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Sibirskii Zhurnal Industrial'noi Matematiki
2011, vol. 14, No 2 (46)
Contents
SibJIM 2(46), 2011.
UDC 519.626.1
Aleksandrov V. M., Dykhta V. A.
An approximate solution to the minimization problem for expended resources. I. Construction of a quasioptimal controlConsidering linear systems with constrained control and fixed transition time we propose two methods for solving approximately the minimization problem for expended resources. We prove that the switching moments of quasioptimal controls in the resources expended are independent of the initial conditions and constant for autonomous systems. We find a region of initial conditions for which the constraints on the control are never violated.
Keywords: optimal control, quasioptimal control, resource expenditure, linear system, switching moments, conjugate system, admissible region.
Pp. 3–14.Aleksandrov Vladimir Mikhauilovich
Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyuga prospect, Novosibirsk 630090, Russia, E-mail: vladalex@math.nsc.ru
Dykhta Vladimir Aleksandrovich
Institute of Systems Dynamics and Control SB RAS, 134 Lermontova str. Irkutsk 664033, Russia E-mail: dykhta@icc.ru
UDC 517.958
Alekseev G. V., Romanov V. G.
On a class of nonscattering acoustic shells for a model of anisotropic acousticsWe consider a model of anisotropic acoustics ehich describes the diffraction of a sound wave on a local anisotropic inhomogeneity. We study the existence of inhomogeneities yielding the zero scattered field, which would have to appear as a field generated by exterior compactly distributed sources encounters the inhomogeneity. We show that inhomogeneities of this type exist. We give a method for constructing a class of nonscattering anisotropic media depending on an arbitrary function of one variable. We exhibit explicit formulas determining the main shell parameters.
Keywords: acoustics equations, anisotropic media, nonscattering shells, compactly distributed sources.
Pp. 15–20.Alekseev Gennadiui Valentinovich
Institute of Applied Mathematics FEB RAS, 7 Radio str. Vladivostok 690041, Russia. E-mail: alekseev@iam.dvo.ru
Romanov Vladimir Gavrilovich
Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyuga prospect, Novosibirsk 630090. Russia. E-mail: romanov@math.nsc.ru
UDC 517.958
Anikonov D. S., Nazarov V. G., Prokhorov I. V.
The problem of single angle probing of an unknown mediumWe propose an algorithm for determining the shadows of unknown bodies inside an arbitrary medium probed by a beam of radiation registered only in one direction. We demonstrate the successful work of the algorithm in the case that the probed bodies are indistinguishable on the visualization line (a photograph). We study the problem of imposing the shadows of some bodies on those of others and conclude that this circumstance may both improve or worsen the reconstruction quality depending on the radiation characteristics of the medium. The included results of simulations show a good agreement between the theoretical and computer approaches.
Keywords: radiation transfer equation, X-ray tomography.
Pp. 21–27.Anikonov Dmitriui Sergeevich
Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyuga prospect, Novosibirsk 630090. Russia. E-mail: anik@math.nsc.ru
Nazarov Vasiliui Gennad’evich
Prokhorov Igor' Vasil'evich
Institute of Applied Mathematics FEB RAS. 7 Radio str. Vladivostok 690041. Russia. E-mail: naz@iam.dvo.ru;prh@iam.dvo.ru
UDC 517.9
Anikonov Yu. E., Neschadim M. V.
On analytical methods in the theory of inverse problems for hyperbolic equations. IIWe give new representations of solutions to and coefficients of mathematical physics equations in the form of differential-algebraic identities. These representations are partially of use in studying one-dimensional inverse problems.
Keywords: inverse problems of mathematical physics, analytical solution methods.
Pp. 28–33.Anikonov Yuriui Evgen'evich
Neschadim Mikhail Vladimirovich
Sobolev Institute of Mathematics SB RAS. 4 Acad. Koptyuga prospect.
Novosibirsk State University, 2 Pirogova str. Novosibirsk 630090, Russia. E-mail: anikon@math.nsc.ru ; neshch@math.nsc.ru
UDC 517.956.6
Apakov Yu. P.
The three-dimensional analog of the Tricomi problem for a parabolic-hyperbolic equationFor a parabolic-hyperbolic equation we study the three-dimensional analog of the Tricomi problem with a noncharacteristic plane where the equation type changes We prove the uniqueness of solution to the problem by the method of a priori estimates and reduce the existence of a solution to that for a Volterra integral equation of the second kind.
Keywords: parabolic-hyperbolic equation, Tricomi problem, Fourier transform, maximum principle, uniqueness, existence, integral equation.
Pp. 34–44.Apakov Yusupzhon Pulatovich
Namangan Engineering and Pedagogical Institute. 12 Dustlik prospect, Namagan 160103. Uzbekistan. E-mail: apakov.1956@mail.ru
UDC 519.24:519.86
Artem'ev S. S., Aschepkova Yu. I., Yakunin M. A.
A credit risk estimate for long-term financial flows basing on statistical modelingWe consider a mathematical model of long-term financial flows as the sum of a random number of random variables. For the particular case of flows in retirement funds we obtain distributions of gains and losses at a given moment in the distant future. We present the results of simulations using the statistical modeling of financial flows. We describe a program for estimating the credit risk of a retirement fund for various development scenarios of the world and regional economies.
Keywords: payment flow, sum of random variables, probability density, statistical modeling.
Pp. 45–54.Artem'ev Sergeui Semenovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Acad. Lavrent'eva prospect
Novosibirsk State University, 2 Pirogova str. Novosibirsk 630090. Russia. E-mail: ssa@osmf.sscc.ru
Aschepkova Yulia Ivanovna
Novosibirsk State University, 2 Pirogova str. Novosibirsk 630090. Russia. E-mail: yulya.aschepkova@ngs.ru
Yakunin Mikhail Aleksandrovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Acad. Lavrent'eva prospect. Novosibirsk 630090. Russia. E-mail: yma@osmf.sscc.ru
UDC 519.8
Bulavskiui V. A.
On a game of influence on system parametersWe study a game with finitely many players. The gain functions depend not only on the players' strategy profiles, but also on a certain vector state parameter of the system. The relation between a strategy profile and the value of the state parameter is defined quite generally as a correspondence. A player's decision on the best response to the system state and the other players' strategy profiles is made under the additional assumption that the variation of a player's strategy influences the variation of the system parameter. We consider both the version with constant assumptions and the version in which these assumptions by the players are in some correspondence with the strategy profiles and the value of the system parameter. We generalize the concept of equilibrium to this case and prove a theorem of its existence.
Keywords: game theory, nonsmooth model of game theory, equilibrium with assumed variation.
Pp. 55–62.Bulavskiui Vladimir Aleksandrovich
Central Economics and Mathematics Institute RAS, 47 Nakhimovskiui prospect, Moscow 117418, Russia. E-mail: lapissa@hotbox.ru
UDC 519.852.6
Zabinyako G. I.
On use of assignment algorithms to re-construct inverse matricesWe consider the questions of re-construction for the matrices inverse to the basis matrices of the revised simplex method. In order to choose the indices of pivots we use certain rules to form an auxiliary matrix from the basis matrix. The list of pivots results from solving assignment problems for the auxiliary matrix. On numerical examples of high dimension we analyze the efficiency of algorithms for solving assignment problems.
Keywords: revised simplex method, sparse matrices, assignment problems.
Pp. 63–69.Zabinyako Gerard Idel'fonovich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Acad. Lavrent'eva prospect, Novosibirsk 630090, E-mail: zabin@rav.sscc.ru
UDC 519.853.62:544.341
Il'in V. P., Morgunov K. G., Chauiko A. N.
On the interior point method for solving thermodynamic equilibrium problemsWe consider the problem of thermodynamic equilibrium in multicomponent systems with a given total composition as a traditional inverse problem. We propose a modification of the corresponding optimization problem for which, considering the specifics of the target functional, and basing on the interior point method we develop a two-level iterative process and sos find a solution satisfying the Karush–Kuhn–Tucker conditions. The results of simulations confirm the convergence of the method and demonstrate its wider applicability in comparison with analogous methods.
Pp. 69–77.Il'in Valeriui Pavlovich
Chauiko Aleksandr Nikolaevich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Acad. Lavrent'eva prospect, Novosibirsk 630090. Russia E-mail: ilin@sscc.ru; alexander.chaiko@gmail.com
Morgunov Konstantin Grigor'evich
Institute of Geology and Mineralogy SB RAS, 3 Acd. Koptyuga prospect, Novosibirsk 630090. Russia
UDC 519.8
Maergouiz L. S., Sidorova T. Yu., Khlebopros R. G.
A mathematical algorithm for redistributing the emission of greenhouse gasesIn connection with the current climate change problem, to reach an agreement between the emitents of greeenhouse gases (on the scale of the world, separate country, its region, a megapolis) we propose a mathematical algorithm for redistributing the emission of these gases.
Keywords: mathematical algorithm, extremal problem, greeenhouse effect, differentiated responsibility principle.
Pp. 78–83.Maergouiz Lev Sergeevich, E-mail: bear.lion@mail.ru
Sidorova Tat'yana Yur'evna, E-mail: tatiana-sidorova@mail.ru
Siberian Federal University
Khlebopros Rem Grigor'evich, E-mail: olikru@yandex.ru
Siberian Federal University, Presidium of KSC SB RAS, 82 Svobodnyui prospect, Krasnoyarsk 660041 Russia
UDC 519.245:57.022
Pertsev N. V., Pichugin B. Yu., Loginov K. K.
Statistical modeling of the dynamics of populations affected by toxic pollutantsWe present a stochastic model describing the dynamics of competing populations whose individua are affected by toxic pollutants. In order to construct the model, we use a probabilistic analog of the Lotka–Volterra model as a multidimensional inhomogeneous nonlinear birth and death process. We complement the postulates of the birth and death process with a description of the mechanism how? the toxic substance affects the death rate. We construct recurrences describing the dynamics of the population and the quantity of the toxic substance in the environment. We develop an algorithm for modeling the dynamics of populations and the quantity of the toxic substance basing on the Monte-Carlo method. We present the results of simulations studying degeneration conditions for one of the two competing populations, as well as conditions which guarantee that their numbers are maintained at nonzero stationary levels. In order to study analytically the behavior of the expected values of the populations we construct an auxiliary model as a system of nonlinear differential equations.
Pp. 84–94.Pertsev Nikolaui Viktorovich
Pichugin Boris Yur'evich
Loginov Konstantin Konstantinovich
Omsk Branch, Sobolev Institute of Mathematics SB RAS, 13 Pevtsova str. Omsk 644099, Russia E-mail: homlab@ya.ru
UDC 519.866.2
Rapoport E. O.
On some problems of ground rent modeling in a mixed economyWe study the problems of formalizing the concept of land annuity. Basing on the constructed mathematical economics models we study the questions of changing the price of product as the demand and investment into various plots change, while the need is fixed, and in the case of a mixed economy, when the production level is determined in a competitive environment. We study the patterns of variation of the prices and production levels as various coalitions form.
Keywords: mixed economy, ground rent, production functions, prices, coalitions, mathematical economics modeling.
Pp. 94–105.Rapoport Ernest Osherovich
Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyuga prospect,
Novosibirsk State University, 2 Pirogova str. Novosibirsk 630090. Russia, E-mail: rapoport@math.nsc.ru
UDC 537.81
Savchenko A. O., Savchenko O. Ya.
The flow of a harmonic coaxial vector field around an ellipsoid of revolutionWe give an integral equation determining the coaxial magnetic field near the surface of an axially symmetric superconducting body and the fluid velocity near the surface of an axially symmetric body coaxial with the flow of an ideal fluid. Using this equation in the case that prior to the introduction of an ellipsoid of revolution the axially symmetric magnetic field was changing along the axis as a polynomial, we determine analytically the density of the surface current and the force which the magnetic field exerts on the ellipsoid. We determine analytically the velocity of the fluid near the surface of the ellipsoid of revolution and the force acting on an ellipsoid placed coaxially into the flow of an ideal fluid when the velocity of the fluid prior to the mantling of an ellipsoid was changing along the symmetry axis of the flow as a polynomial.
Keywords: magnetic field, body of revolution, superconductivity, ellipsoid, surface currents, velocity, ideal fluid, polynomial.
Pp. 106–111.Savchenko Aleksandr Oliverovich
Savchenko Oliver Yakovlevich
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Lavrent'eva str.
Novosibirsk State University. Novosibirsk 630090 Russia. E-mail: savch@ommfao1.sscc.ru
UDC 517.956
Chirkunov Yu. A.
Systems of linear differential equations whose principal Lie algebra is not x-autonomicFor a system of first order linear differential equations with constant coefficients we obtain necessary and sufficient conditions for its principal Lie algebra not to be x-autonomous. Using this results, we establish that the principal Lie algebras of some systems of mathematical physics are x-autonomous.
Keywords: x-autonomicity of the principal Lie algebra, canonical systems of linear differential equations, x-autonomicity criteria.
Pp. 112–123.Chirkunov Yuriui Aleksandrovich
Novosibirsk State Technical University, 20 Karl Marx prospect, Novosibirsk 630092, Russia
Institute of Computational Technologies SB RAS, 6 Acad. Lavrent'eva prospect, Novosibirsk 630090, Russia, E-mail: chr01@rambler.ru
UDC 519.865.3
Shmyrev V. I.
A linear model of production-exchange. Polyhedral complexes and a test for an equilibriumWe study an economic model of exchange whose participants include both consumers and companies producing commodities. The production capabilities of the companies are constrained by the expenses of a certain single resource. We study in detail the version of the model with fixed budgets of the participants. We show that an equilibrium exists and describe an original approach of polyhedral complementarity, which enables us to obtain a test for an equilibrium state. Thus, we can propose a finite algorithm for finding an equilibrium.
Pp. 124–131.Shmyrev Vadim Ivanovich
Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyuga prospect Novosibirsk 630090 Russia E-mail: shvi@math.nsc.ru
UDC 519.63:550.832
Epov M. I., Kabanikhin S. I., Shishlenin M. A., Mironov V. L., Muzalevskiui K. V.A comparative analysis of two methods for calculating electromagnetic fields in the near-well space of oil and gas collectors
We make a comparative analysis of the discrete source method and the finite difference method in simulating electromagnetic fields in the subnanosecond range. We show a good agreement betweenthe results of modeling. We indicate directions for developing the finite difference method in the case of more complicated models.
Keywords: Maxwell's equations, discrete sources method, finite differences method.
Pp. 132–138.Epov Mikhail Ivanovich
Institute of Oil and Gas Geology and Geophysics SB RAS, 3 Acad. Koptyuga prospect, Novosibirsk 630090 Russia E-mail: epovmi@ipgg.nsc.ru
Kabanikhin Sergeui Igorevich
Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyuga
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, 6 Acad. Lavrent'eva prospect, Novosibirsk 630090 Russia E-mail: kabanikhin@sscc.ru
Shishlenin Makism Aleksandrovich
Sobolev Institute of Mathematics SB RAS Novosibirsk 630090 Russia E-mail: mshishlenin@ngs.ru
Mironov Valeriui Leonidovich
Muzalevskiui Konstantin Viktorovich
Institute of Physics SB RAS Siberian State Aerospace University, 50, bldg. 38, Akademgorodok, Krasnoyarsk 660036 Russia E-mail: rsdvm@ksc.krasn.ru; rsdkm@ksc.krasn.ru
UDC 533:517.958
Yulmukhametova Yu. V.
Submodels of gas motion with a linear velocity field in a degenerate caseWe seek solutions as a linear velocity field to gasodynamic equations with arbitrary state equation. We find all possible state equations and the corresponding differential equations of submodels when the auxiliary matrix is degenerate.
Keywords: gasodynamics, state equation, linear velocity field.
Pp. 139–150.Yulmukhametova Yulmukhametova Valer'evna
Ufa State Aviation Technical University, 12 Karl Marx str. Ufa 450077 Russia E-mail: taryv@yandex.ru