Vol 21, No 6 (2015)

Articles

Vladimir Alexandrovich Kondratiev (on the80 anniversary from the date of birth)

Astashova I., Pulkina L.

Abstract

Vladimir Alexandrovich Kondratiev (on the80 anniversary from the date of birth)
 
Vladimir Alexandrovich Kondratiev (on the80 anniversary from the date of birth)
Vestnik of Samara University. Natural Science Series. 2015;21(6):9-11
pages 9-11 views

ON OSCILLATION OF SOLUTIONS TO QUASI-LINEAR EMDEN – FOWLER TYPE HIGHER-ORDER DIFFERENTIAL EQUATIONS

Astashova I.

Abstract

Existence and behavior of oscillatory solutions to nonlinear equations with regular and singular power nonlinearity are investigated. In particular, the existence of oscillatory solutions is proved for the equation y(n) + P(x; y; y ′ ; : : : ; y(n−1))|y|k sign y = 0; n > 2; k ∈ R; k > 1; P ̸= 0; P ∈ C(Rn+1): A criterion is formulated for oscillation of all solutions to the quasilinear even-order differential equation y(n) + nΣ−1 i=0 aj(x) y(i) + p(x) |y|ksigny = 0; p ∈ C(R); aj ∈ C(R); j = 0; : : : ; n − 1; k > 1; n = 2m; m ∈ N; which generalizes the well-known Atkinson’s and Kiguradze’s criteria. The existence of quasi-periodic solutions is proved both for regular (k > 1) and singular (0 < k < 1) nonlinear equations y(n) + p0 |y|ksigny = 0; n > 2; k ∈ R; k > 0; k ̸= 1; p0 ∈ R; with (−1)np0 > 0: A result on the existence of periodic oscillatory solutions is formulated for this equation with n = 4; k > 0; k ̸= 1; p0 < 0:

Vestnik of Samara University. Natural Science Series. 2015;21(6):12-22
pages 12-22 views

ESTIMATES OF POSITIVE NONTRIVIAL SOLUTIONS OF A DIFFERENTIAL EQUATION WITH POWER NONLINEARITY

Bezukhov D.

Abstract

Differential equations in paper with power nonlinearity are considered. Solutions which are defined in some neighborhood of plus infinity are called proper solutions. It is proved that proper
solution to the equation is kneser solution, which means that solution and it’s quasiderivatives change their signs and tend to zero. The integral representation for proper solutions is proved. Upper estimates for solution and it’s quasiderivatives for proper solutions with maximal interval of existence is positive semiaxis to the equation with quasiderivative are proved. Upper and lower
estimates of solution and it’s derivatives for proper solutions with maximal interval of existence is positive semiaxis to the equation with derivative are proved
Differential equationsy[n] = rn(x)ddx(rn
Vestnik of Samara University. Natural Science Series. 2015;21(6):23-26
pages 23-26 views

Integral representations of solutionsof Riquier for polyharmonic equations in n-dimensional ball

Borodacheva E., Sokolovskiy V.

Abstract

The solution of Riquier’s problem - the problem of finding in n-dimensional ball of solving k + 1 - harmonic equation for given values on the boundary of the desired solution u and powers of the Laplacian from one to k inclusive of this decision is obtained. The first part provides an exact statement of the problem, the main result (form of the solution of it), and the idea of this proof is stated. The second part introduces a family of some differential and integral operators in the space of harmonic functions in the ball used in the proof of the main result; some properties of these operators are set. The content of the third part is the proof of the main result. It is based on the properties of operators introduced in the second part.
Vestnik of Samara University. Natural Science Series. 2015;21(6):27-39
pages 27-39 views

ON REPRESENTATION OF MODULAR FORMS AS HOMOGENEOUS POLYNOMIALS

Voskresenskaya G.

Abstract

In the article we study the spaces of modular forms such that each element of them is a homogeneous polynomial of modular forms of low weights of the same level. It is a classical fact that it is true for the level 1. N. Koblitz point out that it is true for cusp forms of level 4. In this article we show that the analogous situation takes place for the levels corresponding to the eta-products with multiplicative coefficients. In all cases under consideration the base functions are eta-products. In each case the base functions are written explicitly. Dimensions of spaces are calculated by the Cohen - Oesterle formula, the orders in cusps are calculated by the Biagioli formula.
Vestnik of Samara University. Natural Science Series. 2015;21(6):40-49
pages 40-49 views

Asymptotic classification of solutionsto the second-order Emden - Fowler type differential equation with negativepotential

Dulina K., Korchemkina T.

Abstract

Consider the second-order differential equation of Emden - Fowler type with negative potential y′′ - p (x, y, y′) |y|sgn y = 0: The function p (x; y; y′) is assumed positive, continuous, and Lipschitz continuous in y, y′: In the case ofsingular nonlinearity (0 < k < 1) the solutions to above equation can behavein a special way not only near the boundaries of their domains but also near internal points of the domains. This is why a notion of maximally uniquely extended solutions is introduced. Asymptotic classification of non-extensible solutions to above equation in case of regular nonlinearity (k > 1) and classification of maximally uniquely extended solutions to above equation in case of singular nonlinearity (0 < k < 1) are obtained.
Vestnik of Samara University. Natural Science Series. 2015;21(6):50-56
pages 50-56 views

ON A MINIMIZATION PROBLEM FOR A FUNCTIONAL GENERATED BY THE STURM – LIOUVILLE PROBLEM WITH INTEGRAL CONDITION ON THE POTENTIAL

Ezhak S.

Abstract

In this article we consider the minimization problem of the functional generated by a Sturm — Liouville problem with Dirichlet boundary conditions and with an integral condition on the potential. Estimation of the infimum of functional in some class of functions y and Q(x) is
reduced to estimation of a nonlinear functional non depending on the potential Q(x). This leads to related parameterized nonlinear boundary value problem. Upper and lower estimates are obtained for different values of parameter.

Vestnik of Samara University. Natural Science Series. 2015;21(6):57-61
pages 57-61 views

Inverse problems for the heat equation

Zaynullov A.

Abstract

The inverse problem of finding initial conditions and the right-hand side had been studied for the inhomogeneous heat equation on the basis of formulas for the solution of the first initial-boundary value problem. A criterion of uniqueness of solution of the inverse problem for finding the initial condition was found with Spectral analysis. The right side of the heat equation is represented as a product of two functions, one of which depends on the spatial coordinates and the other from time. In one task, along with an unknown solution is sought factor on the right side, depending on the time, and in another - a factor that depends on the spatial coordinates. For these tasks, we prove uniqueness theorems, the existence and stability of solution.
Vestnik of Samara University. Natural Science Series. 2015;21(6):62-75
pages 62-75 views

Finding of a numerical solution tothe Cauchy - Dirichlet problem for Boussinesq - Lo`ve equation using finitedifferences method

Zamyshlyaeva A., Surovtsev S.

Abstract

The article is devoted to the numerical investigation of Boussinesq - L`ove mathematical model. Algorithm for finding numerical solution to the Cauchy - Dirichlet problem for Boussinesq - Lo`ve equation modeling longitudinal oscillations in a thin elastic rod with regard to transverse inertia was obtained on the basis of phase space method and by using finite differences method. This problem can be reduced to the Cauchy problem for Sobolev type equation of the second order, which is not solvable for arbitrary initial values. The constructed algorithm includes additional check if initial data belongs to the phase space. The algorithm is implemented as a program in Matlab. The results of numerical experiments are obtained both in regular and degenerate cases. The graphs of obtained solutions are presented in each case.
Vestnik of Samara University. Natural Science Series. 2015;21(6):76-81
pages 76-81 views

One generalization of Marchaud inequality onsignsensitive weights

Ibragimova B.

Abstract

At the proof of a classical Marсhaud inequality for equidistant moduli of continuity of the highest degree the reduction of their definition for arbitrary sign of a step of a finite difference to positive values of this step is used. In case of moduli of continuity with a weight such reduction reduces definitions of moduli of continuity to restriction. Consequently for determination of properties of moduli of continuity with a weight other approach of reasoning is required. Unlike usual weight signsensitive weight allows to consider not only an absolute value of an increment of function, but also a sign of this increment. In the work for metrics with signsensitive weight an analogue of Marchaud inequality on estimation of modulus of continuity of given degree over modulus of continuity of a higher degree is obtained.
Vestnik of Samara University. Natural Science Series. 2015;21(6):82-88
pages 82-88 views

On the maximum principle for a class of nonlinear parabolicequations

Kon’kov A.

Abstract

In this paper, we consider solutions of nonlinear parabolic equations in the half-space.It is well-known that, in the case of linear equations, one needs to impose additional conditions on solutions for the validity of the maximum principle. The most famous of them are the conditions of Tikhonov and T¨acklind. We show that such restrictions are not needed for a wide class of nonlinear equations. In so doing, the coefficients of lower-order derivatives can grow arbitrarily as the spatial variables tend to infinity.We give an example which demonstrates an application of the obtained re- sults for nonlinearities of the Emden - Fowler type.
Vestnik of Samara University. Natural Science Series. 2015;21(6):89-92
pages 89-92 views

Numerical investigation of thegeneralized Hoff model

Manakova N., Vasyuchkova K.

Abstract

The work is devoted to the numerical investigation of the generalized Hoff model. Hoff equation models the dynamics of buckling construction of I-beams under a constant load. Result of existence and uniqueness of solution to the Showalter - Sidorov problem for the investigated model is formulated. This equation is a semilinear Sobolev type equation. Sobolev type equations constitute a vast area of non-classical equations of mathematical physics. Based on the theoretical results there was developed the algorithm of numerical solution of the problem.
Vestnik of Samara University. Natural Science Series. 2015;21(6):93-97
pages 93-97 views

GERSTEN COMPLEX FOR SHEAVES WITH TRANSFERS FOR NOETHERIAN SCHEMES

Mingazov A.

Abstract

V. Voevodsky introduced Gersten complex for sheaves with transfers in one of the first papers where the category of motives was constructed. Becides he proved Gersten conjecture which states that the Gersten complex for the local ring of point of smooth variety over field k is resolution of the group of global sections over this ring. Because of this fundamental fact Gersten complex can be used for calculations of cohomologies of sheaves with transfers over smooth k -varieties. In this paper we construct Gersten complex for sheaves with trans- fers, which defined on the category of noetherian k -schemes, where char k = 0. After this we proof the Gersten conjecture in case of local noetherian ring over field k . This generalises Voevodsky’s result.
Vestnik of Samara University. Natural Science Series. 2015;21(6):98-101
pages 98-101 views

UNCERTAINTY PRINCIPLES FOR GROUPS AND RECONSTRUCTION OF SIGNALS

Novikov S., Fedina M.

Abstract

Uncertainty principles of harmonic analysis and their analogues for finite abelian groups are considered in the paper. Special attention is paid to the recent results of T. Tao and coauthors about cyclic groups of prime order. It is shown, that indicator functions of subgroups of finite Abelian groups are analogues of Gaussian functions. Finite-dimensional version of Poisson summation formula is proved. Opportunities of application of these results for reconstruction of discrete signals with incomplete number of coefficients are suggested. The principle of partial isometric whereby we can determine the minimum number of measurements for stable recovery of the signal are formulated.
Vestnik of Samara University. Natural Science Series. 2015;21(6):102-109
pages 102-109 views

ON SOLUTIONS OF TRAVELING WAVE TYPE FOR A NONLINEAR PARABOLIC EQUATION

Pikulin S.

Abstract

We consider the Kolmogorov — Petrovsky — Piskunov equation which is
a quasilinear parabolic equation of second order appearing in the flame propagation
theory and in modeling of certain biological processes. An analytical
construction of self-similar solutions of traveling wave kind is presented for the
special case when the nonlinear term of the equation is the product of the
argument and a linear function of a positive power of the argument. The approach
to the construction of solutions is based on the study of singular points
of analytic continuation of the solution to the complex domain and on applying
the Fuchs — Kovalevskaya — Painlev´e test. The resulting representation of the
solution allows an efficient numerical implementation.

Vestnik of Samara University. Natural Science Series. 2015;21(6):110-116
pages 110-116 views

ON THE EXISTENCE OF SOLUTIONS WITH PRESCRIBED NUMBER OF ZEROS TO REGULAR NONLINEAR EMDEN - FOWLER TYPE THIRD-ORDER EQUATION WITH VARIABLE COEFFICIENT

Rogachev V.

Abstract

A third order Emden - Fowler type equation is considered. Existence of solution with given number of zeros on given interval is proved. This theorem extends previous results, related to Emden - Fowler type equation with constant coefficient, in case of variable coefficient.
Vestnik of Samara University. Natural Science Series. 2015;21(6):117-123
pages 117-123 views

ON THE UPPER ESTIMATES FOR THE FIRST EIGENVALUE OF A STURM - LIOUVILLE PROBLEM WITH A WEIGHTED INTEGRAL CONDITION

Telnova M.

Abstract

In this paper a problem for which the origin problem was a problem known as the Lagrange problem or the problem on finding the form of the firmest column of the given volume is viewed. The Lagrange problem was the source for different extremal eigenvalue problems, among them for eigenvalue problems for the second-order differential equations, with an integral condition on the potential. In this paper the problem of that kind is considered under the con- dition that the integral condition contains a weight function. The method of finding the sharp upper estimates for the first eigenvalue of a Sturm - Liouville problem with Dirichlet conditions for some values of parameters in the integral condition was found and attainability of those estimates was proved. In this paper a problem for which the origin problem was a problem known as the Lagrange problem or the problem on finding the form of the firmest column of the given volume is viewed. The Lagrange problem was the source for different extremal eigenvalue problems, among them for eigenvalue problems for the second-order differential equations, with an integral condition on the potential. In this paper the problem of that kind is considered under the con- dition that the integral condition contains a weight function. The method of finding the sharp upper estimates for the first eigenvalue of a Sturm - Liouville problem with Dirichlet conditions for some values of parameters in the integral condition was found and attainability of those estimates was proved.
Vestnik of Samara University. Natural Science Series. 2015;21(6):124-129
pages 124-129 views

ON ASYMPTOTIC PROPERTIES OF SOLUTIONS, DEFINED ON THE HALF OF AXIS OF ONE SEMILINEAR ODE

Filimonova I., Khachlaev T.

Abstract

The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential. The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential.
Vestnik of Samara University. Natural Science Series. 2015;21(6):130-134
pages 130-134 views

ON THE ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF THE BOUNDARY VALUE PROBLEM WITH A PARAMETER

Filinovskiy A.

Abstract

The paper presents the investigation of an eigenvalue problem for the Laplace operator with Robin boundary condition in a bounded domain with smooth boundary. The case of boundary condition containing a real parameter is con- sidered. It is proved that multiplicity of the eigenvalue to the Robin problem for all values of the parameter greater than some number does not exceed the mul- tiplicity of the corresponding eigenvalue to the Dirichlet problem for the Laplace operator. For simple eigenvalue of the Dirichlet problem the convergence of eigen- function of the Robin problem to the eigenfunction of the Dirichlet problem for unlimited increase of the parameter is proved. The formula for derivative on the parameter for eigenvalues of the Robin problem is established. This formula is used to justify the asymptotic expansions of eigenvalues of the Robin problem for large positive values of the parameter.
Vestnik of Samara University. Natural Science Series. 2015;21(6):135-140
pages 135-140 views

ABOUT NUMERICAL MODELLING OF THOMSON SELF-OSCILLATORY SYSTEMS

Zaitsev V., Karlov A., Karlov A.

Abstract

The algorithm of numerical integration of a task of Cauchy for the equations of the movement of self-oscillatory systems of Thomson type is offered. The algorithm is based on the use of samples of impulse response of linear resonant system as discretization sequences at the transition to the discrete time in the integral form of the equations of motion. Estimates of an error of numerical decisions are given. Transformation of finite difference algorithm in object of nonlinear dynamics in discrete time is discussed. Version of discrete mapping of Van der Pol oscillator is proposed. The algorithm of numerical integration of a task of Cauchy for the equations of the movement of self-oscillatory systems of Thomson type is offered. The algorithm is based on the use of samples of impulse response of linear resonant system as discretization sequences at the transition to the discrete time in the integral form of the equations of motion. Estimates of an error of numerical decisions are given. Transformation of finite difference algorithm in object of nonlinear dynamics in discrete time is discussed. Version of discrete mapping of Van der Pol oscillator is proposed. The algorithm of numerical integration of a task of Cauchy for the equations of the movement of self-oscillatory systems of Thomson type is offered. The algorithm is based on the use of samples of impulse response of linear resonant system as discretization sequences at the transition to the discrete time in the integral form of the equations of motion. Estimates of an error of numerical decisions are given. Transformation of finite difference algorithm in object of nonlinear dynamics in discrete time is discussed. Version of discrete mapping of Van der Pol oscillator is proposed.
Vestnik of Samara University. Natural Science Series. 2015;21(6):141-150
pages 141-150 views

DYNAMICS OF TWO-LEVEL ATOMS INTERACTING WITH THERMAL FIELD FOR ENTANGLED INITIAL ATOMIC STATES

Bashkirov E., Solovieva A., Mastyugin M.

Abstract

In this paper the entanglement dynamics for system of two two-level atoms interacting with a mode of thermal electromagnetic field in lossless cavity has been investigated. Initial atomic states are the entangled Bell type states. Using the full set of eigenvectors of the Hamiltonian of the considered model we have derived the exact solution for the density matrix for the whole system. On its basis the reduced atomic density matrix and Peres - Horodetcki parameter have been calculated. Calculating of entanglement parameter shows the possibility of high degree of entanglement even for large intensity of thermal field. Thus there is a possibility of maintenance and control over the degree of entanglement.
Vestnik of Samara University. Natural Science Series. 2015;21(6):151-160
pages 151-160 views

MONTE-CARLO CALCULATIONS OF PHASE TRANSITION TEMPERATURE IN THE ISING MODEL WITH LONG RANGE INTERACTIONS

Biryukov A., Degtyarova Y.

Abstract

The article deals with two-dimensional and three-dimensional Ising models with the long-range spin interactions. The intensity of interaction between the spins relies decreasing with distance r as a power law r-d-σ with dimensional d and parameter σ. The research are conducted by Monte-Carlo method with Metropolis algorithm using parallel computing techniques. On the basis of nu- merical simulation the dependence of the phase transition temperature on the parameter σ is found. It is shown that at phase transition temperature decreases with increasing σ.
Vestnik of Samara University. Natural Science Series. 2015;21(6):161-170
pages 161-170 views

CONSTRUCTION OF SYSTEMS OF PROTECTION FROM UNAUTHORIZED ACCESS FOR INFORMATION SYSTEMS TAMPER LOCATED IN THE INDUSTRIAL USE

Krutov A.

Abstract

The article discusses the question of construction of the systems of protection against unauthorized access to information systems that are in commercial operation. It offers two concepts of development of the system from unauthorized access with minimal modifications in the already developed information system. As a means of building an effective system of protection in the process means of careful monitoring of access database DBMS Oracle (Fine Grained Access Control) are used. Depending on the size of the source database and the possibility to modify the table structure of the information system it is offered to use one or the other method for constructing a system of protection against unauthorized access. Developed protection system is relatively indepen- dent module that can be implemented in as needed.
Vestnik of Samara University. Natural Science Series. 2015;21(6):171-177
pages 171-177 views

FEATURES OF FUNCTIONAL PROGRAMMING OF INTERACTIVE GRAPHICAL APPLICATIONS

Taranchuk V.

Abstract

In the article methodical and technical solutions which essentially expand capabilities of creation of the electronic intelligent educational resources contain- ing mathematical notation of any level of complexity and graphics illustrations of all types and categories are discussed. Base units of program modules, key constructions of codes, functions and options of language of the system of com- puter algebra Mathematica are explained. Main rules of preparation of freely distributed interactive program applications of CDF format are noted. Exam- ples from practice of preparation of teaching materials of discipline ”Computer Graphics” are given. User interface and results of execution of program modules are illustrated.
Vestnik of Samara University. Natural Science Series. 2015;21(6):178-189
pages 178-189 views

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