focusing on particle detection. The main objective of the research is to develop an algorithm capable of efficiently detecting multiple instances of objects in various scenarios, while maintaining specificity for structures of interest. The algorithm is based on using extremal regions as candidates for detection, followed by evaluating these regions with trained parameters. A key element of the algorithm is its built-in non-overlapping constraint, which enables effective handling of particle clustering. Experimental results on various microscopy datasets confirm the method's robustness to changes in image intensity, particle density, and size. The proposed algorithm serves as a valuable tool in the development of object detection methods for microscopy images and can be applied in both scientific and medical research.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):116-127
116-127
A novel method for generating the optimal routing matrix of queuing networks with batch service
Abstract
In this paper, we consider a large-scale open queuing network. The arrival process in the queueing network is Poissonian. Сustomer transitions between nodes are described by the routing matrix. Each node consists of a single server and an infinite waiting queue. Customers are served as a unique batch of a given size with exponentially distributed service time. After the completion of service, customers are routed between nodes one at a time, independently of each other. We assume that, at any node, the number of destinations is much larger than the batch size. We also assume that the transition probabilities of customers between nodes are comparable. In this paper, we propose a method for generating the optimal routing matrix that provides the minimum average sojourn times in each node. We also provide a condition for relative arrival rates, under which the queuing network topology is radial (star-shaped), and expressions for optimal input rates to nodes. Finally, examples of the optimal routing matrix for different values of the overall input rate and in case of link failures are presented.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):128-139
128-139
Mathematical models for processing and interpreting seismic data in a new seismic survey method
Abstract
In the present work, a semi-analytical mathematical model is presented for application in a new seismic survey method with vertically positioned seismic receivers. This model describes the propagation of an acoustic impulse vertically in a layered structure with inclined boundaries of layers, with consideration of the main parameters of rock formations such as layer thicknesses, densities, velocities of acoustic wave propagation in rocks, the quality factors of these layers and the angles of boundary inclination. A comparison is made between the theoretical seismogram based on the semi-analytical model and the results of exact modeling in the Comsol Multiphysics software package. An algorithm for signal filtering in experimental seismograms is developed to extract signals coming vertically. A method of least squares is produced for determining acoustic and geological parameters of rock formations automatically (without the participation of a geologist-interpreter) to search for the global minimum of the objective function, validated by comparing the calculated parameters with data from a specific geological section.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2025;25(1):140-149
140-149