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Sibirskii  Zhurnal  Industrial'noi  Matematiki
2014,  vol. 17,  No 3 (59)

Contents

                   

A.A. Baloev
UDC 517.91
 The matrix-algebraic form of a solution to the system of linear ordinary differential equations with constant coefficients

We propose a form for representing the general solution to a system of linear differential equations with constant coefficients which covers all possible versions of the general solutions corresponding to different multiplicities of the roots of the characteristic equation. The properties and some particular cases of this solution are studied.

Keywords: ordinary differential equation, multiple root.
Pp. 3–12.

Baloev Arnol'd Andreevich
Kazan State Technical University, 10 Marx st., 420111 Kazan. E-mail: a.baloev@mail.ru


 

UDC 517.958:517.956.3:539.3:544.275.7
A.M. Blokhin, D.L. Tkachev
Linear asymptotic instability of a stationary flow of a polymeric medium in a plane channel in the case of periodic perturbations

We study the linear stability of a stationary flow of an incompressible viscoelastic polymeric fluid in a plane infinite channel in the case of periodic perturbations with respect to a variable related to the length of the channel.

Keywords: polymer, rheological relation, stationary solution, Poiseuille flow, well-posedness, spectrum, Lyapunov stability.
Pp. 13–25.

Blokhin Aleksandr Mikhailovich
Tkachev Dmitrii Leonidovich
Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: blokhin@math.nsc.ru; tkachev@math.nsc.ru

 

UDC 517.954
L.N. Bondar
On the solvability of the second boundary value problem for the Stokes system

We consider the second boundary value problem for the Stokes system in the half-space. We prove the theorem on the existence of a solution in Sobolev spaces.

Keywords: elliptic system, boundary value problem, Stokes system, Sobolev space
Pp. 26–39.

Bondar Lina Nikolaevna
 Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: b_lina@ngs.ru

 

UDC 539.374:539.224
A.A. Burenin, Ε.P. Dats,  A.V. Tkachevΰ
On the modeling of the shrink fit technology

We solve a one-dimensional problem of the theory of thermal stresses that models shrink fit on a cylindrical shaft. A distinctive feature in the statement is the account taken of the emerging and developing a plastic flow of the material of the assembly elements because of the nonstationarity of the temperature field and the dependence of the yield strength of the material on  temperature. It is shown that irreversible deformation can significantly reduce the level of the final residual stresses providing the desired tightness.

Kew words: elasticity, plasticity, shrink fit, thermal stress, residual deformation, residual stress.
Pp. 40–47.

Burenin Anatolii Aleksandrovich
Tkacheva Anastasia Valer'evna
 Institute of Engineering and Metallurgy FEB RAS, 1 Metallurgov st. 681005 Komsomolsk-on-Amur. E-mail: burenin@iacp.dvo.ru; 4nansi4@mail.ru
Dats Evgeny Pavlovich,
Vladivostok State University of Economics and Service, 39a Gogol st., 690990 Vladivostok.
E-mail: dats@mail.dvo.ru

 

UDC 517.929.4
G.V. Demidenko,  I. I. Matveeva
On the exponential stability of solutions to one class of differential equations of neutral type

We consider systems of delay differential equations of neutral type. New estimates are obtained that characterize the exponential decay rate of solutions at infinity.

Keywords: delay differential equation, exponential stability, Lyapunov—Krasovskii functional, estimate for solutions.
Pp. 59–70.

Demidenko Gennadii Vladimirovich
Matveeva Inessa Izotovna
 Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: demidenk@math.nsc.ru matveeva@math.nsc.ru

 

UDC 519.242.5
V.I. Denisov, V.S. Timofeev, E.A. Khailenko
Semiparametric reconstruction of the density function based on a generalized lambda-distribution in the problem of identification of regression models

The problem of  estimating the parameters of regression models is considered. We study the method of the adaptive estimation of the parameters of regression models with the use of the semiparametric approach to the estimation of the density distribution function of random errors. The accuracy of the estimation of the parameters of the regression dependencies of this method is compared with the results obtained by the adaptive method on the basis of the generalized  lambda-distribution developed earlier by the authors.

Keywords: regression dependency, adaptive estimation, nuclear estimate, generalized lambda-distribution, maximum likelihood method, identification of a distribution.
Pp. 71–77.

Denisov Vladimir Ivanovich
Timofeev Vladimir Semenovich
Khailenko Ekaterina Alekseevna
 Novosibirsk State Technical University, 20 Karl Marx av. 630073 Novosibirsk.
E-mail: videnis@nstu.ru; netsc@fpm.ami.nstu.ru; ekavka@yandex.ru

 

UDC 330.35.01
Sh.A. Iskenderov
Optimal investment in economic growth with nonlinear amortization function

We consider the neoclassical model of economic growth and find conditions for the attainability and optimal management of investments for this model. Here the production and amortization functions depend nonlinearly on the capital and labor. We prove theorems on the existence and uniqueness of an optimal investment in the economy and the attainability of a given economic level.

Keywords: economic growth, stability, attainability, nonlinearity, optimal control, nonlinear dependence of the amortization on capital and labor.
Pp. 78–85.

Iskenderov Shahin Asafogly
 Institute of Cybernetics of NAS of Azerbaijan, 9 B. Vahavzade st., AZ-1141  Baku, Azerbaijan.
E-mail: office@lsu.edu.az

 

UDC 517.929:614.4
N.V. Pertsev
A continuous-discrete model of the spread and control of tuberculosis

The equations of a continuous-discrete mathematical model of the spread and control of tuberculosis in a certain region are presented. The equations of the model are constructed that account for the reproduction of the population of the region and impulse changes in the numbers of individuals at discrete points in time under the influence of various factors. The results of the investigations of solutions to the model are formulated. The conditions are obtained on the parameters of the model and the initial data under which there exist solutions to the model interpreted as the complete extinction of tuberculosis in a region or the maintenance of the sizes of groups of infected individuals at a certain acceptable level. For analyzing the solutions to the model, we use the method of monotone operators and a comparison system in the form of delay integrodifferential equations, which is a simplified version of the original model.

Keywords: delay integrodifferential equation, asymptotic behavior of solutions, method of monotone operators, epidemiology, tuberculosis.
Pp. 86–97.

Pertsev Nikolai Viktorovich
 Omsk Branch of Sobolev Institute of Mathematics SB RAS, 13 Pevtsov st., 644043 Omsk.
E-mail: homlab@ya.ru

 

UDC 517.968
V.G. Romanov
Recovering jumps in X-ray tomography

The problem of finding the boundaries of the discontinuities and the jumps of a piecewise smooth function is considered for X-ray fan tomography. An algorithm of the reconstruction of unknown values and explicit formulas for their calculation are given. The original problem of X-ray tomography is reduced to a problem for a continuous piecewise differentiable function.

Key words: tomography, determination of discontinuity lines, stability, uniqueness.
Pp. 98–110.

Romanov Vladimir Gavrilovich
 Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk.
E-mail: romanov@math.nsc.ru

 

UDC 517.925.5:517.929
I.A. Uvarova
On a system of nonlinear differential equations of higher dimension

We consider a Cauchy problem for a system of nonlinear differential equations of higher dimension. We prove that for a sufficiently large number of differential equations, the last component of the solution to the Cauchy problem is an approximate solution to the initial value problem for a delay differential equation.

Keywords: system of nonlinear ordinary differential equations of higher dimension, limit theorem, delay differential equation
Pp. 111–121.

Uvarova Irina Alekseevna
 Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: sibirochka@ngs.ru

 

UDC 519.24
P.A. Philonenko, S.N. Postovalov
Study of the influence of the distribution law of censoring times and the censoring degree on the power of homogeneity tests

We expose the results of studying the power of statistical tests for checking the hypothesis of homogeneity in randomly censored data for various situations (of different censoring degrees, alternative hypotheses, and laws of distribution of censoring moments). The results of modeling show that the power of the tests depends on the distribution of the censoring times in the case when the survival functions intersect. If they do not intersect then the distribution law for the censoring moments does not have a statistically significant influence on the power of tests. If the survival functions intersect then the Bagdonavicius – Nikulin tests are the most powerful of all those considered but their power decays rapidly as the censoring degree grows. If the survival functions do not intersect then the rank tests are more powerful than the Bagdonavicius – Nikulin tests.

Keywords: randomly censored data, homogeneity hypothesis, Peto's generalized Wilcoxon test, Gehan's generalized Wilcoxon test, logrank test, Cox – Mantel test, Q-test, Bagdonavicius – Nikulin test (single crossing), Bagdonavicius – Nikulin test (multiple crossing).
Pp. 122–134.

Philonenko Petr Aleksandrovich
Postovalov Sergey Nikolaevich
 Novosibirsk State Technical University, 20 Karl Marx av. 630073 Novosibirsk.
E-mail: petr-filonenko@mail.ru; postovalov@ngs.ru

 

UDK 519.237.5
M.G. Chebunin
Estimation of parameters of probabilistic models which is based on the number of different elements in a sample

We consider a sample from a one-parameter family of distributions on natural numbers. It is assumed that the values of the elements in the sample are unknown and only the number of different elements of the sample is known. From these statistics, we construct estimators of the parameter for a few concrete parametric families. It is shown by an example that such an estimator need not be asymptotically normal.

Keywords: number of different elements, parametric family of distributions, consistent estimator, Gumbel distribution.
Pp. 135–149.

Chebunin Mikhail Georgievich
 Sobolev Institute of Mathematics SB RAS, 4 Koptyug av., 630090 Novosibirsk;
Novosibirsk State University, 2 Pirogova st., 630090 Novosibirsk.
E-mail: chebuninmikhail@gmail.com

 

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