This paper provides an overview of the 23rd International Saratov Winter School “Contemporary Problems of Function Theory and Their Applications” (Saratov, January 27–30, 2026), which was dedicated to the 90th anniversary of Professor A. P. Khromov.
In this study, we consider polygonal lines connecting two given points $A,B\in\mathbb{R}^2$ and consisting of exactly $n+1$ segments (i.e., having $n$ turn points). The absolute value of the turn angle at each interior point is bounded by a given number $\varphi\in(0,\pi/2)$. Under the condition $n\varphi<\pi$, we describe the set to which all interior vertices of such a polygonal line belong. It is proved that for any point $B^{(1)}$ from this set there exists a polygonal line satisfying the given constraints. Based on these results, an explicit formula is obtained that describes the set of all admissible sequences $\bigl(B^{(1)},\ldots,B^{(n)}\bigr)$ of corner points. The resulting description can serve as a foundation for constructing algorithms for enumerating admissible polygonal lines and for solving optimization problems for an objective function that takes into account both the cost of traversing the segments and the cost of making turns.
The presented paper is devoted to completing the development of an asymptotic method for solving boundary value problems of the hyperbolic boundary layer in elastic shells of revolution of arbitrary shape under three types of non-self-balanced end shock loading: longitudinal tangential, bending, and normal. The hyperbolic boundary layer is a component of the decomposition scheme for the non-stationary stress-strain state of thin shells localized in the vicinity of the leading fronts of dilatation and shear waves. The proposed asymptotic method for solving boundary value problems for all three types of stress strain state is based on obtaining an exact solution for the Laplace transforms using the integral transformation method in the case of cylindrical shells. The solution thus obtained, in the form of an expansion in vibration modes, which are an asymptotic representation of the vibration modes for a plane layer described by the Rayleigh – Lamb dispersion equation, is generalized to the case of shells of revolution of arbitrary form using the method of exponential representations. The originals of the transforms are presented in the form of series in Bessel functions. The studies of hyperbolic boundary layers are carried out in special near-front coordinate systems associated with the wave-front geometry, whose asymptotic model is defined by normals rotated to the middle surface.
An open acyclic fork-join queueing network is considered. When a job arrives from a source into the network, it is split to be serviced into a given constant number of associated tasks. The tasks are routed between service-nodes independently of each other and independently of tasks associated with other jobs. Upon leaving the queuing network, all tasks enter a node called the join-node, where associated tasks join into a job. A method is proposed for calculating the mean time spent by a task in the join-node and the mean response time in the fork-join queueing network. For the case of so-called uniform workload distribution in the queueing network, the calculation of the mean response time in the queueing network is described. Uniform workload distribution is determined by equal route lengths of tasks in the queuing network, equal probabilities of these routes, as well as equal service rates in the service-nodes, proportional to the number of tasks associated with jobs. A comparison of numerical results obtained in computational experiments using analytical and simulation models of a fork-join queueing network is presented. The results can be used to evaluate the time characteristics of the operation of multiprocessor systems, cloud computing systems and other stochastic systems modeled by fork-join queueing networks.
The paper formalizes the Planning–Modeling–Prediction (PMP) methodology as a basis for clinical decision support systems used in preoperative planning in traumatology and orthopedics. Two anatomical regions are considered: the spine--pelvic complex and the first ray of the foot, for which computational, software, and methodological results had been obtained previously. Comparison of these implementations made it possible to identify the common elements of the digital workflow: clinical and radiological input data, a set of admissible surgical options, a biomechanical model, computed indicators, criteria of geometric and mechanical admissibility, predictive interpretation, and a rule for comparing treatment options. A formal description of the CDSS architecture is proposed, separating the general sequence of transformations from anatomy-specific content. The implementation for the spine–pelvic complex is shown to be more complete in terms of modules and accumulated data, whereas the implementation for the first ray of the foot follows the same architectural logic with changes in the radiographic basis, intervention parameters, loading scenarios, and interpretation of the result. The proposed scheme can be used to transfer the PMP methodology to other problems in musculoskeletal surgery, provided that features, constraints, and admissibility criteria are specified separately for each new anatomical region.
To investigate the mechanical properties of quasi-2D layered networks composed of chiral multi-walled carbon nanotubes and their bundles, a methodology has been developed for constructing atomistic supercell models based on chiral nanotubes of the $(m,2m)$ type. These networks feature X-shaped van der Waals junctions between nanotubes, with varying periodicity steps along two mutually perpendicular directions. The methodology, formulated as a sequential algorithm of mathematical operations, is universal and can be applied to any configuration of multi-walled carbon nanotubes and their bundles, regardless of the number of constituent nanotubes. The developed approach also enables precise control over the network periodicity — i.e., the size of the window formed by the interconnected tubes. Validation of the methodology was performed using examples of individual double-walled carbon nanotubes and bundles containing both two and three nanotubes.
Identification of mechanical properties within the framework of a transversely isotropic model is a relevant problem in biomechanics and geophysics. The use of one-dimensional rod models for interpreting laboratory test results often leads to significant systematic errors due to neglected end effects and inhomogeneity of the stress-strain state. The aim of this study is to develop an applied theory of deformation for slender prisms and to solve the inverse problem of reconstructing elastic constants from displacement measurements on the lateral surface of the specimen. To address this problem, Lagrange’s variational principle was combined with the Kantorovich method. This approach made it possible to reduce the three-dimensional equations of elasticity to a system of ordinary differential equations and to obtain analytical solutions for specimens of arbitrary cross-section. Two types of specimen orientation are analyzed in detail: longitudinal and transverse relative to the axis of elastic symmetry. Based on the obtained solutions, an algorithm for identifying the five elastic constants is proposed, consisting of two stages. In the first stage, four elastic constants are determined from displacement measurements in the region away from the ends. In the second stage, the shear modulus \(c_{44}\) is calculated using an approximate Saint-Venant relation. The effectiveness of the proposed approach is confirmed by a series of numerical experiments for cortical bone tissue and shale using the finite element method. The results demonstrate good accuracy in parameter reconstruction. The developed model can significantly improve the reliability of interpreting data from standard compression tests on transversely isotropic materials.
The article investigates the fundamental continuation problem for generalized solutions of systems of linear differential equations with constant coefficients P(D)u=0 in a multidimensional parallelepiped $\pi \subset \mathbb{R}^n$. In contrast to classical works, where solutions are considered in spaces of distributions (generalized functions of finite order), here the continuation is constructed within the class of generalized functions of infinite order, which are closely related to analytic functionals and the theory of hyperfunctions. A solution u is considered that is defined only in a neighborhood of the union of $\bigcup\limits_{k=1}^n {{\pi_k}}$ faces of the parallelepiped $\pi \subset \mathbb{R}^n$. The key result is a continuation theorem stating that, under the condition of weak hyperbolicity of the operator P(D) with respect to the normals to the faces and under a certain condition on its characteristic set $N'=\mathrm{char}(P)$, every generalized solution $u$ from the corresponding space of generalized functions of infinite order can be extended to a solution $\mathcal{U}_L^\beta$ in a neighborhood of the entire parallelepiped $\pi$. The proof is based on Palamodov’s representation theory of generalized solutions and on the method of “gluing’’ analytic functionals. The functional $\widetilde\mu$ is represented as n functionals supported by the characteristic set N, which are then combined, using the tools of complex analysis, into a single functional $\widetilde\mu$ on the whole domain. An estimate is also provided for the norm of the extended solution $\upsilon$ in terms of the norm of the original solution $u$. The result constitutes a significant generalization of the classical Goursat and Darboux – Goursat – Baudot problems, transferring them to much broader functional classes, which is of considerable importance for modern mathematical physics and analysis.
Knee menisci provide load redistribution, increase joint congruence, stabilize the joint, and participate in lubrication and nutrition of cartilage. Reliable parameters of meniscal mechanical behavior, accounting for anisotropy, inhomogeneity, nonlinearity, viscoelasticity, and poroelastic effects, are required for biomechanical and numerical modeling of the knee joint. This review considers experimental studies of the mechanical properties of human and animal menisci, with particular attention to testing techniques for tension, compression, shear, relaxation, and indentation. The classes of material models used (linear elastic, biphasic, poroviscoelastic, hyperelastic, and damage models) are analyzed separately. It is shown that the scatter of literature data is largely due to differences in specimen preparation protocols, cutting orientation, hydration conditions, clamping methods, loading rate, and strain measurement techniques. Based on this, a research plan is proposed for developing a testing methodology on bovine menisci to identify their mechanical parameters.
A set of binary relations closed with respect to some collection of operations on them forms an algebra called an algebra of relations. Algebras of relations can be viewed as partially ordered by the set-theoretic inclusion. Theory of algebras of relations is an essential part of modern algebraic logic and has numerous applications in semigroup theory. As a rule, operations on relations are specified using logical formulas. An operation on relations is called conjunctive if it can be defined by a first-order formula containing only conjunctions. The paper is devoted to the study of associative binary conjunctive operations on relations, that is, classes of semigroups of relations with conjunctive operations. A classification of conjunctive operations is carried out, all associative operations are identified, and problems of finding axiomatic characteristics for most of them are solved.
This paper proposes a modification of the inverted series method that expands the classes of exact solutions it identifies for nonlinear differential equations in mathematical physics. The effectiveness of the modified method is demonstrated using examples of solving the integrable Tzitzeica equation and the nonintegrable Fisher equation.
This work is devoted to the numerical simulation of gas suspension oscillations in an acoustic resonator. The mathematical model utilizes a continuum technique for simulating the dynamics of multiphase media in Euler coordinates, accounting for the interaction between the gas and the dispersed phase. The dynamics of the carrier medium are described by a system of Navier – Stokes equations for a compressible, heat-conducting gas, taking into account interphase heat and momentum exchange between the mixture phases. The interphase momentum exchange forces included the aerodynamic drag force, the added mass force, and the dynamic Archimedes force. The dispersed phase dynamics are described by a system of equations including the continuity equation for the mean density, the conservation equations for the spatial components of the dispersed phase momentum, and the thermal energy conservation equation, all written taking into account interphase thermal interaction and momentum exchange between the phases. The system of equations for the dynamics of a multi-velocity, multi-temperature, monodisperse system was integrated using an explicit, second-order finite-difference method. A spatial-direction splitting scheme was used to implement the finite-difference method. A nonlinear correction scheme ensured the monotonicity of the solution. Using a numerical model, the oscillations of a gas suspension in a closed acoustic resonator were studied for various piston stroke amplitudes at a frequency close to the first linear resonance frequency. The numerical results were compared with the physical experiment. The comparison showed acceptable agreement between the numerical solution and the physical experiment data. Furthermore, within the framework of the monodisperse approximation of the mathematical model of gas suspension dynamics, the effect of particle dispersion on the intensity of change in the longitudinal component of the dispersed phase velocity and fluctuations in the dispersed phase concentration was studied. Larger dispersed inclusions have a lower velocity, and it was also found that smaller dispersed inclusions result in smaller amplitudes of carrier medium pressure fluctuations.
Hemodynamic disturbances and the geometric features of blood vessels play an important role in the onset and progression of various vascular pathologies, such as aneurysms. Cerebral aneurysms are particularly dangerous due to their specific location. Experimental and clinical methods often fail to adequately assess a patient's current hemodynamic status or predict disease progression in vivo. Numerical modeling of hemodynamics can become a key tool for assessing the risk of growth and rupture of cerebral aneurysms. The accuracy of such models depends on many factors, including the choice of a rheological blood model. Despite the widespread use of the Newtonian model, its adequacy for cerebral arteries requires verification in comparison with more complex non-Newtonian models that account for shear-dependent viscosity. The aim of this study was to conduct a comparative analysis of the influence of three rheological blood models (Newtonian, Carreau, and Casson) on the hemodynamic characteristics in eight anatomical variants of the Circle of Willis with aneurysms. In this study, the geometry of the cerebral vasculature was obtained by segmenting CT images. Computational fluid dynamics (CFD) methods were used to simulate blood flow. Velocity profiles based on intracranial Doppler ultrasound data were set at the inlets, and a three-element Windkessel model was applied at the outlets. The distributions of velocity, pressure, wall shear stress (WSS), and its oscillatory shear index (OSI) were investigated. The vessel walls were considered rigid. It was found that in the large arteries, the differences between the rheological models in the calculations of velocity, pressure, WSS, and OSI are insignificant. All models showed a similar systematic deviation from clinical Doppler data. Inside the aneurysm domes, blood flow velocities are low, and the profiles for all three rheological models are practically identical in shape, with the Newtonian model tending to overestimate the values. The results indicate that for modeling hemodynamics in the large vessels of the Circle of Willis, the use of a Newtonian model is a permissible simplification.
This article examines some unresolved issues concerning the mechanical properties of the middle ear organs - its tendons, in particular the stapedius tendon. The mechanical properties of biological tissues are a central topic in biomechanics and bioengineering. Mechanical characteristics are important parameters in computer modeling of organs and tissues during their functioning or under external influences. Mechanical properties of the stapedius tendon of the human middle ear are examined within the framework of the most commonly used hyperelastic models in the literature, as well as formally defined deformation models that allow for the description of the experimental curve with minimal error. The calculations were performed in the Mathcad 15.0 computer algebra system using specially developed functionality. The agreement between mechanical test data and model data was assessed using descriptive statistics. The results showed that the polynomial, Veronda-Westmann, and exponential models were the most accurate in terms of fitting the experimental data. The Hill-Drucker criterion E > 0 and the condition 3E/3A > 0 are satisfied by the Ogden, Yeoh, Veronda-Westmann, Fung, and Gent models, as well as one formally defined model (the exponential model). It is not recommended to use the 2parameter Mooney-Rivlin model in the undeformed state and under small deformations due to the loss of mechanical stability of the model in this range A. The results obtained in the work can be used for practical purposes in the creation of a physical model and finite element modeling of the middle ear, as well as in reconstructive surgery in the selection of artificial replacement materials for prosthetics and plastic surgery (stapedoplasty).
We consider a locally compact zero-dimensional group (G,+(center dot)), whose coset orders are arbitrary prime numbers. If the orders of the cosets are constant and equal to some prime number, the multiresolution analysis can be constructed by using a dilation operator. If the orders of the cosets are different, it is not possible to define the dilation operator. In this paper, we describe a method for constructing a multiresolution analysis without using the dilation operator and construct the corresponding wavelet basis.
The article presents a comprehensive methodology for forecasting macroeconomic indicators for long-term planning in project finance. The purpose of the research is to develop a system of stochastic simulations capable of generating plausible scenarios of economic development, taking into account the relationships between various economic parameters. The methodology includes two key components: an algorithm for selecting significant predictors based on sparse graphs and the minimum Steiner tree, and a system of stochastic simulations integrating the CIR++ model with the Monte Carlo method. The author has developed an efficient algorithm for building regression models that takes into account structural relationships between economic indicators. The research material consisted of historical data on a wide range of Russia's macroeconomic indicators: GDP, inflation, interest rates, real estate price indices, and loan delinquency rates. The results of applying the methodology demonstrate high accuracy of forecasting on historical data and intuitively understandable behavior in the long term. Model validation is based on conceptual validity, systematic output analysis, and business logic verification rather than traditional point forecast metrics, which is appropriate for long-term scenario generation. PCST hyperparameter calibration methodology and extreme scenario modeling for tail risk assessment are presented. The system is capable of generating probabilistic scenarios with a horizon of up to 30 years, which allows assessing various aspects of risks, including extreme scenarios. The modular architecture of the system provides flexibility and adaptability to various economic conditions. The results of the research have practical significance for risk management in financial institutions and strategic planning in project finance.
The article substantiates a method for solving linear Diophantine equations using the theory of group actions. The purpose of this paper is to introduce actions of certain groups on the set of linear Diophantine equations and to study their properties related to the set of solutions of these equations. Using group-theoretic methods, we achieve the goal and establish that the actions of symmetry groups of regular n-dimensional polyhedra on the set of equations under study are reduced to a combination of the actions of the symmetric group Sn and the automorphism group of the group of integers Aut(Z) on the same set. The relationship between the actions of a group of parallel transfers on the set of linear Diophantine equations and on the set of their solutions is also studied: for example, the vector of the general solution of an equation obtained as a result of an action can be found as the sum of the vector of the general solution of the equation that was subjected to the action and the vector of parallel transfer. In this article, we continued the formation of a class of linear Diophantine equations. Thus, it became possible to solve more equations using the solution of just one representative.
In this paper, an analytical solution is presented to provide an accurate trajectory of a ray propagating from the known position of the source to the receiver in a two-gradient medium. A system of two linear gradients connects two different layered media when the transition from one to the other occurs at some boundary. Within each medium, refractive indices determine the propagation of waves and, accordingly, the curved trajectories of rays. Different radii of curves make it difficult to track the ray as it propagates from the source to the receiver. Euler's method provides an exact solution for a one-gradient model. However, in the case of two gradients, the accurate solution cannot be obtained because of the underdetermined common system for ray curves and computational complexity. In this paper, a technique is described that combines Euler's method and trigonometric functions to derive direct formulas for calculating key angles responsible for the ray path in both gradient media. An exact solution overcomes the drawbacks of iterative approaches, which are subject to computational errors. The basic formula developed for two-gradient models was tested using a small set of real data by transforming it into a particular case of a one-gradient model. The independence of the evaluations is confirmed by comparing the calculated parameters with those taken from an earlier publication. The derived formulas are essential for solving problems in oil and gas exploration, geothermal exploration, and other challenges related to energetics. The solution can be extended for acoustic, optical, and other tasks.
A split-merge queueing system with two servers, the service times of which are identically distributed and dependent according to different copulas, is considered. The influence of service time dependence on the average sojourn time of a request in the system is studied. Expressions are derived for the expected value of the system response time, its behavior is analyzed for various values of the Kendall and Blomquist correlation coefficients, and formulas are derived for the bounds of the average response time depending on the value of the Blomqvist coefficient.
This paper proposes a mathematical modeling method for the heat conduction process in a porous medium with an ordered macrostructure. Based on the combined use of the minimal representative volume method and computational homogenization, a dependence of the effective thermal conductivity of the porous medium on the geometric characteristics (thickness, height) of the elementary cell — the unit structural element of the studied medium — is obtained. The elementary cell is considered as a triply periodic minimal surface (TPMS) of the Schwarz P type. The derived dependence for determining the values of the effective thermal conductivity coefficient was used in formulating the boundary value problem of heat conduction in a thin porous plate under first-kind boundary conditions. Using an approximate analytical method based on the introduction of additional boundary characteristics and a new unknown function, a simple analytical solution to the formulated problem was obtained. The analysis of the obtained solutions led to the conclusion that TPMS cells can be used for designing materials with specified thermophysical properties. In particular, it is shown that by adjusting the porosity of the plate, the intensity of heat transfer can be increased or decreased, and the required values of the plate’s thermal resistance can be achieved.
