The theorem of uniqueness in the problem of spectral analysis for ordinary differential operators with coincidence of part of eigenvalues of operators for Neiman and Dirichlet boundary problems is proved 

The subject of the abstract potential is considered in the common situation of the linear spaces and its link to the operators used in the abstract sсheme of Gauss's method of elimination is shown. In case of Hilbert's spaces the estimations of the boundaries for mentioned operators are given 

This work deals with problem of non parametrical dividing surface restoration in a direct form in probability statement of the pattern recognition problem. In difference to known suggested approach allows get an optimal asymptotically convergent estimation. The idea consists in extracting from the initial learning samples elements which are laying just near separating surfaces, and then in direct restoration of latter on these elements using the methods of nonparametric regression analysis 

The thin stiffened shell with a slanted edge is considered. The low frequency vibration and buckling are investigated. The optimal shell parameters for the first frequency and external pressure are founded. The mitered pipe joint consisting of two connected circular cylindrical thin shells is also mentioned 

The problem of driving a symmetrical wedge of given weight into a beam is considered. The time-dependent external force is pushing on the wedge. Every half of the cracked beam is considered as console beam with non-constant length. The curve of deflection of the beam is examined as a series of proper functions of static problem of deflection of the beam. The problem is reduced to solution of system 2+n differential equations of second kind (Lagrange equations) if kinetic energy of the system is calculated with n members of the series. The approximations are constructed for n=0,1,2. The conditions for considering only 1 member of the series are found. The influence of the force applied to the wedge and the wedge mass on the dynamics of crack development is investigated 

Dynamics of the sliding elastic strip is considered. The upper foundation of the strip is compressed and moves with the given velocity. The lower foundation of the strip slides along the surface with the friction dependent on sliding velocity. The solution of problem was found for different approximation of friction-velocity characteristic with the using of small parameter method. The modes of fluctuations of various character asymptotically tending to equilibrium or drifting from equilibrium, were revealed in dependence on system's parameters. The conditions of satisfactory using of the different types of friction characteristic approximation were obtained. The conditions, at which it is necessary to use one or another approximation of the characteristic of friction, were obtained 

1. Неканоническое разделение переменных   1.1. Гиперболические задачи   1.2. Эллиптические задачи   1.3. Асимптотическое разделение переменных 2. Локально нелинейные асимптотики   2.1. Некоторые любопытные интегралы   2.2. Метод асимптотических интегральных итераций   2.3. Итерации с расширением области действия 3. Методология асимптотической математики   3.1. Метод порядковых уравнений   3.2. Переходные слои   3.3. Становление новой парадигмы 

Singular perturbed eigenvalue problem for Laplace operator in a two-dimensional domain with frequent alternation of the type of boundary conditions is considered. It's known that under certain relation between the measures of the parts of the boundary with Dirichlet and Neumann conditions the limit (homogenized) problem is Dirichlet problem. Exactly this situation is studied in the paper. On the base of the asymptotics methods we construct and justify the complete asymptotics for the eigenelements of the perturbed problem regarding two parameters, characterizing lengths of the boundary's parts with different types of boundary conditions 

This article describes the possibilities of pseudo-inversion methodology application for feed-forward neural networks learning. Golub and Pereyra's method solving least squares problems with linear-nonlinear structure applied to networks learning is specified. Application of this method to multi-layer and multi-output neural networks is declared. The idea of Golub and Pereyra's method and block pseudo-inversion combining is stated 

The problem of the buckling stability at pure torsion of a thin cylindrical shell with an oblique edge is considered. The asymptotic formulas for a critical load value in the second approximation are obtained. Comparison of accuracy of outcomes of the first and second approximations is carried out. The relation of a critical load valuu to length of a shell and joining angle is demonstrated