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The article is devoted to the analysis of electrodynamic properties elliptical frame structure. Taking into account double symmetry internal problem of electrodynamics for the structure under consideration in the framework of the thin-wire approximation is reduced to four integral Fredholm equations of the first kind, written with respect to independent current functions. A study of spectral characteristics of the integral operators of the corresponding integral equations for various values of the electrical length and ellipticity of the frame. It is shown that the eigenfunctions of integral operators for close values of these parameters have a high degree of correlation, with In this case, the eigenfunctions are close in form to trigonometric functions. Features of the frequency dependence of the eigenvalues integral operators. The conclusion is made about the resonant nature of these dependences, what makes an elliptical frame structure in many respects similar to the previously considered tubular vibrator and spherical spiral particle. The results presented in the article form an in-depth understanding of the processes occurring in the structure under consideration, and also serve as a guideline in the construction of approximation models for solving the internal tasks.

In this paper, we study the issue of boundary control of the temperature field of a rod with a selected point. The main purpose of the work is to clarify the conditions for the existence of a boundary control that ensures the transition of the temperature field from the initial state to the final state. Relations connecting the boundary controls with the initial and final states, as well as with the external temperature field are found. Such boundary controls, generally speaking, constitute an infinite set. For an unambiguous choice of the boundary control, a strictly convex objective functional is chosen. We are looking for a boundary control that minimizes the selected target functional. To do this, we first investigate the existence and uniqueness of solutions to the initial boundary value problem and the conjugate problem. And also, we present the derivation of a system of linear Fredholm integral equations of the second kind, which are satisfied by an optimal boundary control that minimizes a strictly convex target functional on a convex set. Along the way, the linear part of the increment of the target functional is highlighted. Necessary and sufficient conditions for the minimum of a smooth convex functional on a convex set are established. The difference between the results of this work and the available ones is that in the proposed work, the temperature field is given by the heat conduction equation with a loaded term. As a result, the conjugate problem has a slightly different domain of definition than the domain of the conjugate problem in the case of no load.

We study a sequence of differential operators of high even order whose potentials converge to the Dirac delta-function. One of the types of separated boundary conditions is considered. At the points of potential discontinuity, it is necessary to study the conditions of gluing for the correct determination of the corresponding differential equations solutions. For large values of the spectral parameter, asymptotic solutions of differential equations are furnished by the Naimark method. The conditions of gluing are studied, the boundary conditions are investigated, the equation for the eigenvalues of the considered differential operator is derived. The method of successive approximations is used to find the asymptotics of spectrum of studied differential operators, the limit of which determines a spectrum of operator with a delta-function potential.

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Background and Objectives: The Zakharov–Shabat spectral problem allows to find soliton solutions of the nonlinear Schrodinger equation. Solving the Zakharov–Shabat problem gives both a discrete set of eigenvalues λj and a continuous one. Each discrete eigenvalue corresponds to an individual soliton with the real part Re(λj) providing the soliton velocity and the imaginary part Im(λj) determining the soliton amplitude. Solitons can be used in optical communication lines to compensate both non-linearity and dispersion. However, a direct use of solitons in return-to-zero signal encoding is inhibited. The interaction between solitions leads to the loss of transmitted data. The problem of soliton interaction can be solved using eigenvalues. The latter do not change when the solitons obey the nonlinear Schrodinger equation. Eigenvalue communication was realized recently using electronic signal processing. To increase the transmission speed the all-optical method for controlling eigenvalues should be developed. The presented research is useful to develop optical methods for the transformation of the eigenvalues. The purpose of the current paper is twofold. First, we intend to clarify the issue of whether the dispersion perturbation can not only split a bound soliton state but join solitons into a short oscillating period breather. The second goal of the paper is to describe the complicated dynamics and mutual interaction of complex eigenvalues of the Zakharov–Shabat spectral problem. Materials and Methods: Pulse propagation in single-mode optical fibers with a variable core diameter can be described using the nonlinear Schrödinger equation (NLSE) which coefficients depends on the evolution coordinate. The NLSE with the variable dispersion coefficient was considered. The dispersion coefficient was described using a hyperbolic tangent function. The NLSE and the Zakharov– Shabat spectral problem were solved using the split-step method and the layer-peeling method, respectively. Results: The results of numerical analysis of the modification of soliton pulses under the effect of variable dispersion coefficient are presented. The main attention is paid to the process of transformation of eigenvalues of the Zakharov–Shabat problem. Collision of two in-phase solitons, which are characterized by two complex eigenvalues is considered. When the coefficients of the nonlinear Schrodinger equation change, the collision of the solitons becomes inelastic. The inelastic collision is characterized by the change of the eigenvalues. It is shown that the variation of the coefficients of the NLSE allows to control both real and imaginary parts of the eigenvalues. Two scenarios for the change of the eigenvalues were identified. The first scenario is characterized by preserving the zero real part of the eigenvalues. The second one is characterized by the equality of their imaginary parts. The transformation of eigenvalues is most effective at the distance where the field spectrum possesses a two-lobe shape. Variation of the NLSE coefficient can introduce splitting or joining of colliding soliton pulses. Conclusion: The presented results show that the eigenvalues can be changed only with a small variation of the NLSE coefficients. On the one hand, a change in the eigenvalues under the effect of inelastic soliton collision is an undesirable effect since the inelastic collision of solitons will lead to unaccounted modulation in soliton optical communication links. On the other hand, the dependence of the eigenvalues on the parameters of the colliding solitons allows to modulate the eigenvalues using all-fiber optical devices. Currently, the modulation of the eigenvalues is organized using electronic devices. Therefore, the transmission of information is limited to nanosecond pulses. For picosecond pulse communication, the development of all-optical modulation methods is required. The presented results will be useful in the development of methods for controlling optical solitons and soliton states of the Bose–Einstein condensate.

The article proposes an option for transforming a mathematical model of the object, formed by the extended nodal method in the time-domain solution for modal analysis. Since finding the eigenvalues ​​and eigenvectors for systems of ordinary equations given in the Cauchy normal form is possible, calculations are presented that allow us to obtain a system of equations in the Cauchy normal form from a mathematical model in a differential-algebraic form through linearization. The extended nodal method contains derivatives of state variables in the vector of unknown, and the Jacobi matrix obtained at each Newton iteration of each step of numerical integration can be used to obtain a linearized mathematical model, but the equilibrium equations, as a rule, contain several derivatives with respect to time. By introducing additional variables, it is possible to reduce the linearized mathematical model to the Cauchy normal form, while the Jacobi matrix structure remains essentially unchanged.The proposed solution is implemented in the mathematical core of the PRADIS Gen2 PA-8 software package, which made it possible to expand its functionality by an operator of modal analysis.The presented calculations of test schemes have shown the correctness of the method proposed.

With the development of information technology, the requirements for steganography systems are growing. Today, the steganography system must resist steganalysis attacks using the latest mathematical bases, attacks and countermeasures, which can be easily and efficiently implemented in any modern software environment, providing significant bandwidth when organizing a secret communication channel, and so on. All possible means must be used to meet these requirements. One of the ways to meet a certain requirement is to choose the cover as which the digital image is considered in the study. The cover deciding is to be done in order to provide the best characteristics of stego-object in a sense. The aim of the study is to provide the possibility to choose a cover image, which results in the stego-object that is robust to various image processing attacks. The following results are obtained in the study: the expediency of using perturbations of eigenvectors of the cover image matrix as indicators of security of secret message embedded into the cover is theoretically substantiated and practically confirmed; the quantitative estimation of the perturbation of the eigenvector matrix of the digital image as a result of embedding is proposed, which allows separating the sensitive eigenvector from the insensitive one, adequately reflects the perturbation of the eigenvector regardless of the angle of rotation; the proposed condition for the selection of eigenvectors protected from influence, taking into account the individual characteristics of the cover image; the formula for calculating the amount of protected additional information is proposed; a cover image selection method from the available set of potential containers is developed, which provides stego-object robustness to various image processing attacks. [ABSTRACT FROM AUTHOR]

Determination of thermal regimes of composite cylindrical bodies by analytical methods leads to the appearance of complex characteristic equations, the solution of which is the determination of eigenvalues. The article considers a relatively simple approximate analytical method for determining the eigenvalues of characteristic equations for a two-layer cylindrical body under boundary conditions of the third kind. This method can also be easily used in more complex formulations of heat conduction problems.

The singular integral equation of a thin tubular vibrator is considered. This paper analyzed the dependence of the eigenvalues of the integral operator of the singular integral equation on the frequency and geometric dimensions of the vibrator with the determination of their asymptotic behavior at small lengths and small radii of the vibrator. It is shown that these dependences admit a simple polynomial approximation, and it is shown that an approximation in harmonic functions can be used for the eigenfunctions of an integral operator. Results can be used as the basis for constructing a approximate solutions of current on the tubular vibrator.

The mathematical substantiation of the algorithm for synthesis of the proper transformation and finding the eigenvalue formulae of a persymmetric matrix of dimension N = 2 k ( k =1, 4 ) based on orthogonal rotation operators is given. The proposed algorithm made it possible to improve the author's approach to calculating eigenvalues based on numerical examples for the maximal dimension of matrices 64×64, resulting the possibility to obtain analytical relations for calculating the eigenvalues of the persymmetric matrix. It is shown that the proper transformation has a factorized structure in the form of a product of rotation operators, each of which is a direct sum of elementary Givens and Jacobian rotation matrices.

The overall objective is to describe the behavior of the eigenvalues of approximating operators and figuring out how to limit one turns oneʼs own importance. Earlier we have done the following: built approximation expression L0u = −Du + a(e)δu = f operators of finite rank; explicit form approximating the resolvent family; resolutions and found the limit cases of resonance highlighted. In this article, we will continue to address this problem and set out a step associated with the description of the spectrum constructed limit operators and study the behavior of the eigenvalues of approximating operators, using Newtonʼs diagram method. As a result of eigenvalues of the operator were found.

The eigenvalues and eigenfunctions of two boundary value problems for three-dimensional equations of elliptic types with singular coefficients with lower terms are found.

We consider an arbitrary periodic operator with a finite number of distant perturbations in an arbitrary domain of a multidimensional space. The perturbations are arbitrary localized operators. We introduce distant perturbations through shift operators, perturbing operators, and certain weight functions that satisfy a set of the certain conditions.The main aim of the paper is to study the spectrum of a perturbed operator when the distances between domains in which perturbations are located tend to infinity. The formulation of theses problem is quite general and has not been investigated before. Inprevious papers only the spectral properties of differential operators with distant perturbations were considered.The main results are as follows:invariance of the essential spectrum of the perturbed operator with regard to distant perturbations;the existence of a simple and isolated eigenvalue of a perturbed operator that converges to a simple and isolated eigenvalue of the limit operator.complete asymptotic series for the perturbed eigenvalue and the perturbed eigenfunction.proof of uniform convergence of these series and derivation of explicit formulas for their coefficients.The technique used to obtain the results is to reduce the eigenvalue equation to a regularly perturbed operator equation in a special Hilbert space. The perturbation smallness was described by two small characteristic parameters. The adapted version of the Birman-Schwinger method then applied allows us to reduce the problem to the analysis of an operator equation and to search of some holomorphic function zeros.

The article considers asymptotic distribution of characteristic constants in periodic and antiperiodic boundary-value problems for the second-order linear equation with periodic coefficients. It allows getting asymptotics of stability and instability zones of solutions. It was shown that in the absence of turning points (\(r(t) > 0\)) the instability zones lengths converge to zero with their number increasing, and the stability zones lengths converge to a positive quantity. The situation, when (\(r(t) \geqslant 0\)) and there are zeroes \(r(t),\) results in the fact that the lengths of stability and instability zones have a finite nonzero bound at an unbounded increase of the number of the corresponding zone. But if the function \(r(t)\) is alternating, the lengths of all stability zones converge to zero, and the lengths of instability zones converge to some finite quantities. These conclusions allowed to formulate a series of interesting criteria of stability and instability of solutions of the second-order equation with periodic coefficients. The results given are illustrated by a substantial example. The methods of investigation are based on a detailed study of the so-called special standard equations and the consequent reduction of original equations to any particular type of standard equations. Here, asymptotic methods of the theory of singular perturbance, as well as certain properties of a series of special functions are used.

The question of constructing a solution of the boundary layer type for a rectangular orthotropic double-layer plate is discussed, on the face planes of which mixed homogeneous conditions are specified. Using the asymptotic method, solutions of the boundary layer type are constructed, a transcendental equation is derived to determine the eigenvalues, which characterize the decay rates of the found solution. It is shown that for orthotropic materials the solution of the type of the boundary layer disintegrated into flat and anti-flat boundary layers. [ABSTRACT FROM AUTHOR]

The work objective is to describe the numerical solution method for the stationary Schrödinger equation based on the application of the integral equation identical to the Schrödinger equation. The structure of this integral equation is close to the structure of the Fredholm equation of second kind and allows obtaining the problem numerical solution. The method under study allows finding the energy eigenvalues and eigensolutions to the quantum-mechanical problems of various dimensions. The test results of the solving problems method for one-dimensional and two-dimensional quantum oscillators are obtained. The found numerical values of eigenenergy and eigenfunctions of the oscillator are compared to the known analytical solutions, and then the error of result is evaluated. The highest accuracy of the solution is obtained for the first energy levels. The numerical solution error increases with the number of the energy eigenvalue. For the subsequent energy level, the error increases almost by an order of magnitude. The solution error for the fourth energy level is less than 2% if the integration domain contains 500 elements. If the energy level is degenerate, it is possible to obtain all eigenfunctions corresponding to the given level. For this purpose, various auxiliary functions the symmetry of which is coherent with the eigenfunction symmetry are used.

A method of problem solution of the eigenvalues and eigenfunctions for the Helmholtz equation in the domains with arbitrary configuration is worked out. In developing the approach of the numerical solution of problems, the point-source method (PSM) is used. The proposed method is based on the analysis of the condition number of the PSM system or error of the problem numerical solution. The concept of “eigenvalues criterion” is introduced. The research result is a developed effective method - an algorithm for solving problems of eigenvalues and eigenfunctions for the Helmholtz equation. It is shown that at the approach of the Helmholtz parameter to the problem eigenvalue, the condition number of the PSM system and the error of the numerical solution rise sharply. Therefore, the dependence of the condition number of the PSM system or the error of the problem numerical solution can be calculated from the Helmholtz parameter. Then, according to the position of the maximum of the obtained dependences, the eigenvalues of the Helmholtz equation in a given domain are found. It allows searching the eigenvalues. After finding the eigenvalues, it is possible to proceed to the determination of the eigenfunctions. At that, if the eigenvalue appears degenerate, that is some eigenfunctions correspond to it, then it is possible to find all the eigenfunctions taking into account the symmetry of the solution domain. The two-dimensional and three-dimensional test problems are solved. Upon the results obtained, the conclusion about the efficiency of the proposed method is made.

For singularly perturbed second order equations the dependence of eigenvalues of the first boundary problem on a small parameter at the highest derivative is studied. The main assumption is that the coefficient at the first derivative in the equation is the sign of the variable. This leads to the emerging of so-called turning points. Asymptotic expansions on the small parameter are obtained for all eigenvalues of the considered boundary problem. It turns out that the expansions are defined by the behavior of coefficients in a neighborhood of turning points only

For a second order equation with a small factor at the highest derivative the asymptotic behavior of all eigenvalues of periodic and antiperiodic problems is studied. The main assumption is that the coefficient at the first derivative in the equation is the sign of the variable so that turning points exist an algorithm for computing all coefficients of asymptotic series for every considered eigenvalue is developed. It turns out that the values of these coefficients are defined by coefficient values of the original equation only in a neighborhood of turning points. Asymptotics for the length of Lyapunov zones of stability and instability was obtained. In particular, the problem of stability of solutions of second order equations with periodic coefficients and small parameter at the highest derivative was solved

We consider a linear differential equation of second order with a small factor at the highest derivative. We study the problem of the asymptotic behavior of the eigenvalues of the first boundary value problem (task Dirichlet) in situation when the turning points (points where the coefficient at the first derivative equals to zero) exist. It is shown that only the behavior of coefficients of the equation in a small neighborhood of the turning points is essential. The main result is a theorem on the limit values of the eigenvalues of the first boundary value problem.

In this paper we studied the spectral problem for an ordinary second order differential equation on a finite interval with a discontinuous coefficient of the highest derivative. At the ends of the segment the boundary conditions of the first kind are given. We found eigenvalues with their asymptotic behavior as the roots of the transcendental equation. The system of eigenfunctions is the trigonometric sine on one half of the segment, and the hyperbolic sine on the other. The system of eigenfunctions is not orthogonal in the space of square integrable functions. The corresponding biorthogonal system of functions was built as a solution to the dual problem. In the proof of the completeness of the biorthogonal system we used well known Keldysh theorem about the completeness of the eigenfunctions system of a nonselfadjoint operator.

Renewable energy resources, for example, photovoltaic elements and wind turbines draw the increasing attention due to energy shortage and depletion of energy sources. Such energy sources usually give low voltage output. So the elementary photovoltaic electric power supply based on silicon gives about 0,7 V EMF, and based on germanium - 0,4 V. However the possibilities of sequential accumulation of output potential are limited due to the need of increase in parallel branches of solar panels. The solution of this problem led to development of power supply systems on the basis of pulse-width boost converters. There is a task on improving the efficiency of this type of converters. In particular, to determine stability and research of dynamics in mathematical models of installations of power electronics the Jacobi's method is widely applied. While developing the power supply systems of different function the use of the Jacobi matrix increases their energy parameters. At the same time application of function of Poincare for iterative display of the Jacobi method makes it possible to determine the operating modes of the converter for specific points after changing the parameters of the system. Thus, Jacobi's matrix, being an dimensionless size, allows linearizing the characteristic in the fixed points for their analysis. Therefore, partial differentiation of nonlinear system gives the opportunity to obtain the fixed values, being the characteristic animator of the system - eigenvalues of a matrix. All the studies were carried out on the example of two-phase boost dc-dc converter with pulse frequency modulation. The peculiarity of the paper consists in use of the Jacobi matrix for linearizing non-linear systems when studying dynamics to determine steady and unstable modes of functioning of energy converters. Such mathematical approach allows increasing the efficiency and reliability of the voltage converter by identification of the abnormal dynamic operation modes in the device, reducing energy losses, i. e. increasing energy efficiency of a power supply system. The aim of the research is to develop an alternative method of linearization by using Jacobi's matrix to analyze stability of operation of various converters. The circuit of the studied converter assumes energy accumulation in inductive elements for shutdown at zero current or zero voltage. Results will be introduced in the form of display of Poincare through the Newton-Rafson-Jacobi method. Methods: review of technical publications, concerned the need of using Jacobi matrixes to analyze various converters stability. The control algorithm of two-phase pulse converter is simulated in LTSpice and MATLAB. The analysis of appropriate characteristics of TPBC includes Jacobi and Newton-Rafson methods. The latter has rather fast convergence and fine precision of discrete values. It is shown below in chaos and bifurcation modes. Results. The obtained eigenvalues define the operation modes of management system of TPBC, developed on the basis of the twophase pulse converter. These results will help to reveal those parameters of the system in case of which it functions in normal or abnormal statuses, thereby will allow separating one modes of functioning from others, and thus increasinge reliability and efficiency of the system.

The purpose of this work is to find solutions for the system of equations Oberbeck– Boussinesq flat convection Bénard–Marangoni a viscous incompressible fluid. In this viscous incompressible fluid the radial component of the temperature gradient may become zero. It is shown that the initial system may be reduced to the system of equations of ordinary differential equations of the eleventh order. We obtain the exact solution at the point of the extremum of the temperature (at zero including Grasgof’s). Integration of equations is carried out in dimensionless variables, which are non-classical way: put the scale factor for each variable, and not by linear characteristic size of the layer. The solution is the initial approximation to the solution of convection Bénard–Marangoni in numbers Grasgof’s, the big zero.

A spectral boundary problem of special type containing a spectral parameter in the boundary condition is completely solved in this paper. The characteristic equation for spectrum points determination is obtained, the energy innerproduct is derived, and the orthonormal system of the eigenfunctions is built up.

Developed steganographic algorithm is compressive strength algorithms for arbitrary cover images including attacks with low quality factor. Cover is digital image stored in arbitrary format. Development of proposed algorithm based on existing analogues that based on general approach to the analysis of the status and technology functioning of information systems. The main advantage of the new steganographic algorithm is 6 times increase of its capacity, compared with the algorithms which it is based. The results of computational experiment confirming the high performance decoding of embedded information by compression attacks with low quality factor. [ABSTRACT FROM AUTHOR]

It were investigated folowing things norm of vector under periodic and aperiodic action matrixes from finite set M of matrix with rational elements is bounded, AZR and PZR for action of such set. A difficult case is considered, when every matrix from M has eigenvalues such that some from them are greater than 1 and other less then 1. A lower spectral radius (LSR) is studied for such set M by opening of question about that, whether is M characteristics of AZR and PZR. The conducted research witnessed of the eventual set of matrices, M̃ which satisfies certain terms, has not PAS and AAS, it is proofed that ρ̄(M̃)≥ 1. Conditions of AZR and PZR is holds for M̃[√] and vectors from √". It is established existing of the systems of matrices M̃ are above Q for which takes place ES. [ABSTRACT FROM AUTHOR]