Краткие сообщения. Рубрика в журнале - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Research of default risk level of Russian energy
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The liberalization of the electricity market intensifies the competition for investors in the Russian energy market, for which the financial stability of energy companies is an important criterion. The paper presents the mathematical modelling of integrated assessment of the default risk level of Russian energy companies, taking into account their specific characteristics: type of the energy business, form of ownership, and regional specific. The research is based on the industry approach to the diagnostics of default risk level of energy companies. The approach includes the logit-model and assessment of the coefficients significance. The complexity of this model is conditioned by the study of external and internal financial and economic indicators, as well as qualitative criteria based on the introduction of dummy-variables. Four groups of default risk level of Russian energy companies are proposed. The use of mathematical modelling tools increases the accuracy of assessing the financial insolvency of energy companies in comparison with the traditional methods. Therefore, the proposed approach is novel, relevant, and practically significant. Research veracity is confirmed by the practical implementation. We recommend the use of the proposed methodology in assessment of the current state and development strategy of Russian energy companies, as well as by investors and analysts to make financial decisions.
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We propose development of examination methodology based on a sequential application of the MAI method (i.e., the hierarchy analysis method) and associative training of neural networks. The proposed method is an alternative to the usual methods to solve a direct examination problem. We present a methodological approach to the examination problem. The approach allows to save information about all objects and consider their indicators in total. Therefore, there is the soft maximum principle (softmax), based on the model of expert evaluations mixing. This approach allows different interpretations of the examination results, which save quality unchanged overall picture of the examination object indicators ratio, and to get more reliable examination results, especially in cases where the objects characteristics are very different.
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Smooth approximation of the quantile function derivatives
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In this paper, a smooth approximation of the second-order derivatives of quantile function is provided. The convergence of approximations of the first and second order derivatives of quantile function is studied in cases when there exists a deterministic equivalent for the corresponding stochastic programming problem. The quantile function is one of common criteria in stochastic programming problems. The first-order derivative of quantile function can be represented as a ratio of partial derivatives of probability function. Using smooth approximation of probability function and its derivatives we obtain approximations of these derivatives in the form of volume integrals. Approximation of the second-order derivative is obtained directly as derivative of the first-order derivative. A numerical example is provided to evaluate the accuracy of the presented approximations.
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Software for the mobile robot spatial orientation system
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Ensuring the orientation of the mobile robot in space requires solving the problem of its precise positioning. For this purpose, it is proposed to include an additional measuring complex in the control system of the mobile robot. The complex is designed to determine the orientation of the robot in space by the values of the angles of course, roll and pitch. The analysis of the mobile robot positioning solutions allows to justify the feasibility of using inertial navigation systems based on microelectromechanical sensors to obtain navigation information about the orientation of the mobile robot in space. The key element of the developed functional scheme of the mobile robot control system is the software of the mobile robot spatial orientation system. The software implements separate sections of the code that determine the orientation in space using inertial sensors in parallel to the main algorithm of the mobile robot. The result of the developed software is a string containing up-to-date information about the three orientation angles of the robot: the string is sent to the server to form control actions to correct the spatial orientation of the mobile robot. To improve the accuracy of determining the robot orientation in space based on the values of the angles of course, roll and pitch, the developed software eliminates the systematic error of microelectromechanical sensors and corrects the magnetometer readings taking into account the displacement of the magnetic field along its three axes. The developed software of the mobile robot spatial orientation system provides a significant increase in the positioning accuracy of the mobile robot designed for use in a room with an area of 30 to 200 m with a known layout and a priori set starting point.
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The existence theorem for systems of nonlinear Volterra integral equations kernels of the first kind with piecewise continuous is proved. Such equations model evolving dynamical systems. A numerical method for solving nonlinear Volterra integral equations of the first kind with piecewise continuous kernels is proposed using midpoint quadrature rule. Also numerical method for solution of systems of linear Volterra equations of the first kind is described. The examples demonstrate efficiency of proposed algorithms. The accuracy of proposed numerical methods is O(N-1).
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Spectral problems on compact graphs
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The method of finding the eigenvalues and eigenfunctions of abstract discrete semi-bounded operators on compact graphs is developed. Linear formulas allowing to calculate the eigenvalues of these operators are obtained. The eigenvalues can be calculates starting from any of their numbers, regardless of whether the eigenvalues with previous numbers are known. Formulas allow us to solve the problem of computing all the necessary points of the spectrum of discrete semibounded operators defined on geometric graphs. The method for finding the eigenfunctions is based on the Galerkin method. The problem of choosing the basis functions underlying the construction of the solution of spectral problems generated by discrete semibounded operators is considered. An algorithm to construct the basis functions is developed. A computational experiment to find the eigenvalues and eigenfunctions of the Sturm - Liouville operator defined on a two-ribbed compact graph with standard gluing conditions is performed. The results of the computational experiment showed the high efficiency of the developed methods.
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Stability of a stationary solution to one class of non-autonomous Sobolev type equations
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The article is devoted to the study of the stability of a stationary solution to the Cauchy problem for a non-autonomous linear Sobolev type equation in a relatively bounded case. Namely, we consider the case when the relative spectrum of the equation operator can intersect with the imaginary axis. In this case, there exist no exponential dichotomies and the second Lyapunov method is used to study stability. The stability of stationary solutions makes it possible to evaluate the qualitative behavior of systems described using such equations. In addition to introduction, conclusion and list of references, the article contains two sections. Section 1 describes the construction of solutions to non-autonomous equations of the class under consideration, and Section 2 examines the stability of a stationary solution to such equations.
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Stochastic Leontieff-type equations with multiplicative effect in spaces of complex-valued «noises»
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We consider a Leontieff-type stochastic equation, that is, a system of differential equations implicit with respect to the time derivative in the spaces of random processes. The concepts previously introduced for the spaces of differentiable «noise» using the Nelson-Gliklikh derivative carry over to the case of complex-valued «noise»; in addition, the right-hand side of the equation is subject to multiplicative effect of a special form. We construct a solution to the Showalter-Sidorov problem for Leontieff-type equations with multiplicative effect of a complex-valued process of special form. Aside from the introduction and references, the article consists of two parts. In the first part we carry over various concepts of the space of real-valued differentiable «noise» to the complex-valued case. In the second part we construct a Showalter-Sidorov solution to a Leontieff-type equation with multiplicative effect of a complex-valued process of special form. The list of references is not intended to be complete and reflects only the authors'' personal preferences.
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Stochastic inclusions with forward mean derivatives having decomposable right-hand sides
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In this paper, we prove a theorem on the existence of solutions for stochastic differential inclusions given in terms of the forward mean derivatives and the quadratic mean derivatives. These derivatives present information on the drift and the diffusion coefficient, respectively. The forward mean derivatives were introduced by E. Nelson for the needs of the so-called stochastic mechanics (a version of quantum mechanics), while the quadratic mean derivatives were introduced by Yu.E. Gliklich and S.V. Azarina. In the case of both the forward mean derivatives and the quadratic mean derivatives, we assume that the right-hand side is set-valued and lower semi-continuous, but not necessarily convex. Instead of this, we assume that the right-hand side is decomposable. Such inclusions naturally arise in many models of physical processes.
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Stochastic model of optimal dynamic measurements
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Under consideration is the stochastic model of optimal dynamic measurements. To solve this problem, the theory of optimal dynamic measurements which has actively been developing for the deterministic problems is extended to the stochastic case. The main purpose of the model is to restore a dynamically distorted input signal from a given observation using methods of the theory of dynamic measurements and the optimal control theory for Leontief type systems. Based on the results obtained by the authors earlier it is shown that optimal dynamic measurement as a minimum point of the cost functional doesn't depend on stochastic interference such as resonances in chains and random interference at the output of measuring transducer.
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Synthesis of surface h-polarized currents on an unclosed cylindrical surface
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The article describes the inverse problem of diffraction of electromagnetic waves, finding surface H-polarized currents on an unclosed cylindrical surface according to a given radiation pattern. The work is based on modelling an operator equation with a small parameter. The operator is represented as the sum of a positive-definite, continuously invertible operator and a compact positive operator. The positive-definite operator exactly coincides with the main operator of the corresponding direct problem of diffraction of electromagnetic waves. Due to this fact, the solution to the simulated equation satisfies the necessary boundary conditions. And this is the novelty and difference of the approach developed in this work from the methods known in the scientific literature. We develop a theory of an operator equation with a small parameter and a numerical method based on Chebyshev polynomials with weights that take into account the behavior at the boundary. The efficiency of the numerical method is shown.
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The Barenblatt-Zheltov-Kochina model on the segment with Wentzell boundary conditions
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In terms of the theory of relative p-bounded operators, we study the Barenblatt-Zheltov-Kochina model, which describes dynamics of pressure of a filtered fluid in a fractured-porous medium with general Wentzell boundary conditions. In particular, we consider spectrum of one-dimensional Laplace operator on the segment [0, 1] with general Wentzell boundary conditions. We examine the relative spectrum in one-dimensional Barenblatt-Zheltov-Kochina equation, and construct the resolving group in the Cauchy-Wentzell problem with general Wentzell boundary conditions. In the paper, these problems are solved under the assumption that the initial space is a contraction of the space L2(0, 1).
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The Lyapunov stability of the Cauchy-Dirichlet problem for the generalized Hoff equation
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We consider the initial boundary value problem with homogeneous Dirichlet boundary conditions for the generalized Hoff equation in a bounded domain. This equation models the dynamics of buckling of a double-tee girder under constant load and belongs to a large class of Sobolev type semilinear equations (We can isolate the linear and non-linear parts of the operator acting on the original function). The paper addresses the stability of zero solution of this problem. There are two methods in the theory of stability: the first one is the study of stability by linear approximation and the second one is the study of stability by Lyapunov function. We use the second Lyapunov''s method adapted to the case of incomplete normed spaces. The main result of this paper is a theorem on the stability and asymptotic stability of zero solution to this problem.
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We study conditions for the loss of stability in a plastic deformation of a layer of weaker material in a sheet specimen. The layer is not collinear with the exterior forces acting in the sheet plane, which are orthogonal to each other and have opposite signs. The parameters of the problem are: the angle between the layer and the direction of exterior forces; the ratio of stresses due to exterior forces; the ratio of strengths of the layer material and the main material of the sheet specimen; the strengthening law of the layer material; the ratio of thicknesses of the layer and the specimen. Basing on Swift's plastic instability criterion for a deformation of the layer material, we obtain an algorithm for calculating critical stress in the layer and critical exterior loading in dependence on the indicated parameters. When contact strengthening of the layer is absent, our results have explicit analytic expressions. We find conditions under which the layer does not lower the strength of the specimen. We find conditions for the stressed state of the layer to be a pure shear and study this case.
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The existence of a unique solution to a mixed control problem for Sobolev-type equations
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This article studies a mixed control problem for Sobolev-type equations in the case of a relatively radial operator. We use the Showalter-Sidorov initial condition. The difference in the statement of our problem from those studied previously by other researchers amounts to the form of the quality functional, which, in the authors' opinion, is more adequate to model applications in economics and technology. We prove an existence and uniqueness theorem for the solution to this problem.
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The multipoint initial-final value condition for the Navier - Stokes linear model
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The Navier - Stokes system models the dynamics of a viscous incompressible fluid. The problem of existence of solutions of the Cauchy - Dirichlet problem for this system is included in the list of the most serious problems of this century. In this paper it is proposed to consider the multipoint initial-final conditions instead of the Cauchy conditions. It should be noted that nowadays the study of solvabilityof initial-final value problems is a new and actively developing direction of the Sobolev type equations theory. The main result of the paper is the proof of unique solvability of the stated problem for the system of Navier - Stokes equations.
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We consider the problem of identifying the trajectory of a mobile point source described by the Delta function in a one-dimensional linear convective transport equation under a given additional boundary condition. To solve this problem, the Delta function is approximated by a continuous function and a discrete analog of the problem is constructed using finite-difference approximations in the form of an implicit difference scheme. To solve the resulting difference problem, we propose a special representation that allows to split the problem into two mutually independent linear first-order difference problems at each discrete value of a time variable. The result is an explicit formula for determining the position of a mobile point source for each discrete value of a time variable. Based on the proposed computational algorithm, numerical experiments were performed for model problems.
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The rate of convergence of hypersingular equations numerical computation
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Numerical methods for solving hypersingular equations based on Chebyshev polynomials of the second kind with a weight taking into account the Meixner physical conditions on the edge are developed. We obtained estimates of the rate of convergence using the analytical form of the matrix of an integral operator with a logarithmic singularity. Authors considered a delta function model, and its inapplicability in diffraction problems and vibrator antennas are shown. Previously, a numerical-analytical method for solving the excitation problems of vibrator antennas was proposed, but in the present work, the rationale for the numerical-analytical method is given for the first time. Unlike the reduction method, the numerical-analytical method demonstrates reliable convergence, not only in diffraction problems but also in antenna excitation problems. The specific feature of the excitation problems is that the right-hand side of the hypersingular equation is localized in a small region, in comparison with the characteristic dimensions of the antenna. Mathematically, this means that the right-hand side of the hypersingular equation decomposes into a slowly-convergent series. A similar property is also possessed by the solution of the equation. That is why the method of reduction is not effective enough. An example of a numerical solution is considered. Estimates of the rate of convergence are obtained. The applicability of developed methods for investigating a wide range of diffraction problems is shown.
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Алгоритмы нахождения асимптотических формул собственных чисел дискретных полуограниченных операторов
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Методы нахождения асимптотических формул собственных чисел дискретных полуограниченных операторов, определенных на компактных множествах, в каждом случае индивидуальны. Поэтому возникает необходимость разработать алгоритмы, позволяющие находить асимптотические формулы собственных значений любых дискретных полуограниченных операторов, определенных на компактных множествах. Это значительно упростит их нахождение и позволит написать программы для получения асимптотических формул. Данные алгоритмы помогут находить асимптотические формулы собственных значений вектор-операторов, заданных на конечных связанных графах. В статье, на основе разработанных раннее методов создан алгоритм, позволяющий находить асимптотические формулы собственных чисел с любым порядковым номером дискретных полуограниченных операторов, определенных на компактных множествах. Приведены примеры сравнения асимптотических формул, найденных по разработанной методике и по известным формулам, полученных ранее другими авторами, которые хорошо согласуются между собой.
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Впервые изучены детерминированная и стохастическая системы Вентцеля уравнений Баренблатта - Желтова - Кочиной, описывающих процесс фильтрации влаги в трехмерном шаре и на его границе. В детерминированном случае установлена однозначная разрешимость начальной задачи для системы Вентцеля в специфическом построенном гильбертовом пространстве. В случае стохастической системы используется теория производной Нельсона - Гликлиха и строится стохастическое решение, которое позволяет определять прогнозы количественного изменения геохимического режима грунтовых вод при безнапорной фильтрации. Отметим, что для изучаемой системы фильтрации рассматривалось неклассическое условие Вентцеля, поскольку оно представлено уравнением с оператором Лапласа - Бельтрами, заданным на границе области, понимаемой как гладкое компактное риманово многообразие без края, причем внешнее воздействие представлено нормальной производной функции, заданной в области.
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