Mathematics

About the Classical Solution of the Mixed Problem for the Wave Equation

Authors: Khromov A. P.

The classic solution of the mixed problem for a wave equation with a complex potential and minimal smoothness of initial data is established by the Fourier method. The resolvent approach consists of constructing formal solution with the help of the Cauchy – Poincaré method of integrating the resolvent of the corresponding spectral problem over spectral parameter. The method requires no information about eigen and associated functions and uses only the main part of eigenvalues asymptotics. Krylov’s idea of accelerating the convergence of Fourier series is essentially employed.

The Solution of the Homogeneous Riemann Boundary Value Problem with a Countable Set of Points of Discontinuity of the First Kind its Coefficient

Authors: Salimov R. B.

We consider the Riemann homogeneous boundary value problem with a countable set of points of discontinuity of the first kind in the case, when it is required to find two functions, analytic, respectively, in the upper and lower half-plane, for a given linear boundary condition on the real axis, connecting the boundary values of the unknown functions.

Almost Periodic at Infinity Solutions of Difference Equations

Authors: Ryzhkova A. A., Trishina I. A.

A class of sequences almost periodic at infinity is introduced and studied. The necessity to consider such sequences is based on the fact that they appear in difference equations under consideration. The main results relate to the proof of almost periodicity at infinity of solutions of difference equations.

Generalized Characters Over Numerical Fields and a Counterpart of Chudakov Hypothesis

Authors: Matveev V. A., Matveev O. A.

The well-known Chudakov hypothesis for numeric characters, conjectured by Chudakov in 1950, suggests that finite-valued numeric character h(n), which satisfies the following conditions: 1) h(p)  ≠ 0 for almost all prime p; 2) S(x) = Ʃn≤x h(n) = αx + O(1),is a Dirichlet character. A numeric character which satisfies these conditions is called a generalized character, principal if α ≠ 0 and non-principal otherwise. Chudakov hypothesis for principal characters was proven in 1964, but for non-principal ones thus far it remains unproved.

About Differential Operators and Matrices of the Second Order

Authors: Duplishcheva A. Y.

Differential operators of the second order are studied. Conditions of their invertibility are obtained. The main results are obtained on the comparison of the operator matrix of the second order with the researching operator.

SYSTEMS OF DIFFERENTIAL EQUATIONS ON THE LINE WITH REGULAR SINGULARITIES

Authors: Gorbunov O. V., Shieh C., Yurko V. A.

Non-selfadjoint second order differential systems on the line having a non-integrable regular singularity are studied. We construct special fundamental systems of solutions with prescribed analytic and asymptotic properties. Asymptotics of the corresponding Stockes multipliers is established.

Comparison Standards Method for Solving of the Multi-criterion Discrete Optimization Problems

Authors: Budaeva A. A.

Research results of management and planning problems show that in real statement these problems are multi-criterion. For effective solution to these problems it is necessary to construct multi-criterion mathematical model and then it is necessary to optimize it, beforehand selecting the most appropriate method for this purpose. Proposed approach for multi criteria discrete optimization problems is based on the concepts of measurement standards and distances. With the help of this method the multi-criterion discrete optimization problem solution is considered.

On Variety of Semigroups of Reletions with Operation of Reflexive Double Cylindrification

Authors: Bredikhin D. A., Popovich A. V.

In the paper, the basis of identities for the variety generated by semigroups of relations with the operation of reflexive double cylindrification is found.

Weakly Ill-posed Problems of Integral Geometry witch Perturbation on Polygonal Lines

Authors: Begmatov A. H., Pirimbetov A. O., Seidullaev A. K.

We study a problem of reconstruction of a function in a strip from their given integrals with known weight function along polygonal lines. We obtained two simply inversion formulas for the solution to the problem. Using these representations we prove uniqueness and existence theorems for solutions and obtain stability estimates of a solution to the problem in Sobolev’s spaces and thus show their weak ill-posedness. Then we consider integral geometry problems with perturbation. The uniqueness theorems are proved and stability estimates of solutions in Sobolev spaces are obtained.